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## **Meet the editor**

Radostina Petrova holds an MSc in Civil Engineering from the University of Architecture, Civil Engineering and Geodesy of Sofia, Bulgaria, and a PhD in Applied Mechanics from the Technical University of Sofia. She has more than 25 years of experience working for the industry and the academia. Over the last 10 years, she was an associate professor in Dynamics, Strength

and Reliability of Machines, Devices, and Systems at the Technical University of Sofia. Previously, she has been invited to teach at the Vienna University of Technology, Austria, and University of Stavanger, Norway. Radostina has an active professional practice as a structural engineer. Radostina Petrova is an internationally recognized expert in the field of multibody dynamic simulation of mechanical systems, nonlinear structural analysis, structural modeling and analysis using FEM, CAD/CAE (FEA) design of structures and mechanical systems, wind engineering, etc. She is awarded an "Ernst Mach" research grant, international research grants of the EEA, and a Fulbright grant, which is to be implemented at the Cornell University, USA. She has an award for outstanding contributions in reviewing for Elsevier publishers.

She is an author/editor of four books, several book chapters, and more than a hundred scientific articles.

## Contents

## **Preface XI**


## Preface

Chapter 8 **The Role of Finite Element Analysis in Studying Potential**

Raymond C.W. Wong, John S.P. Loh and I. Islam

Chapter 9 **Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis 213**

Chapter 10 **Finite Elements Method in Implant Prosthetics 231**

Iulia Roateşi

**VI** Contents

**Failure of Mandibular Reconstruction Methods 191**

Yutaka Inaba, Hiroyuki Ike, Masatoshi Oba and Tomoyuki Saito

The finite element method (FEM) is a numerical technique for finding approximate solutions to different numerical problems. It subdivides a large domain into smaller parts, called finite elements (FEs). Each FE is modeled by simple equations. The assembly of all FE equations into a large global system gives the solution to the original complex problem. Thus, instead of tackling a complicated problem directly, FEM allows us to divide it into simpler and easi‐ ly solvable subproblems before working our way backward in order to combine the solu‐ tions and obtain a unique result for the system as a whole.

FEM is an extremely powerful technique, but we should bear in mind that it works via dis‐ cretization, which in some cases might present an inadequate approximation of the behavior of a real physical structure. Therefore, we must always ensure that the discrete model mesh is dense enough, to produce accurate outputs.

The practical applications of FEM are best known as finite element analysis (FEA). FEA's applications in the field of engineering always include the use of a software program that carries out the necessary computations for the FEM algorithm. FEA is a good choice for ana‐ lyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution is not smooth.

The first three chapters of this book contribute to the development of new FE techniques by providing an insight into a few technical handicaps of the FEM and by proposing new com‐ putational techniques and algorithms to overcome them.

Prof. Grzegorz's chapter titled "Problems of Hierarchical Modeling and hp-Adaptive Finite Element Analysis in Elasticity, Dielectricity and Piezoelectricity" provides a universal meth‐ odology, in which applicability is demonstrated in the context of three physical problems. The chapter presents the author's hitherto accomplishments in hierarchical models and hp approximations of linear elasticity onto dielectricity and piezoelectricity, including an error estimation and an adaptivity control.

The second chapter by Prof. Mahboub "The discontinuous Galerkin Finite Element Method for Ordinary Differential Equations: A priori and A posteriori Error Estimates, Superconver‐ gence, and Adaptivity" demonstrates the application of the original discontinuous Galerkin finite element method for nonlinear ordinary differential equations with upwind fluxes. A simple, computationally efficient, and asymptotically exact error estimator is developed, in‐ cluding a local adaptive mesh refinement procedure. The applicability of the proposed method is illustrated with several numerical examples.

In the next chapter, Prof. Seyed and his colleague Khan study the natural frequencies of double-walled carbon nanotubes for different boundary conditions. The nanotubes are mod‐ eled as both local and nonlocal Euler-Bernoulli beams coupled with van der Waals interac‐ tion forces. The universality of the proposed model is proven by its application to a doubleelastic Euler-Bernoulli beam. This formulation paves the way for FEM analysis of multiwalled carbon nanotubes.

It is really important to develop new efficient computer algorithms for solving complex problems and to demonstrate their practical applications in the modeling of different physic phenomena. The following chapters highlight this close connection between the develop‐ ment of new techniques and their implementations.

First, Associate Prof. Ezgi investigates the influence of interface parameters on critical stress transfer factors of material under tensile loading by using an existing commercial software for FEA. Initially, a polypropylene short fiber is embedded into a polypropylene coethylene cylindrical interface and later into the matrix material.

Next, Dr. Kurahashi and his team demonstrate numerical examples of a shallow water flow estimation based on the Kalman filter FEM. They explore the influence of differently set ob‐ servation points on the final numerical results using lumped matrixes for triangular FEs and adopting the Galerkin method.

Subsequently, Dr. Asatur and his colleagues analyze the nonuniform bending of single layer cells of epithelial tissues, surrounding a number of organs in the human body. They model thin epithelial tissues as membranes and the total elastic energy as a combination of stretch‐ ing and bending energies. Tissues of different shapes under specific assumptions are ex‐ plored using finite element discretization.

Finally, Dr. Sarkar together with his colleagues simulates contact instability, including crack opening in soft thin films adopting FEM. They utilize FE techniques to overcome the hur‐ dles in the atomistic level and the analytical continuum simulations.

The state-of-the-art tools for solving problems via FEA allow practitioners to produce defor‐ mation and strength visualizations and to control the complexity of both modeling and anal‐ ysis of systems in simulations, as well as fine-tune numerous other options with very little effort, thereby optimizing the investments in labor and resources for each analysis. In partic‐ ular, for engineering applications, the desired level of accuracy and the associated computa‐ tional time requirements can be managed simultaneously. Thus, FEA allows entire designs to be constructed, refined, and optimized before manufacturing.

This book also marks the described above advantages of FEA implemented in the industrylevel software packages via examples from the field of biomechanical engineering.

Dr. Wong and his colleagues demonstrate the advantages of FEA in studying a potential failure of mandible through reconstruction methods. They review the basic knowledge of the mandible necessary to perform FEA, the challenges in studying this problem, and sever‐ al studies investigating it. The authors focus on FEA as a new method for mandibular recon‐ struction and implant evaluation.

The chapter "Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis" by Dr. Yutaka and his colleagues compares equivalent stress and strain energy density to bone mineral density in the femur after total hip arthroplasty using subject-specific FEA. They show that FEA can predict adaptive bone remodeling after me‐ chanical loading changes. Furthermore, they evaluate the effects of two different types of stem geometries on load distribution and on bone mineral density with FEA. Their analysis reveals that a large tapered wedge-type stem and stovepipe femur may be associated with significant proximal bone mineral density loss.

Associate Prof. Roatesi studies the behavior of functional loadings for implant prosthetics rehabilitation. She calculates stresses, displacements, and strains in the implant and the sur‐ rounding bone and thus assesses risk factors from a biomechanical perspective. Her analysis determines the factors affecting the implant's stability and succeeds in calculating safety fac‐ tors for different load cases.

Finally, we can conclude that this book undoubtedly states the power and numerous advan‐ tages of FEM for various applications. Current research points to the vitality and potential of finite elements for further development in terms of both more efficient numerical techniques and areas of application. FEM is many practitioners' tool of choice since it allows for an ac‐ celerated initial prototype design and subsequent design refinements via optimization on performance in numerical tests and simulations. Additionally, state-of-the-art algorithms al‐ low for short computation times for an affordable investment which further strengthens FEM's positions in the modern engineering world.

## **Acknowledgment**

eled as both local and nonlocal Euler-Bernoulli beams coupled with van der Waals interac‐ tion forces. The universality of the proposed model is proven by its application to a doubleelastic Euler-Bernoulli beam. This formulation paves the way for FEM analysis of

It is really important to develop new efficient computer algorithms for solving complex problems and to demonstrate their practical applications in the modeling of different physic phenomena. The following chapters highlight this close connection between the develop‐

First, Associate Prof. Ezgi investigates the influence of interface parameters on critical stress transfer factors of material under tensile loading by using an existing commercial software for FEA. Initially, a polypropylene short fiber is embedded into a polypropylene coethylene

Next, Dr. Kurahashi and his team demonstrate numerical examples of a shallow water flow estimation based on the Kalman filter FEM. They explore the influence of differently set ob‐ servation points on the final numerical results using lumped matrixes for triangular FEs and

Subsequently, Dr. Asatur and his colleagues analyze the nonuniform bending of single layer cells of epithelial tissues, surrounding a number of organs in the human body. They model thin epithelial tissues as membranes and the total elastic energy as a combination of stretch‐ ing and bending energies. Tissues of different shapes under specific assumptions are ex‐

Finally, Dr. Sarkar together with his colleagues simulates contact instability, including crack opening in soft thin films adopting FEM. They utilize FE techniques to overcome the hur‐

The state-of-the-art tools for solving problems via FEA allow practitioners to produce defor‐ mation and strength visualizations and to control the complexity of both modeling and anal‐ ysis of systems in simulations, as well as fine-tune numerous other options with very little effort, thereby optimizing the investments in labor and resources for each analysis. In partic‐ ular, for engineering applications, the desired level of accuracy and the associated computa‐ tional time requirements can be managed simultaneously. Thus, FEA allows entire designs

This book also marks the described above advantages of FEA implemented in the industry-

Dr. Wong and his colleagues demonstrate the advantages of FEA in studying a potential failure of mandible through reconstruction methods. They review the basic knowledge of the mandible necessary to perform FEA, the challenges in studying this problem, and sever‐ al studies investigating it. The authors focus on FEA as a new method for mandibular recon‐

The chapter "Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis" by Dr. Yutaka and his colleagues compares equivalent stress and strain energy density to bone mineral density in the femur after total hip arthroplasty using subject-specific FEA. They show that FEA can predict adaptive bone remodeling after me‐ chanical loading changes. Furthermore, they evaluate the effects of two different types of

level software packages via examples from the field of biomechanical engineering.

dles in the atomistic level and the analytical continuum simulations.

to be constructed, refined, and optimized before manufacturing.

multiwalled carbon nanotubes.

VIII Preface

adopting the Galerkin method.

plored using finite element discretization.

struction and implant evaluation.

ment of new techniques and their implementations.

cylindrical interface and later into the matrix material.

The editor would like to express her gratitude to her loving family and to InTech publishers, especially the responsible Publishing Process Manager Ms. Martina Usljebrka, without whose support this book would not be possible.

> **Assoc. Prof. Eng. Radostina Petrova, PhD** Bulgaria

## **Problems of Hierarchical Modelling and** *hp***-Adaptive Finite Element Analysis in Elasticity, Dielectricity and Piezoelectricity Problems of Hierarchical Modelling and** *hp***-Adaptive Finite Element Analysis in Elasticity, Dielectricity and Piezoelectricity**

Grzegorz Zboiński Grzegorz Zboiński

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64892

#### **Abstract**

In this chapter, we consider theoretical and implementation difficulties in application of the hierarchical modelling and *hp*-adaptive finite element approach to elasticity, dielectricity and piezoelectricity. The main feature of the applied methodology is its generalizing character which is reflected by application of the same or analogous algorithms to three mentioned physical problems, including multi-physics problem of piezoelectricity, simple and complex physical description as well as simple and complex geometries. In contrast to the most common approaches dealing with a single physical phenomenon, described by a single physical model, within a single geometrical part, this chapter presents the ideas which brake and overcome such a simplicity. This presented chapter generalizes author's hitherto accomplishments, in hierarchical models and *hp*-approximations of linear elasticity, onto dielectricity and piezoelectricity. The same refers to error estimation and adaptivity control. In this context, the main similarities and differences of three physical problems are of interest in this work.

**Keywords:** physical complexity, elasticity, dielectricity, piezoelectricity, geometrical complexity, model complexity, hierarchical modelling, finite elements, hierarchical approximations, error estimation, *hp*-adaptivity

## **1. Introduction**

In this chapter of the book we extend our hitherto propositions concerning 3D-based hierarchical models of liner elasticity onto 3D-based linear dielectric and piezoelectric media. In the case of hierarchical models of linear elasticity we apply 3D-elasticity model, hierarchical

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

shell models, first-order shell model and the solid-to-shell or shell-to-shell transition models. In the case of dielectricity we utilize 3D-dielectricity model, and the 3D-based hierarchical dielectric models as well. The piezoelectric case needs combination of two mentioned mechanical and electric hierarchies, so as to generate the hierarchy of 3D-based piezoelectric models. Any combination of the mentioned elastic and dielectric models is possible. As far as the *hp*-discretization is concerned we extend the ideas of hierarchical approximations, constrained approximations and the transition approximations of the displacement field onto electric potential field of dielectricity or the coupled electro-mechanical field of piezoelectricity. The mentioned approximations allow *p*-adaptivity (three-dimensional or longitudinal), *q*adaptivity (transverse one), *h*-adaptivity (three-dimensional or two-dimensional ones) and *M*-adaptivity (model adaptivity). The error assessment in three classes of problems is based on the equilibrated residual methods (ERM) applied to total and approximation error estimations. The modelling error is obtained as a difference of the previous two errors. The estimated error values are utilized for adaptivity control. The adaptive procedures for dielectricity and piezoelectricity are obtained through the generalization of the three- or fourstep strategies applied so far to the elasticity case. The difficulties in generalization of the above-mentioned methods of hierarchical modelling, *hp*-approximations, error estimation and adaptivity control onto electrical and electro-mechanical problems are addressed in this chapter.

## **2. Considered problems**

In the chapter we consider five problems. The first three correspond to stationary problems of mechanical, electric and electro-mechanical equilibrium of the elastic, dielectric and piezoelectric media, respectively. The last two problems deal with free vibration problems of linear elasticity and linear piezoelectricity.

#### **2.1. Elastostatic problem**

Let us start with the standard formulation of the linear elasticity [1]. The problem local equations include equilibrium, constitutive and geometric relations:

$$\begin{aligned} \sigma\_{ij,j} + f\_i &= 0, \\ \sigma\_{ij} &= D\_{ijkl} \varepsilon\_{kl}, \\ \varepsilon\_{ij} &= 1 / 2(u\_{i,j} + u\_{j,i}), \quad \mathbf{x} \in V \end{aligned} \tag{1}$$

where , , , *i,j,k,l* = 1,2,3 are the elasticity constants tensor, and the stress and strain tensors, respectively. The given vector of mass load is denoted as *fi* , while *ui* is the unknown vector of displacements. The above equations hold in volume *V* of the body.

The standard boundary conditions for stresses and displacements are:

Problems of Hierarchical Modelling and *hp*-Adaptive Finite Element Analysis in Elasticity, Dielectricity... http://dx.doi.org/10.5772/64892 3

$$\begin{aligned} \sigma\_y n\_j &= p\_i, \qquad \mathbf{x} \in S\_p, \\ \mu\_i &= w\_i, \qquad \mathbf{x} \in S\_w \end{aligned} \tag{2}$$

where *nj* denotes components of the normal to the body surface *S*, composed of its loaded *SP* and supported *SW* parts: = ∪ . The vectors *pi* and *wi* represent the given stresses and displacements on *SP* and *SW*, respectively.

The equivalent variational formulation results from minimization of the potential energy of the elastic body:

$$\int\_{V} D\_{ijkl} \mathbf{v}\_{i,j} \mu\_{k,l} \, dV = \int\_{V} \mathbf{v}\_{i} f\_{i} \, dV + \int\_{S\_{P}} \mathbf{v}\_{i} p\_{i} \, dS \tag{3}$$

where *vi* represent admissible displacements conforming to the displacement boundary conditions.

The above variational functional can be utilized in the derivation of the global finite element equations of the form:

$$K\_M \, q^{q,hp} = F\_V + F\_S \tag{4}$$

where , , , are the stiffness matrix within the mechanical equilibrium problem, and the

vectors of the nodal forces due to volume and surface loadings. The term , ℎ stands for the nodal displacement degrees of freedom (dofs). The applied hierarchical *q,hp*-approximation of displacements will be addressed in the next sections.

#### **2.2. Electrostatic problem of dielectrics**

shell models, first-order shell model and the solid-to-shell or shell-to-shell transition models. In the case of dielectricity we utilize 3D-dielectricity model, and the 3D-based hierarchical dielectric models as well. The piezoelectric case needs combination of two mentioned mechanical and electric hierarchies, so as to generate the hierarchy of 3D-based piezoelectric models. Any combination of the mentioned elastic and dielectric models is possible. As far as the *hp*-discretization is concerned we extend the ideas of hierarchical approximations, constrained approximations and the transition approximations of the displacement field onto electric potential field of dielectricity or the coupled electro-mechanical field of piezoelectricity. The mentioned approximations allow *p*-adaptivity (three-dimensional or longitudinal), *q*adaptivity (transverse one), *h*-adaptivity (three-dimensional or two-dimensional ones) and *M*-adaptivity (model adaptivity). The error assessment in three classes of problems is based on the equilibrated residual methods (ERM) applied to total and approximation error estimations. The modelling error is obtained as a difference of the previous two errors. The estimated error values are utilized for adaptivity control. The adaptive procedures for dielectricity and piezoelectricity are obtained through the generalization of the three- or fourstep strategies applied so far to the elasticity case. The difficulties in generalization of the above-mentioned methods of hierarchical modelling, *hp*-approximations, error estimation and adaptivity control onto electrical and electro-mechanical problems are addressed in this

In the chapter we consider five problems. The first three correspond to stationary problems of mechanical, electric and electro-mechanical equilibrium of the elastic, dielectric and piezoelectric media, respectively. The last two problems deal with free vibration problems of linear

Let us start with the standard formulation of the linear elasticity [1]. The problem local

, ,

*uu V*

where , , , *i,j,k,l* = 1,2,3 are the elasticity constants tensor, and the stress and strain

+ Î*x*

(1)

is the unknown

, while *ui*

= 0, = ,

= 1/ 2( ),

*ij i j ji*

vector of displacements. The above equations hold in volume *V* of the body.

 e

equations include equilibrium, constitutive and geometric relations:

,

tensors, respectively. The given vector of mass load is denoted as *fi*

The standard boundary conditions for stresses and displacements are:

s

s

e

*ij j i ij ijkl kl*

+

*f D*

chapter.

**2. Considered problems**

2 Perusal of the Finite Element Method

elasticity and linear piezoelectricity.

**2.1. Elastostatic problem**

The standard local formulation of linear dielectricity [2] consists of the Gauss law, here corresponding to the lack of volume charges, the constitutive relation and the electric field definition:

$$\begin{aligned} d\_{i,j} &= 0, \\ d\_i &= \gamma\_y E\_j, \\ E\_j &= -\phi\_{,j}, \qquad \mathbf{x} \in V \end{aligned} \tag{5}$$

Above, , *i,j* = 1,2,3 stands for the dielectric constants tensor, while denotes the electric displacement vector. The scalar term represents the electric potential field, searched in the volume of the dielectric.

The boundary conditions for the electric displacements and electric potential read:

$$\begin{aligned} d\_i n\_i &= -c, \quad \mathbf{x} \in \mathcal{S}\_{\varrho}, \\ \phi &= \mathcal{X}, \quad \mathbf{x} \in \mathcal{S}\_{\mathbb{F}} \end{aligned} \tag{6}$$

where and are the given scalar values of the surface charge and electric potential on the parts and , respectively, of the surface of the dielectric body ( = ∪ ).

The corresponding variational formulation which reflects minimization of the potential electric energy can be described as follows:

$$\int\_{V} \gamma\_{\psi} \, \psi\_{,i} \, \phi\_{,j} \, dV = \int\_{S\_{Q}} \psi \mathbf{c} \, dS \tag{7}$$

with being the admissible electric potential conforming to the second boundary condition of Eq. (6).

The finite element formulation can be expressed as follows:

$$K\_E \text{ } \mathfrak{g}^{\rho, h\pi} = F\_{\underline{\mathcal{Q}}} \tag{8}$$

where is the characteristic matrix of dielectricity, denotes the characteristic nodal vector of electric charges and , ℎ stands for the unknown nodal vector of electric potentials. The

hierarchical *ρ,hπ*-approximation of the potential will be explained later on in this chapter.

#### **2.3. Stationary electro-mechanical problem**

Formally, the local formulation of the piezoelectric problem of electro-mechanical equilibrium [3] can be treated as a combination of the linear elasticity and linear dielectricity Eqs. (1) and (5):

$$\begin{aligned} \sigma\_{y\_{i,j}} + f\_i &= 0, \\ d\_{i,j} &= 0, \\ \sigma\_{\dot{y}} &= D\_{\dot{y}kl} \varepsilon\_{kl} - C\_{k\dot{y}} E\_k, \\ d\_i &= C\_{ikl} \varepsilon\_{kl} + \gamma\_{\dot{y}} E\_{,i}, \\ \sigma\_{\dot{y}} &= 1 / 2(\mu\_{i,j} + \mu\_{j,i}), \\ E\_j &= -\phi\_{,j}, \quad \mathbf{x} \in V \end{aligned} \tag{9}$$

What couples both sets of equations are the modified constitutive relations, where the coupling piezoelectric constants tensor appears.

The boundary conditions of the coupled problem are:

The boundary conditions for the electric displacements and electric potential read:

*dn c S*

*x*

, *i i Q*

parts and , respectively, of the surface of the dielectric body ( = ∪ ).

*ij i j* , , <sup>=</sup> *V SQ* gyf

> , *<sup>h</sup> E Q* r p *K F* j

The finite element formulation can be expressed as follows:

**2.3. Stationary electro-mechanical problem**

piezoelectric constants tensor appears.

f c *S* =- Î = Î

energy can be described as follows:

4 Perusal of the Finite Element Method

of Eq. (6).

, ,

*x*

*F*

where and are the given scalar values of the surface charge and electric potential on the

The corresponding variational formulation which reflects minimization of the potential electric

with being the admissible electric potential conforming to the second boundary condition

where is the characteristic matrix of dielectricity, denotes the characteristic nodal vector of electric charges and , ℎ stands for the unknown nodal vector of electric potentials. The hierarchical *ρ,hπ*-approximation of the potential will be explained later on in this chapter.

Formally, the local formulation of the piezoelectric problem of electro-mechanical equilibrium [3] can be treated as a combination of the linear elasticity and linear dielectricity Eqs. (1) and (5):

= 0,

= , = , = 1/ 2( ),

*D CE*

*ij ijkl kl kij k i ikl kl ij j ij i j ji*

 e


, ,

What couples both sets of equations are the modified constitutive relations, where the coupling

*u u*

,

f*x*

*E V*

*dC E*

e g


= ,

*j j*

, ,

*d*

s

e

s

= 0,

+

*f*

*ij j i i i*

 y

*dV c dS* ò ò (7)

= (8)

(6)

(9)

$$\begin{aligned} \sigma\_{\psi} n\_{j} &= p\_{i}, & \mathbf{x} \in S\_{F}, \\ d\_{i} n\_{i} &= -\mathbf{c}, & \mathbf{x} \in S\_{\mathcal{Q}}, \\ u\_{i} &= w\_{i}, & \mathbf{x} \in S\_{W}, \\ \phi &= \mathcal{K}, & \mathbf{x} \in S\_{F} \end{aligned} \tag{10}$$

The variational functional of the electro-mechanical problem consists of the terms of functionals (3) and (7) completed with the terms describing the piezoelectric coupling through the tensor , i.e.

$$\begin{aligned} \int\_{V} D\_{jkl} \mathbf{v}\_{i,j} \mu\_{k,l} \, dV + \int\_{V} C\_{jk} \mathbf{v}\_{i,j} \boldsymbol{\phi}\_{,k} \, dV + \int\_{V} \mathbf{v}\_{i} f\_{i} \, dV + \int\_{S\_{P}} \mathbf{v}\_{i} p\_{i} \, dS \\ + \int\_{V} \mathbf{v}\_{k} \boldsymbol{\eta}\_{,l} \boldsymbol{\phi}\_{,k} \, dV + \int\_{V} C\_{il} \boldsymbol{\eta}\_{,l} \boldsymbol{\mu}\_{k,l} \, dV + \int\_{S\_{Q}} \boldsymbol{\eta} c \, dS = 0 \end{aligned} \tag{11}$$

The above variational formulation leads to the following finite element equations

$$\begin{aligned} \mathbf{K}\_M \neq \mathbf{q}^{q, hp} - \mathbf{K}\_C \underbrace{\mathbf{q}^{\rho, h\pi}}\_{=} &= F\_V + F\_S\\ \mathbf{K}\_C^T \neq \mathbf{q}^{q, hp} + \mathbf{K}\_E \underbrace{\mathbf{q}^{\rho, h\pi}}\_{=} &= F\_Q \end{aligned} \tag{12}$$

where is the characteristic matrix of piezoelectric coupling.

The above finite element equations correspond to a very general case when both the direct and inverse piezoelectric phenomena are present. Substitution of the second Eq. (12) into the first one leads to the single combined equation from which the nodal displacements , ℎ can be calculated. The opposite substitution gives the combined equation from which the nodal electric potentials , ℎ can be extracted. The first situation corresponds to the so-called actuation action of the piezoelectric, while the second one to the sensing action of the piezoelectric body. These two modes of action can be associated with the direct and inverse piezoelectric phenomena, respectively.

#### **2.4. Mechanical problem of free vibration**

The local formulation of the free vibration problem of linear elasticity is composed of the following equations

$$\begin{aligned} \sigma\_{\stackrel{\circ}{\vee},j} &= \varrho \ddot{u}\_i, \\ \sigma\_{\stackrel{\circ}{\vee}} &= D\_{\stackrel{\circ}{\vee}k\!} \varepsilon\_{kl}, \\ \varepsilon\_{\stackrel{\circ}{\vee}} &= 1 / 2(u\_{i,j} + u\_{j,i}), \quad \mathbf{x} \in V \end{aligned} \tag{13}$$

where is a density of the elastic body, while ¨ represents the acceleration vector. The displacements are of harmonic character, i.e. *ui* = *ai* sin *ωt*, with standing for the unknown natural frequencies of the body and denoting the searched displacement amplitudes.

The boundary conditions are:

$$\begin{aligned} \sigma\_{\psi} u\_{\rangle} &= 0, \qquad \mathbf{x} \in S\_{F}, \\ u\_{\neq} &= 0, \qquad \mathbf{x} \in S\_{W} \end{aligned} \tag{14}$$

The variational formulation of the free vibration problem takes advantage of the Hamilton's principle and reads

$$\int\_{V} D\_{ij\mathbf{d}} \mathbf{v}\_{i,j} a\_{k,l} \, dV - \alpha^2 \int\_{V} \varrho \mathbf{v}\_i a\_i \, dV = 0 \tag{15}$$

The finite element formulation derived from the above variational functional represents a set of uniform algebraic equations. Such a set possesses a solution if the following characteristic equation is fulfilled:

$$\det(\mathbf{K}\_M - \alpha^2 \mathbf{M}) = 0 \tag{16}$$

From this equation natural frequencies , = 1,2,…, can be calculated, where is the total number of degrees of freedom of the vibrating elastic body. Above, *M* represents the mass (or inertia) matrix.

For each natural frequency the nodal vector of displacement amplitudes , ℎ can be determined with use of the below finite element equations:

$$\begin{aligned} \left(\mathbf{K}\_M - \alpha\_n^2 \mathbf{M}\right) \mathbf{q}\_n^{q, hp} &= \mathbf{0}, \\ \left(\mathbf{q}\_n^{q, hp}\right)^T \mathbf{M} \; \mathbf{q}\_n^{q, hp} &= \mathbf{1} \end{aligned} \tag{17}$$

The second relation above is the normalization condition, completing 1 geometrically independent finite element equations of the first relation.

#### **2.5. Coupled problem of free vibration**

,

s

s

e

where is a density of the elastic body, while ¨

displacements are of harmonic character, i.e. *ui*

natural frequencies of the body and

The boundary conditions are:

6 Perusal of the Finite Element Method

principle and reads

equation is fulfilled:

inertia) matrix.

= , = , = 1/ 2( ),

ñ&&

*u D*

 e

*ij j i ij ijkl kl ij i j ji*

, ,

+ Î

*uu V*

*x*

= *ai*

= 0, ,

Î *x x*

The variational formulation of the free vibration problem takes advantage of the Hamilton's

2 , , - = 0 ò ò <sup>ñ</sup> *ijkl i j k l i i V V D v a dV v a dV* w

The finite element formulation derived from the above variational functional represents a set of uniform algebraic equations. Such a set possesses a solution if the following characteristic

number of degrees of freedom of the vibrating elastic body. Above, *M* represents the mass (or

,

,

1

=

*q hp*

<sup>2</sup> det( ) 0 *K M <sup>M</sup>* - = w

For each natural frequency the nodal vector of displacement amplitudes

, ,

w

*q hp q h n n <sup>T</sup> n n <sup>p</sup>*

)

*q Mq*

*M*

2 ,

)

*K Mq* **0**


From this equation natural frequencies , = 1,2,…

determined with use of the below finite element equations:

( (

*n S u S*

Î

= 0, *ij j P i W*

s

(13)

(14)

represents the acceleration vector. The

sin *ωt*, with standing for the unknown

(15)

(16)

can be calculated, where is the total

, ℎ can be

(17)

denoting the searched displacement amplitudes.

We start here with the local (strong) formulation of the undamped vibration problem of the piezoelectric medium

$$\begin{aligned} \sigma\_{\vec{y},j} + f\_i &= \varrho \ddot{u}\_i, \\ d\_{i,i} &= 0, \\ \sigma\_{\vec{y}} &= D\_{\dot{y}kl} \varepsilon\_{kl} - C\_{kl} E\_k, \\ d\_i &= C\_{kl} \varepsilon\_{kl} + \chi\_{\vec{y}} E\_j, \\ \sigma\_{\vec{y}} &= 1/2(\mu\_{i,j} + \mu\_{j,i}), \\ E\_j &= -\phi\_{,j}, \quad \mathbf{x} \in V \end{aligned} \tag{18}$$

completed with the following boundary conditions of the coupled electromechanical field

$$\begin{aligned} \sigma\_{\psi} n\_{j} &= p\_{i}, & \mathbf{x} \in S\_{p}, \\ d\_{i} n\_{i} &= -\mathbf{c}, & \mathbf{x} \in S\_{\mathcal{Q}}, \\ u\_{i} &= w\_{i}, & \mathbf{x} \in S\_{\mathcal{W}}, \\ \phi &= \chi, & \mathbf{x} \in S\_{F} \end{aligned} \tag{19}$$

In the case of stationary mass = () and surface = () loadings and charges = (), and the stationary displacement = () and electric potential = () boundary conditions as well, the problem converts into two independent ones. The first of them is exactly the stationary task of the electro-mechanical equilibrium defined with the local formulation (9) and (10). The corresponding variational and finite element formulations are exactly described with Eqs. (11) and (12), respectively. The solution in displacements = () to this stationary problem determines the equilibrium state around which the free vibrations of the piezoelectric are performed. This solution allows the determination of the initial stresses = (*u*) which are taken into account in the second problem of free vibration.

The local formulation of the mentioned free vibration problem of the piezoelectric can now be determined in the following way:

$$\begin{aligned} \sigma\_{\psi\_{i,l}} &= \varrho i i\_{i}, \\ d\_{i,l} &= 0, \\ \sigma\_{\psi} &= \mathfrak{c}\_{\psi} + D\_{\psi k} \mathfrak{c}\_{kl} - C\_{kj} E\_{k,l} \\ d\_{i} &= C\_{ik} \mathfrak{c}\_{kl} + \mathfrak{y}\_{\psi} E\_{j}, \\ \sigma\_{\psi} &= 1 / 2(\mathfrak{u}\_{i,j} + \mathfrak{u}\_{j,i}) \\ E\_{j} &= -\phi\_{,j}, \quad \mathbf{x} \in V \end{aligned} \tag{20}$$

where the displacement and the coupled potential fields are: *ui* = *ai* sin *ωt* and *ϕ* = *α* sin *ωt*, respectively, with and standing for the displacement and potential amplitudes.

The above set of differential equations has to be completed with the boundary conditions of the form

$$\begin{aligned} \sigma\_{ij} n\_j &= 0, & \mathbf{x} \in S\_p, \\ d\_i n\_i &= 0, & \mathbf{x} \in S\_{\mathcal{Q}}, \\ u\_i &= w\_i, & \mathbf{x} \in S\_{\mathcal{W}}, \\ \phi &= \mathcal{K}, & \mathbf{x} \in S\_F \end{aligned} \tag{21}$$

The equivalent variational formulation of the problem reads

$$\begin{aligned} \int\_{V} \mathsf{c}\_{\overline{\eta}} \boldsymbol{\nu}\_{i,l} \boldsymbol{a}\_{k,j} \, dV - \int\_{V} \boldsymbol{D}\_{\overline{\eta}kl} \boldsymbol{\nu}\_{i,j} \boldsymbol{a}\_{k,l} \, dV + \int\_{V} \mathsf{C}\_{\overline{\eta}k} \boldsymbol{\nu}\_{i,j} \boldsymbol{a}\_{k,k} \, dV + o\overline{\nu}\_{l}^{2} \int\_{V} \boldsymbol{\varrho} \boldsymbol{\nu}\_{i} \boldsymbol{a}\_{i} \, dV \\ + \int\_{V} \boldsymbol{\nu}\_{i\overline{k}} \boldsymbol{\nu}\_{i\overline{k}} \boldsymbol{\alpha}\_{,k} \, dV + \int\_{V} \boldsymbol{C}\_{i\overline{\boldsymbol{k}}} \boldsymbol{\nu}\_{i\overline{k}} \boldsymbol{a}\_{k,l} \, dV = 0 \end{aligned} \tag{22}$$

while the corresponding finite element equation describing free vibration of the initially stressed piezoelectric medium is

$$\begin{aligned} \mathbf{K} (\mathbf{K}\_M + \mathbf{K}\_G - \alpha^2 \mathbf{M}) \mathbf{q}^{q, hp} - \mathbf{K}\_C \mathbf{q}^{\rho, h\pi} &= \mathbf{0} \\ \mathbf{K}\_C^T \mathbf{q}^{q, hp} + \mathbf{K}\_E \mathbf{q}^{\rho, h\pi} &= \mathbf{0} \end{aligned} \tag{23}$$

where , ℎ and , ℎ represent the nodal, displacement and electric potential, amplitude degrees of freedom, while stands for the so-called geometric stiffness matrix due to the initial stresses.

As the above set of linear algebraic equations is homogeneous, the solution to it can be obtained if and only if the following characteristic equation:

Problems of Hierarchical Modelling and *hp*-Adaptive Finite Element Analysis in Elasticity, Dielectricity... http://dx.doi.org/10.5772/64892 9

$$\det(\mathbf{K}\_M + \mathbf{K}\_G + \mathbf{K}\_C \mathbf{K}\_E^{-1} \mathbf{K}\_C^T - o^2 \mathbf{M}) = 0\tag{24}$$

is fulfilled. This equation has been obtained after substitution of the second relation (23) into the first one. This allows to remove electric potential amplitudes , ℎ from the combined equation. From this equation = 1,2,…, natural frequencies can be obtained, where is the total number of degrees of freedom within the mechanical field.

The corresponding normalized mode shapes can be obtained from

$$\begin{aligned} \left( (\mathbf{K}\_M + \mathbf{K}\_G + \mathbf{K}\_C \mathbf{K}\_E^{-1} \mathbf{K}\_C^T - \alpha\_n^2 \mathbf{M} \right) \mathbf{q}\_n^{q, hp} &= \mathbf{0}, \\ \left( \mathbf{q}\_n^{q, hp} \right)^T \mathbf{M} \mathbf{q}\_n^{q, hp} &= 1 \end{aligned} \tag{25}$$

where the normalization condition, the same as in (17), has been applied.

## **3. Complexity of the modelling**

, ,

*i i*

*d*

s

e

where the displacement and the coupled potential fields are: *ui*

respectively, with

8 Perusal of the Finite Element Method

ς

stressed piezoelectric medium is

initial stresses.

the form

s

,

and standing for the displacement and potential amplitudes.

= *ai*

2

ñ

 w

**0**

= 0

a

(20)

(21)

(22)

(23)

sin *ωt* and *ϕ* = *α* sin *ωt*,

, ,

The above set of differential equations has to be completed with the boundary conditions of

= 0, , = 0, , =, ,

*x x x x*

*ij j P i i Q ii W*

*n S dn S uw S*

Î Î Î Î

*F*

*S*

,

f

= ,

s

f c

, , , , , ,

, ,

*T q hp h C E*

+

*Kq K*

if and only if the following characteristic equation:

, , , ,

()=

*M G C*

+- w

j

*K K Mq K*

r p

*ik i k ikl i k l V V*

òò ò ò

*ij k i k j ijkl i j k l ijk i j k i i VV V V*

*v a dV D v a dV C v dV v a dV*


*dV C a dV*

y

while the corresponding finite element equation describing free vibration of the initially

2, ,

**0**

where , ℎ and , ℎ represent the nodal, displacement and electric potential, amplitude degrees of freedom, while stands for the so-called geometric stiffness matrix due to the

As the above set of linear algebraic equations is homogeneous, the solution to it can be obtained

*q hp h*

r p

j

=

The equivalent variational formulation of the problem reads

gy a

+ +

ò ò

*E V*

= ,

*j j*

,

*u*

ñ

= , = 1 / 2( ) ,

*i ikl kl ij j ij i j ji*

*dC E*

e g


=V + - + +

*ij ij ijkl kl kij k*

 e

*u u*

*x*

*D CE*

0,

= =

&& *ij j i*

There are three types of complexity considered in this chapter. The first one deals with physical complexity which consists in the presence of more than one physical phenomenon in the problem. The second complexity refers to geometry of the domain under consideration. The geometry is regarded as a complex one if more than one type of geometry is applied. One may deal with a three-dimensional geometry, thin-walled geometry and transition geometry, for example. The third type of complexity is model complexity. In this case, one employs more than one model for description of at least one physical phenomenon under consideration. Combination of these three types of complexity can be regarded as a unique feature of the presented research.

The examples of such complex modelling are electro-mechanical systems composed of geometrically complex elastic structures, joined with the geometrically complex piezoelectric actuators or sensors. In the general case of arbitrary geometry, such systems may require complex mechanical and electro-mechanical description.

#### **3.1. Physical complexity**

There are two physical sub-systems present in the considered electro-mechanical systems. The first sub-system concerns bodies subject to elastic deformation and representing structural or machine elements, while the second one concerns piezoelectric bodies acting as actuators or sensors, where the direct or inverse piezoelectric phenomena take place.

#### **3.2. Geometrical complexity**

In both, mechanical and piezoelectric, sub-domains we deal with the complex geometry of the structural and piezoelectric elements. In the case of the structural elements, they can be threedimensional bodies, bounded with surfaces, thin- or thick-shell bodies [4] and solid-to-shell transition bodies. In the case of the piezoelectric members, they can be of three-dimensional, symmetric-thickness or transition character. The shell or symmetric-thickness elements are defined by means of the mid-surface and thickness concepts. In the case of both transition members, we deal with three-dimensional geometry, bounded with surfaces, apart from the boundary to be joined with the shell or symmetric-thickness elements. On this superficial boundary part the mid-surface and the symmetric thickness function have to be defined.

## **3.3. Model complexity**

In the case of the mechanical sub-system complex geometry, the mechanical description may include: three-dimensional elasticity model, hierarchical shell models, the first-order shell model and the solid-to-shell or shell-to-shell transition models. The latter two models allow joining the first-order shell domains with the 3D elasticity and hierarchical shell ones, respectively. In the case of the piezoelectric sub-system, the dielectric model can either represent three-dimensional dielectricity or hierarchical symmetric-thickness dielectric models. The piezoelectric model can be any combination of the listed elastic and dielelectric models.

## **4. The applied methodology**

There are five related aspects of the presented methodology of adaptive hierarchical modelling and adaptive ℎ-finite element analysis of elastic, dielectric and piezoelectric bodies. The first of them is the 3D-based approach proposed in [5]. The second issue, i.e. hierarchical mechanical models were initiated in [6], further developed in [7] and finalized in 3D-based version in [8], while the electric models were introduced in [9] for laminated piezoelectrics and in [10] for dielectrics. General rules of the next aspect of hierarchical approximations were given in [11]. Such approximations for hierarchical shells were developed in [7] and for complex structures in [8]. The latter work generalizes the former attempts given in [12–15]. Approximations for piezoelectric problems were elaborated in [9, 16]. The next issue of error estimation by the equilibrated residual method [17] for 3D elasticity was addressed in [18], for the hierarchical shells in [19], for the 3D-based first-order shells in [20, 21] and for the 3D-based complex structures in [22]. Application of the method in electric problems was proposed in [23]. Finally, adaptivity control with the three-step strategy for simple structures was presented in [24], while adaptivity for simple piezoelectrics was introduced in [25]. The first work was extended onto the 3D-based complex structures in [22].

#### **4.1. 3D-based approach**

The applied 3D-based approach [5, 8] lies in application of only three-dimensional degrees of freedom (dofs) regardless of the applied mechanical or electric models. This means that the conventional mid-surface dofs of the shell models, i.e. mid-surface displacements, rotations and other generalized displacement dofs of the mid-surface, are replaced with the equivalent through-thickness displacement dofs similar to the three-dimensional dofs of the 3D elasticity model. Also the mid-surface dofs of the two-dimensional dielectric theory are replaced with the through-thickness electric potential dofs of the three-dimensional dielectrics.

The equivalence of the displacement mid-surface dofs and the through-thickness dofs can be expressed through:

$$\mu\_{j}^{\prime} = \sum\_{m=0}^{l} \xi\_{3}^{\prime m} s\_{j}^{\prime m} = \sum\_{m=0}^{l} t\_{m}(\xi\_{3}^{\prime}) \mu\_{j}^{\prime m} \tag{26}$$

where represents the local, tangent ( = 1,2) and normal ( = 3), displacement fields. The terms and *tm* stand for the *m*th power of the local, normal coordinate ( = 0 on the mid-surface) and the *m*th polynomial through-thickness function of this coordinate, respectively. The mid-surface and through-thickness displacement dof functions are denoted as and , respectively, with *m* = 1,2,…,*I* and being the order of the shell theory.

In the case of the electric potential field, the analogous equivalence can be seen in:

$$\phi = \sum\_{m=0}^{J} \xi\_3^{\prime \prime m} \delta^m = \sum\_{m=0}^{J} t\_m(\xi\_3^{\prime}) \phi^m \tag{27}$$

where denotes the scalar field of the potential, while and represent *m*th (*m* = 1,2,…,*J*) mid-surface and through-thickness potential dof functions, while is the order of the twodimensional dielectric theory.

#### **4.2. Hierarchical models**

dimensional bodies, bounded with surfaces, thin- or thick-shell bodies [4] and solid-to-shell transition bodies. In the case of the piezoelectric members, they can be of three-dimensional, symmetric-thickness or transition character. The shell or symmetric-thickness elements are defined by means of the mid-surface and thickness concepts. In the case of both transition members, we deal with three-dimensional geometry, bounded with surfaces, apart from the boundary to be joined with the shell or symmetric-thickness elements. On this superficial boundary part the mid-surface and the symmetric thickness function have to be defined.

In the case of the mechanical sub-system complex geometry, the mechanical description may include: three-dimensional elasticity model, hierarchical shell models, the first-order shell model and the solid-to-shell or shell-to-shell transition models. The latter two models allow joining the first-order shell domains with the 3D elasticity and hierarchical shell ones, respectively. In the case of the piezoelectric sub-system, the dielectric model can either represent three-dimensional dielectricity or hierarchical symmetric-thickness dielectric models. The piezoelectric model can be any combination of the listed elastic and dielelectric models.

There are five related aspects of the presented methodology of adaptive hierarchical modelling and adaptive ℎ-finite element analysis of elastic, dielectric and piezoelectric bodies. The first of them is the 3D-based approach proposed in [5]. The second issue, i.e. hierarchical mechanical models were initiated in [6], further developed in [7] and finalized in 3D-based version in [8], while the electric models were introduced in [9] for laminated piezoelectrics and in [10] for dielectrics. General rules of the next aspect of hierarchical approximations were given in [11]. Such approximations for hierarchical shells were developed in [7] and for complex structures in [8]. The latter work generalizes the former attempts given in [12–15]. Approximations for piezoelectric problems were elaborated in [9, 16]. The next issue of error estimation by the equilibrated residual method [17] for 3D elasticity was addressed in [18], for the hierarchical shells in [19], for the 3D-based first-order shells in [20, 21] and for the 3D-based complex structures in [22]. Application of the method in electric problems was proposed in [23]. Finally, adaptivity control with the three-step strategy for simple structures was presented in [24], while adaptivity for simple piezoelectrics was introduced in [25]. The first work was extended

The applied 3D-based approach [5, 8] lies in application of only three-dimensional degrees of freedom (dofs) regardless of the applied mechanical or electric models. This means that the conventional mid-surface dofs of the shell models, i.e. mid-surface displacements, rotations and other generalized displacement dofs of the mid-surface, are replaced with the equivalent through-thickness displacement dofs similar to the three-dimensional dofs of the 3D elasticity

**3.3. Model complexity**

10 Perusal of the Finite Element Method

**4. The applied methodology**

onto the 3D-based complex structures in [22].

**4.1. 3D-based approach**

In the case of the mechanical elastic models the hierarchy M of the 3D or 3D-based models is [5, 8]:

$$M \in \mathcal{M}, \ M = \{\Im D, MI, \, RM, \Im D \nmid RM, MI \mid RM\} \tag{28}$$

where 3*D* represents three-dimensional elasticity, denotes hierarchical higher-order shell models, is the first-order Reissner-Mindlin shell model, while 3/ and / denote the solid-to-shell or shell-to-shell transition models. It is worth mentioning that the second and last models form the following sub-hierarchies:

$$\begin{aligned} MI &= \{M\,\ $, M\,\$ , M\,4, \ldots\}, \\ MI &\mid RM = \{M\,\ $/\,RM, \,M\,\$ /\,RM, \,M\,4/\,RM, \ldots\} \end{aligned} \tag{29}$$

with being the order of each particular theory.

The hierarchical character of the above models results from the mentioned order of the applied 3D-based theories, i.e.

$$M = RM \Longrightarrow I = 1, K \equiv I$$

$$M = M\,\, 2 \,\, \, RM, \,\, M\,\, 3 \,\, \, RM, \,\, M\,\, 4 \,\, \, RM, \,\, \ldots \,\, \, I = 2, 3, 4, \,\ldots, \,\, K = 1$$

$$M = M\,\, 2, \,\, M\,\, 3, \,\, M\,\, 4, \,\ldots \,\, \, I = 2, 3, 4, \,\ldots, \,\, K \equiv I \tag{30}$$

$$M = \exists D \,\, \, R M \,\, \Rightarrow I \to \infty, \, K = 1$$

$$M = \exists D \,\, \, \, I \to \infty, \, K \equiv I$$

Note that for the pure models ( , 3) , while for the transition ones (/, 3/ .

The hierarchy of 3D-based models possesses the following property:

$$\lim\_{K=1,J} \left( \lim\_{I \to \ast} \left\| \boldsymbol{\mu}^{I/K(M)} \right\|\_{U,V} \right) = \left\| \boldsymbol{\mu}^{3D} \right\|\_{U,V} \tag{31}$$

guaranteeing that the solutions /() obtained with the subsequent models tend in the limit to the solution 3 of the three-dimensional elasticity (the highest model of the hierarchy), when ∞. In the above relation the norm of the strain energy is applied in order to compare the solutions. The norm is defined as follows:

$$\left\|\boldsymbol{\mu}^{I/K(M)}\right\|\_{U,\boldsymbol{V}} = \frac{1}{2}\int\_{\boldsymbol{V}} \boldsymbol{\mathfrak{o}}^{\boldsymbol{T}}(\boldsymbol{\mu}^{I/K(M)}) \,\,\mathfrak{o}(\boldsymbol{\mu}^{I/K(M)}) \,\,\,dV\tag{32}$$

where and are the stress and strain vectors, respectively.

In the case of the dielectric theories the hierarchy E of the 3D-based models :

$$E \in \mathcal{E}, \ \mathcal{E} = \{\Im D, EJ\} \tag{33}$$

is composed of the three-dimensional theory 3D and the set:

$$EJ = \{E1, E2, E3, \ldots\} \tag{34}$$

of the 3D-based hierarchical dielectric models , where is the order of the corresponding dielectric theory.

The hierarchy can be ordered with respect to the order in the following way:

Problems of Hierarchical Modelling and *hp*-Adaptive Finite Element Analysis in Elasticity, Dielectricity... http://dx.doi.org/10.5772/64892 13

$$\begin{aligned} E &= E1, \, E2, \, E3, \dots \Rightarrow J = 1, 2, 3, \dots, L \equiv J \\ E &= 3D \Rightarrow J \to \infty, \, L \equiv J \end{aligned} \tag{35}$$

and is characterized with the following property:

The hierarchical character of the above models results from the mentioned order of the

= 2 / , 3 / , 4 / , = 2,3,4, , = 1

Note that for the pure models ( , 3) , while for the transition ones (/, 3/

to the solution 3 of the three-dimensional elasticity (the highest model of the hierarchy), when ∞. In the above relation the norm of the strain energy is applied in order to compare

<sup>1</sup> = ( )( ) <sup>2</sup>

of the 3D-based hierarchical dielectric models , where is the order of the corresponding

e

( ) /( ) <sup>3</sup> , , =1, lim lim = *IKM <sup>D</sup> U V U V K II* ®¥

/( ) /( ) /( )

*IKM T IKM IKM U V <sup>V</sup> dV* ò *u uu* s

In the case of the dielectric theories the hierarchy E of the 3D-based models :

The hierarchy can be ordered with respect to the order in the following way:

*M M RM M RM M RM I K*

The hierarchy of 3D-based models possesses the following property:

= = 1,

Þ

*M RM I K I*

Þ º

K K K K

Þ º

*u u* (31)

obtained with the subsequent models tend in the limit

*E* Î = E, E {3 , } *D EJ* (33)

*EJ E E E* = { 1, 2, 3, } K (34)

(32)

(30)

=3 / , =1 =3 ,

*M D I KI*

Þ ®¥ Þ ®¥ º

= 2, 3, 4, = 2,3,4, ,

*MMMM I KI M D RM I K*

applied 3D-based theories, i.e.

12 Perusal of the Finite Element Method

guaranteeing that the solutions /()

the solutions. The norm is defined as follows:

,

where and are the stress and strain vectors, respectively.

is composed of the three-dimensional theory 3D and the set:

.

dielectric theory.

$$\lim\_{L=1,J} \left( \lim\_{J \to \circ} \left\| \phi^{J/L(E)} \right\|\_{W,V} \right) = \left\| \phi^{3D} \right\|\_{W,V} \tag{36}$$

which says that the solutions *ϕJ/L(E)* based on the subsequent models of the hierarchy give in the limit ( ∞) the solution *ϕ*<sup>3</sup>*<sup>D</sup>* of the highest model of the hierarchy, i.e. the model of threedimensional dielectricity. In the case of the applied pure models, and 3*D, L ≡ J*.

The norm applied for the model comparisons is based on the electrostatic energy , i.e.

$$\left\|\boldsymbol{\phi}^{J/L(E)}\right\|\_{\boldsymbol{W},\boldsymbol{V}} = \frac{1}{2}\int\_{\boldsymbol{V}} \boldsymbol{d}^{\top}(\boldsymbol{\phi}^{J/L(E)}) \cdot \boldsymbol{E}(\boldsymbol{\phi}^{J/L(E)}) \cdot \boldsymbol{dV} \tag{37}$$

with and *E* being the electric displacement and electric field vectors, respectively.

#### **4.3. Hierarchical approximations**

The hierarchical approximations [6, 11] applied to the hierarchy of the 3D-based elastic models can be defined as follows [5, 8]:

$$M = RM, \ I \equiv K = 1 \Longrightarrow \mathfrak{u}^{q(M), hp} = \mathfrak{u}^{bp}$$

$$M \in MI \mid RM, \ I \geq 2, \ K = 1 \Longrightarrow \mathfrak{u}^{q(M), hp} = \mathfrak{u}^{bpq \wedge hp}$$

$$M \in MI, \ I \equiv K \geq 2 \Longrightarrow \mathfrak{u}^{q(M), hp} = \mathfrak{u}^{bpq} \tag{38}$$

$$M = \mathfrak{X}D \mid RM, \ I \to \infty, \ K = 1 \Longrightarrow \mathfrak{u}^{q(M), hp} = \mathfrak{u}^{bpq \wedge hp}$$

$$M = \mathfrak{X}D, \ I \equiv K \to \infty \Longrightarrow \mathfrak{u}^{q(M), hp} = \mathfrak{u}^{bpq}$$

Above we applied the general notation, (),ℎ, valid for any hierarchical model, and the equivalent symbols of the approximations for each particular model (on the right side of the above relations), where ℎ represents the generalized size of the element, denotes the longitudinal order of approximation and () is the transverse order of approximation, equivalent to the order of the theory, i.e. () (). The specific approximations are either two-dimensional (ℎ) or three-dimensional (ℎ, ℎ) or mixed (ℎ/ℎ, ℎ/ℎ).

The above approximations lead in the limit (1/ℎ ∞, ∞) to the exact solution of the appropriate mechanical model of the order /, i.e.

$$\lim\_{\nu\_{1/h,p\to\nu}} \left\| \mu^{q(M),hp} \right\|\_{U,\downarrow^{\tilde{V}}} = \left\| \mu^{I/K(M)} \right\|\_{U,V} \tag{39}$$

Combination of the models defined with (30) and the corresponding approximations of (38) leads to the hierarchical numerical models of the following property:

$$\lim\_{M \to 1, I} \left| \lim\_{I \to \epsilon} \left( \lim\_{1/h, p \to \epsilon} \left\| \boldsymbol{\mu}^{q(M), hp} \right\|\_{U, \boldsymbol{\omega}^{\boldsymbol{\prime}}} \right) \right| = \lim\_{K \to 1, I} \left( \lim\_{I \to \epsilon} \left\| \boldsymbol{\mu}^{I/K(M)} \right\|\_{U, \boldsymbol{\mathcal{V}}} \right) = \left\| \boldsymbol{\mu}^{\boldsymbol{\varepsilon} D} \right\|\_{U, \boldsymbol{\mathcal{V}}} \tag{40}$$

As it can be seen above, the solutions to these models tend in the limit to the exact solution of the highest mechanical model (3).

By analogy, the numerical approximations of the 3D-based dielectric models are:

$$\begin{aligned} E \in EJ, J \equiv L \ge 2 \implies \phi^{\rho(E), \hbar x} = \phi^{\hbar \eta \rho} \\ E = 3D, J \equiv L \to \infty \implies \phi^{\rho(E), \hbar x} = \phi^{\hbar \pi x} \end{aligned} \tag{41}$$

Above, for the sake of further considerations, the longitudinal approximation order is denoted as and the transverse one as (), where () (). The principal property of the above electric potential approximations reads

$$\lim\_{\nu\_{\perp}, \mu\_{\perp} \to \ast} \left\| \phi^{\rho(E), \hbar x} \right\|\_{w, \square^{\prime}} = \left\| \phi^{J^{\prime} L(E)} \right\|\_{w, V} \tag{42}$$

while for the hierarchical numerical dielectric models, being the combination of the electric models (35) and the above approximations (41), one has

$$\lim\_{L=1,J} \left[ \lim\_{J \to \alpha} \left( \lim\_{l \ne k, x \to \alpha} \left\| \phi^{\circ (E), hx} \right\|\_{W, \cup \stackrel{\circ}{V}} \right) \right] = \lim\_{L=1,J} \left( \lim\_{J \to \alpha} \left\| \phi^{J/L(E)} \right\|\_{W, \stackrel{\circ}{V}} \right) = \left\| \phi^{1D} \right\|\_{W, \vee} \tag{43}$$

where the limit solutions and 3*<sup>D</sup>* are present.

#### **4.4. Error estimation**

It is well known [17–19, 22] that the equilibrated residual methods of error estimation require solution of the following approximated local (element) problems of mechanical equilibrium:

$$
\stackrel{\circ}{\text{B}}(\mathfrak{u}^{\mathcal{Q},HP},\mathfrak{v}^{\mathcal{Q},HP}) - \stackrel{\circ}{\text{L}}(\mathfrak{v}^{\mathcal{Q},HP}) - \stackrel{\circ}{\int\_{S^{i}\,(S\_{P}\cup\mathcal{S}\_{D})}} (\mathfrak{v}^{\mathcal{Q},HP})^{\top} \left< \mathfrak{r}(\mathfrak{u}^{\mathcal{q},bp}) \right> \stackrel{\circ}{\text{dS}} = 0 \tag{44}
$$

where B and L are the bilinear and linear forms representing virtual strain energy and the virtual work of the external forces. The last term stands for the virtual work of the inter-element stress fluxes . The element size in the local problems is denoted as , the longitudinal approximation order is , while the transverse one is ().

The above three terms can be defined as follows

( ), /( ) , , 1/ , lim *<sup>e</sup>* = *q M hp IKM*

Combination of the models defined with (30) and the corresponding approximations of (38)

( ) ( ) ( ), /( ) <sup>3</sup> 1, 1/ , , 1, , , lim lim lim *<sup>e</sup>* = lim lim <sup>=</sup> *q M hp IKM <sup>D</sup> K I I hp* = ®¥ ®¥ *U V* È *K II* = ®¥ *U V U V*

As it can be seen above, the solutions to these models tend in the limit to the exact solution of

,2 = =3 , =

( ), ( ),

r p

f

f

Above, for the sake of further considerations, the longitudinal approximation order is denoted as and the transverse one as (), where () (). The principal property

> ( ), /() , , 1/ , lim *<sup>e</sup>* = *E h J LE w V W V h*

while for the hierarchical numerical dielectric models, being the combination of the electric

( ) ( ) ( ), /() <sup>3</sup> =1, 1/ , , =1, , , lim lim lim *<sup>e</sup>* = lim lim <sup>=</sup> *E h J LE <sup>D</sup> L JJ h W V L JJ W V W V*

It is well known [17–19, 22] that the equilibrated residual methods of error estimation require solution of the following approximated local (element) problems of mechanical equilibrium:

> ,, , , , \( ) B( , ) L( ) ( ) ( ) = 0 *<sup>e</sup> P D*

*e e e e Q HP Q HP Q HP Q H P T q hp*

ê ú ë û (43)

*SS S dS* <sup>È</sup> - - áñ ò *uv v v ru* (44)

 f

r p

*Eh h Eh h*

 f

 f  pr

> pp

®¥ <sup>È</sup> (42)

ff

(41)

By analogy, the numerical approximations of the 3D-based dielectric models are:

º ®¥Þ

Î º ³Þ

r p

f

®¥ ®¥ È ®¥

p

models (35) and the above approximations (41), one has

p

where the limit solutions and 3*<sup>D</sup>* are present.

**4.4. Error estimation**

r p

f

é ù

leads to the hierarchical numerical models of the following property:

*E EJ J L E DJ L*

of the above electric potential approximations reads

é ù

the highest mechanical model (3).

14 Perusal of the Finite Element Method

*U V U V h p*®¥ <sup>È</sup> *u u* (39)

ê ú ë û *u uu* (40)

$$\stackrel{\varepsilon}{\mathbf{B}}(\mathfrak{u}^{Q,HP},\mathfrak{v}^{Q,HP}) = \int\_{\stackrel{\varepsilon}{V}} \mathfrak{e}^{T}(\mathfrak{v}^{Q,HP}) \, \mathbf{D}\mathfrak{e}(\mathfrak{u}^{Q,HP}) \, \stackrel{\varepsilon}{dV} = \left(\stackrel{\varepsilon}{\mathbf{\tilde{v}}}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{\mathbf{\tilde{k}}}\_{M} \stackrel{\varepsilon}{\mathbf{q}}^{Q,HP},$$

$$\stackrel{\varepsilon}{\mathbf{\tilde{L}}}(\mathfrak{v}^{Q,HP}) = \int\_{\stackrel{\varepsilon}{V}} \left(\mathfrak{v}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{V} \, \stackrel{\varepsilon}{dV} + \int\_{\stackrel{\varepsilon}{S}} \left(\mathfrak{v}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{\mathbf{p}} \, \stackrel{\varepsilon}{dS} = \left(\stackrel{\varepsilon}{\mathbf{\tilde{v}}}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{\mathbf{\tilde{f}}}\_{M} + \stackrel{\varepsilon}{f}\_{S}), \tag{45}$$

$$\int\_{\stackrel{\varepsilon}{S}(\vee P \cup \mathcal{S}\_{W})} \left(\mathfrak{v}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{\mathbf{\tilde{f}}}(\mathfrak{v}^{q,hp}) \stackrel{\varepsilon}{dS} = \left(\stackrel{\varepsilon}{\mathbf{\tilde{v}}}^{Q,HP}\right)^{T} \stackrel{\varepsilon}{f}\_{R}$$

with being the elasticity constants matrix. Thus, Eq. (44) can be written in the language of finite elements:

$$\stackrel{c}{k}\_{M}\stackrel{c}{q}^{Q,HP} = \stackrel{c}{f}\_{V} + \stackrel{c}{f}\_{S} + \stackrel{c}{f}\_{R} \tag{46}$$

where denote element stiffness matrix, and element forces vectors due to mass, surface and inter-element stress loadings, while and represent solution and admissible displacement dof vectors in the local problem. The solution vector is then utilized in the error estimation. The collection of the element solutions obtained this way leads to the global error estimate which constitutes the upper bound of the true error [18, 19, 22].

The above approach can be extended onto the mechanical problem of free vibration by replacing the linear form of (44) by the virtual work of the inertia forces [26]. These forces are: , where is the element mass (or inertia) matrix, while the natural frequencies and the vector of the element displacement amplitudes are taken from the global problem solution. These forces replace and in (45) and (46). Note that now the element solutions, analogous to this obtained from (46), do not guarantee the upper boundedness of the global error by its residual estimate.

In the case of electrostatic dielectricity, the local problems of the equilibrated residual method take the form [23]:

$$\stackrel{\circ}{\text{b}}(\phi^{\text{p},H\text{II}},\psi^{\text{p},H\text{II}}) - \stackrel{\circ}{\text{l}}(\psi^{\text{p},H\text{II}}) - \int\_{S^{\circ}(\cup\_{\mathbb{Q}} \cup \cup\_{\mathbb{F}})} \psi^{\text{p},H\text{II}} \langle \stackrel{\circ}{h}(\phi^{\text{p},h\text{r}}) \rangle \stackrel{\circ}{dS} = 0 \tag{47}$$

with the bilinear and linear forms, and , representing virtual electrostatic energy and the virtual work of external charges. The right term above is equal to the virtual work of the equilibrated charge flux . The indices , Π, P ≡ P() stand for the element size and the element longitudinal and transverse approximation orders.

Taking into considerations the below definitions:

$$\begin{aligned} \,^\ell \boldsymbol{\Phi}^\ell(\boldsymbol{\phi}^{\mathrm{P,H}\boldsymbol{\Pi}}, \boldsymbol{\psi}^{\mathrm{P,H}\boldsymbol{\Pi}}) &= \int\_V \,^\ell \boldsymbol{F}^\ell(\boldsymbol{\psi}^{\mathrm{P,H}\boldsymbol{\Pi}}) \, \mathbf{y} \mathbf{E} (\boldsymbol{\phi}^{\mathrm{P,H}\boldsymbol{\Pi}}) \, ^\ell \boldsymbol{\tilde{V}} &= \left(\boldsymbol{\tilde{\Psi}}^{\mathrm{P,H}\boldsymbol{\Pi}}\right)^T \, ^\ell \boldsymbol{\tilde{k}}\_E \, ^\ell \boldsymbol{\tilde{\Psi}}^{\mathrm{P,H}\boldsymbol{\Pi}} \, \mathbf{y} \\ \,^\ell \boldsymbol{\tilde{\Psi}}^{\mathrm{P,H}\boldsymbol{\Pi}}) &= \int\_{\boldsymbol{\tilde{\mathcal{S}}}\_{\boldsymbol{Q}}} \, \boldsymbol{\psi}^{\mathrm{P,H}\boldsymbol{\Pi}} \, \boldsymbol{c} \, \stackrel{\ell}{\operatorname{\mathbf{S}}} &= \left(\boldsymbol{\tilde{\Psi}}^{\mathrm{P,H}\boldsymbol{\Pi}}\right)^T \boldsymbol{f}\_{\boldsymbol{Q}'} \\ \int\_{\boldsymbol{\tilde{\mathcal{S}}} (\boldsymbol{\mathcal{S}}\_{\boldsymbol{Q}} \boldsymbol{\psi}^{\mathrm{P,H}})} \, \boldsymbol{\psi}^{\mathrm{P,H}\boldsymbol{\Pi}} \, \left(\boldsymbol{\tilde{\mathcal{S}}}^{\mathrm{P,H}\boldsymbol{\Pi}}\right)^T \boldsymbol{f}\_{\boldsymbol{H}} \end{aligned} \tag{48}$$

where *γ* is the dielectricity constants matrix, one can transform (47) into the following local finite element equation

$$
\stackrel{\circ}{k}\_E \stackrel{\circ}{\Phi}^{\mathbb{P}, H \Pi} = \stackrel{\circ}{f}\_Q + \stackrel{\circ}{f}\_H \tag{49}
$$

Above denote element dielectricity matrix, and element vectors due to surface and inter-element charges, while and are the solution and admissible electric potential dof vectors of the local problem. This element solution is applied to the error estimation of the dielectric problems. The global error estimate obtained with use of the above element solutions upper-bounds the true global error.

#### **4.5. Adaptive strategy**

The adaptive strategy for the complex problems of elasticity is based on the Texas Three-Step Strategy [24]. The latter strategy consists in solution of the global problem thrice on the socalled initial, intermediate and final meshes. The intermediate mesh is obtained from the initial one based on the initial mesh estimated values of element errors and the *hp* convergence theory relating these errors to the discretization parameter ℎ. Thanks to this the *h*-adaptation is performed. The final mesh is obtained from the intermediate one in the analogous way through *p*-refinement. This process takes advantage of the intermediate mesh estimated errors and the relation between these errors and the discretization parameter *p*. The original strategy can be enriched with the fourth step [5, 27], called the modification one, which is performed on the initial mesh and is applied when the unpleasant numerical phenomena due to the improper solution limit, numerical locking or boundary layers appear in the mechanical problem. The purpose of this additional step is to get rid of the numerical consequences of the mentioned phenomena before the error-controlled *hp*-adaptivity is started. The control of this step is based on the sensitivity analysis of the local (element) problems solutions to these three phenomena.

The above four-step strategy can be easily extended onto the complex dielectric problems. It has to account only for the boundary layer phenomenon within the modification step, as two other phenomena do not appear in the problems of dielectrics. Other steps do not change.

## **5. Problems within the methodology**

with the bilinear and linear forms, and , representing virtual electrostatic energy and the virtual work of external charges. The right term above is equal to the virtual work of the

equilibrated charge flux . The indices , Π, P ≡ P() stand for the element size and

where *γ* is the dielectricity constants matrix, one can transform (47) into the following local

Above denote element dielectricity matrix, and element vectors due to surface and

inter-element charges, while and are the solution and admissible electric potential dof vectors of the local problem. This element solution is applied to the error estimation of the dielectric problems. The global error estimate obtained with use of the above element solutions

The adaptive strategy for the complex problems of elasticity is based on the Texas Three-Step Strategy [24]. The latter strategy consists in solution of the global problem thrice on the socalled initial, intermediate and final meshes. The intermediate mesh is obtained from the initial one based on the initial mesh estimated values of element errors and the *hp* convergence theory relating these errors to the discretization parameter ℎ. Thanks to this the *h*-adaptation is performed. The final mesh is obtained from the intermediate one in the analogous way through *p*-refinement. This process takes advantage of the intermediate mesh estimated errors and the relation between these errors and the discretization parameter *p*. The original strategy can be enriched with the fourth step [5, 27], called the modification one, which is performed on the initial mesh and is applied when the unpleasant numerical phenomena due to the improper solution limit, numerical locking or boundary layers appear in the mechanical problem. The purpose of this additional step is to get rid of the numerical consequences of the mentioned

(48)

(49)

the element longitudinal and transverse approximation orders.

Taking into considerations the below definitions:

finite element equation

16 Perusal of the Finite Element Method

upper-bounds the true global error.

**4.5. Adaptive strategy**

In this section of the chapter we consider the main difficulties in generalization of the hierarchical models, hierarchical approximations, equilibration residual method of error estimation and three- or four-step adaptive strategies, presented in the previous sections for the problems of elasticity and dielectricity, onto coupled problems of piezoelectricity.

## **5.1. Hierarchical model and approximation issues**

The first task here is to compose the elastic and dielectric models, defined with (28) and (33), respectively, into one consistent hierarchy of the piezoelectric media. Our proposition on how to perform this task follows from the main feature of both component hierarchies which are characterized with the independently changing orders and of the elastic and dielectric models, as shown in (30) and (35). Because of this we can propose the definition

$$P \in \mathcal{P}, \ P = \left\{ \left( M, E \right) \colon \ M \in \mathcal{M}, \ E \in \mathcal{E} \right\} \tag{50}$$

which determines the hierarchy P of piezoelectric models as composed of all combinations (*M,E*) of the elastic and dielectric models. Even though the following property:

$$\lim\_{\ell \to 1, l} \left\langle \lim\_{I \to \alpha} \left[ \lim\_{L \to 1, J} \left( \lim\_{J \to \alpha} \left\| (\mathfrak{u}^{I/K(M)}, \phi^{J/L(E)}) \right\|\_{\mathcal{X}, V} \right) \right] \right\rangle = \left\| (\mathfrak{u}^{\text{3D}}, \phi^{\text{3D}}) \right\|\_{\mathcal{X}, V} \tag{51}$$

is valid in the limit, the ordering of the coupled electro-mechanical solutions measured in the energy norm is not unique due to different signs of the strain (or elastostatic) , electrostatic and coupling parts of the co-energy: = + , i.e.

$$\left\| \left( \mathfrak{u}^{I/K(M)}, \mathfrak{d}^{J/L(E)} \right) \right\|\_{\mathcal{X}, \mathcal{V}} = \frac{1}{2} \int\_{\mathcal{V}} \left| \left. E^{\top} \mathfrak{Y} E - \mathfrak{s}^{\top} \mathbf{D} \mathfrak{s} + 2 \mathfrak{s}^{\top} \mathbf{C} E \right| dV \right| dV \tag{52}$$

where = (*I/K*(*M*) ) and *E = E(ϕJ/L(E))*, while *C* is the coupling (piezoelectric) constants matrix.

The most general case of the ℎ-approximation within the coupled field of displacements and electric potential may include totally independent ℎ and approximations within the mechanical and electric fields. However, such an approach requires the vice-versa projections of the displacements and potential solutions between the independent ℎ-meshes of each field. The approach results in the additional projection error which should be included in the error estimation. Because of this we propose the simplified approach which consists in application of the common ℎ mesh and the independent and approximations within the displacements and electric potential fields. This assumption leads to the following limit property of the coupled solution

$$\lim\_{\varepsilon \searrow 1/h, \rho \to \varepsilon} \left( \lim\_{1/h, \pi \to \varepsilon} \left\| (\mathbf{u}^{\rho(M), h\rho}, \phi^{\rho(E), h\pi}) \right\|\_{\mathcal{X}, \overset{\varepsilon}{\to} V} \right) = \left\| (\mathbf{u}^{I/K(M)}, \phi^{J/L(E)}) \right\|\_{\mathcal{X}, V} \tag{53}$$

It should be noticed that the monotonic character of the co-energy of the consecutive approximate solutions is not guaranteed here due to the coupled character of the electro-mechanical field.

Practical realization of the above concepts of hierarchical modelling and approximations is implemented by means of the ℎ- and ℎ-adaptive piezoelectric finite elements, presented in **Figure 1**. As said before, ℎ represents the assumed characteristic element size, common for both fields, while and denote independent longitudinal approximation orders within the displacements and electric potential fields. The transverse approximation orders and of the mechanical and electric fields, respectively, are equivalent to the independent hierarchical orders and of the elastic and dielectric models, i.e. , .

The normalized versions of the prismatic adaptive elements are presented above, where the 3D-based solid (or hierarchical shell), transition (an example of) and first-order shell mechanical elements are combined with the three-dimensional (or 3D-based symmetric-thickness) dielectric elements. In the figure the normalized coordinates are defined as 1,2,3 ∈ (0,1), while the vertex, mid-edge, mid-base, mid-side and middle nodes of mechanical character are denoted as either 1, 2, …, 21 or 1, 2, …, 18 or 1, 2, …, 14, while the corresponding electric ones are marked with 1, 2, …, 21.

#### **5.2. Problems within error estimation**

In order to generalize the equilibrated residual method for piezoelectricity the local elastic (44) and dielectric (47) problems have to be replaced with the coupled stationary problems describing the electro-mechanical equilibrium:

$$\begin{aligned} \overset{\epsilon}{\mathsf{B}}(\mathsf{u}^{\mathcal{Q},HP},\mathsf{v}^{\mathcal{Q},HP}) - \overset{\epsilon}{\mathsf{C}}(\mathsf{\phi}^{\mathcal{P},HP},\mathsf{v}^{\mathcal{Q},HP}) &= \overset{\epsilon}{\mathsf{L}}(\mathsf{v}^{\mathcal{Q},HP}) + \int\_{\mathcal{S}\,\,(\mathcal{S}\_{\mathcal{P}}\cup\mathcal{S}\_{\mathcal{D}})}^{\epsilon} (\mathsf{v}^{\mathcal{Q},HP})^{\mathsf{T}} \langle \mathsf{r} \rangle \overset{\epsilon}{\mathsf{d}} \overset{\epsilon}{\mathsf{S}} \\ \overset{\epsilon}{\mathsf{C}}(\mathsf{u}^{\mathcal{Q},HP},\mathsf{v}^{\mathcal{P},HP}) + \overset{\epsilon}{\mathsf{b}}(\mathsf{\phi}^{\mathcal{P},HP},\mathsf{v}^{\mathcal{P},HP}) &= \overset{\epsilon}{\mathsf{I}}(\mathsf{v}^{\mathcal{P},HP}) + \int\_{\mathcal{S}\,\,(\mathcal{S}\_{\mathcal{Q}}\cup\mathcal{S}\_{\mathcal{F}})}^{\epsilon} \mathsf{v}^{\mathcal{P},HP} \langle \mathsf{h} \rangle \overset{\epsilon}{\operatorname{d}} \end{aligned} \tag{54}$$

Problems of Hierarchical Modelling and *hp*-Adaptive Finite Element Analysis in Elasticity, Dielectricity... http://dx.doi.org/10.5772/64892 19

chanical and electric fields. However, such an approach requires the vice-versa projections of the displacements and potential solutions between the independent ℎ-meshes of each field. The approach results in the additional projection error which should be included in the error estimation. Because of this we propose the simplified approach which consists in application of the common ℎ mesh and the independent and approximations within the displacements and electric potential fields. This assumption leads to the following limit property of the

> ( ) ( ), ( ), / ( ) /() 1/ , 1/ , , , lim lim ( , ) ( , ) *<sup>e</sup>*

*hp h Z V Z V*

r p

f

orders and of the elastic and dielectric models, i.e. , .

electric ones are marked with 1, 2, …, 21.

describing the electro-mechanical equilibrium:

**5.2. Problems within error estimation**

y

*u*

p *q M hp E h IKM J LE*

It should be noticed that the monotonic character of the co-energy of the consecutive approximate solutions is not guaranteed here due to the coupled character of the electro-mechanical

Practical realization of the above concepts of hierarchical modelling and approximations is implemented by means of the ℎ- and ℎ-adaptive piezoelectric finite elements, presented in **Figure 1**. As said before, ℎ represents the assumed characteristic element size, common for both fields, while and denote independent longitudinal approximation orders within the displacements and electric potential fields. The transverse approximation orders and of the mechanical and electric fields, respectively, are equivalent to the independent hierarchical

The normalized versions of the prismatic adaptive elements are presented above, where the 3D-based solid (or hierarchical shell), transition (an example of) and first-order shell mechanical elements are combined with the three-dimensional (or 3D-based symmetric-thickness) dielectric elements. In the figure the normalized coordinates are defined as 1,2,3 ∈ (0,1), while the vertex, mid-edge, mid-base, mid-side and middle nodes of mechanical character are denoted as either 1, 2, …, 21 or 1, 2, …, 18 or 1, 2, …, 14, while the corresponding

In order to generalize the equilibrated residual method for piezoelectricity the local elastic (44) and dielectric (47) problems have to be replaced with the coupled stationary problems

\( )

È

*SS S*

*e P D*

ò

*e Q F*

ò

\( )

È

 y *dS*

(54)

*h dS*

*SS S*

, , P, , , ,

*<sup>e</sup> <sup>e</sup> <sup>e</sup> e e Q HP Q HP H Q HP Q HP Q HP T*

*uv v v v r*

*<sup>e</sup> <sup>e</sup> <sup>e</sup> e e Q HP H H H <sup>H</sup> <sup>H</sup>*

 y

P PP P P


+ + á ñ

B( , ) C( , ) = L( ) ( )

C( , ) b( , ) = l( )

P

f

 fy

, P, P, P, P, P,

 f

®¥ ®¥ <sup>È</sup> *u u* <sup>=</sup> (53)

coupled solution

18 Perusal of the Finite Element Method

field.

**Figure 1.** Piezoelectric solid/three-dimensional (top), transition/three-dimensional (middle), shell/three-dimensional (bottom) elements.

where and , while the above coupling forms are determined as follows:

$$\begin{aligned} \stackrel{\circ}{\mathbf{C}}(\boldsymbol{\phi}^{\mathbb{P},H\Pi},\boldsymbol{\nu}^{\mathbb{Q},H\mathbf{P}}) &= \int\_{\stackrel{\circ}{\boldsymbol{V}}} \boldsymbol{\mathfrak{e}}^{\top}(\boldsymbol{\nu}^{\mathbb{Q},H\mathbf{P}}) \, \mathbf{C} \, \boldsymbol{E} (\boldsymbol{\phi}^{\mathbb{P},H\Pi}) \, \stackrel{\circ}{dV} = (\overset{\circ}{\boldsymbol{\nu}^{\mathbb{Q},H\mathbf{P}}})^{\top} \stackrel{\circ}{\mathbf{k}}\_{\mathcal{C}} \stackrel{\circ}{\boldsymbol{\Phi}}^{\mathbb{P},H\Pi} \\ \stackrel{\circ}{\mathbf{C}}(\boldsymbol{\mu}^{\mathbb{Q},H\mathbf{P}},\boldsymbol{\nu}^{\mathbb{P},H\Pi}) &= \int\_{\stackrel{\circ}{\boldsymbol{V}}} \boldsymbol{E}^{\top}(\boldsymbol{\nu}^{\mathbb{P},H\Pi}) \, \mathbf{C}^{\top} \boldsymbol{\varepsilon} (\boldsymbol{\mu}^{\mathbb{Q},H\mathbf{P}}) \, \stackrel{\circ}{dV} = (\overset{\circ}{\boldsymbol{\nu}^{\mathbb{P},H\Pi}})^{\top} \stackrel{\circ}{\mathbf{k}}^{\mathbb{P}}\_{\mathcal{C}} \stackrel{\circ}{\mathbf{q}}^{\mathbb{Q},H\mathbf{P}} \end{aligned} \tag{55}$$

Definitions (45), (48) and (55) allow to rewrite (54) in the finite element language:

$$\begin{aligned} \stackrel{\varepsilon}{k}\_M \stackrel{\varepsilon}{q}^{Q,HP} - \stackrel{\varepsilon}{k}\_C \stackrel{\varepsilon}{\mathfrak{p}}^{P,H\Pi} &= \stackrel{\varepsilon}{f}\_V + \stackrel{\varepsilon}{f}\_S + \stackrel{\varepsilon}{f}\_R\\ \stackrel{\varepsilon}{k}\_C \stackrel{\varepsilon}{q}^{Q,HP} + \stackrel{\varepsilon}{k}\_E \stackrel{\varepsilon}{\mathfrak{p}}^{P,H\Pi} &= \stackrel{\varepsilon}{f}\_Q + \stackrel{\varepsilon}{f}\_H \end{aligned} \tag{56}$$

The main disadvantage of the local solutions and obtained from the above set of equations is that they do not lead to the upper bound of the global approximation error of the global problem (12) due to the coupled character of piezoelectricity. Note that the upper bound property is present in the cases of pure elasticity and pure dielectricity. In these circumstances we propose the simplified approach which consists in decoupling of the local mechanical and electric fields:

$$\begin{aligned} \stackrel{\varepsilon}{k}\_M \stackrel{\varepsilon}{q}^{Q,HP} &= \stackrel{\varepsilon}{k}\_C \stackrel{\varepsilon}{\phi}^{\rho,h\pi} + \stackrel{\varepsilon}{f}\_V + \stackrel{\varepsilon}{f}\_S + \stackrel{\varepsilon}{f}\_R\\ \stackrel{\varepsilon}{k}\_E \stackrel{\varepsilon}{\phi}^{P,H\Pi} &= \ -\stackrel{\varepsilon}{k}\_C \stackrel{\varepsilon}{q}^{\rho,hp} + \stackrel{\varepsilon}{f}\_O + \stackrel{\varepsilon}{f}\_H \end{aligned} \tag{57}$$

The simplified solutions of the above two decoupled problems is suggested for serving the approximated error assessment of the coupled field.

In the case of the coupled free-vibration problem we propose to apply the same approach as for the purely elastic vibrations. This means that the inertia forces introduced in Subsection 4.4 have to replace the volume and surface forces in the first equations (56) and (57). The element constitutive stiffness matrix in both equations has to be changed for the following sum: + , where stands for the element geometric stiffness matrix due to the initial stresses which correspond to the electro-mechanical equilibrium. Because of this one may skip the electric charges in the second equations (56) and (57).

#### **5.3. Adaptivity control matters**

The main problem with the piezoelectricity in the context of adaptivity control of the three- or four-step ℎ-adaptive procedure is that the convergence theorem for the finite element approximation of the coupled piezoelectric field is not at one's disposal. In these circumstances we propose to use the ℎ-convergence exponents of the elasticity and dielectricity problems for the mechanical and electric components of the coupled field.

With this assumption in mind we generalize the adaptive scheme applied successfully for the complex elastic structures [28]. It can also be adopted for the dielectricity problems. The generalization is presented in **Figure 2** in the form of a block diagram, where = 1,2,3,4 represent the consecutive steps of the algorithm—the initial, modification, intermediate and final ones.

**Figure 2.** Four-step *hp*-adaptive scheme for complex piezoelectric or piezoelectric-elastic problems.

## **6. Conclusions**

where and , while the above coupling forms are determined as

The main disadvantage of the local solutions and obtained from the above set of equations is that they do not lead to the upper bound of the global approximation error of the global problem (12) due to the coupled character of piezoelectricity. Note that the upper bound property is present in the cases of pure elasticity and pure dielectricity. In these circumstances we propose the simplified approach which consists in decoupling of the local mechanical and

The simplified solutions of the above two decoupled problems is suggested for serving the

In the case of the coupled free-vibration problem we propose to apply the same approach as for the purely elastic vibrations. This means that the inertia forces introduced in Subsection 4.4 have to replace the volume and surface forces in the first equations (56) and (57). The

element constitutive stiffness matrix in both equations has to be changed for the following

sum: + , where stands for the element geometric stiffness matrix due to the initial stresses which correspond to the electro-mechanical equilibrium. Because of this one may skip

The main problem with the piezoelectricity in the context of adaptivity control of the three- or four-step ℎ-adaptive procedure is that the convergence theorem for the finite element

approximated error assessment of the coupled field.

the electric charges in the second equations (56) and (57).

**5.3. Adaptivity control matters**

Definitions (45), (48) and (55) allow to rewrite (54) in the finite element language:

(55)

(56)

(57)

follows:

20 Perusal of the Finite Element Method

electric fields:

In the presented chapter we showed how to generalize hierarchical modelling, ℎ-adaptive hierarchical approximations, the equilibrated residual method of error estimation and the fourstep adaptive procedure, originally applied to elasticity and possible also in dielectricity, for the coupled pieozoelectric problems.

We suggest to combine the hierarchical models for complex elastic structures with the analogous models of dielectricity in order to obtain all coupled combinations of the component mechanical and electric models. In the case of the approximations, we suggest to apply the common ℎ-mesh and independent *p* and approximations for the displacement and electric potential fields.

In the error estimation, we propose to decouple the mechanical and electric fields in the local problems of the residual approach.

Our four-step ℎ-adaptive algorithm is based on the convergence theorems for the purely elastic and purely dielectric problems. These theorems are applied to the intermediate and final meshes generation.

## **Acknowledgements**

The support of the Polish Scientific Research Committee (now the National Science Centre) under the research grant no. N N504 5153 040 is thankfully acknowledged.

## **Nomenclature**

 – displacement amplitude vector components ( = 1,2,3),

 – a node with displacement degrees of freedom (dofs), = 1,2,…,21,

b – a bilinear form representing the element virtual electrostatic energy,

 – a node with electric potential dofs, = 1,2,…,21,

B – a bilinear form representing the element virtual strain energy,

*c* – the electric surface charge,

– a bilinear form representing the element coupling electro-mechanical energy,

– components of the piezoelectricity tensor (*i,k,l* = 1,2,3),

– the piezoelectric constants matrix,

 – components of the electric displacement vector ( = 1,2,3),

– the electric displacement vector,

– elasticity tensor components, *i,j,k,l* = 1,2,3,


step adaptive procedure, originally applied to elasticity and possible also in dielectricity, for

We suggest to combine the hierarchical models for complex elastic structures with the analogous models of dielectricity in order to obtain all coupled combinations of the component mechanical and electric models. In the case of the approximations, we suggest to apply the common ℎ-mesh and independent *p* and approximations for the displacement and electric

In the error estimation, we propose to decouple the mechanical and electric fields in the local

Our four-step ℎ-adaptive algorithm is based on the convergence theorems for the purely elastic and purely dielectric problems. These theorems are applied to the intermediate and

The support of the Polish Scientific Research Committee (now the National Science Centre)

under the research grant no. N N504 5153 040 is thankfully acknowledged.

– a node with displacement degrees of freedom (dofs), = 1,2,…,21,

– a bilinear form representing the element virtual electrostatic energy,

– a bilinear form representing the element coupling electro-mechanical energy,

– a bilinear form representing the element virtual strain energy,

– components of the piezoelectricity tensor (*i,k,l* = 1,2,3),

– components of the electric displacement vector ( = 1,2,3),

– displacement amplitude vector components ( = 1,2,3),

– a node with electric potential dofs, = 1,2,…,21,

the coupled pieozoelectric problems.

22 Perusal of the Finite Element Method

problems of the residual approach.

final meshes generation.

**Acknowledgements**

*c* – the electric surface charge,

– the piezoelectric constants matrix,

– the electric displacement vector,

– elasticity tensor components, *i,j,k,l* = 1,2,3,

**Nomenclature**

b 

B 

potential fields.


– the global piezoelectric (coupling) matrix,

– the global dielectric matrix,

– the global geometric stiffness matrix,

– the global (constitutive) stiffness matrix,

– a linear form equal to the element virtual work of the external charges,

– a linear form representing the element virtual work of the external forces,

– the element mass (inertia) matrix,

– the number of a dof function,

– mechanical model,

M – the hierarchy of mechanical models,

– the global mass (inertia) matrix,

– the number of a natural frequency,

 – vector components ( = 1,2,3) of the normal to the body surface,

– the global number of dofs (degrees of freedom),

 – the element approximation order or the longitudinal order of approximation for displacements,

 – surface load vector components,

– the longitudinal order of approximation for element displacements,

– piezoelectric model,

P – the hierarchy of piezoelectric models,

– the surface load vector,

– the transverse order of approximation for displacements,

, ℎ – the global displacement (or displacement amplitude) dofs vector,

– the element displacement (or displacement amplitude) dofs vector,

– the local problem transverse order of approximation for displacements,

– a vector of the equilibrated inter-element stresses,

– a mid-surface displacement dof function,


– the global piezoelectric (coupling) matrix,

– the global geometric stiffness matrix,

– the element mass (inertia) matrix,

M – the hierarchy of mechanical models,

– the global mass (inertia) matrix,

– the number of a natural frequency,

– surface load vector components,

P – the hierarchy of piezoelectric models,

– piezoelectric model,

– the surface load vector,

– the global number of dofs (degrees of freedom),

– the number of a dof function,

– mechanical model,

ments,

– the global (constitutive) stiffness matrix,

– a linear form equal to the element virtual work of the external charges,

– vector components ( = 1,2,3) of the normal to the body surface,

– the longitudinal order of approximation for element displacements,

, ℎ – the global displacement (or displacement amplitude) dofs vector,

– the element displacement (or displacement amplitude) dofs vector,

– the local problem transverse order of approximation for displacements,

– the transverse order of approximation for displacements,

– a vector of the equilibrated inter-element stresses,

– a mid-surface displacement dof function,

– the element approximation order or the longitudinal order of approximation for displace-

– a linear form representing the element virtual work of the external forces,

– the global dielectric matrix,

24 Perusal of the Finite Element Method


– the longitudinal approximation order of electric potential,

Π – the element longitudinal approximation order of electric potential,

– the transverse approximation order of electric potential,

– mass density of the body,

P – the element transverse approximation order of electric potential,

 – initial stress tensor components (*i,j* = 1,2,3),

– stress tensor components (*i,j* = 1,2,3),

– the stress vector,

, ℎ – the global electric potential (or potential amplitude) dofs vector,

– the local (element) electric potential dofs vector,

– the electric potential,

– the *m*th through-thickness electric potential dof function,

– the given electric potential on the body surface,

– the admissible electric potential,

– the admissible electric potential dofs vector of an element,

– a natural frequency,

– the *n*th natural frequency

## **Author details**

#### Grzegorz Zboiński

Address all correspondence to: zboi@imp.gda.pl

1 Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdańsk, Poland

2 Faculty of Technical Sciences, University of Warmia and Mazury, Olsztyn, Poland

## **References**

– the longitudinal approximation order of electric potential,

– the transverse approximation order of electric potential,

– initial stress tensor components (*i,j* = 1,2,3),

– stress tensor components (*i,j* = 1,2,3),

– mass density of the body,

26 Perusal of the Finite Element Method

– the stress vector,

– the electric potential,

– a natural frequency,

**Author details**

Grzegorz Zboiński

– the *n*th natural frequency

Π – the element longitudinal approximation order of electric potential,

P – the element transverse approximation order of electric potential,

, ℎ – the global electric potential (or potential amplitude) dofs vector,

– the admissible electric potential dofs vector of an element,

1 Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdańsk, Poland

2 Faculty of Technical Sciences, University of Warmia and Mazury, Olsztyn, Poland

– the local (element) electric potential dofs vector,

– the *m*th through-thickness electric potential dof function,

– the given electric potential on the body surface,

Address all correspondence to: zboi@imp.gda.pl

– the admissible electric potential,


Diez P, Wiberg NE, editors. Adaptive Modeling and Simulation 2005. Barcelona: CIMNE; 2005. pp. 321–330.

[28] Zboiński G. Unresolved problems of adaptive hierarchical modelling and *hp*-adaptive analysis within computational solid mechanics. In: Kuczma M, Wilmański K, editors. Advanced Structural Materials: Vol. 1. Computer Methods in Mechanics. Berlin: Springer-Verlag; 2010. Ch. 7, pp. 111–145.

[14] Zboiński G. 3D-based *hp*-adaptive first order shell finite element for modelling and analysis of complex structures: Part 2. Application to structural analysis. Int. J. Numer.

[15] Zboiński G, Ostachowicz W. An algorithm of a family of 3D-based solid-toshell transition, *hpq/hp*-adaptive finite elements. J. Theoret. Appl. Mech. 2000;38:791–

[16] Lammering R, Mesecke-Rischmann S. Multi-field variational formulation and related finite elements for piezoelectric shells. Smart Mat. Struct. 2003;12:904–

[17] Ainsworth M, Oden JT. A Posteriori Error Estimation in Finite Element Analysis. New

[18] Ainsworth M, Oden JT, Wu W. A posteriori error estimation for *hp*-approximation in

[19] Cho JR, Oden JT. A priori error estimations of *hp*-finite element approximations for hierarchical models of plate and shell-like structures. Comput. Methods Appl. Mech.

[20] Zboiński G. A posteriori error estimation for *hp*-approximation of the 3D-based first order shell model: Part I. Theoretical aspects. Appl. Math. Informat. Mech. 2003;8(1):

[21] Zboiński G. A posteriori error estimation for *hp*-approximation of the 3D-based first order shell model: Part II. Implementation aspects. Appl. Math. Informat. Mech.

[22] Zboiński G. Adaptive *hpq* finite element methods for the analysis of 3D-based models of complex structures: Part 2. A posteriori error estimation. Comput. Methods Appl.

[23] Zboiński G. Application of the element residual methods to dielectric and piezoelectric problems. In: Kleiber M, et al., editors. Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues. London: CRC Press;

[24] Oden JT. Error estimation and control in computational fluid dynamics: The O. C. Zienkiewicz lecture. In: Proc. Math. of Finite Elements (MAFELAP VIII). Uxbridge:

[25] Vokas C, Kasper M. Adaptation in coupled problems. Int. J. Comput. Math. Electr.

[26] Jasiński M, Zboiński G. On some *hp*-adaptive finite element method for natural

[27] Zboiński G. Numerical tools for a posteriori detection and assessment of the improper solution limit, locking and boundary layers in analysis of thin-walled structures. In:

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28 Perusal of the Finite Element Method

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2016. pp. 605–609.

#### **The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations**

Mahboub Baccouch Mahboub Baccouch

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64967

#### **Abstract**

We present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is \$(p + 1) \$th order convergent in the \$L^2\$-norm, when the space of piecewise polynomials of degree \$p\$ is used. A \$ (2p+1) \$th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order \$p+2\$ to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of the DG error is proportional to the \$ (p + 1) \$-degree right Radau polynomial. These results allow us to develop a residual-based *a posteriori* error estimator which is computationally simple, efficient, and asymptotically exact. The proposed *a posteriori* error estimator is proved to converge to the actual error in the \$L^2\$-norm with order \$p+2\$. Computational results indicate that the theoretical orders of convergence are optimal. Finally, a local adaptive mesh refinement procedure that makes use of our local *a posteriori* error estimate is also presented. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement.

**Keywords:** discontinuous Galerkin finite element method, ordinary differential equations, a priori error estimates, superconvergence, a posteriori error estimates, adaptive mesh refinement

## **1. Introduction**

In this chapter, we introduce and analyze the discontinuous Galerkin (DG) method applied to the following first-order initial-value problem (IVP)

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$\frac{d\vec{u}}{dt} = \vec{f}(t, \vec{u}), \quad t \in [0, T], \quad \vec{u}(0) = \vec{u}\_{0'} \tag{1}$$

where : [0, ] ℝ, <sup>0</sup> ∈ ℝ, and : [0, ]×ℝ ℝ. We assume that the solution exists and is unique and we would like to approximate it using a discontinuous piecewise polynomial space. According to the ordinary differential equation (ODE) theory, the condition ∈ 1([0, ]×ℝ) is sufficient to guarantee the existence and uniqueness of the solution to (1). We note that a general th-order IVP of the form () = (, , ′, …, ( 1)) with initial conditions (0) = 0, ′(0) = 1, …, ( 1)(0) = <sup>1</sup> can be converted into a system of equations in the form (1), where = [, ′, …, ( 1)] , (, )=[2, 3, …, , (, 1, …, )] , and <sup>0</sup> = [0, 1, …, <sup>1</sup>] .

The high-order DG method considered here is a class of finite element methods (FEMs) using completely discontinuous piecewise polynomials for the numerical solution and the test functions. The DG method was first designed as an effective numerical method for solving hyperbolic conservation laws, which may have discontinuous solutions. Here, we will discuss the algorithm formulation, stability analysis, and error estimates for the DG method solving nonlinear ODEs. DG method combines the best proprieties of the classical continuous finite element and finite volume methods such as consistency, flexibility, stability, conservation of local physical quantities, robustness, and compactness. Recently, DG methods become highly attractive and popular, mainly because these methods are high-order accurate, nonlinear stable, highly parallelizable, easy to handle complicated geometries and boundary conditions, and capable to capture discontinuities without spurious oscillations. The original DG finite element method (FEM) was introduced in 1973 by Reed and Hill [1] for solving steady-state first-order linear hyperbolic problems. It provides an effective means of solving hyperbolic problems on unstructured meshes in a parallel computing environment. The discontinuous basis can capture shock waves and other discontinuities with accuracy [2, 3]. The DG method can easily handle adaptivity strategies since the ℎ refinement (mesh refinement and coarsening) and the refinement (method order variation) can be done without taking into account the continuity restrictions typical of conforming FEMs. Moreover, the degree of the approximating polynomial can be easily changed from one element to the other [3]. Adaptivity is of particular importance in nonlinear hyperbolic problems given the complexity of the structure of the discontinuities and geometries involved. Due to local structure of DG methods, physical quantities such as mass, momentum, and energy are conserved locally through DG schemes. This property is very important for flow and transport problems. Furthermore, the DG method is highly parallelizable [4, 5]. Because of these nice features, the DG method has been analyzed and extended to a wide range of applications. In particular, DG methods have been used to solve ODEs [6–9], hyperbolic [5, 6, 10–19] and diffusion and convection diffusion [20–23] partial differential equations (PDEs), to mention a few. For transient problems, Cockburn and Shu [17] introduced and developed the so-called Runge-Kutta discontinuous Galerkin (RKDG) methods. These numerical methods use DG discretizations in space and combine it with an explicit Runge-Kutta time-marching algorithm. The proceedings of Cockburn et al. [24] and Shu [25] contain a more complete and current survey of the DG method and its applications. Despite the attractive advantages mentioned above, DG methods have some drawbacks. Unlike the continuous FEMs, DG methods produce dense and ill-conditioned matrices increasing with the order of polynomial degree[23].

Î <sup>r</sup> <sup>r</sup> r rr

equations in the form (1), where = [, ′, …, ( 1)]

 .

and <sup>0</sup> = [0, 1, …, <sup>1</sup>]

32 Perusal of the Finite Element Method

where : [0, ] ℝ, <sup>0</sup> ∈ ℝ, and : [0, ]×ℝ ℝ. We assume that the solution exists and is unique and we would like to approximate it using a discontinuous piecewise polynomial space. According to the ordinary differential equation (ODE) theory, the condition ∈ 1([0, ]×ℝ) is sufficient to guarantee the existence and uniqueness of the solution to (1). We note that a general th-order IVP of the form () = (, , ′, …, ( 1)) with initial conditions (0) = 0, ′(0) = 1, …, ( 1)(0) = <sup>1</sup> can be converted into a system of

The high-order DG method considered here is a class of finite element methods (FEMs) using completely discontinuous piecewise polynomials for the numerical solution and the test functions. The DG method was first designed as an effective numerical method for solving hyperbolic conservation laws, which may have discontinuous solutions. Here, we will discuss the algorithm formulation, stability analysis, and error estimates for the DG method solving nonlinear ODEs. DG method combines the best proprieties of the classical continuous finite element and finite volume methods such as consistency, flexibility, stability, conservation of local physical quantities, robustness, and compactness. Recently, DG methods become highly attractive and popular, mainly because these methods are high-order accurate, nonlinear stable, highly parallelizable, easy to handle complicated geometries and boundary conditions, and capable to capture discontinuities without spurious oscillations. The original DG finite element method (FEM) was introduced in 1973 by Reed and Hill [1] for solving steady-state first-order linear hyperbolic problems. It provides an effective means of solving hyperbolic problems on unstructured meshes in a parallel computing environment. The discontinuous basis can capture shock waves and other discontinuities with accuracy [2, 3]. The DG method can easily handle adaptivity strategies since the ℎ refinement (mesh refinement and coarsening) and the refinement (method order variation) can be done without taking into account the continuity restrictions typical of conforming FEMs. Moreover, the degree of the approximating polynomial can be easily changed from one element to the other [3]. Adaptivity is of particular importance in nonlinear hyperbolic problems given the complexity of the structure of the discontinuities and geometries involved. Due to local structure of DG methods, physical quantities such as mass, momentum, and energy are conserved locally through DG schemes. This property is very important for flow and transport problems. Furthermore, the DG method is highly parallelizable [4, 5]. Because of these nice features, the DG method has been analyzed and extended to a wide range of applications. In particular, DG methods have been used to solve ODEs [6–9], hyperbolic [5, 6, 10–19] and diffusion and convection diffusion [20–23] partial differential equations (PDEs), to mention a few. For transient problems, Cockburn and Shu [17] introduced and developed the so-called Runge-Kutta discontinuous Galerkin (RKDG) methods. These numerical methods use DG discretizations in space and combine it with an explicit Runge-Kutta time-marching algorithm. The proceedings of Cockburn et al. [24] and

<sup>0</sup> = ( , ), [0, ], (0) = , *du f tu t T u u dt* (1)

, (, )=[2, 3, …, , (, 1, …, )]

,

Related theoretical results in the literature including superconvergence results and error estimates of the DG methods for ODEs are given in [7–9, 26–28]. In 1974, LaSaint and Raviart [9] presented the first error analysis of the DG method for the initial-value problem (1). They showed that the DG method is equivalent to an implicit Runge-Kutta method and proved a rate of convergence of (*hp* ) for general triangulations and of (*hp*+1) for Cartesian grids. Delfour et al. [7] investigated a class of Galerkin methods which lead to a family of one-step schemes generating approximations up to order 2 + 2 for the solution of an ODE, when polynomials of degree are used. In their proposed method, the numerical solution ℎ at the discontinuity

point is defined as an average across the jump, i.e., ℎ( −) + (1 − )ℎ( +). By choosing special values of , one can obtain the original DG scheme of LeSaint and Raviart [9] and Euler's explicit, improved, and implicit schemes. Delfour and Dubeau [27] introduced a family of discontinuous piecewise polynomial approximation schemes. They presented a more general framework of one-step methods such as implicit Runge-Kutta and Crank-Nicholson schemes, multistep methods such as Adams-Bashforth and Adams-Moulton schemes, and hybrid methods. Later, Johnson [8] proved new optimal *a priori* error estimates for a class of implicit one-step methods for stiff ODEs obtained by using the discontinuous Galerkin method with piecewise polynomials of degree zero and one. Johnson and Pitkaränta [29] proved a rate of convergence of (*hp*+1/2) for general triangulations and Peterson [19] confirmed this rate to be optimal. Richter [30] obtained the optimal rate of convergence (*hp*+1) for some structured twodimensional non-Cartesian grids. We also would like to mention the work of Estep [28], where the author outlined a rigorous theory of global error control for the approximation of the IVP (1). In [6], Adjerid et al. showed that the DG solution of one-dimensional hyperbolic problems exhibit an (*hp*+2) superconvergence rate at the roots of the right Radau polynomial of degree + 1. Furthermore, they obtained a (2 + 1)th order superconvergence rate of the DG approximation at the downwind point of each element. They performed a local error analysis and showed that the local error on each element is proportional to a Radau polynomial. They further constructed implicit residual-based *a posteriori* error estimates but they did not prove their asymptotic exactness. In 2010, Deng and Xiong [31] investigated a DG method with interpolated coefficients for the IVP (1). They proved pointwise superconvergence results at Radau points. More recently, the author [12, 15, 26, 32–39] investigated the global convergence of the several residual-based *a posteriori* DG and local DG (LDG) error estimates for a variety of linear and nonlinear problems.

This chapter is organized as follows: In Section 2, we present the discrete DG method for the classical nonlinear initial-value problem. In Section 3, we present a detailed proof of the optimal *a priori* error estimate of the DG scheme. We state and prove our main superconvergence results in Section 4. In Section 5, we present the *a posteriori* error estimation procedure and prove that these error estimates converge to the true errors under mesh refinement. In Section 6, we propose an adaptive algorithm based on the local *a posteriori* error estimates. In Section 7, we present several numerical examples to validate our theoretical results. We conclude and discuss our results in Section 8.

## **2. The DG scheme for nonlinear IVPs**

The error analysis of nonlinear scalar and vector initial-value problems (IVPs) having smooth solutions is similar. For this, we restrict our theoretical discussion to the following nonlinear initial-value problem (IVP)

$$
\mu' = f(t, \mu), \quad t \in [0, T], \quad u(0) = \mu\_{0'} \tag{2}
$$

where (, ): [0,]×ℝ ℝ is a sufficiently smooth function with respect to the variables and . More precisely, we assume that (, ) ≤ 1 on the set = [0,]×ℝ ⊂ ℝ2, where 1 is a positive constant. We note that the assumption (, ) ≤ 1 is sufficient to ensure that (, ) satisfies a Lipschitz condition in on the convex set with Lipschitz constant 1

$$<\langle f(t,\mathbf{u}) - f(t,\mathbf{v}) \mid \mathbf{u} \rangle \le M\_1 \left\| \mathbf{u} - \mathbf{v} \right\|, \quad \text{for any (t,u) and (t,v)} \in \mathcal{D}.\tag{3}$$

Next, we introduce the DG method for the model problem (2). Let 0 = <sup>0</sup> < 1 <⋯< = be a partition of the interval = [0,]. We denote the mesh by = [ 1, ], = 1, …, . We denote the length of by ℎ = <sup>1</sup>. We also denote ℎ = max 1 ≤ ≤ ℎ and ℎmin = min 1 ≤ ≤ ℎ as the length of the largest and smallest subinterval, respectively. Here, we consider regular meshes, that is, ℎ ℎmin ≤ , where 1 is a constant (independent of ℎ) during mesh refinement. If = 1, then the mesh is uniformly distributed. In this case, the nodes and mesh size are defined by = ℎ, = 0, 1, …, , ℎ = / .

Throughout this work, we define ( ) and ( +) to be the left limit and the right limit of the function at the discontinuity point , i.e., ( ) = lim 0 ( + ) and ( +) = lim 0+ ( + ). To simplify the notation, we denote by []( ) = ( +) ( ) the jump of at the point .

If we multiply (2) by an arbitrary test function , integrate over the interval , and integrate by parts, we get the DG weak formulation

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations http://dx.doi.org/10.5772/64967 35

$$\int\_{I\_{j}} \mathbf{v}' \mathbf{u} dt + \int\_{I\_{j}} f(t, \mathbf{u}) \mathbf{v} dt - \mathfrak{u}(t\_{j}) \mathbf{v}(t\_{j}) + \mathfrak{u}(t\_{j-1}) \mathbf{v}(t\_{j-1}) = \mathbf{0}. \tag{4}$$

We denote by ℎ the finite element space of polynomials of degree at most in each interval , i.e.,

$$V\_h^p = \{ \upsilon : \upsilon \upharpoonright\_{\iota\_{\dot{\jmath}}} \in P^p(I\_{\dot{\jmath}})\_{\iota} \ \dot{\jmath} = 1, \ldots, N \}\_{\iota}$$

where ( ) denotes the set of all polynomials of degree no more than on . We would like to emphasize that polynomials in ℎ are allowed to have discontinuities at the nodes .

Replacing the exact solution () by a piecewise polynomial ℎ() ∈ ℎ and choosing ∈ ℎ , we obtain the DG scheme: Find ℎ ∈ ℎ such that ∀ ∈ ℎ and = 1, …, ,

$$
\hat{\Box}\_{I\_j} \mathbf{v}' \boldsymbol{u}\_h \mathbf{d}t + \int\_{I\_j} f(t, \boldsymbol{u}\_h) \mathbf{v} \mathrm{d}t - \hat{\boldsymbol{u}}\_h(t\_j) \mathbf{v}(t\_j^-) + \hat{\boldsymbol{u}}\_h(t\_{j-1}) \mathbf{v}(t\_{j-1}^+) = \mathbf{0},\tag{5a}
$$

where ℎ( ) is the so-called numerical flux which is nothing but the discrete approximation at the node = . We remark that ℎ is not necessarily continuous at the nodes.

To complete the definition of the DG scheme, we still need to define ℎ on the boundaries of . Since for IVPs, information travel from the past into the future, it is reasonable to take ℎ as the classical upwind flux

$$
\hat{u}\_h(t\_0) = u\_0, \quad \text{and} \quad \hat{u}\_h(t\_j) = u\_h(t\_j^-), \quad j = 1, \dots, N. \tag{5b}
$$

#### **2.1. Implementation**

propose an adaptive algorithm based on the local *a posteriori* error estimates. In Section 7, we present several numerical examples to validate our theoretical results. We conclude and

The error analysis of nonlinear scalar and vector initial-value problems (IVPs) having smooth solutions is similar. For this, we restrict our theoretical discussion to the following nonlinear

where (, ): [0,]×ℝ ℝ is a sufficiently smooth function with respect to the variables and . More precisely, we assume that (, ) ≤ 1 on the set = [0,]×ℝ ⊂ ℝ2, where 1 is a positive constant. We note that the assumption (, ) ≤ 1 is sufficient to ensure that (, )


by ℎ = <sup>1</sup>. We also denote ℎ = max

as the length of the largest and smallest subinterval, respectively. Here,

satisfies a Lipschitz condition in on the convex set with Lipschitz constant 1

Next, we introduce the DG method for the model problem (2). Let 0 =

ℎmin

during mesh refinement. If = 1, then the mesh is uniformly distributed. In this case, the

) and (

( + ). To simplify the notation, we denote by [](

If we multiply (2) by an arbitrary test function , integrate over the interval

be a partition of the interval = [0,]. We denote the mesh by

nodes and mesh size are defined by = ℎ, = 0, 1, …, , ℎ = / .

of the function at the discontinuity point

.

′ ∈ <sup>0</sup> *u f tu t T u u* = ( , ), [0, ], (0) = , (2)

<sup>0</sup> < 1 <⋯< =

1 ≤ ≤

], = 1, …, . We

ℎ and

( + ) and

, and integrate

) the

+) (

= [ 1,

≤ , where 1 is a constant (independent of ℎ)

, i.e., (

+) to be the left limit and the right limit

) = lim 0

) = (

discuss our results in Section 8.

34 Perusal of the Finite Element Method

initial-value problem (IVP)

denote the length of

ℎ

we consider regular meshes, that is, ℎ

Throughout this work, we define (

by parts, we get the DG weak formulation

1 ≤ ≤

ℎmin = min

(

+) = lim 0+

jump of at the point

**2. The DG scheme for nonlinear IVPs**

The DG solution ℎ() can be efficiently obtained in the following order: first, we compute ℎ() in the first element <sup>1</sup> using (5a) and (5b) with = 1 since ℎ(0 −) = 0 is known. Then, we can find ℎ() in <sup>2</sup> since ℎ() in <sup>1</sup> is already available. We can repeat the same process to compute ℎ() in <sup>3</sup>, …, . More specifically, ℎ() can be obtained locally for each using the following two steps: (i) express ℎ() as a linear combination of orthogonal basis , (), = 0, …, , where , is the th degree Legendre polynomial on , i.e., ℎ() = ∑ = 0 , , (), ∈ , where

, () = 0 = is a local basis of ( ), and (ii) choose the test functions =, (), = 0, …, . Thus, on each , we get a ( + 1) × ( + 1) system of nonlinear algebraic equations, which can be solved for the unknown coefficients 0,, …, , using, e.g., Newton's method for nonlinear systems. Once we obtain the DG solution on all elements , = 1, …, , we get the DG solution which is a piecewise discontinuous polynomial of degree ≤ . We refer to [7–9] for more details about DG methods for ODEs as well as their properties and applications.

#### **2.2. Linear stability for the DG method**

Let us now establish a stability result for the DG method applied to the linear case, i.e., (, ) = . Taking =ℎ in the discrete weak formulation (5a), we get

$$\frac{\mu\_h^2(t\_{\boldsymbol{\beta}}^-)}{2} + \frac{\mu\_h^2(t\_{\boldsymbol{\beta}-1}^+)}{2} - \mu\_h(t\_{\boldsymbol{\beta}-1}^-)\mu\_h(t\_{\boldsymbol{\beta}-1}^+) = \mathcal{A} \int\_{I\_{\boldsymbol{\beta}}} \mu\_h^2 dt\_{\boldsymbol{\beta}}$$

which is equivalent to

$$\frac{\mu\_h^2(t\_{\nearrow}^-)}{2} - \frac{\mu\_h^2(t\_{\nearrow}^-)}{2} + \frac{1}{2} \left(\mu\_h(t\_{\nearrow}^-) - \mu\_h(t\_{\nearrow}^+) \right)^2 = \mathcal{A} \int\_{I\_{\nearrow}} \mu\_h^2 dt.$$

Summing over all elements, we get the equality

$$\frac{\mu\_h^2(T^-)}{2} - \frac{\mu\_0^2}{2} + \frac{1}{2} \sum\_{j=1}^N \left(\mu\_h(t\_{j-1}^-) - \mu\_h(t\_{j-1}^+)\right)^2 = \lambda \int\_{\Omega} \mu\_h^2 dt.$$

Consequently, ℎ 2(−) <sup>2</sup> <sup>−</sup> 0 2 <sup>2</sup> <sup>≤</sup> ∫ ℎ 2, which gives the stability result ℎ 2(−) ≤ 0 2 provided that 0.

## **3.** *A priori* **error analysis**

We begin by defining some norms that will be used throughout this work. We define the 2 inner product of two integrable functions, and , on the interval = [ − 1, ] as (, ) <sup>=</sup> ∫ ()(). Denote 0, = (, ) 1/2 to be the standard 2 norm of on . Moreover, the standard ∞ norm of on is defined by

 ∞, = sup () . Let ( ), where = 0, 1, …, denote the standard Sobolev space of square integrable functions on with all derivatives (), = 0, 1, …, being square integrable on , i.e., ( ) = : ∫ |()() 2 < ∞, 0 <sup>≤</sup> ≤ , and equipped with the norm , <sup>=</sup> ∑ = 0 () 0, <sup>2</sup> 1/2 . The ( ) seminorm of a function on is given by , <sup>=</sup> () 0, . We also define the norms on the whole computational domain as follows:

, () = 0

Thus, on each

which is equivalent to

Consequently, ℎ

that 0.

= [ − 1,

2 norm of on

= is a local basis of (

**2.2. Linear stability for the DG method**

systems. Once we obtain the DG solution on all elements

about DG methods for ODEs as well as their properties and applications.

(, ) = . Taking =ℎ in the discrete weak formulation (5a), we get

− +

2 2

− −

*hj hj*

*ut ut*

−

Summing over all elements, we get the equality

*h*

0 2

<sup>2</sup> <sup>≤</sup> ∫

ℎ

2(−) <sup>2</sup> <sup>−</sup>

] as (, )

 <sup>=</sup> ∫ 

**3.** *A priori* **error analysis**

*ut ut*

*hj hj*

36 Perusal of the Finite Element Method

), and (ii) choose the test functions =, (), = 0, …, .

, = 1, …, , we get the DG solution

, we get a ( + 1) × ( + 1) system of nonlinear algebraic equations, which can

λ

*j*

λ

λ

= (, )

2(−) ≤ 0

1/2 to be the standard

is defined by

2 provided

*j*

be solved for the unknown coefficients 0,, …, , using, e.g., Newton's method for nonlinear

which is a piecewise discontinuous polynomial of degree ≤ . We refer to [7–9] for more details

Let us now establish a stability result for the DG method applied to the linear case, i.e.,

− − + − − + − ∫

− − + − − −+ − ∫ 2 2 <sup>2</sup> <sup>1</sup> <sup>2</sup>

− − <sup>Ω</sup> −+ − <sup>∑</sup> <sup>∫</sup> 2 2 <sup>2</sup> <sup>0</sup> <sup>2</sup>

*uT u u t u t u dt*

We begin by defining some norms that will be used throughout this work. We define the 2 inner product of two integrable functions, and , on the interval

()(). Denote 0,

. Moreover, the standard ∞ norm of on

( ) <sup>1</sup> ( ) ( )= . 2 22

=1

*j*

*N*

() ( ) <sup>1</sup> ( ) ( )= . 2 22

1 2 1 1 () ( ) ( ) ( )= , 2 2

( )

( )

− +

1 1

*hj hj h*

2, which gives the stability result ℎ

1 1

*hj hj h I*

*u t u t u dt*

*hj hj h I*

*u t u t u dt*

$$\left\|\boldsymbol{\mu}\right\|\_{0,\Omega} = \left(\sum\_{j=1}^N \left\|\boldsymbol{\mu}\right\|\_{0,I\_j}^2\right)^{1/2}, \quad \left\|\boldsymbol{\mu}\right\|\_{o,\Omega} = \max\_{1 \le j \le N} \left\|\boldsymbol{\mu}\right\|\_{o,I\_j}, \quad \left\|\boldsymbol{\mu}\right\|\_{s,\Omega} = \left(\sum\_{j=1}^N \left\|\boldsymbol{\mu}\right\|\_{s,I\_j}^2\right)^{1/2}.$$

The seminorm on the whole computational domain is defined as , <sup>=</sup> ∑ = 1 , <sup>2</sup> 1/2 . We note that if (), = 1, 2, …, then the norm , on the whole computational domain is the standard Sobolev norm ∑ = 0 () 0, <sup>2</sup> 1/2 . For convenience, we use and to denote 0, and 0,, respectively.

For <sup>1</sup>, we consider two special projection operators, ℎ ±, which are defined as follows: For a smooth function , the restrictions of ℎ + and ℎ − to are polynomials in ( ) satisfying

$$\int\_{I\_j} (P\_h^- u - u)\mathbf{v}dt = 0, \forall \; \mathbf{v} \in P^{\rho - 1}(I\_j), \quad \text{and} \quad (P\_h^- u - u)(t\_j^-) = 0,\tag{6a}$$

$$\int\_{I\_j} (P\_h^+ u - u)\text{v}dt = 0, \forall \text{ v} \in P^{\rho - 1}(I\_j), \quad \text{and} \quad (P\_h^+ u - u)(t\_{j-1}^+) = 0. \tag{6b}$$

These two particular Gauss-Radau projections are very important in the proofs of optimal 2 error estimates and superconvergence results. We note that the special projections ℎ ± are mainly utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy [22].

For the projections mentioned above, it is easy to show that for any + 1( ) with = 1, …, , there exists a constant independent of the mesh size ℎ such that (see, e.g., [40])

$$\left\|\mu - P\_h^{\pm}u\right\|\_{l\_j} \le \mathrm{Ch}\_{\boldsymbol{\beta}}^{\nu+1} \left\|u\right\|\_{\boldsymbol{p}+1,l\_j}, \quad \left\|(\mu - P\_h^{\pm}u)'\right\|\_{l\_j} \le \mathrm{Ch}\_{\boldsymbol{\beta}}^{\nu} \left\|u\right\|\_{\boldsymbol{p},l\_j}.\tag{7}$$

Moreover, we recall the inverse properties of the finite element space ℎ that will be used in our error analysis: For any ℎ ℎ , there exists a positive constant independent of ℎ and ℎ, such that, ∀ = 1, …, ,

$$\left\lVert \boldsymbol{\nu}\_{h}^{(k)} \right\rVert\_{\boldsymbol{l}\_{j}} \leq \boldsymbol{Ch}\_{j}^{-k} \left\lVert \left\lVert \boldsymbol{\nu}\_{h} \right\rVert\right\rVert\_{\boldsymbol{l}\_{j}}, \quad k \geq 1, \quad \left| \boldsymbol{\nu}\_{h} \left(\boldsymbol{t}\_{j-1}^{+} \right) \right| + \left| \boldsymbol{\nu}\_{h} \left(\boldsymbol{t}\_{j}^{-} \right) \right| \leq \boldsymbol{Ch}\_{j}^{-1/2} \left\lVert \left\lVert \boldsymbol{\nu}\_{h} \right\rVert\_{\boldsymbol{l}\_{j}}.\tag{8}$$

From now on, the notation , 1, 2, etc. will be used to denote generic positive constants independent of ℎ, but may depend upon the exact solution of (1) and its derivatives. They also may have different values at different places.

Throughout this work, let us denote = ℎ to be the error between the exact solution of (2) and the DG solution defined in (5a) and (5b), = ℎ to be the projection error, and <sup>=</sup> ℎ ℎ to be the error between the projection of the exact solution ℎ and the DG solution ℎ. We observe that the actual error can be written as = ( ℎ )+(ℎ ℎ) = + .

Now, we are ready to prove our optimal error estimates for in the 2 and 1 norms.

**Theorem 3.1.** *Suppose that the exact solution of (2) is sufficiently smooth with bounded derivatives, i.e.,* + 1, *is bounded. We also assume that* (, ) ≤ 1 *on* = [0,]×ℝ*. Let* <sup>0</sup> *and* ℎ *be the DG solution of (5a) and (5b), then, for sufficiently small* ℎ*, there exists a positive constant independent of* ℎ *such that,*

$$\left\| e \right\| \le C \left\| h^{\nu+1} \right\| \tag{9}$$

$$\sum\_{j=1}^{N} \left\| \boldsymbol{e}^{\prime} \right\|\_{\boldsymbol{l}\_{j}}^{2} \le \text{Ch}^{2p} \; \prime \quad \left\| \boldsymbol{e} \right\|\_{\mathbf{l},\Omega} \le \text{Ch}^{p} \; \tag{10}$$

*Proof.* We first need to derive some error equations which will be used repeatedly throughout this and the next sections. Subtracting (5a) from (4) with ℎ and using the numerical flux (5b), we obtain the following error equation: ∀ℎ ,

$$\int\_{I\_{j}} \boldsymbol{v}^{\prime} \boldsymbol{e} dt + \int\_{I\_{j}} (f(t, \boldsymbol{u}) - f(t, \boldsymbol{u}\_{h})) \boldsymbol{v} dt + \boldsymbol{e}(t\_{j-1}^{-}) \boldsymbol{v}(t\_{j-1}^{+}) - \boldsymbol{e}(t\_{j}^{-}) \boldsymbol{v}(t\_{j}^{-}) = 0. \tag{11}$$

By integration by parts, we get

For the projections mentioned above, it is easy to show that for any + 1(

± ± + <sup>+</sup> −≤ − ≤ ′ <sup>1</sup>

Moreover, we recall the inverse properties of the finite element space ℎ

( ) 1/2 <sup>1</sup> || || , 1, ( ) ( ) || || . <sup>−</sup> <sup>+</sup> − − ≤ ³ + <sup>−</sup> ≤ *k k <sup>h</sup> j hI h j h j j hI <sup>I</sup> j j <sup>j</sup>*

From now on, the notation , 1, 2, etc. will be used to denote generic positive constants independent of ℎ, but may depend upon the exact solution of (1) and its derivatives. They also

Throughout this work, let us denote = ℎ to be the error between the exact solution of (2)

solution ℎ. We observe that the actual error can be written as

**Theorem 3.1.** *Suppose that the exact solution of (2) is sufficiently smooth with bounded derivatives,*

*i.e.,* + 1, *is bounded. We also assume that* (, ) ≤ 1 *on* = [0,]×ℝ*. Let* <sup>0</sup> *and* ℎ *be the DG solution of (5a) and (5b), then, for sufficiently small* ℎ*, there exists a positive constant*

<sup>+</sup> ≤ <sup>1</sup>

, .

<sup>Ω</sup> ∑ ′ ≤ ≤ <sup>2</sup> <sup>2</sup> 1, =1

*I j j*

*<sup>N</sup> p p*

ℎ to be the error between the projection of the exact solution ℎ

Now, we are ready to prove our optimal error estimates for in the 2 and 1 norms.

our error analysis: For any ℎ ℎ

may have different values at different places.

and the DG solution defined in (5a) and (5b), = ℎ

ℎ) = + .

ℎ, such that, ∀ = 1, …, ,

38 Perusal of the Finite Element Method

<sup>=</sup> ℎ

= ( ℎ

)+(ℎ

*independent of* ℎ *such that,*

= 1, …, , there exists a constant independent of the mesh size ℎ such that (see, e.g., [40])

*h j h j I I p I p I j j j j*

1, , ,( ) . *p p*

*u P u Ch u u P u Ch u* (7)

*v Ch v k v t v t Ch v* (8)

, there exists a positive constant independent of ℎ and

) with

that will be used in

to be the projection error, and

, *<sup>p</sup> e Ch* (9)

*e Ch e Ch* (10)

and the DG

$$\int\_{\mathbb{T}\_{\bar{\boldsymbol{t}}\_{j}}} e^{\boldsymbol{v}} vdt - \int\_{\mathbb{T}\_{\bar{\boldsymbol{t}}\_{j}}} \left( f(t, \boldsymbol{u}) - f(t, \boldsymbol{u}\_{h}) \right) vdt + [e](t\_{j-1}) v(t\_{j-1}^{\*}) = 0. \tag{12}$$

Applying Taylor's series with integral remainder in the variable and using the relation ℎ = , we write

$$f(t, \boldsymbol{\mu}) - f(t, \boldsymbol{\mu}\_{\boldsymbol{h}}) = \boldsymbol{\theta}(\boldsymbol{\mu} - \boldsymbol{\mu}\_{\boldsymbol{h}}) = \boldsymbol{\theta}\boldsymbol{\varepsilon}, \quad \text{where } \boldsymbol{\theta} = \int\_{0}^{1} f\_{\boldsymbol{\mu}}(t, \boldsymbol{\mu} + \mathbf{s}(\boldsymbol{\mu}\_{\boldsymbol{h}} - \boldsymbol{\mu})) d\mathbf{s} = \int\_{0}^{1} f\_{\boldsymbol{\mu}}(t, \boldsymbol{\mu} - \boldsymbol{\text{se}}) d\mathbf{s}. \tag{13}$$

Substituting (13) into (12), we arrive at

$$\int\_{I\_{j}} \left( e' - \theta e \right) v dt + \left[ e \right] \left( t\_{j-1} \right) v \left( t\_{j-1}^{\*} \right) = 0, \quad \forall \ v \in V\_{h}^{p} . \tag{14}$$

To simplify the notation, we introduce the bilinear operator (; ) as

$$\mathcal{A}\_{\rangle}(e;V) = \int\_{I\_{\neq}} (e' - \theta e)Vdt + [e](t\_{\neq -1})V(t\_{\neq -1}^{+}).\tag{15}$$

Thus, we can write (14) as

$$\mathcal{A}\_{\rangle}(e;\upsilon) = 0, \quad \forall \, \upsilon \in V\_h^{\upsilon}. \tag{16}$$

A direct calculation from integration by parts yields

$$\mathcal{A}\_{\rangle}(e;V) = \int\_{I\_{\neq}} (-V' - \theta V)e dt + e(t\_{\neq}^{-})V(t\_{\neq}^{-}) - e(t\_{\neq-}^{-})V(t\_{\neq-}^{+}).\tag{17}$$

On the other hand, if we add and subtract ℎ + to then we can write (15) as

$$\mathcal{A}\_{\rangle}(e;V) = \mathcal{A}\_{\rangle}(e;V - P\_{\hbar}^{+}V) + \mathcal{A}\_{\rangle}(e;P\_{\hbar}^{+}V). \tag{18}$$

Combining (18) and (16) with =ℎ + ( ) and applying the property of the projection ℎ <sup>+</sup>, i.e., ( ℎ +)( <sup>1</sup> <sup>+</sup> )=0, we obtain

$$\mathcal{A}\_{\rangle}(e;V) = \int\_{I\_{\underline{j}}} (e' - \theta e)(V - P\_{\underline{h}}^{+}V)dt + [e](t\_{j-1})(V - P\_{\underline{h}}^{+}V)(t\_{j-1}^{+}) = \int\_{I\_{\underline{j}}} (e' - \theta e)(V - P\_{\underline{h}}^{+}V)dt. \tag{19}$$

If is a polynomial of degree at most then ′ is a polynomial of degree at most 1. Therefore, by the property of the projection ℎ +, we immediately see

$$\int\_{I\_{\slash}} v'(V - P\_h^+ V)dt = 0, \quad \forall \ v \in P^\nu(I\_{\slash}). \tag{20}$$

Substituting the relation =+ into (19) and invoking (20) with =, we get

$$\mathcal{A}\_{\rangle}(\varepsilon; V) = \int\_{I\_{j}} (\varepsilon' - \theta \varepsilon)(V - P\_{\hbar}^{+}V)dt + \int\_{I\_{j}} \overline{\varepsilon}(V - P\_{\hbar}^{+}V)dt = \int\_{I\_{j}} (\varepsilon' - \theta \varepsilon)(V - P\_{\hbar}^{+}V)dt. \tag{21}$$

Now, we are ready to prove the theorem. We construct the following auxiliary problem: find such that

$$-\phi' - \theta \phi = e, \quad t \in [0, T] \quad \text{subject to} \quad \phi(T) = 0. \tag{22}$$

where = () = ∫0 1 (, () ()). Clearly, the exact solution to (22) is given by the explicit formula

$$\varphi(t) = \frac{1}{\Theta(t)} \int\_{t}^{T} \Theta(y) e(y) dy, \quad \text{where} \quad \Theta(t) = \exp\left(-\int\_{t}^{T} \theta(s) ds\right). \tag{23}$$

Next, we prove some regular estimates which will be needed in our error analysis. Using the assumption (, ) ≤ 1, we see that (), [0, ] is bounded by 1

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations http://dx.doi.org/10.5772/64967 41

$$\left| \left| \theta(t) \right| \leq \int\_{0}^{1} \left| f\_{\ast}(t, u(t) - \mathrm{se}(t)) \right| \, d\mathrm{s} \leq \int\_{0}^{1} M\_{\mathrm{l}} \mathrm{ds} \equiv M\_{\mathrm{l}}, \quad \forall \, t \in [0, T]. \tag{24a}$$

Using the definition of and the estimate (24a), we have

$$0 \le \Theta(t) \le \exp\left(\int\_0^T |\, \partial(\mathbf{s}) \, | \, \mathbf{ds}\right) \le \exp\left(\int\_0^T M\_1 \, \mathbf{ds}\right) = \exp\left(M\_1 T\right) = C\_1. \tag{24b}$$

Similarly, we can easily estimate 1 () as follows

On the other hand, if we add and subtract ℎ

Combining (18) and (16) with =ℎ

+)( <sup>1</sup>

by the property of the projection ℎ

ε θ

θ

<sup>+</sup> )=0, we obtain

*j*

−− ∈

ϕ θϕ

1

ℎ

<sup>+</sup>, i.e., ( ℎ

40 Perusal of the Finite Element Method

such that

where = () = ∫0

ϕ

explicit formula

+ to then we can write (15) as

+ + ( ; ) = ( ; ) ( ; ). − + *j j h jh eV eV PV ePV* (18)

<sup>+</sup> + + <sup>+</sup>

*e V e e V P V dt e t V P V t e e V P V dt* (19)

If is a polynomial of degree at most then ′ is a polynomial of degree at most 1. Therefore,

+, we immediately see

<sup>+</sup> ′ <sup>−</sup> ∀ ∈ ∫ ( ) = 0, ( ). *<sup>p</sup> h j I*

 ++ + ′ ′ −− + − −− ∫ ∫∫ ( ; ) = ( )( ) '( ) = ( )( ) . *j hh h I II j jj*

Now, we are ready to prove the theorem. We construct the following auxiliary problem: find

′ = , [0, ] subject to ( ) = 0. *et T*

 Θ Θ− <sup>Θ</sup> ∫ ∫ <sup>1</sup> ( ) = ( ) ( ) , where ( ) = exp ( ) . ( ) *T T t t <sup>t</sup> y e y dy <sup>t</sup> s ds*

Next, we prove some regular estimates which will be needed in our error analysis. Using the

assumption (, ) ≤ 1, we see that (), [0, ] is bounded by 1

*e V e V P V dt e V P V dt e V P V dt* (21)

Substituting the relation =+ into (19) and invoking (20) with =, we get

) and applying the property of the projection

 θ

*v V P V dt v P I* (20)

ε θ

ϕ

(, () ()). Clearly, the exact solution to (22) is given by the

*<sup>t</sup>* (23)

( )

 θ

*T* (22)

+ (

− − ′ ′ −− + − −− ∫ ∫ 1 1 ( ; ) = ( )( ) [ ]( )( )( ) = ( )( ) . *j h j hj h I I j j*

$$0 \le \frac{1}{\Theta(t)} = \exp\left(\int\_{t}^{r} \theta(s) ds\right) \le \exp\left(\int\_{0}^{r} |\theta(s)| \, ds\right) \le \exp\left(\int\_{0}^{r} M\_{l} ds\right) = \exp\left(M\_{l}T\right) = C\_{l}.\tag{24c}$$

Applying the estimates (24b), (24c), and the Cauchy-Schwarz inequality, we get

$$\left|\varphi(t)\right| \leq \frac{1}{\Theta(t)} \int\_{t}^{\top} \Theta(y) \left|e(y)\right| dy \leq \mathbb{C}\_{1} \int\_{t}^{\top} \mathbb{C}\_{1} \left|e(y)\right| dy \leq \mathbb{C}\_{1}^{2} \int\_{0}^{\top} \left|e(y)\right| dy \leq \mathbb{C}\_{1}^{2} T^{1/2} \left\|e\right\|\_{\prime} \quad t \in [0, T].$$

Squaring both sides and intergrading over yields

$$\left\|\rho\right\|^2 \le C\_1^4 T^2 \left\|e\right\|^2 = C\_2 \left\|e\right\|^2. \tag{25a}$$

We also need to obtain an estimate of 1,. Using (22) and (24a) gives

$$\mid \phi' \vdash \mid \theta \phi + e \mid \leq M\_1 \mid \phi \mid + \mid e \mid \quad t \in [0, T] \dots$$

Squaring both sides, applying the inequality ( + )<sup>2</sup> ≤ 22 + 22, integrating over the computational domain , and using (25a), we get

$$\left\|\boldsymbol{\rho}\right\|\_{\mathrm{l},\Omega}^{2} = \int\_{0}^{\boldsymbol{\tau}} |\boldsymbol{\rho}\prime|^{2} \, d\boldsymbol{t} \leq \mathfrak{L}(\boldsymbol{M}\_{\mathrm{l}}^{2} \left\|\boldsymbol{\rho}\right\|^{2} + \left\|\boldsymbol{e}\right\|^{2}) \leq \mathfrak{L}(\boldsymbol{M}\_{\mathrm{l}}^{2}\boldsymbol{C}\_{2} + \mathrm{l}) \left\|\boldsymbol{e}\right\|^{2} \leq \boldsymbol{C}\_{3} \left\|\boldsymbol{e}\right\|^{2}. \tag{25b}$$

Applying the projection result and the estimate (3.20b) yields

$$\left\|\boldsymbol{\varphi} - \boldsymbol{P}\_{h}^{\*}\boldsymbol{\varphi}\right\| \leq C\_{4}h\left\|\boldsymbol{\varphi}\right\|\_{\mathrm{l},\Omega} \leq C\_{\mathrm{s}}h\left\|\boldsymbol{e}\right\|.\tag{25c}$$

Now, we are ready to show (9). Using (17) with = and (22), we obtain

$$\mathcal{A}\_{\rangle}(e;\phi\mathfrak{p}) = \int\_{I\_{\underline{j}}} (-\phi' - \theta\phi\mathfrak{p}) e dt - e(t\_{\underline{j}-1}^{-})\phi(t\_{\underline{j}-1}) + e(t\_{\underline{j}}^{-})\phi(t\_{\underline{j}}) = \int\_{I\_{\underline{j}}} e^{2} dt - e(t\_{\underline{j}-1}^{-})\phi(t\_{\underline{j}-1}) + e(t\_{\underline{j}}^{-})\phi(t\_{\underline{j}}).$$

Summing over the elements and using the fact that () = (0 −)=0 yields

$$\sum\_{j=1}^{N} \mathcal{A}\_{j}(e; \phi) = \left\| e \right\|^{2} - e(t\_{0}^{-})\phi(t\_{0}^{-}) + e(T^{-})\phi(T^{-}) = \left\| e \right\|^{2}. \tag{26}$$

On the other hand, if we choose = in (21) then we get

$$\mathcal{A}\_{\rangle}(e;\varphi) = \int\_{I\_{\neq}} (\varepsilon' - \theta e)(\varphi - P\_{\hbar}^{\ast}\varphi)dt. \tag{27}$$

Summing over all elements and applying the Cauchy-Schwarz inequality, we get

$$\sum\_{j=1}^N \mathcal{A}\_j(e; \varphi) \le \left( \left\| \varepsilon' \right\| + M\_1 \left\| e \right\| \right) \left\| \varphi - P\_h^\* \varphi \right\|.$$

Using the estimate (25c), we deduce that

$$\sum\_{j=1}^{N} \mathcal{A}\_{j}(\boldsymbol{e}; \boldsymbol{\varphi}) \le (\mathbb{C}\_{\boldsymbol{v}} h^{\boldsymbol{v}} \left\| \boldsymbol{u} \right\|\_{\boldsymbol{\rho} + 1, \Omega} + M\_{1} \left\| \boldsymbol{e} \right\| \mathbb{C}\_{1} h \left\| \boldsymbol{e} \right\| \le \mathbb{C} (h^{\boldsymbol{v} + 1} + h \left\| \boldsymbol{e} \right\|) \left\| \boldsymbol{e} \right\|. \tag{28}$$

Combining (26) and (28), we conclude that

$$\left\|e\right\| \le Ch^{p+1} + Ch \left\|e\right\|.\tag{29}$$

Thus, (1 − ℎ) ≤ ℎ + 1, where is a positive constant independent of ℎ. Therefore, for sufficiently small ℎ, e.g., ℎ ≤ <sup>1</sup> 2, we obtain <sup>1</sup> <sup>2</sup> ≤ (1 − ℎ) ≤ ℎ + 1, which yields ≤ 2 ℎ + 1 for ℎ small. Thus, we completed the proof of (9).

To show (10), we use = + , the classical inverse inequality (8), the estimate (9), and the projection result (7) to obtain

ε ε ( ) ε − − ′′ ′ +≤ + ≤ + ≤ + + ≤ + ≤ 1 1 1 21 2 32 =' ' . *<sup>p</sup> p p pp e e e C h e C h C h e C h C h C h Ch*

We note that 1, <sup>2</sup> <sup>=</sup> 2 <sup>+</sup> ′ <sup>2</sup> . Applying (9) and the estimate ′ ≤ ℎ yields 1, 2 ≤ 1ℎ2 + 2 + 2ℎ2 <sup>=</sup> (*h*<sup>2</sup>*<sup>p</sup>* ), which completes the proof of the theorem.

## **4. Superconvergence error analysis**

Now, we are ready to show (9). Using (17) with = and (22), we obtain

 ϕ

*j j*

Summing over the elements and using the fact that () = (0

∑ − +

ϕ

=1

<sup>0</sup> 1 1 1, =1

≤ 2 ℎ + 1 for ℎ small. Thus, we completed the proof of (9).

*j*

*N*

ϕ

On the other hand, if we choose = in (21) then we get

=1

Using the estimate (25c), we deduce that

ϕ

Combining (26) and (28), we conclude that

sufficiently small ℎ, e.g., ℎ ≤ <sup>1</sup>

projection result (7) to obtain

*<sup>j</sup> <sup>p</sup> <sup>j</sup>*

*N j j*

− − − − −− − + − + ′ ∫ ∫ <sup>2</sup>

 εθϕ

Summing over all elements and applying the Cauchy-Schwarz inequality, we get

ϕε

 <sup>+</sup> + Ω ∑ ≤ + ≤+<sup>1</sup>

*<sup>N</sup> p p*

(; ) ( ) ( ).

Thus, (1 − ℎ) ≤ ℎ + 1, where is a positive constant independent of ℎ. Therefore, for

To show (10), we use = + , the classical inverse inequality (8), the estimate (9), and the

2, we obtain <sup>1</sup>

∑ ≤+ − ′ <sup>1</sup>

(; ) ( ) .

*j h*

*e Me P*

<sup>+</sup> ′ − − ∫ ( ; ) = ( )( ) . *j h I j*

0 0

( ; )= ( ) ( ) ( ) ( )= .

 ϕ

1 1 1 1 ( ; ) = ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ). *j j j jj j j jj I I*

> ϕϕ−− − −

2 2

 ϕ

> ϕϕ+

*e Ch u M e Ch e Ch h e e* (28)

<sup>+</sup> ≤ +<sup>1</sup> . *<sup>p</sup> e Ch Ch e* (29)

<sup>2</sup> ≤ (1 − ℎ) ≤ ℎ + 1, which yields

*e edt e t t e t t e dt e t t e t t*

−− −−

*e e et t eT T e* (26)

*e e P dt* (27)

 ϕ

−)=0 yields

 ϕ

ϕ

42 Perusal of the Finite Element Method

 ϕ θϕ

> In this section, we study the superconvergence properties of the DG method. We first show a (2 + 1)th order superconvergence rate of the DG approximation at the downwind point of each element. Then, we apply this superconvergence result to show that the DG solution converges to the special projection of the exact solution ℎ − at (*hp*+2). This result allows us to prove that the leading term of the DG error is proportional to the ( + 1) degree right Radau polynomial.

> First, we define some special polynomials. The th degree Legendre polynomial can be defined by Rodrigues formula [41]

$$
\tilde{L}\_p(\xi) = \frac{1}{2^p p!} \frac{d^p}{d\xi^p} \left( (\xi^2 - 1)^p \right)\_{\prime} \quad -1 \le \xi \le 1.
$$

It satisfies the following important properties: (1) = 1, ( − 1) = ( − 1), and the orthogonality relation

$$\int\_{-1}^{1} \tilde{L}\_{\rho}(\xi) \tilde{L}\_{q}(\xi) d\xi = \frac{2}{2p+1} \delta\_{pq'} \quad \text{where } \delta\_{pq} \text{ is the Kronecker symbol.} \tag{30}$$

One can easily write the ( + 1) degree Legendre polynomial on [ − 1,1] as

$$
\tilde{L}\_{p+1}(\xi) = \frac{(2p+2)!}{2^{p+1}(\{p+1\}!)^{2}} \xi^{p+1} + \tilde{q}\_{p}(\xi), \quad \text{where } \tilde{q}\_{p} \in P^{p}([-1,1]).
$$

The ( + 1) degree right Radau polynomial on [ − 1,1] is defined as + 1() = + 1() − (). It has + 1 real distinct roots, −1 < 0 <⋯< = 1.

Mapping into the reference element [ − 1,1] by the linear transformation = + − <sup>1</sup> <sup>2</sup> <sup>+</sup> ℎ <sup>2</sup> , we obtain the shifted Legendre and Radau polynomials on :

$$L\_{p\circ 1,j}(t) = \tilde{L}\_{p\circ 1} \left(\frac{2t - t\_j - t\_{j-1}}{h\_j}\right), \quad R\_{p\circ 1,j}(t) = \tilde{R}\_{p\circ 1} \left(\frac{2t - t\_j - t\_{j-1}}{h\_j}\right).$$

Next, we define the monic Radau polynomial, + 1,(), on as

$$\left\|\boldsymbol{\nu}\_{p+1,j}(t) = \frac{h\_{j}^{p+1}[(p+1)!]^{2}}{(2p+2)!} \mathbf{R}\_{p+1,j}(t) = c\_{p}h\_{j}^{p+1} \mathbf{R}\_{p+1,j}(t), \quad \text{where } c\_{p} = \frac{((p+1)!)^{2}}{(2p+2)!}.\tag{31}$$

Throughout this work the roots of + 1,() are denoted by , <sup>=</sup> + − <sup>1</sup> <sup>2</sup> <sup>+</sup> ℎ <sup>2</sup> , = 0, 1, …, .

In the next lemma, we recall the following results which will be needed in our error analysis [32].

**Lemma 4.1.** *The polynomials* , *and* + 1, *satisfy the following properties*

$$\left\| L\_{p,j} \right\|\_{l\_j}^2 = \frac{h\_j}{2p+1}, \quad \int\_{l\_j} \left\| \nu\_{p+1,j}^{\prime} \nu\_{p+1,j} \right\| dt = -k\_1 h\_j^{2p+2}, \quad \left\| \nu\_{p+1,j} \right\|\_{l\_j}^2 = (2p+2)k\_2 h\_j^{2p+3}, \tag{32}$$

*where* 1 = 2 <sup>2</sup>, 2 <sup>=</sup> 1 (2 + 1)(2 + 3) , *and* <sup>=</sup> (( + 1)!)2 (2 + 2)! .

Now, we are ready to prove the following superconvergence results.

**Theorem 4.1.** *Suppose that the assumptions of Theorem 1 are satisfied. Also, we assume that is sufficiently smooth with respect to and (for example,* ℎ() = (, ()) ∈ ([0, ]) *is enough.). Then there exists a positive constant such that*

$$\left| e(t\_k^{-}) \right| \le C h^{2^{p\_0+1}}, \quad k = 1, \dots, N\_\prime \tag{33}$$

$$\left|\overline{e}\left(t\_k^{-}\right)\right| \le Ch^{2\rho+1}, \quad k = 1, \dots, N,\tag{34}$$

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations http://dx.doi.org/10.5772/64967 45

$$\left\| \overline{\mathcal{e}}' \right\| \leq C \, h^{p \ast 1} \, \text{ } \tag{35}$$

$$\left\|\overline{\boldsymbol{e}}\right\| \leq C \; h^{\nu+2}.\tag{36}$$

*Proof.* To prove (33), we proceed by the duality argument. Consider the following auxiliary problem:

$$\mathcal{W}' + \theta \mathcal{W} = 0, \quad t \in [0, t\_k] \quad \text{subject to} \quad \mathcal{W}(t\_k) = 1,\tag{37}$$

where <sup>1</sup> ≤ <sup>≤</sup> and = () = ∫0 1 (, () − . The exact solution of this problem is () = exp ∫ () , = [0,] . Using the assumption ℎ() = (, ()) ([0, ]) and the estimate (24a), we can easily show that there exists a constant such that

$$\left\|\mathcal{W}\right\|\_{p\,\,\,\mathrm{+}\,\Omega\_{\mathrm{k}}}\leq\mathrm{C}.\tag{38}$$

Using (17) and (37), we get

Mapping

ψ

, <sup>=</sup> + − <sup>1</sup> <sup>2</sup> <sup>+</sup>

[32].

*where* 1 = 2

 = + − <sup>1</sup> <sup>2</sup> <sup>+</sup>

44 Perusal of the Finite Element Method

ℎ

+

*p*

ℎ <sup>2</sup>

*h*

*p*

*Then there exists a positive constant such that*

<sup>2</sup>, 2 <sup>=</sup>

into the reference element [ − 1,1] by the linear transformation

 − − − − 1 1

*h h*

*tt t tt t*

*j j*

− −

 as  :

(31)

(32)

<sup>2</sup> , we obtain the shifted Legendre and Radau polynomials on

2 2 ( )= , ( )= . *j j j j*

+

*h p <sup>p</sup> <sup>t</sup> R t ch R t c*

[( 1)!] (( 1)!) ( )= ( ) = ( ), where = . (2 2)! (2 2)!

*p p*

Throughout this work the roots of + 1,() are denoted by

In the next lemma, we recall the following results which will be needed in our error analysis

ψ

(2 + 2)! .

+ + <sup>+</sup> ′ − + <sup>+</sup> <sup>∫</sup> 2 2 2 2 2 3 , 1, 1, <sup>1</sup> 1, <sup>2</sup> = , = , = (2 2) , 2 1 *j p p*

**Theorem 4.1.** *Suppose that the assumptions of Theorem 1 are satisfied. Also, we assume that is*

*sufficiently smooth with respect to and (for example,* ℎ() = (, ()) ∈ ([0, ]) *is enough.).*

*p j p jp j j p j <sup>j</sup> I I <sup>I</sup> j j <sup>j</sup>*

*L dt k h p kh*

(2 + 1)(2 + 3) , *and* <sup>=</sup> (( + 1)!)2

Now, we are ready to prove the following superconvergence results.

+ +

<sup>−</sup> <sup>+</sup> *e t Ch k N* ( ) , = 1, , , *<sup>k</sup>* ≤ 2 1 *<sup>p</sup>* (33)

<sup>−</sup> <sup>+</sup> *e t Ch k N* ( ) , = 1, , , *<sup>k</sup>* ≤ 2 1 *<sup>p</sup>* (34)

+ + + + 1 2 2 1

+ + + +

*pj p pj p*

*L tL R tR*

*p j p j pj p j p*

**Lemma 4.1.** *The polynomials* , *and* + 1, *satisfy the following properties*

Next, we define the monic Radau polynomial, + 1,(), on

+ + +

, = 0, 1, …, .

ψ ψ

1

1, 1, 1,

*j p*

1, 1 1, 1

$$\mathcal{A}\_{\rangle}(\mathcal{e}; \mathcal{W}) = \int\_{\mathbf{i}\_{\rangle}} (-\mathcal{W}' - \theta \mathcal{W}) \mathrm{e}dt + -e(\mathbf{i}\_{\rangle-1}^{-})\mathcal{W}(\mathbf{t}\_{\rangle-1}) + e(\mathbf{i}\_{\rangle}^{-})\mathcal{W}(\mathbf{t}\_{\rangle}) = -e(\mathbf{i}\_{\rangle-1}^{-})\mathcal{W}(\mathbf{t}\_{\rangle-1}) + e(\mathbf{i}\_{\rangle}^{-})\mathcal{W}(\mathbf{t}\_{\rangle}).$$

Summing over the elements , = 1, …, , using ()=1, and the fact that (0 −)=0, we obtain

$$\sum\_{j=1}^{k} \mathcal{A}\_{j}(e; \mathcal{W}) = -e(t\_{0}^{-})\mathcal{W}(t\_{0}) + e(t\_{k}^{-})\mathcal{W}(t\_{k}) = e(t\_{k}^{-}).\tag{39}$$

Now, taking = in (21) yields

$$\mathcal{A}\_{\rangle}(e;W) = \int\_{I\_{\slash}} (\varepsilon' - \theta e)(W - P\_{\hbar}^{\,\*}W)dt...$$

Summing over all elements , = 1, …, with = 1, …, and applying (39), we arrive at

$$e(t\_k^-) = \sum\_{j=1}^k \int\_{I\_j} (\varepsilon' - \theta e)(W - P\_h^+ W)dt.$$

Using (24a) and applying the Cauchy-Schwarz inequality, we obtain

$$\left| \left| e(t\_k^-) \right| \le \left( \left\| \boldsymbol{\varepsilon}' \right\|\_{0,\Omega\_k} + M\_1 \left\| \boldsymbol{\varepsilon} \right\|\_{0,\Omega\_k} \right) \left\| \boldsymbol{\mathcal{W}} - \boldsymbol{P}\_h^+ \boldsymbol{\mathcal{W}} \right\|\_{0,\Omega\_k} \le \left( \left\| \boldsymbol{\varepsilon}' \right\| + M\_1 \left\| \boldsymbol{\varepsilon} \right\| \right) \left\| \boldsymbol{\mathcal{W}} - \boldsymbol{P}\_h^+ \boldsymbol{\mathcal{W}} \right\|\_{0,\Omega\_k}.$$

Invoking the estimates (7), (9), and (38), we conclude that

$$\left| e(t\_k^-) \right| \le \left( \mathbb{C}\_v h^p \left| \mu \right|\_{p+1,\Omega} + M\_1 \mathbb{C}\_1 h^{p+1} \right) \mathbb{C}\_2 h^{p+1} \left| \mathcal{W} \right|\_{p+1,\Omega\_{\hat{k}}} \le \mathbb{C} (h^p + h^{p+1}) h^{p+1} = \mathcal{O} \left( h^{2^{p+1}} \right),\tag{40}$$

for all = 1, …, , which completes the proof of (33).

In order to prove (34), we use the relation = + , the property of the projection ℎ −, i.e., ( −)=0, and the estimate (33) to get

$$\left| \overline{e}(t\_k^-) \right| = \left| e(t\_k^-) - \varepsilon(t\_k^-) \right| = \left| e(t\_k^-) \right| = \mathcal{O}(h^{2^{\rho\_k+1}}).$$

Next, we will derive optimal error estimate for ′ . By the property of ℎ −, we have

$$\int\_{I\_{\neq}} \varepsilon \upsilon' dt = 0, \quad \forall \ v \in P^{\mathbb{P}}(I\_{\neq}), \quad \text{and} \quad \varepsilon(t\_{\neq}^{-}) = 0, \quad j = 1, \ldots, N. \tag{41}$$

Using the relation =+, applying (41) and (11) yields

$$\int\_{I\_{j}} \boldsymbol{v} \cdot \overline{\boldsymbol{\varepsilon}} \, dt + \int\_{I\_{j}} (f(t, \boldsymbol{u}) - f(t, \boldsymbol{u}\_{h})) \boldsymbol{v} \, dt + \overline{\boldsymbol{\varepsilon}} (t\_{j-1}^{-}) \boldsymbol{v} (t\_{j-1}^{+}) - \overline{\boldsymbol{\varepsilon}} (t\_{j}^{-}) \boldsymbol{v} (t\_{j}^{-}) = 0.$$

By integration by parts on the first term, we obtain

$$\int\_{I\_{j}} \left(\overline{e} - f(t, \mu) + f(t, \mu\_{h})\right) \upsilon dt + [\overline{e}](t\_{j-1})\upsilon(t\_{j-1}^{\*}) = 0. \tag{42}$$

Choosing () = ′() − ( − 1)′( <sup>−</sup> <sup>1</sup> <sup>+</sup> ), () ∈ ( ) in (42), we have, by the property ( − 1) = ( − 1) and the orthogonality relation (30), ( <sup>−</sup> <sup>1</sup> <sup>+</sup> )=0 and

$$\begin{split} \int\_{I\_{\underline{j}}} (\overline{\boldsymbol{c}})^2 dt &= (-1)^{\overline{\boldsymbol{c}}} \overline{\boldsymbol{c}}(t\_{\underline{j-1}}^{+}) \int\_{I\_{\underline{j}}} \mathcal{L}\_{\underline{p},\boldsymbol{\overline{c}}} \overline{\boldsymbol{c}} dt + \int\_{I\_{\underline{j}}} (f(t, \boldsymbol{u}) - f(t, \boldsymbol{u}\_{\underline{h}})) (\overline{\boldsymbol{c}} - \boldsymbol{\{-1\}}^{\boldsymbol{v}} \overline{\boldsymbol{c}}(t\_{\underline{j-1}}^{+}) \mathcal{L}\_{\underline{p},\boldsymbol{\overline{c}}}) dt \\ &= \int\_{I\_{\underline{j}}} (f(t, \boldsymbol{u}) - f(t, \boldsymbol{u}\_{\underline{h}})) (\overline{\boldsymbol{c}} - \boldsymbol{\{-1\}}^{\boldsymbol{v}} \overline{\boldsymbol{c}}(t\_{\underline{j-1}}^{+}) \mathcal{L}\_{\underline{p},\boldsymbol{\overline{c}}}) dt. \end{split} \tag{43}$$

Using (3) and applying the Cauchy-Schwarz inequality gives

$$\begin{split} \left\lVert \overline{\boldsymbol{\varepsilon}} \right\rVert\_{\boldsymbol{l}\_{j}}^{2} &\leq \int\_{\boldsymbol{l}\_{j}} \left| f(\boldsymbol{t}, \boldsymbol{u}) - f(\boldsymbol{t}, \boldsymbol{u}\_{h}) \right| \left( \left| \overline{\boldsymbol{\varepsilon}} \right| + \left| \overline{\boldsymbol{\varepsilon}} \left( \boldsymbol{t}\_{j-1}^{+} \right) \right| \left| \boldsymbol{L}\_{p\_{>}} \right| \right) dt \leq M\_{1} \int\_{\boldsymbol{l}\_{j}} \left| \boldsymbol{e} \right| \left( \left| \overline{\boldsymbol{\varepsilon}} \right| + \left| \overline{\boldsymbol{\varepsilon}} \left( \boldsymbol{t}\_{j-1}^{+} \right) \right| \left| \boldsymbol{L}\_{p\_{>}} \right| \right) dt \\ &\leq M\_{1} \left\lVert \boldsymbol{e} \right\rVert\_{\boldsymbol{l}\_{j}} \left( \left\lVert \overline{\boldsymbol{\varepsilon}} \right\rVert\_{\boldsymbol{l}\_{j}} + \left| \overline{\boldsymbol{\varepsilon}} \left( \boldsymbol{t}\_{j-1}^{+} \right) \right| \left\lVert \boldsymbol{L}\_{p\_{>}} \right\rVert\_{\boldsymbol{l}\_{j}} \right). \end{split} \tag{44}$$

Combining (44) with (8) and (32), we obtain

ε θ − + ∑ ′ − − ∫ =1 ( ) = ( )( ) .

*k h I*

 − ++ Ω Ω Ω Ω ≤ + − ≤+ − ′ ′ 1 1 0, 0, 0, 0, () ( ) ( ) . *k hh k k k k e t M e W PW M e W PW*

− + + ++ + + Ω + Ω ≤ + ≤ + 1 1 1 1 21 <sup>0</sup> 11 2 1, 1, () ( ) ( ) = ( ), *<sup>p</sup> p p ppp p*

In order to prove (34), we use the relation = + , the property of the projection ℎ

ε − −− − <sup>+</sup> − 2 1 ( ) = ( ) ( ) = ( ) = ( ). *<sup>p</sup> k kk k e t et t et h*

 <sup>−</sup> ′ ∀ ∈ ∫ = 0, ( ), and ( ) = 0, = 1, , . *<sup>p</sup> j j <sup>I</sup>*

 ε

− − ′ +− + − ∫ ∫ 1 1 ( ( , ) ( , )) ( ) ( ) ( ) ( ) = 0. *h j j jj I I*

*v edt f t u f t u vdt e t v t e t v t*

( ) <sup>+</sup> − − −+ + ∫ 1 1 ' ( , ) ( , ) [ ]( ) ( ) = 0. *h jj <sup>I</sup>*

*v dt v P I t jN* (41)

− + −−

*e f t u f t u vdt e t v t* (42)

Next, we will derive optimal error estimate for ′ . By the property of ℎ

*e t Ch u MCh C h W C h h h h* (40)

*e t e W P W dt*

 ε

−, i.e.,

−, we have

*k*

Using (24a) and applying the Cauchy-Schwarz inequality, we obtain

Invoking the estimates (7), (9), and (38), we conclude that

*<sup>k</sup> p p <sup>k</sup>*

for all = 1, …, , which completes the proof of (33).

−)=0, and the estimate (33) to get

ε

Using the relation =+, applying (41) and (11) yields

*j j*

*j*

By integration by parts on the first term, we obtain

*j*

(

ε

46 Perusal of the Finite Element Method

*j j*

$$\left\| \overline{\boldsymbol{e}}' \right\|\_{l\_j}^2 \le \mathcal{M}\_1 \left\| \boldsymbol{e} \right\|\_{l\_j} \left( \left\| \overline{\boldsymbol{e}}' \right\|\_{l\_j} + \left( \mathbf{C}\_1 h\_j^{-1/2} \left\| \overline{\boldsymbol{e}}' \right\|\_{l\_j} \right) \left( \frac{h\_j^{1/2}}{(2p+1)^{1/2}} \right) \right) \le \mathbf{C} \left\| \boldsymbol{e} \right\|\_{l\_j} \left\| \overline{\boldsymbol{e}}' \right\|\_{l\_j}.$$

Consequently, ′ ≤ . Taking the square of both sides, summing over all elements, and using (9), we conclude that

$$\left\|\overline{\boldsymbol{e}}\right\|^2 \le \mathbb{C} \left\|\boldsymbol{e}\right\|^2 \le \mathcal{C} h^{2^{p\_s+2}}.\tag{45}$$

Finally, we will estimate . Using the fundamental theorem of calculus, we write

$$\mathbb{I}\mid\overline{\boldsymbol{e}}\{\boldsymbol{t}\}\mid\vdash\|\overline{\boldsymbol{e}}\{\boldsymbol{t}\_{\boldsymbol{\cdot}}^{-}\}+\int\_{\boldsymbol{t}\_{\boldsymbol{\cdot}}}^{\boldsymbol{t}}\overline{\boldsymbol{e}}\{\operatorname{s}\}\operatorname{ds}\mid\operatorname{\mathbb{L}}\|\overline{\boldsymbol{e}}\{\operatorname{t}\_{\boldsymbol{\cdot}}^{-}\}\|+\int\_{\boldsymbol{I}\_{\boldsymbol{\cdot}}}\|\overline{\boldsymbol{e}}\{\operatorname{s}\}\|\operatorname{ds},\quad\forall\ \boldsymbol{t}\in\boldsymbol{I}\_{\boldsymbol{\cdot}}.$$

Taking the square of both sides, applying the inequality ( + )2 ≤ 22 + 22, and applying the Cauchy-Schwartz inequality, we get

$$2\|\overline{e}(t)\|^2 \le 2\|\overline{e}(t\_{\rangle}^-)\|^2 + 2\left(\int\_{I\_{\underline{j}}} \|\overline{e}\langle \mathbf{s}\rangle\| \, ds\right)^2 \le 2\|\overline{e}(t\_{\rangle}^-)\|^2 + 2h\int\_{I\_{\underline{j}}} \|\overline{e}\langle \mathbf{s}\rangle\|^2 \, ds = 2\left\|\overline{e}(t\_{\rangle}^-)\|^2 + 2h\left\|\overline{e}'\right\|\_{\underline{j}}^2.$$

Integrating this inequality with respect to and using the estimate (34), we get

$$\left\| \overline{\boldsymbol{\varepsilon}} \right\|\_{\boldsymbol{l}\_{\boldsymbol{j}}}^2 \leq \mathfrak{D} \boldsymbol{h}\_{\boldsymbol{j}} \left\| \overline{\boldsymbol{\varepsilon}} \{ \boldsymbol{t}\_{\boldsymbol{j}}^{-} \} \right\|^2 + \mathfrak{D} \boldsymbol{h}\_{\boldsymbol{j}}^2 \left\| \overline{\boldsymbol{\varepsilon}} \right\|\_{\boldsymbol{l}\_{\boldsymbol{j}}}^2 \leq \mathfrak{D} \boldsymbol{h}\_{\boldsymbol{j}}^{4^{p+3}} + \mathfrak{D} \boldsymbol{h}\_{\boldsymbol{j}}^2 \left\| \overline{\boldsymbol{\varepsilon}} \right\|\_{\boldsymbol{l}\_{\boldsymbol{j}}}^2.$$

Summing over all elements and using the estimate (35) and the fact that ℎ = maxℎ , we obtain

$$\left\|\overline{\boldsymbol{e}}\right\|^2 \le \mathbb{C}\_{\mathbf{r}} h^{4\boldsymbol{\uprho}+2} + 2h^2 \left\|\overline{\boldsymbol{e}}\right\|^2 \le \mathbb{C}\_{\mathbf{r}} h^{4\boldsymbol{\uprho}+2} + 2\mathbb{C}\_{\mathbf{r}} h^{2\boldsymbol{\uprho}+4} = \mathcal{O}(h^{2\boldsymbol{\uprho}+4}),\tag{46}$$

where we used 4 + 2 ≥ 2 + 4 for ≥ 1. This completes the proof of the theorem.

The previous theorem indicates that the DG solution ℎ is closer to ℎ − than to the exact solution . Next, we apply the results of Theorem 2 to prove that the actual error can be split into a significant part, which is proportional to the ( + 1) degree right Radau polynomial, and a less significant part that converges at (*hp*+2) rate in the 2 norm. Before we prove this result, we introduce two interpolation operators and . The interpolation operator is defined as follows: For smooth = (), ∈ ( ) and interpolates at the roots of the ( + 1) degree right Radau polynomial shifted to , i.e., at , , = 0, 1, …, ,. The interpolation operator satisfies ∈ + 1( ) and interpolates at , , = 0, 1, …, , and at an additional point in with ≠ , , = 0, 1, …, . The choice of the additional point is not important and can be chosen as = − <sup>1</sup>.

Next, we recall the following results from [12] which will be needed in our analysis.

**Lemma 4.2.** *If* ∈ + 2( ), *then interpolation error can be split as*

$$
\mu \mathbf{u} - \boldsymbol{\pi} \mathbf{u} = \boldsymbol{\phi}\_j + \boldsymbol{\gamma}\_j, \quad \text{on} \ I\_j,\tag{47a}
$$

*where*

$$
\mu \phi\_j(t) = \alpha\_j \nu\_{\,\_{p+1,j}}(t), \quad \nu \nu\_{\,\_{p+1,j}}(t) = \prod\_{i=0}^{p} (t - t\_{\,\_{j,i}}), \quad \gamma\_{\,\_{j}} = u - \hat{\pi}u,\tag{47b}
$$

*and is the coefficient of* + 1 *in the* ( + 1) *degree polynomial . Furthermore,*

$$\left\|\phi\_{j}\right\|\_{s,I\_{j}} \leq C h\_{j}^{p+1-s} \left\|u\right\|\_{p+1,I\_{j}}, \quad s = 0,...,p,\tag{47c}$$

$$\left\|\boldsymbol{\mathcal{V}}\_{j}\right\|\_{s,\boldsymbol{I}\_{j}} \leq Ch\_{j}^{p+2-s} \left\|\boldsymbol{u}\right\|\_{p+2,\boldsymbol{I}\_{j}}, \quad \text{s} = \boldsymbol{0}, \ldots, p+1. \tag{47d}$$

*Finally,*

, we obtain

− than to the exact

− −− ≤+ ≤+ ′ ′′ <sup>+</sup>

<sup>−</sup> <sup>+</sup> ≤ +≤+ ′ ′ 2 22 2 2 4 3 <sup>2</sup> 2 | ( )| 2 2 2 . *<sup>p</sup> jj j j j I II j jj e h e t h e Ch h e*

<sup>1</sup> 1 2 2 ' 2 = ( ), *<sup>p</sup> p pp e Ch h e Ch Ch h* (46)

) and interpolates at the roots of the ( + 1) degree

+ (47a)

 p

(47b)

, i.e., at , , = 0, 1, …, ,. The interpolation operator

interpolates at , , = 0, 1, …, , and at an additional point

<sup>2</sup> 2 2 2 22 | ( )| 2| ( )| 2 | ( )| 2| ( )| 2 | ( )| = 2| ( )| 2 . *<sup>j</sup> j j j j I I <sup>I</sup> j j <sup>j</sup> e t e t e s ds e t h e s ds e t h e*

∫ ∫ 2

Integrating this inequality with respect to and using the estimate (34), we get

Summing over all elements and using the estimate (35) and the fact that ℎ = maxℎ

where we used 4 + 2 ≥ 2 + 4 for ≥ 1. This completes the proof of the theorem.

The previous theorem indicates that the DG solution ℎ is closer to ℎ

∈ (

Next, we recall the following results from [12] which will be needed in our analysis.

=, , *jj j u u on I* −

1, 1, ,

*t t t tt u u* + + Õ − −

*j jp j p j ji j*

 y

=0 ( ) = ( ), ( ) = ( ), = , ˆ *p*

g

*i*

), *then interpolation error can be split as*

p fg

 

) and

follows: For smooth = (),

 

48 Perusal of the Finite Element Method

be chosen as = − <sup>1</sup>.

**Lemma 4.2.** *If* ∈ + 2(

f

 ay

satisfies

 in 

*where*

right Radau polynomial shifted to

∈ + 1(

<sup>+</sup> + ++ ≤+ ≤+ 2 2 4 2 <sup>2</sup> 42 24 24

solution . Next, we apply the results of Theorem 2 to prove that the actual error can be split into a significant part, which is proportional to the ( + 1) degree right Radau polynomial, and a less significant part that converges at (*hp*+2) rate in the 2 norm. Before we prove this result, we introduce two interpolation operators and . The interpolation operator is defined as

with ≠ , , = 0, 1, …, . The choice of the additional point is not important and can

$$\left\|\pi\mu - P\_h^-\mu\right\|\_{l\_j} \le Ch\_j^{p+2} \left\|\mu\right\|\_{p+2,I\_j}.\tag{48}$$

*Proof.* The proof of this lemma can be found in [12], more precisely in its Lemma 2.1.

The main global superconvergence result is stated in the following theorem.

**Theorem 4.2.** *Under the assumptions of Theorem 2, there exists a constant independent of* ℎ *such that*

$$\left\|\mu\_{h} - \pi u\right\| \le Ch^{p \times 2}.\tag{49}$$

*Moreover, the true error can be divided into a significant part and a less significant part as*

$$
\alpha e(t) \equiv \alpha\_{/} \boldsymbol{\mu}\_{\,\, p \, s \, 1, \, j}(t) + \alpha\_{/}(t), \quad \text{on} \quad I\_{\,\, j}, \tag{50a}
$$

*where*

$$
\mu \boldsymbol{\alpha}\_{j} = \boldsymbol{\mathcal{V}}\_{j} + \boldsymbol{\pi} \boldsymbol{u} - \boldsymbol{u}\_{h}, \tag{50b}
$$

*and*

$$\sum\_{j=1}^{N} \left\| \boldsymbol{o}\_{j} \right\|\_{\boldsymbol{I}\_{j}}^{2} \leq C h^{2(\rho+2)}, \qquad \sum\_{j=1}^{N} \left\| \boldsymbol{o}\_{j^{\*}} \right\|\_{\boldsymbol{I}\_{j}}^{2} \leq C h^{2(\rho+1)}.\tag{50c}$$

*Proof.* Adding and subtracting ℎ − to ℎ − , we write ℎ − = ℎ − ℎ −+ℎ −− <sup>=</sup> − + ℎ − − . Taking the 2 norm and using the triangle inequality, we get

$$\left\|\boldsymbol{\mu}\_{\boldsymbol{h}} - \boldsymbol{\pi}\boldsymbol{\mu}\right\| \leq \left\|\overline{\boldsymbol{e}}\right\| + \left\|\boldsymbol{P}\_{\boldsymbol{h}}^{\cdot}\boldsymbol{\mu} - \boldsymbol{\pi}\boldsymbol{\mu}\right\|.$$

Applying the estimates (36) and (48), we deduce (49). Next, adding and subtracting to , we write = − <sup>+</sup> −ℎ. Moreover, one can split the interpolation error − on as in (47a) to get

$$e = \phi\_{\rangle} + \gamma\_{\rangle} + \pi\mu - \mu\_{\hbar} = \phi\_{\rangle} + o\nu\_{\rangle}, \quad \text{where} \quad o\nu\_{\rangle} = \gamma\_{\rangle} + \pi\mu - \mu\_{\hbar}. \tag{51}$$

Next, we use the Cauchy-Schwarz inequality and the inequality <sup>≤</sup> <sup>1</sup> 2( <sup>2</sup> <sup>+</sup> 2) to write

$$\begin{split} \left\lVert \boldsymbol{\alpha}\_{\boldsymbol{\beta}} \right\rVert\_{\boldsymbol{I}\_{\boldsymbol{\beta}}}^{2} &= \left( \boldsymbol{\gamma}\_{\boldsymbol{\beta}} + \boldsymbol{\pi}\boldsymbol{\mu} - \boldsymbol{u}\_{h}, \boldsymbol{\gamma}\_{\boldsymbol{\beta}} + \boldsymbol{\pi}\boldsymbol{u} - \boldsymbol{u}\_{h} \right)\_{\boldsymbol{I}\_{\boldsymbol{\beta}}} = \left\lVert \boldsymbol{\gamma}\_{\boldsymbol{\beta}} \right\rVert\_{\boldsymbol{\mathcal{I}}\_{\boldsymbol{\beta}}}^{2} + 2 \left( \boldsymbol{\gamma}\_{\boldsymbol{\beta}}, \boldsymbol{\pi}\boldsymbol{u} - \boldsymbol{u}\_{h} \right)\_{\boldsymbol{\mathcal{I}}\_{\boldsymbol{\beta}}} + \left\lVert \boldsymbol{\pi}\boldsymbol{u} - \boldsymbol{u}\_{h} \right\rVert\_{\boldsymbol{\mathcal{I}}\_{\boldsymbol{\beta}}}^{2} \\ &\leq 2 \left( \left\lVert \boldsymbol{\mathcal{V}}\_{\boldsymbol{\beta}} \right\rVert\_{\boldsymbol{\mathcal{I}}\_{\boldsymbol{\beta}}}^{2} + \left\lVert \boldsymbol{\pi}\boldsymbol{u} - \boldsymbol{u}\_{h} \right\rVert\_{\boldsymbol{\mathcal{I}}\_{\boldsymbol{\beta}}}^{2} \right). \end{split}$$

Summing over all elements and applying (47d) with = 0 and (49) yields the first estimate in (50c).

Using the Cauchy-Schwarz inequality and the inequality <sup>≤</sup> <sup>1</sup> 2( <sup>2</sup> <sup>+</sup> 2), we write

$$\left\|\boldsymbol{\alpha}\_{\boldsymbol{\cdot}^{\cdot}}\right\|\_{\boldsymbol{\mathbb{I}}\_{\boldsymbol{\cdot}}}^{2} = \left(\boldsymbol{\gamma}\_{\boldsymbol{\cdot}^{\cdot}} + (\boldsymbol{\pi}\boldsymbol{\mu} - \boldsymbol{\mu}\_{h})^{\prime}\_{\ast}\boldsymbol{\gamma}\_{\boldsymbol{\cdot}^{\cdot}} + (\boldsymbol{\pi}\boldsymbol{\mu} - \boldsymbol{\mu}\_{h})^{\prime}\right)\_{\boldsymbol{\cdot}\_{\boldsymbol{\cdot}}} \leq 2\left(\left\|\boldsymbol{\gamma}\_{\boldsymbol{\cdot}^{\cdot}}\right\|\_{\boldsymbol{\mathbb{I}}\_{\boldsymbol{\cdot}}}^{2} + \left\|(\boldsymbol{\pi}\boldsymbol{\mu} - \boldsymbol{\mu}\_{h})^{\prime}\right\|\_{\boldsymbol{\mathbb{I}}\_{\boldsymbol{\cdot}}}^{2}\right). \tag{52}$$

Using the inverse inequality (8), i.e., (−ℎ)′ ≤ <sup>ℎ</sup>−1 (−ℎ) , we obtain the estimate

$$\left\| \left\| \boldsymbol{\alpha}\_{\boldsymbol{\beta}^\*} \right\|\_{\boldsymbol{I}\_{\boldsymbol{\beta}}}^2 \le \mathbb{C} \left( \left\| \boldsymbol{\mathcal{V}}\_{\boldsymbol{\beta}^\*} \right\|\_{\boldsymbol{I}\_{\boldsymbol{\beta}}}^2 + h^{-2} \left\| \boldsymbol{\pi} \boldsymbol{\mu} - \boldsymbol{\mu}\_h \right\|\_{\boldsymbol{I}\_{\boldsymbol{\beta}}}^2 \right).$$

Summing over all elements and applying (49) and the estimate (47d) with = 1, we establish the second estimate in (50c).

## **5.** *A posteriori* **error estimation**

*Proof.* Adding and subtracting ℎ

−− <sup>=</sup> − + ℎ

π

 φω

Next, we use the Cauchy-Schwarz inequality and the inequality <sup>≤</sup> <sup>1</sup>

= , = 2,

 γ π

 π

2 .

 ≤ +− 

2 2

*u u*

Using the Cauchy-Schwarz inequality and the inequality <sup>≤</sup> <sup>1</sup>

′′ ′ ( ) ′

ωγ

′ ′

 γπ

Using the inverse inequality (8), i.e., (−ℎ)′

*j h <sup>I</sup> <sup>I</sup> <sup>j</sup> <sup>j</sup>*

<sup>−</sup> − ≤+ − . *h h u u e Pu u*

Applying the estimates (36) and (48), we deduce (49). Next, adding and subtracting to , we write = − <sup>+</sup> −ℎ. Moreover, one can split the interpolation error − on

> γ

2 2 2

*j j hj h j j h h <sup>I</sup> I I I I <sup>j</sup> j j j j*

Summing over all elements and applying (47d) with = 0 and (49) yields the first estimate in

+− +− ≤ + − ′′ ′

2 2 2 = ( ), ( ) 2 ( ) . *j j hj h j h <sup>I</sup> I I <sup>I</sup> <sup>j</sup> j j <sup>j</sup>*

+− +− + − + −

*uu uu uu uu*

 γ

 π<sup>−</sup>

 ≤ +− 2 2 <sup>2</sup> <sup>2</sup> . *jj h <sup>I</sup> I I <sup>j</sup> j j C h uu*

 π

*uu uu u u* (52)

( ) ( )

−+ℎ

= φγ

> γ π

> > γ

ω

 π

ℎ − = ℎ − ℎ

50 Perusal of the Finite Element Method

inequality, we get

in (47a) to get

(50c).

estimate

ω

 γπ − to ℎ − , we write

2(

 π

<sup>2</sup> <sup>+</sup> 2), we write

, we obtain the

2(

≤ <sup>ℎ</sup>−1 (−ℎ)

<sup>2</sup> <sup>+</sup> 2) to write

 as

− − . Taking the 2 norm and using the triangle

 π

 ωγ

++ − + = , where = . + − *jj hj j jj h e uu u u* (51)

 γπ  π In this section, we use the superconvergence results from the previous section to construct a residual-based *a posteriori* error estimator which is computationally simple, efficient, and asymptotically exact. We will also prove its asymptotic exactness under mesh refinement. First, we present the weak finite element formulation to compute *a posteriori* error estimate for the nonlinear IVP (2).

In order to obtain a procedure for estimating the error , we multiply (2) by arbitrary smooth function and integrate over the element to obtain

$$\int\_{I\_{\underline{j}}} \mu' v dt = \int\_{I\_{\underline{j}}} f(t, \mu) v dt. \tag{53}$$

Replacing by ℎ + and choosing = + 1,(), we obtain

$$\int\_{I\_{\underline{j}}} e' \boldsymbol{\nu}\_{p+1,\uparrow} dt = \int\_{I\_{\underline{j}}} \left( f(t, \boldsymbol{\mu}\_h + \boldsymbol{e}) - \boldsymbol{\mu}\_h' \right) \boldsymbol{\nu}\_{p+1,\uparrow} dt. \tag{54}$$

Substituting (50a), i.e., () = + 1,() + (), into the left-hand side of (54) yields

$$\alpha \left\{ \,\_1 \psi\_{p+1,\cdot} \vert \nu \nu\_{p+1,\cdot} \right\} dt = \int\_{\mathbb{T}\_j} \left( f(t, \mathfrak{u}\_h + \varepsilon) - \mathfrak{u}\_h' - \alpha \mathfrak{e}\_{\rangle}' \right) \vert \nu \nu\_{p+1,\cdot} \rangle dt. \tag{55}$$

Using (32) and solving for , we obtain

$$\alpha\_{j} = -\frac{1}{k\_{1}h\_{j}^{2p+2}} \int\_{I\_{j}} \left( f(t\_{\prime}\boldsymbol{\mu}\_{h} + \boldsymbol{\varepsilon}) - \boldsymbol{\mu}\_{h}^{\prime} - \alpha\_{j}^{\prime} \right) \boldsymbol{\upmu}\_{p+1,j} dt. \tag{56}$$

Our error estimate procedure consists of approximating the true error on each element by the leading term as

$$e(t) \approx E(t) \equiv a\_{/} \mathcal{Y}\_{p+1, \backslash}(t), \quad t \in I\_{\backslash}, \tag{57a}$$

where the coefficient of the leading term of the error, , is obtained from the coefficient defined in (56) by neglecting the terms and , i.e.,

$$a\_j = -\frac{1}{k\_1 h\_j^{2\rho+2}} \int\_{I\_j} (f(t, u\_h) - u\_{h^\*}) \boldsymbol{\upnu}\_{\boldsymbol{\upnu}+1, j} dt. \tag{57b}$$

We remark that our *a posteriori* error estimate is obtained by solving local problems with no boundary condition.

The global effectivity index, defined by = , is an important criterion for evaluating the quality of an error estimator. The main results of this section are stated in the following theorem. In particular, we prove that the error estimate , in the 2 norm, converges to the actual error . Moreover, we show that our *a posterior* error estimate is asymptotically exact by showing that the global effectivity index 1 as ℎ 0.

**Theorem 5.1.** *Suppose that the assumptions of Theorem 2 are satisfied. If* () = + 1,(), , *where* , = 1, …, , *are defined in (57b), then*

$$\left\| e - E \right\|^2 \le C \left\| \begin{array}{c} h^{2\rho + 4} . \end{array} . \tag{58}$$

*Thus, the post‐processed approximation* ℎ + *yields* (ℎ + 2) *superconvergent solution, i.e.,*

$$\left\|\mu - (\mu\_h + E)\right\|^2 = \sum\_{j=1}^N \left\|\mu - (\mu\_h + a\_j \nu\_{p+1,j})\right\|\_{l\_j}^2 \le C \left\|h\right\|^{2p+4}.\tag{59}$$

*Furthermore, then there exists a positive constant independent of* ℎ *such that*

$$\left\| \left\| e \right\|^2 - \left\| E \right\|^2 \right\| \le C \left\| h^{2^{p+4}} \right\| \tag{60}$$

*Finally, if there exists a constant* = ()>0 *independent of* ℎ *such that*

<sup>+</sup> ≥ <sup>1</sup> , *<sup>p</sup> e Ch* (61)

*then the global effectivity index in the* 2 *norm converges to unity at* (*h*) *rate, i.e.,*

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations http://dx.doi.org/10.5772/64967 53

$$\left\| \frac{\left\| E \right\|}{\left\| e \right\|} = 1 + \mathcal{O}(h). \tag{62}$$

*Proof.* First, we will prove (58) and (59). Since = + 1, + and = + 1, on , we have

$$\left\|\boldsymbol{\varepsilon} - \boldsymbol{E}\right\|\_{\boldsymbol{l}\_{j}}^{2} = \left\|(\boldsymbol{\alpha}\_{j} - \boldsymbol{a}\_{j})\boldsymbol{\nu}\_{p+1,j} + \boldsymbol{o}\boldsymbol{o}\_{j}\right\|\_{\boldsymbol{l}\_{j}}^{2} \leq 2(\boldsymbol{\alpha}\_{j} - \boldsymbol{a}\_{j})^{2} \left\|\boldsymbol{\nu}\_{p+1,j}\right\|\_{\boldsymbol{l}\_{j}}^{2} + 2\left\|\boldsymbol{o}\_{j}\right\|\_{\boldsymbol{l}\_{j}}^{2} \leq 2$$

where we used the inequality ( + )2 ≤ 22 + 22. Summing over all elements yields

$$\left\|e - E\right\|^2 = \sum\_{j=1}^{N} \left\|e - E\right\|\_{l\_j}^2 \le 2\sum\_{j=1}^{N} (\alpha\_j - a\_j)^2 \left\|\nu\_{p\*1,j}\right\|\_{l\_j}^2 + 2\sum\_{j=1}^{N} \left\|\alpha\_j\right\|\_{l\_j}^2. \tag{63}$$

Next, we will derive upper bounds for ∑ = 1 ( − ) <sup>2</sup> + 1, 2 . Subtracting (56) from (57b), we obtain

$$
\alpha\_{\rangle} - \alpha\_{\rangle} = \frac{1}{k\_{\text{l}} \hbar\_{\text{j}}^{2g+2}} \int\_{\mathbf{l}\_{\text{j}}} \left( f(t, \mu\_{\text{h}} + \mathbf{e}) - f(t, \mu\_{\text{h}}) - \alpha\_{\text{j}} \right) \mathbb{1}\_{\mathbf{p} \sim \text{1}, \text{j}} dt. \tag{64}
$$

Thus,

where the coefficient of the leading term of the error,

The global effectivity index, defined by =

, = 1, …, , *are defined in (57b), then*

1

and , i.e.,

2 2 ( ) 1,

We remark that our *a posteriori* error estimate is obtained by solving local problems with no

uating the quality of an error estimator. The main results of this section are stated in the following theorem. In particular, we prove that the error estimate , in the 2 norm, converges to the actual error . Moreover, we show that our *a posterior* error estimate is asymptotically exact by showing that the global effectivity index

**Theorem 5.1.** *Suppose that the assumptions of Theorem 2 are satisfied. If* () =

<sup>+</sup> - ≤

*Thus, the post‐processed approximation* ℎ + *yields* (ℎ + 2) *superconvergent solution, i.e.,*

<sup>+</sup> - <sup>+</sup> ∑ - + ≤ <sup>2</sup> <sup>2</sup> 2 4

<sup>+</sup> ≥ <sup>1</sup>

*<sup>h</sup> h jp j <sup>I</sup> <sup>j</sup> <sup>j</sup>*

( )= ( ) .

=1

*Furthermore, then there exists a positive constant independent of* ℎ *such that*

*Finally, if there exists a constant* = ()>0 *independent of* ℎ *such that*

*then the global effectivity index in the* 2 *norm converges to unity at* (*h*) *rate, i.e.,*

y

*<sup>N</sup> <sup>p</sup>*

1,

<sup>+</sup>

*u u E u u a Ch* (59)

y

<sup>1</sup> <sup>=</sup> (, ) . *<sup>j</sup> <sup>p</sup> h h pj <sup>I</sup> <sup>j</sup> <sup>j</sup> <sup>a</sup> f t u u dt k h* - - <sup>+</sup> ò ¢

defined in (56) by neglecting the terms

boundary condition.

52 Perusal of the Finite Element Method

1 as ℎ 0.

*where*

, is obtained from the coefficient

<sup>+</sup> (57b)

, is an important criterion for eval-

<sup>2</sup> 2 4 . *<sup>p</sup> e E Ch* (58)

<sup>+</sup> - <sup>≤</sup> 2 2 2 4 . *<sup>p</sup> e E Ch* (60)

, *<sup>p</sup> e Ch* (61)

+ 1,(),

,

$$\left| a\_{\rangle} - \alpha\_{\rangle} \right| \leq \frac{1}{k\_{\text{1}} h\_{\text{\textquotedblleft}}^{2^{p+2}}} \int\_{I\_{\text{\textquotedblleft}}} \left( \left| f(t, u\_h + e) - f(t, u\_h) \right| + \left| \alpha\_{\rangle}' \right| \right) \left| \nu\_{p+1, \text{\textquotedblright}} \right| dt. \tag{65}$$

Using the Lipschitz condition (3) and applying the Cauchy-Schwarz inequality yields

$$\left| \left| a\_{j} - \alpha\_{j} \right| \leq \frac{1}{k\_{\mathrm{i}} h\_{j}^{2^{p+2}}} \int\_{I\_{j}} \left( M\_{1} \left| e \right| + \left| \alpha'\_{j} \right| \right) \left| \nu\_{\nu+1,j} \right| dt \leq \frac{\left\| \nu\_{\nu+1,j} \right\|\_{I\_{j}}}{k\_{\mathrm{i}} h\_{j}^{2^{p+2}}} \left( M\_{1} \left\| e \right\|\_{I\_{j}} + \left\| \alpha'\_{j} \right\|\_{I\_{j}} \right). \tag{66}$$

Applying the inequality ( + )<sup>2</sup> ≤ 2(2 + 2), we obtain

$$\left\|(a\_{\rangle}-\alpha\_{\rangle})^{2}\right\| \leq \frac{2\left\|\nu\_{p+1,\boldsymbol{\prime}}\right\|\_{\boldsymbol{I}\_{j}}^{2}}{k\_{1}^{2}h\_{\boldsymbol{\prime}}^{4p+4}}\left(M\_{1}^{2}\left\|e\right\|\_{\boldsymbol{I}\_{j}}^{2} + \left\|o\boldsymbol{\prime}\_{j}\right\|\_{\boldsymbol{I}\_{j}}^{2}\right). \tag{67}$$

Multiplying by + 1, <sup>2</sup> and using (32), i.e., + 1, <sup>2</sup> = (2 + 2)2ℎ 2 + 3 yields

$$\left\|(a\_j - \alpha\_j)^2\right\|\left\|\nu\_{p+1,j}\right\|\_{l\_j}^2 \le \frac{2\left\|\nu\_{p+1,j}\right\|\_{l\_j}^4}{k\_1^2 h\_j^{4^{p+4}}} \left(M\_1^2 \left\|\boldsymbol{\varepsilon}\right\|\_{l\_j}^2 + \left\|\boldsymbol{\alpha}\_j'\right\|\_{l\_j}^2\right) \le k\_3 h\_j^2 \left(\left\|\boldsymbol{\varepsilon}\right\|\_{l\_j}^2 + \left\|\boldsymbol{\alpha}\_j'\right\|\_{l\_j}^2\right) \tag{68}$$

where 3 <sup>=</sup> 2(2 + 2)22 2 1 <sup>2</sup> max(1 2, 1) is a constant independent of the mesh size.

Summing over all elements and using ℎ = max 1 ≤ ≤ ℎ , we arrive at

$$\sum\_{j=1}^N (a\_j - \alpha\_j)^2 \left\| \boldsymbol{\nu}\_{p+1,j} \right\|\_{\boldsymbol{I}\_j}^2 \le k\_3 h^2 \left[ \left\| \boldsymbol{e} \right\|^2 + \sum\_{j=1}^N \left\| \boldsymbol{o}\_j' \right\|\_{\boldsymbol{I}\_j}^2 \right].$$

Combining this estimate with (9) and (50c), we establish

$$\sum\_{j=1}^{N} \left( a\_j - \alpha\_j \right)^2 \left\| \boldsymbol{\nu}\_{p+1,j} \right\|\_{l\_j}^2 \le C h^{2^{p\_0+4}}.\tag{69}$$

Now, combining (63) and the estimates (50c) and (69) yields

$$\left\| e - E \right\|^2 \le 2C\_1 h^{2^{p+4}} + 2C\_2 h^{2^{p+4}} = C h^{2^{p+4}},$$

which completes the proof of (58). Using the relation = ℎ and the estimate (58), we obtain

$$\sum\_{j=1}^N \left\| \mu - (\mu\_h + a\_j \nu\_{p+1,j}) \right\|\_{I\_j}^2 = \left\| \mu - (\mu\_h + E) \right\|^2 = \left\| c - E \right\|^2 \le C h^{2^{p+4}}.$$

Next, we will prove (60). Using the reverse triangle inequality, we have

The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations http://dx.doi.org/10.5772/64967 55

$$\left\| \left\| E \right\| - \left\| e \right\| \right\| \le \left\| E - e \right\|,\tag{70}$$

which, after applying the estimate (58), completes the proof of (60).

In order to show (62), we divide (70) by to obtain | 1 | ≤ . Applying the estimate (58) and the inverse estimate (61), we arrive at

$$|\sigma - 1| \le Ch.$$

Therefore, <sup>=</sup> =1+ (*h*), which establishes (62).

ψ

2

1

*a M e k h*

+ +

− ≤ + ′

2 1, <sup>2</sup> <sup>2</sup> 2 2 2 4 4 1

( ) . *p j <sup>I</sup> j j j <sup>p</sup> <sup>j</sup> <sup>I</sup> <sup>I</sup> <sup>j</sup> <sup>j</sup> <sup>j</sup>*

<sup>2</sup> and using (32), i.e., + 1,

− ≤ +≤ + ′ ′

( ) ,

*j j pj <sup>p</sup> <sup>j</sup> <sup>j</sup> <sup>j</sup> I I <sup>I</sup> I I j j <sup>j</sup> j j <sup>j</sup>*

1, 2 4 4 1 3

1, <sup>2</sup> 2 2 2 2 2 22

1 ≤ ≤

− ≤+ ′ ∑ ∑ 2 2 <sup>2</sup> 2 2 1, 3 =1 =1

( ) .

<sup>+</sup>

+ ++ −≤ + <sup>2</sup> 24 24 24 1 2 2 2 =, *p pp e E C h C h Ch*

which completes the proof of (58). Using the relation = ℎ and the estimate (58), we obtain

<sup>+</sup> ∑ −+ −+ − ≤ <sup>2</sup> 2 2 2 4

*u u a u u E e E Ch*

<sup>+</sup>

( ) = ( )= . *<sup>N</sup> <sup>p</sup>*

*N N j j pj <sup>j</sup> I I j j j j a kh e*

<sup>+</sup> ∑ − ≤ <sup>2</sup> <sup>2</sup> 2 4 1,

*<sup>N</sup> <sup>p</sup>*

 ω

2, 1) is a constant independent of the mesh size.

, we arrive at

.

ω

*a Ch* (69)

ℎ

<sup>2</sup> = (2 + 2)2ℎ

ω

2 + 3 yields

(67)

(68)

ω

α

ψ

1

*k h*

+ +

2

+

α ψ+

( ) α ψ

*j j pj <sup>I</sup> <sup>j</sup> <sup>j</sup>*

*p j <sup>I</sup> j*

4

*a M e kh e*

Multiplying by + 1,

54 Perusal of the Finite Element Method

α ψ

2(2 + 2)22

1

2

<sup>2</sup> max(1

Summing over all elements and using ℎ = max

Combining this estimate with (9) and (50c), we establish

=1

Now, combining (63) and the estimates (50c) and (69) yields

ψ

*h jp j <sup>h</sup> <sup>I</sup> <sup>j</sup> <sup>j</sup>*

=1

1,

Next, we will prove (60). Using the reverse triangle inequality, we have

where 3 <sup>=</sup>

**Remark 5.1.** The previous theorem indicates that the computable quantity converges to at (*hp*+2) rate. This accuracy enhancement is achieved by adding the error estimate to the DG solution ℎ.

**Remark 5.2.** The performance of an error estimator is typically measured by the global effectivity index which is defined as the ratio of the estimated error to the actual error . We say that the error estimator is asymptotically exact if 1 as ℎ 0. The estimate (62) indicates that the global effectivity index in the 2 norm converge to unity at (*h*) rate. Therefore, the proposed estimator is asymptotically exact. We would like to emphasize that is a computable quantity since it only depends on the DG solution ℎ and . It provides an

asymptotically exact *a posteriori* estimator on the actual error . Finally, we would like to point out that our procedure for estimating the error is computationally simple. Furthermore, our DG error indicator is obtained by solving a local problem with no boundary condition on each element. This makes it useful in adaptive computations. We demonstrate this in Section 6.

**Remark 5.3.** Our proofs are valid for any regular meshes and using piecewise polynomials of degree 1. If = 0 then (46) gives = (*h*) which is the same as = (*h*). Thus, our superconvergence results are not valid when using = 0. Also, our error estimate procedure does not apply.

**Remark 5.4.** The assumption (61), which is used to prove the convergence of to unity at (*h*), requires that terms of order (*hp*+1) are present in the error. If not, might not be a good approximation of . We note that the exponent of ℎ in the estimate (9) is optimal in the sense that it cannot be improved. In fact, for the ℎversion finite element method one may show that provided that the ( + 1)th order derivatives of the exact solution do not vanish identically over the whole domain, then an inverse estimate of the form ≥ ()ℎ + 1 is valid for some positive constant () depending only on [42–44].

**Remark 5.** Our results readily extend to nonlinear systems of ODEs of the form

$$\frac{d\vec{u}}{dt} = \vec{f}(t, \vec{u}), \quad t \in [0, T], \quad \vec{u}(0) = \vec{u}\_{0, \prime}$$

where = [1, …, ] : [0, ] <sup>ℝ</sup>, <sup>0</sup> ∈ ℝ, and = [1, …, ] : [0, ]×ℝ <sup>ℝ</sup>. The DG method for this problem consists of finding <sup>ℎ</sup> ∈ <sup>ℎ</sup> <sup>=</sup> : ∈ (( )) , = 1, …, such that: ∀ ∈ <sup>ℎ</sup> and = 1, …, ,

$$\int\_{I\_{\vec{j}}} \langle \vec{\upsilon}^{\prime} \rangle^{l} \vec{u}\_{h} dt + \int\_{I\_{\vec{j}}} \langle \vec{\upsilon} \rangle^{l} \vec{f}(t, \vec{u}\_{h}) dt - \langle \vec{\upsilon} \rangle^{l} (t\_{\vec{j}}^{-}) \vec{u}(t\_{\vec{j}}^{-}) + \langle \vec{\upsilon} \rangle^{l} (t\_{\vec{j}-1}^{+}) \vec{u}(t\_{\vec{j}-1}^{-}) = 0.$$

## **6. Application: adaptive mesh refinement (AMR)**

*A posteriori* error estimates play an essential role in assessing the reliability of numerical solutions and in developing efficient adaptive algorithms. Adaptive methods based on *a posteriori* error estimates have become established procedures for computing efficient and accurate approximations to the solution of differential equations. The standard adaptive FEMs through local refinement can be written in the following loop

$$\text{SOLVE} \rightarrow \text{ESTIMATE} \rightarrow \text{ MARK} \rightarrow \text{ REFINE}.$$

The local *a posteriori* errors estimator of Section 5 can be used to mark elements for refinement.

Next, we present a simple DG adaptive algorithm based on the local *a posteriori* error estimator proposed in the previous section. The adaptive algorithm that we propose has the following steps:

	- **i.** Choose a parameter 0 ≤ ≤ 1. If the estimated global error < max = 1, …, then stop and accept the DG solution on the element .
	- **ii.** Otherwise, reject the DG solution on and divide the element into two uniform elements by adding the coordinate of the midpoint of to the list of nodes.

over the whole domain, then an inverse estimate of the form ≥ ()ℎ + 1 is valid for some

∈ <sup>0</sup> = ( , ), [0, ], (0) = , *du f tu t T u u dt*

∈ ((

))

 

−− + −

<sup>=</sup> :

: [0, ]×ℝ <sup>ℝ</sup>. The DG

, = 1, …, such

: [0, ] <sup>ℝ</sup>, <sup>0</sup> ∈ ℝ, and = [1, …, ]

− − + −+ ∫ ∫ 1 1 ( ') ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) = 0. *tt t t h h jj j j I I*

*A posteriori* error estimates play an essential role in assessing the reliability of numerical solutions and in developing efficient adaptive algorithms. Adaptive methods based on *a posteriori* error estimates have become established procedures for computing efficient and accurate approximations to the solution of differential equations. The standard adaptive FEMs

SOLVE ESTIMATE MARK REFINE. → →→

The local *a posteriori* errors estimator of Section 5 can be used to mark elements for refinement.

Next, we present a simple DG adaptive algorithm based on the local *a posteriori* error estimator proposed in the previous section. The adaptive algorithm that we propose has the following

**1.** Select a tolerance and a maximum bound on the number of interval (say

**2.** Construct an initial coarse mesh with + 1 nodes. For simplicity, we start with a uniform

*v u dt v f t u dt v t u t v t u t*

**Remark 5.** Our results readily extend to nonlinear systems of ODEs of the form

positive constant () depending only on [42–44].

and = 1, …, ,

*j j*

**6. Application: adaptive mesh refinement (AMR)**

through local refinement can be written in the following loop

method for this problem consists of finding <sup>ℎ</sup> ∈ <sup>ℎ</sup>

where = [1, …, ]

56 Perusal of the Finite Element Method

that: ∀ ∈ <sup>ℎ</sup>

steps:

max = 1000). Put = 1.

mesh having = 2 elements.

**3.** While + 1 ≤ max and <sup>≥</sup> do

**Remark 6.1.** *There are many possibilities for selecting the elements to be refined given the local error indicator . In the above algorithm, we used the most popular fixed‐rate strategy which consists of refining the element if* > max = 1, …, *, where* 0 ≤ ≤ 1 *is a parameter provided by the user. Note that the choice* = 0 *gives uniform refinement, while* = 1 *gives no refinement. Also, there are other stopping criteria that may be used to stop the adaptive algorithm.*

## **7. Computational results**

In this section, we present several numerical examples to (i) validate our superconvergence results and the global convergence of the residual-based *a posteriori* error estimates, and (ii) test the above local adaptive mesh refinement procedure that makes use of our local *a posteriori* error estimate.

**Example 7.1.** *The test problem we consider is the following nonlinear IVP*

$$
\mu' = -\mu - \mu^2, \quad t \in [0, 1], \quad \mu(0) = 1.
$$

Clearly, the exact solution is () = <sup>1</sup> 2 − 1 . We use uniform meshes obtained by subdividing the computational domain [0,1] into intervals with = 5, 10, 20, 30, 40, 50. This example is tested by using polynomials with = 0 − 4. **Figure 1** shows the 2 errors and with log-log scale as well as their orders of convergence. These results indicate that = (*hp*+1) and = (*hp*+2). This example demonstrates that our theoretical convergence rates are optimal.

**Figure 1.** Log-log plots of (left) and (right) versus mesh sizes ℎ for Example 7.1 on uniform meshes having = 5, 10, 20, 30, 40, 50 elements using , = 0 to 4.

**Figure 2.** Log-log plots of \* (left) versus ℎ for Example 7.1 using = 5, 10, 20, 30, 40, 50 and , = 0 to 4. Loglog plots of \* (right) versus ℎ using = 5, 10, 20, 30 elements using , = 0 to 3.

Next, we compute the maximum error at the shifted roots of the ( + 1) degree right Radau polynomial on each element and then take the maximum over all elements. For simplicity, we use \* to denote max1 <sup>≤</sup> ≤ max0 ≤ ≤ (, <sup>−</sup> ) , where , are the roots of + 1,(). Similarly, we compute the true error at the downwind point of each element and then we denote \* to be the maximum over all elements , = 1, …, , i.e.,

 \* = max1 ≤ <sup>≤</sup> ( −) . In **Figure 2**, we present the errors \*, \* and their orders of convergence. We observe that \* = (*hp*+2) and \* = (*h*<sup>2</sup>*<sup>p</sup>*+1) as expected. Thus, the error at right Radau points converges at (*hp*+2). Similarly, the error at the downwind point of each element converge with an order 2 + 1. This is in full agreement with the theory.

log-log scale as well as their orders of convergence. These results indicate that = (*hp*+1) and = (*hp*+2). This example demonstrates that our theoretical convergence rates are optimal.

**Figure 1.** Log-log plots of (left) and (right) versus mesh sizes ℎ for Example 7.1 on uniform meshes having

**Figure 2.** Log-log plots of \* (left) versus ℎ for Example 7.1 using = 5, 10, 20, 30, 40, 50 and , = 0 to 4. Log-

Next, we compute the maximum error at the shifted roots of the ( + 1) degree right Radau

+ 1,(). Similarly, we compute the true error at the downwind point of each element and

and then take the maximum over all elements. For simplicity,

<sup>−</sup> ) , where ,

are the roots of

, = 1, …, , i.e.,

log plots of \* (right) versus ℎ using = 5, 10, 20, 30 elements using , = 0 to 3.

then we denote \* to be the maximum over all elements

we use \* to denote max1 <sup>≤</sup> ≤ max0 ≤ ≤ (,

= 5, 10, 20, 30, 40, 50 elements using , = 0 to 4.

58 Perusal of the Finite Element Method

polynomial on each element

Next, we use (57a) and (57b) to compute the *a posteriori* error estimate for the DG solution. The global errors − and their orders of convergence, using the spaces with = 1 − 4, are shown in **Figure 3**. We observe that − = (*hp*+2). This is in full agreement with the theory. This example demonstrates that the convergence rate proved in this work is sharp. Since − <sup>=</sup> <sup>−</sup> (ℎ <sup>+</sup> ) <sup>=</sup> (*hp*+2), we conclude that the computable quantities ℎ <sup>+</sup> converges to the exact solution at (*hp*+2) rate in the 2 norm. We would like to emphasize that this accuracy enhancement is achieved by adding the error estimate to the DG solution ℎ only once at the end of the computation. This leads to a very efficient computation of the postprocessed approximation ℎ <sup>+</sup> .

**Figure 3.** The errors − and their orders of convergence for Example 1 on uniform meshes having = 5, 10, 20, 30, 40, 50 elements using , = 1 to 4.

In **Table 1**, we present the actual *L*<sup>2</sup> errors and the global effectivity indices. These results demonstrate that the proposed *a posteriori* error estimates is asymptotically exact.


**Table 1.** The errors and the global effectivity indices for Example 7.1 on uniform meshes having = 5, 10, 20, 30, 40, 50 elements using , = 1 to 4.

In **Figure 4**, we show the errors = |‖ − ‖| and = − 1 . We see that = (*hp*+2) and = (*h*) as the theory predicts.

**Figure 4.** Convergence rates for (left) and (right) for Example 1 on uniform meshes having =5, 10, 20, 30, 40, 50 elements using , = 1 to 4.

**Example 7.2.** In this example we test our error estimation procedure presented in Section 6 on adaptively refined meshes. We consider the following model problem

$$u' = \beta u, \quad t \in [0, 5], \quad u(0) = 1, 2$$

where the exact solution is simply () = . We apply our adaptive algorithm using = 1 (unstable), = −1 (stable), and = − 20 (stiff). The DG solutions and the sequence of meshes obtained by applying our adaptive algorithm with −2 for = 1 − 4 are shown in **Figures 5**–**7** for = 1, = −1, and = − 20, respectively. As can be expected, the adaptive algorithm refines in the vicinity of the endpoint = 5 with coarser meshes for increasing polynomial degree . Furthermore, we observe that, when is closer to 0, we get more uniform refinement near the portion with high approximation error. When is near 1, we get less uniform refinement near the portion with high approximation error. We also observed that, both for = 0.2 and = 0.9, the optimal convergence rates are achieved asymptotically and that the global effectivity indices converge to unity with increasing polynomial degree . Furthermore, we tested our adaptive algorithm on other problems and observed similar conclusions. These results are not included to save space.

= 1 = 2 = 3 = 4

40, 50 elements using , = 1 to 4.

60 Perusal of the Finite Element Method

= (*h*) as the theory predicts.

50 elements using , = 1 to 4.

where the exact solution is simply () =

 5 4.7637e-03 1.0362 2.7867e-04 1.0531 1.6847e-05 1.0637 1.0386e-06 1.0705 10 1.2750e-03 1.0179 3.7805e-05 1.0271 1.1742e-06 1.0326 3.7481e-08 1.0363 20 3.2849e-04 1.0089 4.8747e-06 1.0136 7.6227e-08 1.0164 1.2290e-09 1.0182 30 1.4736e-04 1.0059 1.4568e-06 1.0090 1.5201e-08 1.0109 1.6369e-10 1.0122 40 8.3262e-05 1.0044 6.1698e-07 1.0068 4.8296e-09 1.0082 3.9026e-11 1.0091 50 5.3429e-05 1.0035 3.1660e-07 1.0054 1.9827e-09 1.0066 1.2820e-11 1.0073

**Table 1.** The errors and the global effectivity indices for Example 7.1 on uniform meshes having = 5, 10, 20, 30,

In **Figure 4**, we show the errors = |‖ − ‖| and = − 1 . We see that = (*hp*+2) and

**Figure 4.** Convergence rates for (left) and (right) for Example 1 on uniform meshes having =5, 10, 20, 30, 40,

**Example 7.2.** In this example we test our error estimation procedure presented in Section 6 on

*u ut u* ′ = , [0,5], (0) = 1,

(unstable), = −1 (stable), and = − 20 (stiff). The DG solutions and the sequence of meshes obtained by applying our adaptive algorithm with −2 for = 1 − 4 are shown in **Figures 5**–**7** for = 1, = −1, and = − 20, respectively. As can be expected, the adaptive

. We apply our adaptive algorithm using = 1

adaptively refined meshes. We consider the following model problem

β∈

**Figure 5.** , ℎ, and final meshes for Example 7.2 with <sup>=</sup> <sup>−</sup> <sup>20</sup> using , = 1 to 4, and = 10−2.

**Figure 6.** *u, uh*, and final meshes for Example 7.2 with β = −1 using P*<sup>p</sup>* , *p* = 1 to 4, and *Tol* = 10−2.

**Figure 7.** *u, uh*, and final meshes for Example 7.2 with β = −20 using P*<sup>p</sup>* , *p* = 1 to 4, and *Tol* = 10−2.

## **8. Concluding remarks**

In this chapter, we presented a detailed analysis of the original discontinuous Galerkin (DG) finite element method for the approximation of initial-value problems (IVPs) for nonlinear ordinary differential equations (ODEs). We proved several optimal error estimates and superconvergence results. In particular, we showed that the DG solution converges to the true solution with order + 1, when the space of piecewise polynomials of degree is used. We further proved the (2 + 1)th superconvergence rate at the downwind points. Moreover, we proved that the DG solution is (*hp*+2) superconvergent toward a particular projection of the exact solution. We used these results and showed that the leading term of the DG error is proportional to the ( + 1) degree right Radau polynomial. This result allowed us to construct computationally simple, efficient, and asymptotically exact *a posteriori* error estimator. It is obtained by solving a local residual problem with no boundary condition on each element. Furthermore, we proved that the proposed *a posteriori* error estimator converges to the actual error in the 2 norm. The order of convergence is proved to be + 2. All proofs are valid for regular meshes and for polynomials with <sup>1</sup>. Finally, we presented a local adaptive procedure that makes use of our local *a posteriori* error estimate. Future work includes the study of superconvergence of DG method for nonlinear boundary-value problems.

#### **Abbreviations**

#### **Symbols**



**Figure 7.** *u, uh*, and final meshes for Example 7.2 with β = −20 using P*<sup>p</sup>*

**8. Concluding remarks**

62 Perusal of the Finite Element Method

**Abbreviations**

AMR Adaptive mesh refinement DG Discontinuous Galerkin

FEM, FEMs Finite element method, finite element methods IVP, IVPs Initial-value problem, initial-value problems

**Symbols**

, *p* = 1 to 4, and *Tol* = 10−2.

In this chapter, we presented a detailed analysis of the original discontinuous Galerkin (DG) finite element method for the approximation of initial-value problems (IVPs) for nonlinear ordinary differential equations (ODEs). We proved several optimal error estimates and superconvergence results. In particular, we showed that the DG solution converges to the true solution with order + 1, when the space of piecewise polynomials of degree is used. We further proved the (2 + 1)th superconvergence rate at the downwind points. Moreover, we proved that the DG solution is (*hp*+2) superconvergent toward a particular projection of the exact solution. We used these results and showed that the leading term of the DG error is proportional to the ( + 1) degree right Radau polynomial. This result allowed us to construct computationally simple, efficient, and asymptotically exact *a posteriori* error estimator. It is obtained by solving a local residual problem with no boundary condition on each element. Furthermore, we proved that the proposed *a posteriori* error estimator converges to the actual error in the 2 norm. The order of convergence is proved to be + 2. All proofs are valid for regular meshes and for polynomials with <sup>1</sup>. Finally, we presented a local adaptive procedure that makes use of our local *a posteriori* error estimate. Future work includes the study

of superconvergence of DG method for nonlinear boundary-value problems.

 ( ) Sobolev space ( ) = : ∫ |() () 2 <sup>&</sup>lt; ∞, 0 <sup>≤</sup> ≤ , Norm of ( ) , ( )-seminorm of on , and ( )-seminorm of on ℎ ± Gauss-Radau projections of onto ℎ , 1, 2, etc Generic positive constants independent of <sup>ℎ</sup> () Exact solution at time ℎ() DG solution at time ℎ( ) ℎ( ) = ℎ( −) True error, <sup>=</sup> −ℎ Projection error, <sup>=</sup> −ℎ − Error <sup>=</sup> ℎ − ℎ −. (), = 0, 1, …, th degree Legendre polynomial on [ − 1,1] (), = 0, 1, …, th degree right Radau polynomial on [ − 1,1] , (), = 0, 1, …, th degree Legendre polynomial on , (), = 0, 1, …, th degree right Radau polynomial on , (), = 0, 1, …, th degree monic Radau polynomial on , = 0, 1, …, Roots of + 1() , , = 0, 1, …, Roots of + 1, () Interpolant, interpolates () at , , = 0, 1, …, , Interpolant, interpolates () at , , = 0, 1, …, , and at an additional point in with ≠ , , = 0, 1, …, − () + 1, (), where is the coefficient of + 1 in the ( + 1) degree polynomial <sup>+</sup> −ℎ , Global effectivity index, = − 1 , Tolerance, maximum bound on the number of interval \* Maximum error at Radau points \* Maximum error at the downwind points

## **Author details**

 ( 

> ,

,

ℎ

ℎ(

,

,

,

,

) Sobolev space

() Exact solution at time ℎ() DG solution at time

True error, <sup>=</sup> −ℎ

Error <sup>=</sup> ℎ − ℎ

, = 0, 1, …, Roots of

, = 0, 1, …, Roots of

−

<sup>+</sup> −ℎ

()

Projection error, <sup>=</sup> −ℎ

) ℎ(

64 Perusal of the Finite Element Method

, and

Norm of

( 

( 

± Gauss-Radau projections of onto ℎ

, 1, 2, etc Generic positive constants independent of <sup>ℎ</sup>

) = ℎ( −)

(), = 0, 1, …, th degree Legendre polynomial on [ − 1,1]

(), = 0, 1, …, th degree Legendre polynomial on

(), = 0, 1, …, th degree right Radau polynomial on

(), = 0, 1, …, th degree monic Radau polynomial on

Interpolant, interpolates () at ,

Interpolant, interpolates () at ,

with ≠ ,

+ 1,

, Global effectivity index, = − 1

\* Maximum error at the downwind points

\* Maximum error at Radau points

+ 1()

 + 1, ()

, = 0, 1, …,

(), where

, Tolerance, maximum bound on the number of interval

(), = 0, 1, …, th degree right Radau polynomial on [ − 1,1]

( 

)-seminorm of on

)-seminorm of on

( ) ) = : ∫

−

−.

 |()

is the coefficient of

, = 0, 1, …, ,

, = 0, 1, …, , and at an additional point

+ 1 in the ( + 1) degree polynomial

 in 

() 2 <sup>&</sup>lt; ∞, 0 <sup>≤</sup> ≤

Mahboub Baccouch

Address all correspondence to: mbaccouch@unomaha.edu

Department of Mathematics, University of Nebraska at Omaha, Omaha, NE, USA

## **References**


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[23] B. Cockburn, C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis 35 (1998): 2440– 2463.

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[19] T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis 28 (1) (1991): 133–140.

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## **On Finite Element Vibration Analysis of Carbon Nanotubes On Finite Element Vibration Analysis of Carbon Nanotubes**

Ishan Ali Khan and Seyed M. Hashemi Ishan Ali Khan and Seyed M. Hashemi

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65358

#### **Abstract**

[35] M. Baccouch, A superconvergent local discontinuous Galerkin method for the secondorder wave equation on Cartesian grids, Computers and Mathematics with Applica-

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[37] M. Baccouch, Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension, International Journal

[38] M. Baccouch, Superconvergence and *a posteriori* error estimates of the DG method for scalar hyperbolic problems on Cartesian grids, Applied Mathematics and Computation

[39] M. Baccouch, B. Johnson, A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations, Journal of Computational and Applied Mathematics

[40] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co.,

[41] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, New York,

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[43] L. Schumaker, Spline Functions: Basic Theory, Cambridge University Press, Cam-

[44] K. Segeth, A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension, Numerische Mathematik 83 (3)

for Partial Differential Equations 31 (2015): 1461–1491.

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308 (2016): 138–165.

Wiley, New York, 2000.

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1965.

Amsterdam/New York/Oxford, 1978.

In this chapter, a finite element formulation is proposed to study the natural frequencies of double-walled carbon nanotubes modeled as, both, local and nonlocal Euler-Bernoulli beams, coupled with van der Waals interaction forces. The formulation uses Galerkinweighted residual approach and employs Hermite cubic polynomial function to derive the linear eigenvalue problem. Natural frequencies are found for clamped-free, clamped-clamped and simply supported-simply supported boundary conditions. The results are in good agreement with the formulations found in the literature. The effect of nonlocal factor on the natural frequencies of the system is found out by comparing local and nonlocal results. Additionally, the universality of the proposed model is proven by application to a double-elastic Euler-Bernoulli beam. This formulation paves way for Finite Element Method (FEM) analysis of multi-walled CNTs—either locally or nonlocally.

**Keywords:** carbon nanotubes, Euler-Bernoulli beam, DWCNTs, finite element analysis, nonlocal continuum mechanics, vibrations

## **1. Introduction**

Carbon nanotubes are tubules of carbon, in the dimensional range of *nanometre* (in the order of 10e−9 m), with atoms arranged in a way that gives them exceptional properties. The high aspect ratio, the ratio of length to its diameter, and chirality are the factors that contribute to the same. Carbon nanotubes are classified as single- and multi-walled depending on the number of tubes held concentrically. A considerable amount of research has been dedicated to the study of mechanical properties of both these types using experimental or theoretical

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

methods. Theoretical methods involve the representation of C**─**C bonds as is, conveniently named as atomistic modelling methods, or as continuous structures such as bar, beam and shell. The latter is usually employed to study the vibrational properties as the use of atomistic models require comparatively large computational time and effort, and are often accurate for a particular time period in the loading cycle. The laws of continuum structures have been found to be at the nanometre scale as well [1].

Continuum structures, by definition, consider the model to be continuous and ignore the interatomic forces. Eringen and Edelen [2] and Eringen [3, 4] proposed a model that would connect the lattice mechanics with the continuum mechanics, named as nonlocal continuum mechanics. This is achieved by introducing nonlocal factors in the governing equations [5]. As will be seen in the following sections, the introduction of these nonlocal factors have shown improved description of the model. Peddieson et al. [6] extended the nonlocal continuum models to nanotechnology and hence paved the way for these theoretical models to represent the properties of nanomaterials with greater accuracy.

Finite element methods have been employed for analysis of continuum structures since 1950s, as the numerical technique is very adaptable in terms of complex geometries and boundary conditions. The method involves the division of the structure into number of 'elements' and using appropriate interpolations functions to express the approximate solutions. The eigenvalue problem hence achieved is solved for system's natural frequencies and mode shapes.

The study of vibrations of carbon nanotubes, modeled as continuum models has gained considerable attention with beams and shells being used extensively. The elastic model has been presented by Ru [7] to study the column buckling of multi-walled carbon nanotubes, where the adjacent tubes were presented to be interacting through van der Waals interaction forces. Ru [8] also studied the transverse vibrations of single- and multi-walled carbon nanotubes by considering them as Euler-Bernoulli beams. Sufficient attention was given to the effect of van der Waals interaction forces on the vibrational frequency. Yoon et al. [9] studied the non-coaxial vibrational modes of multi-walled carbon nanotubes embedded in an elastic medium and deduced that the concentric geometry of carbon nanotubes will be distorted at ultrahigh frequencies. Nonlinear free vibrations of multi-walled carbon nanotubes embedded in an elastic medium were studied by Fu et al. [10]. The relation between nonlinear free vibration frequency, stiffness of the surrounding medium and van der Waals interaction forces was discussed and amplitude curves presented. Vibrational characteristic analysis of doublewalled carbon nanotube (DWCNT) with simply supported boundary condition using multiple Euler-Bernoulli beam model was investigated by Natsuki et al. [11]. It was seen that the amplitudes of inner and outer tubes are non-coaxial and in opposite direction. Natsuki also studied DWCNTs embedded in elastic medium, represented by Winkler spring model, for free supported boundary condition [12] and DWCNTs with different lengths for inner and outer tubes [13]. Xu et al. [14] studied the vibrations of DWCNTs with nonlinear van der Waals interaction forces and concluded that it does not affect the coaxial free vibrations. Elishakoff and Pentaras [15, 16] studied the natural frequencies of DWCNTs and TWCNTs using approximate methods, like Bubnov-Galerkin and Petrov-Galerkin and exact solutions. The coupling between the concentric tubes is considered as van der Waals interaction forces, in both the cases.

methods. Theoretical methods involve the representation of C**─**C bonds as is, conveniently named as atomistic modelling methods, or as continuous structures such as bar, beam and shell. The latter is usually employed to study the vibrational properties as the use of atomistic models require comparatively large computational time and effort, and are often accurate for a particular time period in the loading cycle. The laws of continuum structures have been

Continuum structures, by definition, consider the model to be continuous and ignore the interatomic forces. Eringen and Edelen [2] and Eringen [3, 4] proposed a model that would connect the lattice mechanics with the continuum mechanics, named as nonlocal continuum mechanics. This is achieved by introducing nonlocal factors in the governing equations [5]. As will be seen in the following sections, the introduction of these nonlocal factors have shown improved description of the model. Peddieson et al. [6] extended the nonlocal continuum models to nanotechnology and hence paved the way for these theoretical models to represent the

Finite element methods have been employed for analysis of continuum structures since 1950s, as the numerical technique is very adaptable in terms of complex geometries and boundary conditions. The method involves the division of the structure into number of 'elements' and using appropriate interpolations functions to express the approximate solutions. The eigenvalue problem hence achieved is solved for system's natural frequencies and mode shapes.

The study of vibrations of carbon nanotubes, modeled as continuum models has gained considerable attention with beams and shells being used extensively. The elastic model has been presented by Ru [7] to study the column buckling of multi-walled carbon nanotubes, where the adjacent tubes were presented to be interacting through van der Waals interaction forces. Ru [8] also studied the transverse vibrations of single- and multi-walled carbon nanotubes by considering them as Euler-Bernoulli beams. Sufficient attention was given to the effect of van der Waals interaction forces on the vibrational frequency. Yoon et al. [9] studied the non-coaxial vibrational modes of multi-walled carbon nanotubes embedded in an elastic medium and deduced that the concentric geometry of carbon nanotubes will be distorted at ultrahigh frequencies. Nonlinear free vibrations of multi-walled carbon nanotubes embedded in an elastic medium were studied by Fu et al. [10]. The relation between nonlinear free vibration frequency, stiffness of the surrounding medium and van der Waals interaction forces was discussed and amplitude curves presented. Vibrational characteristic analysis of doublewalled carbon nanotube (DWCNT) with simply supported boundary condition using multiple Euler-Bernoulli beam model was investigated by Natsuki et al. [11]. It was seen that the amplitudes of inner and outer tubes are non-coaxial and in opposite direction. Natsuki also studied DWCNTs embedded in elastic medium, represented by Winkler spring model, for free supported boundary condition [12] and DWCNTs with different lengths for inner and outer tubes [13]. Xu et al. [14] studied the vibrations of DWCNTs with nonlinear van der Waals interaction forces and concluded that it does not affect the coaxial free vibrations. Elishakoff and Pentaras [15, 16] studied the natural frequencies of DWCNTs and TWCNTs using approximate methods, like Bubnov-Galerkin and Petrov-Galerkin and exact solutions. The

found to be at the nanometre scale as well [1].

70 Perusal of the Finite Element Method

properties of nanomaterials with greater accuracy.

Nonlocal continuum mechanics approach has also been used extensively in the recent years. Constitutive relations of nonlocal elasticity to be used in analysis of carbon nanotubes for Euler-Bernoulli, Timoshenko beams and cylindrical shells were presented by Wang and Wang [17]. Extensive attention was given to the shear stress and strain relation for the Timoshenko beams. Wang [18] also studied the wave propagation in carbon nanotubes using nonlocal Euler-Bernoulli and nonlocal Timoshenko beams and concentrated on the effect of nonlocal factor on the mechanical behaviour of carbon nanotubes. Yoon et al. [19] also performed a similar study concluded that the rotary inertia and shear deformation play a significant role when the measured natural frequency is much closer to the critical frequency. One more such study was performed by Wang and Varadan [20], who developed explicit frequency solutions for both single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotube (DWCNTs) on the basis of nonlocal continuum mechanics and employed both the beam theories. The ratio of local and nonlocal frequencies is used to study the effect of nonlocal factor. Arash and Wang [21] reviewed various studies on the nonlocal continuum theory for carbon nanotubes and graphene sheets. They also compared the nonlocal models to the local equivalents. Reddy [22] reformulated governing equations, based on nonlocal theory, and presented analytical solutions for bending, buckling and vibrations of Euler-Bernoulli, Timoshenko, Reddy and Levinson beams. Further, Reddy and Pang [23] also presented the governing equations of Euler-Bernoulli and Timoshenko beams, based on nonlocal theory, specifically for carbon nanotubes and used these equations to study the vibrations, buckling and static bending. Murmu and Pradhan [24] employed the differential quadrature (DQ) method to study the thermo-mechanical vibration of SWCNT embedded in an elastic medium, represented as Winkler-type elastic foundation, using nonlocal elasticity theory. It was found that the difference between local and nonlocal frequencies is particularly high for low temperature changes. Narendar et al. [25] studied the wave propagation in SWCNTs, under magnetic field, using nonlocal Euler-Bernoulli beam theory. De Rosa and Lippiello [26] analysed the free vibration of DWCNTs, modeled as nonlocal Euler-Bernoulli beams, using cell-discretization method (CDM) and a semi-analytical Rayleigh-Schmidt method. A rare study of forced vibrations of carbon nanotubes was performed by Karaoglu and Aydogyu [27] using both local and nonlocal Euler-Bernoulli beam models, for both SWCNTs and DWCNTs. Khosrozadeh and Hajabasi [28] studied the free vibrations of DWCNTs embedded in an elastic medium with nonlinear interlayer van der Waals interaction forces using nonlocal Euler-Bernoulli beam model. Fang et al. [29] performed a similar study for DWCNTs embedded in elastic medium considering it to be Winkler model.

Timoshenko models have also been used extensively as it considers the rotational inertia and transverse shear. Wang et al. [30] considered it and studied the free vibration of MWCNTs using differential quadrature (DQ) method to solve the governing equations. It is seen that the frequencies are over-predicted by Euler-Bernoulli model when the aspect ratio is small. Yang et al. [31] studied the nonlinear free vibration of SWCNT with the von Karman geometric nonlinearity considering them as nonlocal Timoshenko beams. Nonlinear free vibration of embedded DWCNTs modeled as nonlocal Timoshenko beams was presented by Ke et al. [32] where again differential quadrature (DQ) technique was employed, and the surrounding elastic medium was represented by spring-based Winkler model. Pradhan [33] presented a finite element model using nonlocal Timoshenko beams and Galerkin weighted residual technique. The buckling of carbon nanotubes using Timoshenko model was studied by Zhang et al. [34], and various studies on buckling were extensively reviewed by Wang et al. [35] and Shima [36].

A theory similar to nonlocal theory was presented by Park and Gao [37] using modified couple stress theory where internal material length parameter is considered and a variational model is presented basing on minimum total energy potential for Euler-Bernoulli beam model. It found a relation between deflections and beam thickness. Another nonlocal theory, for nanobeams, was presented by Thai [38] in which the governing equations were derived using Hamilton's principle and analytical solutions were presented. The results were found to be comparable to Timoshenko and Reddy beam theories even though the formulation is similar to that of Euler-Bernoulli theory. A generalized nonlocal beam theory, to study properties of nanobeams, was presented by Aydogdu [39]. Various beam models were studied as a special case of this beam theory for simply supported nanobeams, and the numerical results were found to be stand valid. Lu et al. [40] compared various nonlocal beam models that are used for carbon nanotubes and made some interesting observations.

Various variational techniques have also been employed with nonlocal theories to study the vibrational behaviour of carbon nanotubes. Adali [41] employed a semi-inverse method and presented one such model. Alshorbagy et al. [42] developed a nonlocal finite element for static analysis of nanobeams. Eltaher et al. [43] analysed the vibrations of nonlocal Euler-Bernoulli nanobeams using finite element method. A similar study was performed by Phadikar and Pradhan [44], where finite element analysis of nanoplates and nanobeams was presented. All these models are found to be efficient for SWCNTs. Ansari and Hemmatnezhad [45] studied the nonlinear vibrations of MWCNTs embedded in an elastic medium using variational iteration method. A finite element analysis of embedded Timoshenko beam was also performed by Hemmatnezhad and Ansari [46], where carbon nanotubes were modeled as nonlocal beam using statically exact shape functions to achieve an eigenvalue problem. Shakouri et al. [47] studied the transverse vibrations of DWCNTs modeled as nonlocal Euler-Bernoulli beams using Galerkin method. Dinçkal [48], more recently, proposed a finite element model that uses exact dynamic shape functions (trigonometric in nature) to form the frequencydependant dynamic stiffness matrix and hence study the free vibrations of carbon nanotubes, modeled as both nonlocal Euler-Bernoulli and Timoshenko beams. Ehteshami and Hajabasi [49] and Zhang et al. [50] performed an analytical study and presented exact solutions for finding natural frequencies of DWCNTs modeled as nonlocal Euler-Bernoulli beams. The results of both these studies have been considered as benchmark to verify the results obtained in the current study. This chapter exploits the generality of the FEM technique to obtain the natural frequencies of DWCNTs, coupled with van der Waals interaction forces. The model has been proposed and applied to both local (classical) Euler-Bernoulli and nonlocal Euler-Bernoulli beam cases. The results are compared with the values presented in the literature.

## **2. Formulation**

embedded DWCNTs modeled as nonlocal Timoshenko beams was presented by Ke et al. [32] where again differential quadrature (DQ) technique was employed, and the surrounding elastic medium was represented by spring-based Winkler model. Pradhan [33] presented a finite element model using nonlocal Timoshenko beams and Galerkin weighted residual technique. The buckling of carbon nanotubes using Timoshenko model was studied by Zhang et al. [34], and various studies on buckling were extensively reviewed by Wang et al. [35] and

A theory similar to nonlocal theory was presented by Park and Gao [37] using modified couple stress theory where internal material length parameter is considered and a variational model is presented basing on minimum total energy potential for Euler-Bernoulli beam model. It found a relation between deflections and beam thickness. Another nonlocal theory, for nanobeams, was presented by Thai [38] in which the governing equations were derived using Hamilton's principle and analytical solutions were presented. The results were found to be comparable to Timoshenko and Reddy beam theories even though the formulation is similar to that of Euler-Bernoulli theory. A generalized nonlocal beam theory, to study properties of nanobeams, was presented by Aydogdu [39]. Various beam models were studied as a special case of this beam theory for simply supported nanobeams, and the numerical results were found to be stand valid. Lu et al. [40] compared various nonlocal beam models that are used

Various variational techniques have also been employed with nonlocal theories to study the vibrational behaviour of carbon nanotubes. Adali [41] employed a semi-inverse method and presented one such model. Alshorbagy et al. [42] developed a nonlocal finite element for static analysis of nanobeams. Eltaher et al. [43] analysed the vibrations of nonlocal Euler-Bernoulli nanobeams using finite element method. A similar study was performed by Phadikar and Pradhan [44], where finite element analysis of nanoplates and nanobeams was presented. All these models are found to be efficient for SWCNTs. Ansari and Hemmatnezhad [45] studied the nonlinear vibrations of MWCNTs embedded in an elastic medium using variational iteration method. A finite element analysis of embedded Timoshenko beam was also performed by Hemmatnezhad and Ansari [46], where carbon nanotubes were modeled as nonlocal beam using statically exact shape functions to achieve an eigenvalue problem. Shakouri et al. [47] studied the transverse vibrations of DWCNTs modeled as nonlocal Euler-Bernoulli beams using Galerkin method. Dinçkal [48], more recently, proposed a finite element model that uses exact dynamic shape functions (trigonometric in nature) to form the frequencydependant dynamic stiffness matrix and hence study the free vibrations of carbon nanotubes, modeled as both nonlocal Euler-Bernoulli and Timoshenko beams. Ehteshami and Hajabasi [49] and Zhang et al. [50] performed an analytical study and presented exact solutions for finding natural frequencies of DWCNTs modeled as nonlocal Euler-Bernoulli beams. The results of both these studies have been considered as benchmark to verify the results obtained in the current study. This chapter exploits the generality of the FEM technique to obtain the natural frequencies of DWCNTs, coupled with van der Waals interaction forces. The model has been proposed and applied to both local (classical) Euler-Bernoulli and nonlocal Euler-Bernoulli beam cases. The results are compared with the values presented in the literature.

for carbon nanotubes and made some interesting observations.

Shima [36].

72 Perusal of the Finite Element Method

This section gives the formulation for the proposed models. Section 2.1 would present the formulation for local Euler-Bernoulli beam and Section 2.2 for nonlocal Euler-Bernoulli beam. As will be seen ahead, the formulation uses Galerkin weighted residual method and Hermite cubic polynomial functions, in both the cases in interest of generality.

#### **2.1. DWCNTs based on local Euler-Bernoulli beam**

Euler-Bernoulli beam, also known as the classical beam, is known to efficiently represent a carbon nanotube, in the case when the aspect ratio (/ ratio) is sufficiently high. The governing equation for an Euler-Bernoulli beam is given by [51]:

$$
\hat{E}H\vec{\hat{w}}^{\dagger} + \rho A\vec{\hat{w}} = p\left(\mathbf{x}\right) \tag{1}
$$

where is the bending rigidity, is the displacement, is the mass density, is the crosssectional area, and the derivatives of transverse displacement with respect to beam length and time are represented by prime (′) and a dot (.), respectively. Ru [8] used the multiple beam model to derive the governing equations for DWCNTs. It is taken as two single-walled carbon nanotubes coupled by an interaction force, represented by van der Waal's coefficient, which is estimated as:

$$c = \frac{\pi \varepsilon R\_i R\_2 \sigma^6}{a^4} \left[ \frac{100 \, 1 \sigma^6}{3} H^{13} - \frac{1120}{9} H^\gamma \right] \tag{2}$$

where and are the van der Waal's radius and Lennard-Jones potential's well depth, = 0.142 —the C─C bond length and 1 and 2 are the inner and outer radius, respectively, whereas

$$H^m = \left(R\_1 + R\_2\right)^{-m} \int\_0^{\frac{\pi}{2}} \frac{d\theta}{\left(1 - K \cos^2 \theta\right)^{\frac{m}{2}}} \left(m = 7, 13\right) \tag{3}$$

and

$$K = \frac{4R\_1R\_2}{\left(R\_1 + R\_2\right)^2} \tag{4}$$

The governing equations for a DWCNT are [8]:

$$\begin{aligned} E\_1 I\_1 \ddot{\mathbf{w}}\_1^\prime + \rho\_1 A\_1 \ddot{\mathbf{w}}\_1 &= c \mathbf{w}\_2 - c \mathbf{w}\_1 \\\\ E\_2 I\_2 \ddot{\mathbf{w}}\_2^\prime + \rho\_2 A\_2 \ddot{\mathbf{w}}\_2 &= -c \mathbf{w}\_2 + c \mathbf{w}\_1 \end{aligned} \tag{5}$$

where the subscripts 1, 2 for , , , , represent the inner and outer tubes, respectively. Simple harmonic motion is assumed, i.e., , = is assumed to remove time dependency from Eq. (5) and = 1, 2 to represent inner and outer carbon nanotube, respectively, and and represent the circular frequency and time. Using simple harmonic motion in governing equations modifies them as:

$$E\_1 I\_1 W\_1 \stackrel{\text{''}}{+} \left(c - \rho\_1 A\_1 \alpha^2\right) W\_1 - cW\_2 = 0$$

$$E\_2 I\_2 W\_2 \stackrel{\text{'''}}{+} \left(c - \rho\_2 A\_2 \alpha^2\right) W\_2 - cW\_1 = 0\tag{6}$$

Following Galerkin weighted residual method, weight functions are introduced as 1, 2 and the transverse displacements continue to be represented by 1, 2. The weighted-integral form of the governing equations is given as:

$$\begin{aligned} \overline{W}\_1^k &= \underset{\alpha}{\overset{L}{\underset{0}{\rightleftharpoons}}} \left( E\_1 I\_1 W \right)^\* \delta W\_1 + \left( \mathbf{c} - \rho\_1 A\_1 \alpha^2 \right) W\_1 \delta W\_1 - \mathbf{c} W\_2 \delta W\_1 \Big) d\mathbf{x} = \mathbf{0} \\ \overline{W}\_2^k &= \underset{\alpha}{\overset{L}{\underset{0}{\rightleftharpoons}}} \left( E\_2 I\_2 W \right)^\* \delta W\_2 + \left( \mathbf{c} - \rho\_2 A\_2 \alpha^2 \right) W\_2 \delta W\_2 - \mathbf{c} W\_1 \delta W\_2 \Big) d\mathbf{x} = \mathbf{0} \end{aligned} \tag{7}$$

The weak integral form of the governing equations is eliminated by performing integration by parts, twice on Eq. (7), as given below:

$$
\overline{W}\_2^{\text{ik}} = \bigcap\_{0}^{L} \left( E\_2 I\_2 W\_2^{\text{"}} \delta W\_2^{\text{"}} + \left( \mathcal{c} - \rho\_2 A\_2 \rho \boldsymbol{\alpha}^2 \right) W\_2 \delta W\_2 - \mathcal{c} W\_i \delta W\_2 \right) d\mathbf{x} + \left\{ \left( E\_2 I\_2 W\_2^{\text{"}} \right) \delta W\_2 \right\}^L \delta \overline{W}\_2^L
$$

$$
$$

#### On Finite Element Vibration Analysis of Carbon Nanotubes http://dx.doi.org/10.5772/65358 75

$$
\overline{W}\_2^{\perp} = \underset{0}{\stackrel{L}{\underset{0}{\times}}} \left( E\_2 I\_2 W\_2^{\top} \delta W\_{\perp}^{\prime} + \left( \mathbf{c} - \rho\_{\perp} A\_{\pm} \alpha^2 \right) W\_{\perp} \delta W\_{\perp} - \mathbf{c} W\_{\parallel} \delta W\_{\perp} \right) d\mathbf{x} + \left[ \left( E\_{\frac{\nu}{2}} I\_{\frac{\nu}{2}} W\_{\perp}^{\top} \right) \delta W\_{\frac{\nu}{2} \frac{1}{10}} \right. \tag{8}
$$

$$
$$

The boundary terms in Eq. (8) which vanish with the application of system boundary conditions, represent the shear force and moment of the beams are given by = ′′′ and = ′′ and remain similar to those of a classical Euler-Bernoulli beam. The domain length of the DWCNT is now discretized with four nodes and two Degrees of Freedom (DOF) per node (one transverse displacement and one slope per node) such that:

$$\begin{aligned} \overline{W}\_{\text{w}\_{1}}^{k} &= \int\_{0}^{t} \delta W\_{1}^{\prime} \left( E\_{1} I\_{1} W\_{1}^{\prime} \right) d\mathbf{x} + \int\_{0}^{t} \delta W\_{1} \left( c - \rho\_{1} A\_{1} \boldsymbol{\phi}^{2} \right) W\_{1} - \int\_{0}^{t} \delta W\_{1} \boldsymbol{\varepsilon} W\_{2} = 0 \\\ \overline{W}\_{\text{w}\_{2}}^{k} &= \int\_{0}^{t} \delta W\_{2}^{\prime} \left( E\_{2} I\_{2} W\_{2}^{\prime} \right) d\mathbf{x} + \int\_{0}^{t} \delta W\_{2} \left( c - \rho\_{2} A\_{2} \boldsymbol{\phi}^{2} \right) W\_{2} - \int\_{0}^{t} \delta W\_{2} \boldsymbol{\varepsilon} W\_{1} = 0 \end{aligned} \tag{9}$$

The present work uses Hermite type polynomial approximations to derive the finite element formulation such that the interpolation function is given as:

$$\overline{W}\left(\mathbf{x}\right) = \sum\_{i}^{4} N\_{i}\left(\mathbf{x}\_{i}\right) W\_{i} \tag{10}$$

where ( ) is a row vector of cubic shape functions, of the beam elements given by:

$$\begin{aligned} N\_1 &= \frac{1}{L^3} \left( 2\mathbf{x}^3 - 3\mathbf{x}^2 L + L^3 \right) & N\_2 &= \frac{1}{L^3} \left( \mathbf{x}^3 L - 2\mathbf{x}^2 L + \mathbf{x}L \right) \\ N\_3 &= \frac{1}{L^3} \left( -2\mathbf{x}^3 + 3\mathbf{x}^2 L \right) & N\_4 &= \frac{1}{L^3} \left( \mathbf{x}^3 L - \mathbf{x}^2 L^2 \right) \end{aligned} \tag{11}$$

#### **2.2. DWCNTs based on nonlocal Euler-Bernoulli beam**

The governing equations for a DWCNT are [8]:

harmonic motion is assumed, i.e., , =

equations modifies them as:

74 Perusal of the Finite Element Method

of the governing equations is given as:

0

ò

*L k*

*L k*

0

parts, twice on Eq. (7), as given below:

0

*L*

ò

''''

''''

11 1 1 1 1 2 1

&&

+ =-

*E I w A w cw cw*

r

r

22 2 2 2 2 2 1

+ =- +

where the subscripts 1, 2 for , , , , represent the inner and outer tubes, respectively. Simple

from Eq. (5) and = 1, 2 to represent inner and outer carbon nanotube, respectively, and and represent the circular frequency and time. Using simple harmonic motion in governing

+- - =

Following Galerkin weighted residual method, weight functions are introduced as 1, 2 and the transverse displacements continue to be represented by 1, 2. The weighted-integral form

> d

> > d

0 0

 d

> d

 d- +

+- - = (6)

0

0

[( ) ]

d

is assumed to remove time dependency

(5)

(7)

*E I w A w cw cw*

( ) ( )

( ( ) )

= +- - =

*W E I W W c A W W cW W dx*

 rw

*W E I W W c A W W cW W dx*

 rw

 2 1 11 1 11 1 1 2 1

 2 2 22 2 22 2 2 1 2

> rw

( ( ) )

= +- - =

The weak integral form of the governing equations is eliminated by performing integration by

( ( ) ) '' '' <sup>2</sup> '''' 2 22 2 2 22 2 2 1 2 22 20

ò

 d

'' ' 22 2 <sup>0</sup> ( ) *<sup>L</sup>* - é ù *EIW W*dë û = 0

*<sup>k</sup> <sup>L</sup> W E I W W c A W W cW W dx E I W W* = +-

*E I W c A W cW E I W c A W cW* r w

 2 11 1 11 1 2

 2 22 2 22 2 1

r w

''''

''''

''''

d

''''

d

d

&&

The concept of nonlocality, which is accounted for with the introduction of a nonlocal or a small scale factor was extended to the field of nanotechnology by Peddieson et al. [6] where the author applied it to Euler-Bernoulli beam model. The small scale effect or the nonlocal parameter is given as =0 to consider the stress at one point of the system as the function of strain at all points of the system. This is achieved with 0 being the material constant (to be determined for each material) and being the length of C─C bond. Subsequently, researchers like Ehteshami and Hajabasi [49], Zhang et al. [50] and Wang and Varadan [20] developed the model for DWCNTs using nonlocal Euler-Bernoulli beam model. The derivation of governing equations requires the classic Hooke's law for uniaxial stress state to be modified, taking nonlocal elasticity into consideration, as:

$$
\sigma - \left(e\_0 a\right)^2 \frac{\partial^2 \sigma}{\partial \mathbf{x}^2} = E \varepsilon
\tag{12}
$$

with and being axial stress and strain, respectively. The equations of motion, perpendicular to *x* axis, are given by:

$$
\rho \frac{\partial \mathbf{S}}{\partial \mathbf{x}} - \rho A \frac{\partial^2 w}{\partial t^2} = 0; V - \frac{\partial M}{\partial \mathbf{x}} = 0\tag{13}
$$

where is the mass density of material, is the cross-sectional area, is deflection of the beam and and denotes the bending moment and shear force, respectively. Using these in the moment equilibrium conditions, results in the governing equation for nonlocal Euler-Bernoulli beam, given as:

$$p = EI \frac{\partial^4 w}{\partial \mathbf{x}^4} + \rho A \frac{\partial^2 w}{\partial t^2} - \mu^2 \left(\rho A \frac{\partial^4 w}{\partial \mathbf{x}^4} - \frac{\partial^2 p}{\partial \mathbf{x}^2}\right) \tag{14}$$

For nonlocal DWCNTs, the governing equations are derived to be [20, 49, 50]:

$$\frac{1}{2}E\_1 I\_1 \frac{\partial^4 w\_1}{\partial \mathbf{x}^2} + \rho\_1 A\_1 \frac{\partial^2 w\_1}{\partial t^2} - \mu^2 \rho\_1 A\_1 \frac{\partial^4 w\_1}{\partial \mathbf{x}^2 \partial t^2} - c(w\_2 - w\_1) + \mu^2 c \left(\frac{\partial^2 w\_2}{\partial \mathbf{x}^2} - \frac{\partial^2 w\_1}{\partial \mathbf{x}^2}\right) = 0$$

$$E\_2 I\_2 \frac{\partial^4 w\_2}{\partial \mathbf{x}^2} + \rho\_2 A\_2 \frac{\partial^2 w\_2}{\partial t^2} - \mu^2 \rho\_2 A\_2 \frac{\partial^4 w\_2}{\partial \mathbf{x}^2 \partial t^2} - c(w\_1 - w\_2) + \mu^2 c \left(\frac{\partial^2 w\_1}{\partial \mathbf{x}^2} - \frac{\partial^2 w\_2}{\partial \mathbf{x}^2}\right) = 0 \tag{15}$$

The solution of the equations is in the form , = , similar to the local Euler-Bernoulli formulation. The time dependency is removed by considering the transverse displacement with respect to time as circular frequency, 2:

of strain at all points of the system. This is achieved with 0 being the material constant (to be determined for each material) and being the length of C─C bond. Subsequently, researchers like Ehteshami and Hajabasi [49], Zhang et al. [50] and Wang and Varadan [20] developed the model for DWCNTs using nonlocal Euler-Bernoulli beam model. The derivation of governing equations requires the classic Hooke's law for uniaxial stress state to be modified, taking

> ( ) <sup>2</sup> <sup>2</sup> <sup>0</sup> <sup>2</sup> *ea E <sup>x</sup>* s

2 <sup>2</sup> 0; 0 *Sw M A V xt x*

r

r

For nonlocal DWCNTs, the governing equations are derived to be [20, 49, 50]:

 mr

 mr

r

r

 e

with and being axial stress and strain, respectively. The equations of motion, perpendicular

where is the mass density of material, is the cross-sectional area, is deflection of the beam and and denotes the bending moment and shear force, respectively. Using these in the moment equilibrium conditions, results in the governing equation for nonlocal Euler-Bernoulli

> 4 2 4 2 2 4 2 4 2 *w w wp p EI A <sup>A</sup> x t xx*

¶ ¶ ¶¶ æ ö

 mr

4 2 4 2 2 1 1 2 2 1 2 1 1 1 <sup>2</sup> 1 1 <sup>2</sup> 1 1 2 2 2 1 2 2 ( ) <sup>0</sup> *ww w w w EI A A c w w c x t xt x x*

4 2 4 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2

*ww w w w EI A A cw w c*

( )

 <sup>0</sup> ¶¶ ¶ æ ö ¶ ¶ + - - -+ - = ç ÷ ¶ ¶ ¶¶ è ø ¶ ¶

¶¶ ¶ æ ö ¶ ¶ + - - -+ - = ç ÷ ¶ ¶ ¶¶ è ø ¶ ¶

¶ - = ¶ (12)

¶¶ ¶ - =- = ¶¶ ¶ (13)

=+ - - ç ÷ ¶ ¶ ¶¶ è ø (14)

m

m

*x t xt x x* (15)

s

nonlocal elasticity into consideration, as:

to *x* axis, are given by:

76 Perusal of the Finite Element Method

beam, given as:

$$\begin{aligned} \left[E\_1 I\_1 W\_1^\ast - m\_1 \alpha^2 W\_1 - \mu^2 m\_1 \alpha^2 W\_1^\ast - cW\_2 + cW\_1 + \mu^2 cW\_2^\ast - \mu^2 cW\_1^\ast = 0 \\ \left[E\_2 I\_2 W\_2^\ast - m\_2 \alpha^2 W\_2 - \mu^2 m\_2 \alpha^2 W\_2^\ast - cW\_1 + cW\_2 + \mu^2 cW\_1^\ast - \mu^2 cW\_2^\ast = 0\right] \end{aligned} \tag{16}$$

To maintain uniformity in mathematical modelling with the formulation proposed for local Euler-Bernoulli model, Galerkin weighted residual model is used for FEM, with taken as weighted residual. The weak integral form of governing equations takes the form:

$$\begin{cases} (E\_1 W\_1^\* \cdot \delta W\_1 - m\_1 \alpha^2 W\_1 \delta W\_1 - \mu^2 m\_1 \alpha^2 W\_1^\* \cdot \delta W\_1 - c W\_2 \delta W\_1 + c W\_1 \delta W\_1 + \mu^2 c W\_2 \delta W\_1 \\ \\ - \mu^2 c W\_1^\* \cdot \delta W\_1) d\mathbf{x} = 0 \\ \int\_0^L (E\_2 I\_2 W\_2^\* \cdot \delta W\_2 - m\_2 \alpha^2 W\_2 \delta W\_2 - \mu^2 m\_2 \alpha^2 W\_2^\* \cdot \delta W\_2 - c W\_1 \delta W\_2 + c W\_2 \delta W\_2 + \mu^2 c W\_1^\* \delta W\_2 \\ \\ - \mu^2 c W\_2^\* \delta W\_2) d\mathbf{x} = 0 \end{cases} \tag{17}$$

Integration by parts is performed on the above set of equations once, and the weak form vanishes with the application of system boundary conditions,

$$
\begin{bmatrix}
\int\_0^L \left(\left\{\delta W\_1' \left(-E\_1 I\_1 W\_1'\right)\right\} - \left(\delta W\_1 \left(m\_1 \alpha^2 W\_1\right)\right) - \left(\delta W\_1' \left(\mu^2 m\_1 \alpha^2 W\_1'\right)\right) \\
+ \left(\delta W\_1 \left(cW\_1\right)\right) - \left(\delta W\_1 \left(cW\_2\right)\right) - \left(\delta W\_1' \left(\mu^2 cW\_2'\right)\right) + \left(\delta W\_1' \left(\mu^2 cW\_1'\right)\right)
\end{bmatrix} d\mathbf{x} = \mathbf{0}
$$

$$
\begin{bmatrix}
\int\_0^L \left(\left\{\delta W\_2' \left(-E\_2 I\_2 W\_2'\right)\right\} - \left(\delta W\_2 \left(m\_2 \alpha^2 W\_2\right)\right) - \left(\delta W\_2' \left(\mu^2 m\_2 \alpha^2 W\_2'\right)\right) \\
+ \left(\delta W\_2 \left(cW\_2\right)\right) - \left(\delta W\_2 \left(cW\_1\right)\right) - \left(\delta W\_2' \left(\mu^2 cW\_1'\right)\right) + \left(\delta W\_2' \left(\mu^2 cW\_2'\right)\right)
\end{bmatrix} d\mathbf{x} = 0
\tag{18}
$$

The resulting internal shear forces for both the beams are given as:

$$\begin{aligned} V\_1 &= E\_1 I\_1 W\_1^\prime + \mu^2 \left( m\_1 \alpha^2 W\_1^\prime + \mathcal{c} \left( W\_2^\prime - W\_1^\prime \right) \right) \\ V\_2 &= E\_2 I\_2 W\_2^\prime + \mu^2 \left( m\_2 \alpha^2 W\_2^\prime + \mathcal{c} \left( W\_1^\prime - W\_2^\prime \right) \right) \end{aligned} \tag{19}$$

and the resultant bending moments are given as:

$$M\_1 = E\_1 I\_1 W\_1^\ast - \mu^2 \left( m\_1 \alpha^2 W\_1 - c \left( W\_2 - W\_1 \right) \right)$$

$$M\_2 = E\_2 I\_2 W\_2^\ast - \mu^2 \left( m\_2 \alpha^2 W\_2 - c \left( W\_1 - W\_2 \right) \right) \tag{20}$$

Integration by parts is performed once again on the first terms in Eq. (18) and after rearranging for clarity, the following set of equations is obtained:

$$\begin{split} \overline{W}\_{w\_1}^{k} &= E\_1 I\_1 \int\_0^l \left( W\_1^\prime \delta W\_1^\prime \right) d\mathbf{x} - m\_1 \alpha^2 \left( \int\_0^l \left( W\_1 \delta W\_1 \right) + \mu^2 \int\_0^l \left( W\_1^\prime \delta W\_1^\prime \right) \right) d\mathbf{x} \\ &+ c \left[ \int\_0^l \left( W\_1 \delta W\_1 \right) + \mu^2 \int\_0^l \left( W\_1^\prime \delta W\_1^\prime \right) \right] d\mathbf{x} - c \left( \int\_0^l \left( W\_2 \delta W\_1^\prime \right) + \mu^2 \int\_0^l \left( W\_2^\prime \delta W\_1^\prime \right) \right) d\mathbf{x} \\ \overline{W}\_{w\_2}^k &= E\_2 I\_2 \int\_0^l \left( W\_2^\prime \delta W\_2^\prime \right) d\mathbf{x} - m\_2 \alpha^2 \left( \int\_0^l \left( W\_2 \delta W\_2^\prime \right) + \mu^2 \int\_0^l \left( W\_2^\prime \delta W\_2^\prime \right) \right) d\mathbf{x} \\ &+ c \left[ \int\_0^l \left( W\_2 \delta W\_2^\prime \right) + \mu^2 \int\_0^l \left( W\_2^\prime \delta W\_2^\prime \right) \right] d\mathbf{x} - c \left( \int\_0^l \left( W\_1 \delta W\_2^\prime \right) + \mu^2 \int\_0^l \left( W\_1^\prime \delta W\_2^\prime \right) \right) d\mathbf{x} \end{split} \tag{21}$$

Using Hermite cubic polynomial functions, given in Eq. (11), the finite element model is developed. The element mass and stiffness matrices for both the local and nonlocal scenarios are explicitly presented in the Appendix A. An FEM code in MATLAB solves the linear eigenvalue problem which is given as:

$$\det\left(K - \alpha^2 M\right) = 0\tag{22}$$

where represents global stiffness matrix and represents the global mass matrix, respectively, and is achieved after assembly. **Table 1** gives the element matrices and , which are to be assembled to form the global matrices for both local and nonlocal cases.


**Table 1.** Elemental matrices for local and nonlocal DWCNTs modeled as Euler-Bernoulli beams.

## **3. Results and discussion**

and the resultant bending moments are given as:

for clarity, the following set of equations is obtained:

dm

d

d

m

1

*k w*

78 Perusal of the Finite Element Method

2

*k w*

+

<sup>=</sup>

<sup>=</sup>

1 + 3 −4 −3 2 + 4

2

( ) d

*c WW*

eigenvalue problem which is given as:

( ( )) ( ( ))

= - -- (20)

 d

> d

 d

(21)

 d

0 (22)

'' 2 2 1 11 1 11 21 '' 2 2 2 22 2 2 2 12

m

m

*M EIW m W c W W M EIW m W cW W*

= - --

 w

> w

Integration by parts is performed once again on the first terms in Eq. (18) and after rearranging

( ) ( ) ( )

= -+ ç ÷

*W E I W W dx m W W W W dx*

ò òò

 w

*k k k*

*l l l*

'' '' 2 2 '' 11 1 1 1 1 1 1 1 0 0 0

 dm

( ) ( ) ( ) ( )

*c W W W W dx c W W W W dx*

æ ö æ ö ++ - + <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>è</sup> ø è <sup>ø</sup>

00 0 0

òò òò

 d

> d

*kk k k*

*ll l l*

2 '' 2 '' 11 1 1 21 21

æ ö

è ø

 dm

 dm

*W W dx c W W W W dx*

æ ö

è ø

1 + 3 + 5 <sup>−</sup> 4 + 6

0 2 + 4

1 + 3 <sup>0</sup>

<sup>−</sup> 3 + 5 2 + 4 + 6

( ) ( ) ( )

= -+ ç ÷

*W E I W W dx m W W W W dx*

ò òò

 w

*k k kk l l ll*

0 0 00 <sup>æ</sup> ö æ <sup>ö</sup> <sup>ç</sup> + -+ ÷ ç <sup>÷</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>è</sup> ø è <sup>ø</sup> òò òò

> ( ) <sup>2</sup> det *K M* - = w

to be assembled to form the global matrices for both local and nonlocal cases.

<sup>=</sup>

<sup>=</sup>

**Table 1.** Elemental matrices for local and nonlocal DWCNTs modeled as Euler-Bernoulli beams.

**Local Euler-Bernoulli beam Nonlocal Euler-Bernoulli beam**

*k k k*

*l l l*

'' '' 2 2 '' 22 2 2 2 2 2 2 2 0 0 0

 ( ) ( ) ( ) 2 '' 2 '' 2 22 12 1 2

Using Hermite cubic polynomial functions, given in Eq. (11), the finite element model is developed. The element mass and stiffness matrices for both the local and nonlocal scenarios are explicitly presented in the Appendix A. An FEM code in MATLAB solves the linear

where represents global stiffness matrix and represents the global mass matrix, respectively, and is achieved after assembly. **Table 1** gives the element matrices and , which are

 dm The robustness and generality of proposed FEM model is demonstrated by its application to a double-elastic Euler-Bernoulli beam and then to carbon nanotubes modeled as Euler-Bernoulli beam, both locally and nonlocally. Natural frequencies are calculated for clampedfree, clamped-clamped and simply supported-simply supported boundary conditions. The parameters of the local double elastic Euler-Bernoulli beam are taken as:

$$\begin{aligned} \rho\_1 &= \rho\_2 = 1 \,\mathrm{kg} \,\Big/ \mathrm{m}^3 & E\_1 &= E\_2 = 1 \, N \Big/ \mathrm{m}^2 & A\_1 &= 1 \,\mathrm{m}^2 & A\_2 &= 2 \,\mathrm{m}^2\\ I\_1 &= 1 \,\mathrm{m}^4 & I\_2 &= 2 \,\mathrm{m}^4 & \mathcal{c} &= 1 \,\mathrm{0}; \mu = 0.1 & L &= 1 \,\mathrm{m} \end{aligned}$$


**Table 2.** Non-dimentionalized natural frequencies of a local double-elastic Euler-Bernoulli beam.

These parameters are so selected that the double beam replicates the geometry of carbon nanotubes, i.e., both are concentric and are coupled with an interaction forces between the walls. The natural frequencies of the double-elastic system, in this chapter, are normalized using <sup>=</sup> 2 / where <sup>=</sup> 1 <sup>+</sup> 2 and <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>. The values presented by the proposed model converge at about five elements, but the data tabulated, for all cases, are taken for 10 elements, unless mentioned otherwise. The non-dimentionalized natural frequencies for three classic boundary conditions are given in **Table 2**.


**Table 3.** Non-dimentionalized natural frequencies of a nonlocal double-elastic Euler-Bernoulli beam.

The percentage error has been calculated by ((approximate−exact)/exact) × 100 in this study. These values are compared to the first natural frequency obtained using the analytical solutions found in Xu et al. The values are in good agreement and hence the proposed model is found to be efficient to find the natural frequencies of local double-elastic Euler-Bernoulli beam. The applicability of the proposed nonlocal model is demonstrated by the values in **Table 3**. It is to be noted that the values are presented for a single value nonlocal factor, where = 0.1, and it can be confirmed that = 0 would give the same values as presented in **Table 1** as that would reduce the nonlocal formulation to local one. The values presented in **Table 1** are compared to the values obtained using the beam parameters in analytical solutions provided by Zhang et al. [50]. It can be seen the error percentages are small hence validating the proposed model.

**Table 4** gives the error percentage between local and nonlocal double-elastic Euler-Bernoulli beam when / = 10. This error helps to measure the effect of nonlocal factor on the natural frequencies of double-elastic Euler-Bernoulli beams. It is seen that the values obtained using nonlocal model is less than that obtained using local double-elastic Euler-Bernoulli model. This can be attributed to the presence of a nonlocal factor that considers the strain at one point to be function of all the points in the body.


**Table 4.** Error percentage between natural frequencies of local and nonlocal double-elastic Euler-Bernoulli beam.

It is concluded that a novel double-elastic Euler-Bernoulli beam FEM model has been proposed, and its veracity established for both local and nonlocal cases, the same model will now be applied to carbon nanotubes with the parameters used in Xu et al. [14] and Elishakoff and Pentaras [15], given below:

$$\begin{aligned} \rho\_1 &= \rho\_2 = 2.3 \text{ g/cm}^3 & E\_1 &= E\_2 = \text{l Tpa} \\ c &= 71.1 \text{ l GPa} & e\_0 &\approx 0.82; a = 0.142 \text{ nm} \\ r\_{1,i} \text{(inner radius, inner tube)} &= 0.18 \text{ nm} & r\_{1,o} \text{(outer radius, inner tube)} = 0.52 \text{ nm} \\ r\_{2,i} \text{(inner radius, outer tube)} &= 0.53 \text{ nm} & r\_{2,o} \text{(outer radius, outer tube)} = 0.87 \text{ nm} \end{aligned}$$


**Table 5.** Natural frequencies of a local DWCNTs modeled as Euler-Bernoulli beam (1012 rads/s).

The natural frequencies of a local DWCNT, modeled as Euler-Bernoulli beams, are presented in **Table 5**. The nonlocal factor is introduced as a product of material constant and characteristic length. Zhang et al. calculated the material constant as the ratio of local result to nonlocal result for axial buckling strain of SWCNT and arrived at a value 0 ≈ 0.82, and the characteristic length is taken to be the length of C─C bond i.e., = 0.142 nm. The natural frequencies calculated using the proposed model are compared to the values obtained using the analytical formulation presented in Xu et al. [14]. The error percentages in **Table 5** confirm that the proposed formulation holds valid even at nanoscale level. Further, negative error percentage denotes that the values obtained using the present FEM model are a little higher than those presented by Ehteshami and Hajabasi [49] (see **Table 6**).


**Table 6.** Natural frequencies of a nonlocal DWCNT modeled as Euler-Bernoulli beam (1012 rads/s).

to the values obtained using the beam parameters in analytical solutions provided by Zhang et al. [50]. It can be seen the error percentages are small hence validating the proposed model.

**Table 4** gives the error percentage between local and nonlocal double-elastic Euler-Bernoulli beam when / = 10. This error helps to measure the effect of nonlocal factor on the natural frequencies of double-elastic Euler-Bernoulli beams. It is seen that the values obtained using nonlocal model is less than that obtained using local double-elastic Euler-Bernoulli model. This can be attributed to the presence of a nonlocal factor that considers the strain at one point to

**Boundary conditions Local Nonlocal Error percentage**

3

1, 1, 2, 2,

*r r r r*

*i o i o*

**Table 4.** Error percentage between natural frequencies of local and nonlocal double-elastic Euler-Bernoulli beam.

1 2 1 2

**Boundary conditions [14] Present study Error percentage**

**Table 5.** Natural frequencies of a local DWCNTs modeled as Euler-Bernoulli beam (1012 rads/s).

C─F 0.16 0.16 0.01 C─C 1.06 1.06 0 S─S 4.67 4.67 0

*c e a*

 = = = = = » =

The natural frequencies of a local DWCNT, modeled as Euler-Bernoulli beams, are presented in **Table 5**. The nonlocal factor is introduced as a product of material constant and characteristic length. Zhang et al. calculated the material constant as the ratio of local result to nonlocal result for axial buckling strain of SWCNT and arrived at a value 0 ≈ 0.82, and the characteristic length is taken to be the length of C─C bond i.e., = 0.142 nm. The natural frequencies calculated using the proposed model are compared to the values obtained using the analytical formulation presented in Xu et al. [14]. The error percentages in **Table 5** confirm that the

It is concluded that a novel double-elastic Euler-Bernoulli beam FEM model has been proposed, and its veracity established for both local and nonlocal cases, the same model will now be applied to carbon nanotubes with the parameters used in Xu et al. [14] and Elishakoff and

0

= = = =

*E E*

2.3 g/cm 1 Tpa 71.11 GPa 0.82; 0.142 nm (inner radius, inner tube) 0.18 nm (outer radius, inner tube) 0.52 nm (inner radius, outer tube) 0.53 nm (outer radius, outer tube) 0.87 nm

C─F 3.52 3.44 2.25 C─C 22.37 21.10 5.65 S─S 9.87 9.42 4.60

be function of all the points in the body.

80 Perusal of the Finite Element Method

Pentaras [15], given below:

r r


**Table 7.** Error percentage between natural frequencies of local and nonlocal DWCNTs modeled as Euler-Bernoulli beam (1012 rads/s).


**Table 8.** Error percentage between natural frequencies of local and nonlocal DWCNTs modeled as Euler-Bernoulli beam with different / ratios (1012 rads/s).

Employing the FEM model proposed in Section 1.2, the natural frequencies of nonlocal DWCNTs modeled as Euler-Bernoulli beam are given in **Table 6**. The natural frequencies thus obtained are compared to the values obtained using the analytical formulations given by Ehteshami and Hajabasi [49] Zhang et al. [50]. The error percentages, between present study and these two studies, are presented as error percentage 1 and 2, respectively. The effect of nonlocal factor on the natural frequencies is best understood when the error percentage between local results and nonlocal results are compared, presented in **Table 7**. It will be noted that the error percentage is very small compared to the error percentage reported in **Table 4**. This difference is attributed to the value of nonlocal factor—the double-elastic beam model considers it to be between 1 and 2 and DWCNTs have it in the order of e−10, to be exact 1.1644e − 10. It is also seen (**Table 8**) that as / ratio is increased, the natural frequency and the error percentage, between local and nonlocal results, decrease, i.e., as the aspect ratio increases, the effect of nonlocal factor decreases.

It is noted that though many tables in this chapter may have identical values in the columns that are being compared, the indicated percentage error is a nonzero number, or on the contrary, the columns being compared might not have identical values, but the indicated error percentage is zero. A justification for these values stems from the large magnitude of the numbers reported. For the sake of uniformity, we have rounded off the numbers to 2 decimal points.

## **4. Conclusion**

This chapter proposed FEM formulations to find the natural frequencies of DWCNTs modeled as, both, local and nonlocal Euler-Bernoulli beams, coupled with van der Waals interaction forces. It uses Galerkin weighted residual approach and Hermite cubic polynomial functions to form the stiffness and mass matrices. The formulation is applied to double-elastic Euler-Bernoulli beam and to DWCNTs, to check its generality. The natural frequencies thus obtained are in agreement with those of the formulations found in the literature. It is seen that as the / ratio of the system increases, the difference between local and nonlocal formulations reduces. It has also been observed that difference between local and nonlocal system is dependent on the magnitude of the nonlocal factor considered. These formulations would let researchers apply FEM to multi-walled CNTs in order to better understand their behaviour.

## **Nomenclature**


, time

nonlocal factor on the natural frequencies is best understood when the error percentage between local results and nonlocal results are compared, presented in **Table 7**. It will be noted that the error percentage is very small compared to the error percentage reported in **Table 4**. This difference is attributed to the value of nonlocal factor—the double-elastic beam model considers it to be between 1 and 2 and DWCNTs have it in the order of e−10, to be exact 1.1644e − 10. It is also seen (**Table 8**) that as / ratio is increased, the natural frequency and the error percentage, between local and nonlocal results, decrease, i.e., as the aspect ratio

It is noted that though many tables in this chapter may have identical values in the columns that are being compared, the indicated percentage error is a nonzero number, or on the contrary, the columns being compared might not have identical values, but the indicated error percentage is zero. A justification for these values stems from the large magnitude of the numbers reported. For the sake of uniformity, we have rounded off the numbers to 2 decimal

This chapter proposed FEM formulations to find the natural frequencies of DWCNTs modeled as, both, local and nonlocal Euler-Bernoulli beams, coupled with van der Waals interaction forces. It uses Galerkin weighted residual approach and Hermite cubic polynomial functions to form the stiffness and mass matrices. The formulation is applied to double-elastic Euler-Bernoulli beam and to DWCNTs, to check its generality. The natural frequencies thus obtained are in agreement with those of the formulations found in the literature. It is seen that as the / ratio of the system increases, the difference between local and nonlocal formulations reduces. It has also been observed that difference between local and nonlocal system is dependent on the magnitude of the nonlocal factor considered. These formulations would let researchers apply FEM to multi-walled CNTs in order to better understand their behaviour.

increases, the effect of nonlocal factor decreases.

points.

**4. Conclusion**

82 Perusal of the Finite Element Method

**Nomenclature**

, length of the C─C bond

, van der Waal's interaction coefficient

, second area moment of inertia

, cross-sectional area

, Young's Modulus

, bending moment

, shape functions

, shear force


#### **Appendix A. Element matrices**

$$k\_{1} = \begin{bmatrix} \frac{12E\_{1}I\_{1}}{l^{3}} & \frac{6E\_{1}I\_{1}}{l^{2}} & \frac{-12E\_{1}I\_{1}}{l^{3}} & \frac{6E\_{1}I\_{1}}{l^{2}}\\ & \frac{4E\_{1}I\_{1}}{l} & \frac{-6E\_{1}I\_{1}}{l^{2}} & \frac{2E\_{1}I\_{1}}{l}\\ & & \frac{12E\_{1}I\_{1}}{l^{3}} & \frac{-6E\_{1}I\_{1}}{l^{2}}\\ & & & \frac{12E\_{1}I\_{1}}{l^{3}} & \frac{-6E\_{1}I\_{1}}{l^{2}}\\ & & & & \frac{4E\_{1}I\_{1}}{l} \end{bmatrix} \begin{bmatrix} w\_{11} \\ \theta\_{11} \\ w\_{12} \\ \theta\_{12} \end{bmatrix} k\_{3} = \begin{bmatrix} \frac{156c}{420} & \frac{22lc}{420} & \frac{54c}{420} & \frac{-13lc}{420} \\ & \frac{4l^{2}c}{420} & \frac{13lc}{420} & \frac{-3l^{2}c}{420} \\ & \frac{4l^{2}c}{420} & \frac{13lc}{420} & \frac{-3l^{2}c}{420} \\ & & \frac{156c}{420} & \frac{-22lc}{420} \\ & & & \frac{4l^{2}c}{420} \end{bmatrix} \begin{bmatrix} w\_{11} \\ \theta\_{11} \\ w\_{12} \\ \theta\_{12} \end{bmatrix}$$

$$k\_{2} = \begin{bmatrix} \frac{12E\_{2}I\_{2}}{l^{3}} & \frac{6E\_{2}I\_{2}}{l^{2}} & \frac{-12E\_{2}I\_{2}}{l^{3}} & \frac{6E\_{2}I\_{2}}{l^{2}}\\ & \frac{4E\_{2}I\_{2}}{l} & \frac{-6E\_{2}I\_{2}}{l^{2}} & \frac{2E\_{2}I\_{2}}{l}\\ & & \frac{12E\_{2}I\_{2}}{l^{3}} & \frac{-6E\_{2}I\_{2}}{l^{2}}\\ & & & \frac{12E\_{2}I\_{2}}{l^{3}} & \frac{-6E\_{2}I\_{2}}{l^{2}}\\ & & & & \frac{4E\_{2}I\_{2}}{l} \end{bmatrix} \begin{vmatrix} w\_{21} \\ \theta\_{21} \\ w\_{21} \\ \theta\_{22} \end{vmatrix}\_{k\_{4}} = \begin{vmatrix} \frac{156c}{420} & \frac{22lc}{420} & \frac{54c}{420} & \frac{-13lc}{420} \\ & \frac{4l^{2}c}{420} & \frac{13lc}{420} & \frac{-3l^{2}c}{420} \\ & & \frac{4l^{2}c}{420} & \frac{-3l^{2}c}{420} \\ & & \frac{156c}{420} & \frac{-22lc}{420} \\ & & & \frac{4l^{2}c}{420} \end{vmatrix} \begin{vmatrix} w\_{21} \\ w\_{21} \\ w\_{21} \\ w\_{21} \end{vmatrix} = 0$$

$$k\_{5} = \begin{bmatrix} \frac{6\mu^{2}c}{5l} & \frac{\mu^{2}c}{10} & \frac{-6\mu^{2}c}{5l} & \frac{\mu^{2}c}{10} \\\\ \frac{2l\mu^{2}c}{15} & \frac{-\mu^{2}c}{10} & \frac{-\mu^{2}cl}{30} \\\\ \text{Sym} & \frac{6\mu^{2}c}{5l} & \frac{-\mu^{2}c}{10} \\\\ & \frac{2l\mu^{2}c}{15} \end{bmatrix} \begin{bmatrix} \text{w}\_{11} \\\\ \theta\_{11} \\\\ \theta\_{12} \end{bmatrix} k\_{6} = \begin{bmatrix} \frac{6\mu^{2}c}{5l} & \frac{\mu^{2}c}{10} & \frac{-6\mu^{2}c}{5l} & \frac{\mu^{2}c}{10} \\\\ \frac{2l\mu^{2}c}{15} & \frac{-\mu^{2}c}{10} & \frac{-\mu^{2}cl}{30} \\\\ \text{Sym} & \frac{6\mu^{2}c}{5l} & \frac{-\mu^{2}c}{10} \\\\ & \frac{2l\mu^{2}c}{15} & \frac{-\mu^{2}cl}{10} \end{bmatrix} \begin{bmatrix} \text{w}\_{21} \\\\ \theta\_{21} \\\\ \theta\_{22} \end{bmatrix}$$

$$m\_{1} = \begin{bmatrix} \frac{156m\_{1}l}{420} \frac{12l^{2}m\_{1}}{420} & \frac{54m\_{1}l}{420} \frac{-13l^{2}m\_{1}}{420} \\\\ \frac{4l^{3}m\_{1}}{420} \frac{13l^{2}m\_{1}}{420} & \frac{-3l^{3}m\_{1}}{420} \\\\ \text{Sym} & \frac{156m\_{1}l}{420} \frac{-22l^{2}m\_{1}}{420} \end{bmatrix} \begin{Bmatrix} w\_{11} \\\\ \theta\_{11} \\\\ \theta\_{12} \end{Bmatrix}\_{3} = \begin{Bmatrix} \frac{6\mu^{2}m\_{1}}{5l} & \frac{\mu^{2}m\_{1}}{10} & \frac{-6\mu^{2}m\_{1}}{5l} & \frac{\mu^{2}m\_{1}}{10} \\\\ \frac{2\mu^{2}m\_{1}}{15} & \frac{-\mu^{2}m\_{1}}{10} & \frac{-\mu^{2}m\_{1}}{30} \\\\ \text{Sym} & \frac{6\mu^{2}m\_{1}}{5l} & \frac{-\mu^{2}m\_{1}}{10} \\\\ & \frac{2\mu^{2}m\_{1}}{15} \end{Bmatrix} \begin{Bmatrix} w\_{11} \\\\ \theta\_{11} \\\\ \theta\_{12} \end{Bmatrix}\_{3}$$

$$m\_{2} = \begin{bmatrix} \frac{156m\_{2}l}{420} \frac{22l^{2}m\_{2}}{420} & \frac{54m\_{2}l}{420} & \frac{-13l^{2}m\_{2}}{420} \\ & \frac{4l^{3}m\_{2}}{420} \frac{13l^{2}m\_{2}}{420} & \frac{-3l^{3}m\_{2}}{420} \\ & & \frac{156m\_{2}l}{420} \frac{-22l^{2}m\_{2}}{420} \end{bmatrix} \begin{Bmatrix} w\_{21} \\ \theta\_{21} \\ \theta\_{21} \\ w\_{22} \\ \theta\_{22} \end{Bmatrix} m\_{4} = \begin{Bmatrix} \frac{6\mu^{2}m\_{2}}{5l} & \frac{\mu^{2}m\_{2}}{10} & \frac{-6\mu^{2}m\_{2}}{5l} & \frac{\mu^{2}m\_{2}}{10} \\ & & \frac{2\mu^{2}m\_{2}}{15} & \frac{-\mu^{2}m\_{2}}{10} & \frac{-\mu^{2}m\_{2}l}{30} \\ & & \frac{6\mu^{2}m\_{2}}{15} & \frac{-\mu^{2}m\_{2}}{10} & \frac{\mu^{2}m\_{2}}{60} \\ & & & \frac{2\mu^{2}m\_{2}}{15} \end{Bmatrix} \begin{Bmatrix} w\_{21} \\ w\_{21} \\ \theta\_{21} \\ \theta\_{21} \end{Bmatrix}$$

## **Author details**

Ishan Ali Khan and Seyed M. Hashemi\*

\*Address all correspondence to: smhashem@ryerson.ca

Department of Aerospace Engineering, Ryerson University, Toronto, Canada

## **References**

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\*Address all correspondence to: smhashem@ryerson.ca

Department of Aerospace Engineering, Ryerson University, Toronto, Canada


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#### **Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method**

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Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65002

#### **Abstract**

In this chapter, the critical stress transfer factors of interface material have been studied under tensile loading. The polypropylene (PP) short fiber was embedded into the polypropylene co-ethylene (PPE) cylindrical interface first and then into the matrix material. Modified interface PPE material with lower elastic constant value than matrix material was used in our study. In this chapter, interface parameters affecting the stress transfer mechanism have been investigated. Finite element analysis (FEA) package (Ansys) has been used in the numerical modeling by using representative volume element (RVE). Tensile load was applied on one side of the composite cylinder as the other side of the composite is fixed. The critical stress-strain distributions are determined and presented by curves and tables for different fiber and interface diameters. For verification, the equivalent elastic material constants have been compared with the analytical solution and the results have been appropriate.

**Keywords:** FEA, representative volume element, fiber composite, stress transfer, interface, polymer

## **1. Introduction**

There was considerable work on the determination of equivalent elastic constants of fiberreinforced composites in the literature survey. The methods used in these studies can be summarized as follows: (i) numerical methods: finite difference and finite element method; (ii) analytical methods: semi-empirical Halpin-Tsai equation, rule of mixtures (ROM); and (iii) experimental studies in macro-, micro-, and nanoscales.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the literature survey, studies proposing finite difference modeling and equivalent elastic properties of fiber composite structures were calculated in the consideration of the effect of fiber modulus and aspect ratio / [1–6]. These studies were performed on the representative models according to both the multiple and single-fiber-reinforced material compositions.

Finite element method (FEM) is one of the most useful numerical methods in engineering problem-solving. Other methods such as the sublevel numerical procedures were additionally developed and used with FEM in solution phases. One of the main methods used was representative volume element (RVE) that used to predict the mechanical properties of unidirectional fiber composites. By using finite element analysis of RVE, effective elastic modulii of the composite were determined [7]. Experimental studies were also performed to support the numerical results [8]. Two-dimensional (unit cell method) and three-dimensional finite element analysis (FEA) with RVE was performed, and results were compared with the results of experimental studies in the literature [9–23]. Researchers often used the following FE commercial codes: ABAQUS [11, 12, 16, 19], ANSYS [17, 20], NASTRAN [13], and ALGOR-FEAS [9]. The numerically obtained results were compared and discussed with the analytical [2, 5, 14, 15, 21, 23] and experimental data in the literature survey [10, 14, 15, 18].

The aim of this study was to determine the stress and strain distributions in fiber, matrix, and fiber-matrix-interface in discontinuous fiber-matrix composite by using FEA. Each of the FE model consists of a polypropylene co-ethylene (PPE) matrix which contains a polypropylene fiber (PP) in it. In this study, 18 models were used and nine of them had also an interface volume. In this research, following six main concepts were presented by examining the single short fiber cylindrical bar within the matrix material under tensile loading for mechanical response.


## **2. The fiber-reinforced composite material with interface volume: modeling by representative volume element (RVE) method**

In the literature survey, studies proposing finite difference modeling and equivalent elastic properties of fiber composite structures were calculated in the consideration of the effect of fiber modulus and aspect ratio / [1–6]. These studies were performed on the representative models according to both the multiple and single-fiber-reinforced material composi-

Finite element method (FEM) is one of the most useful numerical methods in engineering problem-solving. Other methods such as the sublevel numerical procedures were additionally developed and used with FEM in solution phases. One of the main methods used was representative volume element (RVE) that used to predict the mechanical properties of unidirectional fiber composites. By using finite element analysis of RVE, effective elastic modulii of the composite were determined [7]. Experimental studies were also performed to support the numerical results [8]. Two-dimensional (unit cell method) and three-dimensional finite element analysis (FEA) with RVE was performed, and results were compared with the results of experimental studies in the literature [9–23]. Researchers often used the following FE commercial codes: ABAQUS [11, 12, 16, 19], ANSYS [17, 20], NASTRAN [13], and ALGOR-FEAS [9]. The numerically obtained results were compared and discussed with the analytical [2, 5, 14, 15, 21, 23] and experimental data in the literature survey [10,

The aim of this study was to determine the stress and strain distributions in fiber, matrix, and fiber-matrix-interface in discontinuous fiber-matrix composite by using FEA. Each of the FE model consists of a polypropylene co-ethylene (PPE) matrix which contains a polypropylene fiber (PP) in it. In this study, 18 models were used and nine of them had also an interface volume. In this research, following six main concepts were presented by examining the single short fiber cylindrical bar within the matrix material under tensile loading for mechanical

**1.** By using RVE, testing the effectiveness of the finite element modeling procedures on the results, that is, mesh element types, meshing, definition and implementation of the

**2.** Considering the different volume fractions, testing the effectiveness of the various volume fraction of the fiber on the stress transfer mechanism from matrix to fiber material.

**3.** In terms of three-dimensional analysis, to investigate the influence of definition and

**4.** In terms of three-dimensional analysis, examining the effects of fiber surface area on the

**5.** Establishing the explanations and curves for clarification the stress and strain localization

boundary conditions, application of the tensile loading.

modeling of interface volume between fiber and matrix in 3D.

stress transfer mechanism.

**6.** Calculating the equivalent elastic constant *E*1.

at the ends of the fiber.

tions.

90 Perusal of the Finite Element Method

14, 15, 18].

response.

Cylindrical RVE was used to represent the matrix and fiber all together. The different material properties of fiber, matrix, and interface were selected. The elastic constants of the polymer material were expressed as = 1.05 GPa, = 0.33, = 4.5 GPa, = 0.2, = 0.1 × = 0.105 GPa [4], = 0.33. Here, , , , , , reflected the Young's modulus and Poisson's ratio of the matrix, fiber, and interface materials, respectively. While the fiber material was composed of elastic polyethylene (PE), the matrix and interface materials were composed of bilinear material (PPE). The thickness of the interface region was defined as = 0.025. Here, was the diameter of the fiber. The shear modulus of the interface region was defined as = 0.1 [4]. In 3D composite modeling steps, nine different geometrical properties were used with variable thickness , diameter , and fiber volume fraction as shown in **Table 1**. According to these different geometric values, fiber composite RVE was modeled by FEM and the stress-strain transmission conditions from fiber to interface and stress transfer from interface to matrix of the three types of materials were examined.


**Table 1.** PPE/PP single-fiber-matrix modeling according to the various interface geometric properties and analytical results of continuous = ℓ fiber-matrix composite.

Fiber volume fraction values and the size of the material models were obtained by using the following equation [13];

$$W\_f = \frac{\pi \,\ell \,d^2}{4 \,LS^2} \tag{1}$$

Here, *ℓ*, , and correspond to the fiber spacing, fiber length, and longitudinal fiber spacing, respectively, as shown in **Figure 1**.

**Figure 1.** 2D geometrical parameters of the matrix/interface/fiber combination of PPE/PP microcomposite.

**Figure 2.** One of the typical geometrical descriptions of the polymer composite (matrix/interface/fiber, *d* = 50 μm) and generated volume numbers.

**Figure 3.** Two different RVE fiber composite geometries and applied displacement boundary conditions: (a) fiber, interface, and matrix combination; (b) fiber and matrix combination.

By rotating the 2D model 360° around the *Y*′-axis, 3D material models were obtained as shown in **Figure 2**. Cylindrical geometry was divided into two segments and one segment was used to demonstrate the results (**Figures 2** and **3**). In general, the cylindrical geometry was defined in cylindrical coordinates , , . In Ansys, modeling , , Cartesian coordinate axes refers to the , , in the cylindrical coordinate system. However, in this study, all of the results and Ansys code screen outputs were demonstrated only on the *x*-*y* plane of the projected surface for one half of the cylindrical geometry.

## **2.1. Analytical formulation.**

Here, *ℓ*, , and correspond to the fiber spacing, fiber length, and longitudinal fiber spacing,

**Figure 1.** 2D geometrical parameters of the matrix/interface/fiber combination of PPE/PP microcomposite.

**Figure 2.** One of the typical geometrical descriptions of the polymer composite (matrix/interface/fiber, *d* = 50 μm) and

**Figure 3.** Two different RVE fiber composite geometries and applied displacement boundary conditions: (a) fiber, inter-

By rotating the 2D model 360° around the *Y*′-axis, 3D material models were obtained as shown in **Figure 2**. Cylindrical geometry was divided into two segments and one segment was

respectively, as shown in **Figure 1**.

92 Perusal of the Finite Element Method

generated volume numbers.

face, and matrix combination; (b) fiber and matrix combination.

The aim of the micromechanical approaches was to determine the equivalent elastic constants of composite material considering individual elastic constants constituting the composite material. The equivalent elastic constants 1, 2, 12, 12 of the fiber-reinforced-composite have been determined analytically in terms of the relative volume fractions , and elastic constants , , , , , of fiber and matrix. It may be mentioned two basic analytical approaches in micromechanics of continuous fiber-matrix composite materials in obtaining the equivalent elastic constants. The first one was the rule of mixtures (ROM), and the second one was the Halpin-Tsai semi-empirical model [24, 25]. The first model "rule of mixtures" defined as the mathematical expressions that give the elastic properties of the continuous fiber composite material in terms of original material properties, volume fiber fraction, and arrangement of its constituents: (a) the apparent Young's modulus in the same direction as the fibers (1) (Eq. (2)), (b) the apparent Young's modulus in the direction perpendicular to the fibers (2) (Eq. (3)), (c) the major Poisson's ratio (12) (Eq. (4)), and (d) the in-plane shear modulus (12) of a lamina (Eq. (5)). Rule of mixtures (ROM) equations have been listed and shown as the formulas below:

$$E\_1 = E\_f V\_f + E\_m \left(1 - V\_f\right) \tag{2}$$

$$E\_2 = \frac{E\_f E\_m}{V\_m E\_f + V\_f E\_m} \tag{3}$$

$$
\nu\_{12} = V\_f \nu\_f + \nu\_m \left(1 - V\_f\right) \tag{4}
$$

$$G\_{12} = \frac{G\_m G\_f}{V\_m G\_f + V\_f G\_m} \tag{5}$$

The second model "Halpin-Tsai" included the expressions obtained by curve fitting as: (a) the apparent Young's modulus in the same direction as the fibers (1) (Eq. (6)), (b) the apparent Young's modulus in the direction perpendicular to the fibers (2) (Eq. (7)), (c) the major Poisson's ratio (12) (Eq. (8)), and (d) the in-plane shear modulus (12) (Eq. (9)) of a single layer of layered composite "ply." These related equations have been listed and shown below:

$$E\_1 = E\_f V\_f + E\_m V\_m \tag{6}$$

$$\frac{E\_2}{E\_m} = \left[\frac{1 + \xi \eta V\_f}{1 - \eta V\_f}\right] \quad \eta = \frac{\left[\frac{E\_f}{E\_m}\right] - 1}{\left[\frac{E\_f}{E\_m}\right] + \xi} \tag{7}$$

$$
\nu\_{12} = V\_f \nu\_f + \nu\_m V\_m \tag{8}
$$

$$\frac{G\_{12}}{G\_m} = \frac{1 + \xi \eta V\_f}{1 - \eta V\_f} \quad \eta = \frac{\left[\frac{G\_f}{G\_m}\right] - 1}{\left[\frac{G\_f}{G\_m}\right] + \xi} \tag{9}$$

The reinforcing factor (), measuring reinforcement of the composite, is variable depending on the fiber geometry, packing geometry, and loading condition. It is a measure of reinforcement of the composite. The values of have been obtained by comparing the Eqs. (7) and (8) with exact elasticity solutions, and corresponding curve fitting results. The generalized Hooke's law equation is as follows:

$$
\sigma\_l = C\_{lj} \varepsilon\_j \tag{10}
$$

In this equation, is the stress vector, is the stiffness matrix, and is the strain vector components. In functional form, has been represented by Eq. (11) in terms of the elastic properties of the fiber, matrix, relative volume fractions of fiber and matrix.

$$C\_{ij} = C\_{ij} \left( E\_f, \nu\_f, V\_f, E\_m, \nu\_m, V\_m \right) \tag{11}$$

In most literature studies, the basic analytical approaches have been presented without taking into account the adhesion of the interface across the end faces of the fibers and the stress concentration effects at the fiber ends [2, 26, 27].

## **3. Finite element modeling of short fiber in microcomposite and analyses results**

Young's modulus in the direction perpendicular to the fibers (2) (Eq. (7)), (c) the major Poisson's ratio (12) (Eq. (8)), and (d) the in-plane shear modulus (12) (Eq. (9)) of a single layer of layered composite "ply." These related equations have been listed and shown below:

2

12

is the stress vector,

Hooke's law equation is as follows:

94 Perusal of the Finite Element Method

components. In functional form,

concentration effects at the fiber ends [2, 26, 27].

In this equation,

1 1

n

1 1

*E EV EV* <sup>1</sup> = + *f f mm* (6)

<sup>12</sup> = + *V V f f mm* (8)

*i ij j* = *C* (10)

is the stiffness matrix, and is the strain vector

) (11)

has been represented by Eq. (11) in terms of the elastic

(7)

(9)

1

x

1

x

*f*

*E*

é ù

*m*

ë û

*f*

*G*

é ù

*m*

ê ú + ë û

*G*

*f m*

The reinforcing factor (), measuring reinforcement of the composite, is variable depending on the fiber geometry, packing geometry, and loading condition. It is a measure of reinforcement of the composite. The values of have been obtained by comparing the Eqs. (7) and (8) with exact elasticity solutions, and corresponding curve fitting results. The generalized

> e

h

ê ú - <sup>+</sup> ë û = = - é ù

*E*

*f m*

h

ê ú - é ù <sup>+</sup> ë û = = ê ú ê ú - é ù ë û ê ú <sup>+</sup>

*m f f*

 nn

*m f f*

*G V G G V G*

xh

h

s

properties of the fiber, matrix, relative volume fractions of fiber and matrix.

*C CE VE V ij ij f f f m m m* = ( ,,, ,, n

 n

In most literature studies, the basic analytical approaches have been presented without taking into account the adhesion of the interface across the end faces of the fibers and the stress

*E V E E V E*

xh

h In the literature survey, a detailed examination on fiber-reinforced polymer matrix composites was observed in determining the most suitable fiber-matrix combination model. In the literature, different studies were found about tension/compression and shear loading to the different modeling types of fiber/interface/matrix combinations, and in these studies, the strength of the applied tensile/compression and shear loading types have been investigated. In this research, the stress transfer mechanism and equivalent elastic constants of short fiber composites were tried to be obtained by using new modeling. In FE modeling, the design criteria were as follows: the dimensions of fiber and matrix, the material types of fiber and matrix, the dimensions and material types of interface volume and the application of the tensile load. In the first stage of the study, a single short fiber was studied with ANSYS code in determining the optimum interface thickness by comparing the results obtained. In this step, two different models were tested. The first one was fiber-matrix microcomposite with interface (**Figure 3a**); the second model was fiber-matrix microcomposite without interface (**Figure 3b**). In the second stage of the study, the effects of interface volumes on the values of stress distributions around the tip points of the fiber were attempted to explain. In the calculation of the single-fiber modeling, nine different interface thicknesses and nine different fiber diameters were used = 0.025 , ( = 40, 45, …, 80 m). In this study, the interface volume was defined around all of the faces of the fiber. In the literature, the studies were performed by defining the interface surface or volume along the longitudinal surface (**Figure 3**).


**Table 2.** The used geometric parameters, mesh elements, node numbers, and obtained elastic constants of discontinuous fiber-matrix microcomposite.

In this study, the short single-fiber composite polymer (PPE/PP) material was modeled in microlevel by finite element method (FEM). Single-fiber arrangement in matrix under the axial loading was adequate to explain the overall characteristics of the fiber composite structures. SHELL93 as ANSYS element type was used. This element has 6 degree of freedom per node , , , , , . The generated FE models for nine different geometries have different total number of mesh elements and node numbers. Exact number of elements and nodes corresponding to the composite models has been summarized in **Table 2**.

Composite model consists of three volume sections representing the fiber matrix and interface (**Figure 4**). The first boundary condition of the model was applied by fixing one end of the geometry, and this was performed by setting the 6 degree of freedom of each node to zero (BC\_1).

The second boundary condition of the model was related with the loading condition. The second end of the composite cylinder was pulled with a constant load. The loading magnitude applied to the nodes was = 0.1 N. In this application, because of the unsymmetrical meshing system, the resultant deformation of the bar was obtained as unsymmetrical. In order to overcome this difficulty, second boundary condition was applied to the second end of the cylinder. On this face along the loading application thickness or level , the "unit displacement" = 0.1 m was adopted to each node to obtain the symmetric displacement condition over the whole body <sup>1</sup> m ≤ <sup>≤</sup> <sup>2</sup> m (BC\_2) (**Figure 3**). The stress concentration regions were developed between the fiber and matrix at the sharp edge corners as shown in **Figures 5**–**8**. Fiber-matrix model showed higher stress concentrations than fiber-matrixinterface model. In order to give more explanation to the developing stress distributions on these faces, the third boundary condition was applied (BC\_3). The previously defined firstand second-type boundary conditions were applied over the extremely large sections of the body <sup>20</sup> m ≤ <sup>≤</sup> <sup>26</sup> m , and new stress distribution results were obtained (**Figures 9** and **10**).

**Figure 4.** FEA meshing with SHELL93 element (a) RVE of fiber cylindrical composite model, (b) fiber, (c) matrix, and (d) interface.

Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method http://dx.doi.org/10.5772/65002 97

In this study, the short single-fiber composite polymer (PPE/PP) material was modeled in microlevel by finite element method (FEM). Single-fiber arrangement in matrix under the axial loading was adequate to explain the overall characteristics of the fiber composite structures. SHELL93 as ANSYS element type was used. This element has 6 degree of freedom per node

number of mesh elements and node numbers. Exact number of elements and nodes corre-

Composite model consists of three volume sections representing the fiber matrix and interface (**Figure 4**). The first boundary condition of the model was applied by fixing one end of the geometry, and this was performed by setting the 6 degree of freedom of each node to zero

The second boundary condition of the model was related with the loading condition. The second end of the composite cylinder was pulled with a constant load. The loading magnitude applied to the nodes was = 0.1 N. In this application, because of the unsymmetrical meshing system, the resultant deformation of the bar was obtained as unsymmetrical. In order to overcome this difficulty, second boundary condition was applied to the second end of the

placement" = 0.1 m was adopted to each node to obtain the symmetric displacement

condition over the whole body <sup>1</sup> m ≤ <sup>≤</sup> <sup>2</sup> m (BC\_2) (**Figure 3**). The stress concentration regions were developed between the fiber and matrix at the sharp edge corners as shown in **Figures 5**–**8**. Fiber-matrix model showed higher stress concentrations than fiber-matrixinterface model. In order to give more explanation to the developing stress distributions on these faces, the third boundary condition was applied (BC\_3). The previously defined firstand second-type boundary conditions were applied over the extremely large sections of the

body <sup>20</sup> m ≤ <sup>≤</sup> <sup>26</sup> m , and new stress distribution results were obtained (**Figures 9** and

**Figure 4.** FEA meshing with SHELL93 element (a) RVE of fiber cylindrical composite model, (b) fiber, (c) matrix, and

sponding to the composite models has been summarized in **Table 2**.

cylinder. On this face along the loading application thickness or level

, , . The generated FE models for nine different geometries have different total

, the "unit dis-

, , ,

(BC\_1).

**10**).

(d) interface.

96 Perusal of the Finite Element Method

**Figure 5.** Stress () distribution on the matrix section obtained from FEA nodal solution without interface region ( = 40 m).

**Figure 6.** Stress () distribution on the matrix section obtained from FEA element solution without interface region ( = 40 m).

**Figure 7.** Stress () intensity on the matrix section obtained from FEA element solution without interface region ( = 40 m).

**Figure 8.** Stress () intensity developing on the matrix section (a) with interface region, (b) without interface region ( = 80 m).

**Figure 9.** FE results for the BC\_2, stress localization which is parallel to the applied displacement loading and generated around the sharp corners of the fiber–matrix model without interface region ( = 80 m).

**Figures 5**–**7** have been represented the distribution developing within the composite model in matrix segment with noninterface case for the fiber diameter = 40 m [2]. The stress distributions were defined for both nodal and elemental solutions. The developing stress localizations appeared around the tip sections of the fiber in matrix. The developing maximum stress distributions were intensified around the fiber end sections. At the same region around the neighborhood sections of the fiber segment, stress levels reached to relatively lower stress localizations in comparison with maximum. This stress distribution was wide spread around the maximum stress localization in a definite region. Normal stress value reached to their secondarily largest value around the middle section of the fiber in matrix material. Additionally, nodal solution curves gave better results than elemental solution. In nodal solution, stress localizations appeared in a narrower region with longer segments. In finite element analysis, element solutions were obtained by Gauss quadrature integration points. Nodal solution results were determined at the coordinates of the nodal points by interpolation functions.

Gauss integration points have been defined at the inner section of the elements which were not coincided with nodal coordinates. These element solutions were obtained by using the interpolating functions in the curvilinear coordinate system. Element types varied according to the interpolation functions used. In the generated composite models, the generated total number of elements, nodes, and the obtained results were listed in **Table 2**. When the number of elements increased, the most correct results were obtained. For these processes, highcapacity computer systems were required. Stress distributions developing on the fiber and matrix sections without interface volume were presented in **Figures 11** and **12**. In these figures, stress localizations were developed on both matrix and fiber surfaces.

**Figure 8.** Stress () intensity developing on the matrix section (a) with interface region, (b) without interface region

**Figure 9.** FE results for the BC\_2, stress localization which is parallel to the applied displacement loading and

**Figures 5**–**7** have been represented the distribution developing within the composite model in matrix segment with noninterface case for the fiber diameter = 40 m [2]. The stress distributions were defined for both nodal and elemental solutions. The developing stress localizations appeared around the tip sections of the fiber in matrix. The developing maximum stress distributions were intensified around the fiber end sections. At the same region around the neighborhood sections of the fiber segment, stress levels reached to relatively lower stress localizations in comparison with maximum. This stress distribution was wide spread around the maximum stress localization in a definite region. Normal stress value reached to their secondarily largest value around the middle section of the fiber in matrix material. Additionally, nodal solution curves gave better results than elemental solution. In nodal solution, stress localizations appeared in a narrower region with longer segments. In finite element analysis, element solutions were obtained by Gauss quadrature integration points. Nodal solution results were determined at the coordinates of the nodal points by interpolation functions.

generated around the sharp corners of the fiber–matrix model without interface region ( = 80 m).

( = 80 m).

98 Perusal of the Finite Element Method

**Figure 10.** FE results for the BC\_3, stress localization on the (a) matrix and (b) fiber which are parallel to the applied displacement loading and generated above the sharp corners of the fiber–matrix model without interface region ( = 75μm ).

**Figure 11.** FEA stress distribution developing on the matrix section of the fiber-matrix model without interface region ( = 80 m).

**Figure 12.** Stress () distribution on the fiber section obtained from the FEA without interface region ( = 80 m).

As the fiber diameter decreased, the stress distributions collected at the end surface of the cylindrical fiber material grown locally. When the stress localizations reduced in the interface region between matrix and fiber at the same time, higher stresses in fiber section has emerged. As it has shown by **Figures 13** and **14**, in the polymer fiber model with an interface region, the maximum stress value was = 0.952 × 10−6 N/m2, and in the polymer fiber model without interface, maximum stress developing on the fiber was equal to = 0.732 × 10−6 N/m2 = 40 m . On the other hand, in the model with an interface, matrix sections near to the fiber had the value = 0.265 × 10−6 N/m2, and in the noninterface model, stress value was = 0.429 × 10−6 N/m2 (**Figures 15** and **16**) = 40 m .

Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method http://dx.doi.org/10.5772/65002 101

**Figure 13.** FEA stress distribution developing on the fiber section of the fiber-matrix model without interface region ( = 40 m).

**Figure 11.** FEA stress distribution developing on the matrix section of the fiber-matrix model without interface

**Figure 12.** Stress () distribution on the fiber section obtained from the FEA without interface region ( = 80 m).

As the fiber diameter decreased, the stress distributions collected at the end surface of the cylindrical fiber material grown locally. When the stress localizations reduced in the interface region between matrix and fiber at the same time, higher stresses in fiber section has emerged. As it has shown by **Figures 13** and **14**, in the polymer fiber model with an interface region, the maximum stress value was = 0.952 × 10−6 N/m2, and in the polymer fiber model without interface, maximum stress developing on the fiber was equal to = 0.732 × 10−6 N/m2 = 40 m . On the other hand, in the model with an interface, ma-

trix sections near to the fiber had the value = 0.265 × 10−6 N/m2, and in the noninterface

model, stress value was = 0.429 × 10−6 N/m2 (**Figures 15** and **16**) = 40 m .

region ( = 80 m).

100 Perusal of the Finite Element Method

**Figure 14.** FEA () distribution developing on the fiber section of the fiber-matrix model with interface region ( = 40 m).

**Figure 15.** Stress distribution developing on the matrix section of the fiber-matrix model with interface ( = 40 m).

**Figure 16.** Stress distribution developing on the matrix section of the fiber-matrix model without interface ( = 40 m).

When the models were analyzed in terms of interface thicknesses, in the models with thicker interface, the stress was found to be more. The interface region with the largest diameter had the largest stress and Von Mises stress 2 distributions relative to the smallest one. These were equal to = 0.130 × 10−5 N/m2 (**Figure 17**) and 2 = 0.132 × 10−5 N/m2 (**Figure 18**) for = 40 m and = 0.08967 N/m<sup>2</sup> (**Figure 19**) and 2 = 0.233 × 10−5 N/m2 (**Figure 20**) for = 80 m. Having studied the **Figures 21** and **22**, it was seen that the maximum stress distributions are developed at the tip corner sections of the interface geometries. Nodal solution results of = 0.149 × 10−5 N/m2 for = 40 m and = 0.154 × 10−5 N/m2 for = 80 m were obtained.

**Figure 17.** FEA distribution developing on the interface section of the fiber–matrix model ( = 40 m).

Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method http://dx.doi.org/10.5772/65002 103

**Figure 18.** FEA Von Mises 2 distribution developing on the interface section ( = 40 m).

**Figure 16.** Stress distribution developing on the matrix section of the fiber-matrix model without interface

When the models were analyzed in terms of interface thicknesses, in the models with thicker interface, the stress was found to be more. The interface region with the largest diameter had the largest stress and Von Mises stress 2 distributions relative to the smallest one. These were equal to = 0.130 × 10−5 N/m2 (**Figure 17**) and 2 = 0.132 × 10−5 N/m2 (**Figure 18**)

for = 40 m and = 0.08967 N/m2 (**Figure 19**) and 2 = 0.233 × 10−5 N/m2 (**Figure 20**) for = 80 m. Having studied the **Figures 21** and **22**, it was seen that the maximum stress distributions are developed at the tip corner sections of the interface geometries. Nodal solution results of = 0.149 × 10−5 N/m2 for = 40 m and = 0.154 × 10−5 N/m2 for

**Figure 17.** FEA distribution developing on the interface section of the fiber–matrix model ( = 40 m).

( = 40 m).

102 Perusal of the Finite Element Method

= 80 m were obtained.

**Figure 19.** Stress () distribution on the interface section obtained from FEA nodal solution ( = 80 m).

**Figure 20.** Von Mises 2 stress distribution on the interface section obtained from FEA nodal solution ( = 80 m).

**Figure 21.** Stress intensity on the interface section ( = 40 m).

**Figure 22.** Stress intensity on the interface section ( = 80 m).

For nine different interface geometries, FEA results were represented by curves. These curves included the results for the fiber diameters between 40 and 80 m, and corresponding thicknesses were = 0.025 <sup>40</sup> = 1, …, = 0.025 <sup>80</sup> = 2 (**Table 1**). The overall stress-strain distributions developed in the composite were affected by the thicknesses of the interface region, and these were proportional to the fiber diameter. As it has shown by **Figure 23**, the largest interface stresses were developed in the models with the diameters = 40 m and = 55 m. As the interface grew in diameter, the interface stresses became diminished from 6.9 × 10−7 to 5.5 × 10−7 N/m2. **Figure 23** presents the suddenly changed stress intensity values at the tip points of the fiber. Shear stress distributions and along the main axis ′ (**Figure 2**) were shown with curves. In **Figures 24** and **25**, shear stresses were found to be changed suddenly around the corner sections of the fiber end portion. The largest shearing stresses in fiberreinforced microcomposites have been developed with the largest diameters. In summary,

**Figure 21.** Stress intensity on the interface section ( = 40 m).

104 Perusal of the Finite Element Method

**Figure 22.** Stress intensity on the interface section ( = 80 m).

= 0.025 <sup>40</sup> = 1, …,

nesses were

For nine different interface geometries, FEA results were represented by curves. These curves included the results for the fiber diameters between 40 and 80 m, and corresponding thick-

butions developed in the composite were affected by the thicknesses of the interface region, and these were proportional to the fiber diameter. As it has shown by **Figure 23**, the largest interface stresses were developed in the models with the diameters = 40 m and = 55 m. As the interface grew in diameter, the interface stresses became diminished from 6.9 × 10−7 to 5.5 × 10−7 N/m2. **Figure 23** presents the suddenly changed stress intensity values at the tip points of the fiber. Shear stress distributions and along the main axis ′ (**Figure 2**) were shown with curves. In **Figures 24** and **25**, shear stresses were found to be changed suddenly around the corner sections of the fiber end portion. The largest shearing stresses in fiberreinforced microcomposites have been developed with the largest diameters. In summary,

= 0.025 <sup>80</sup> = 2 (**Table 1**). The overall stress-strain distri-

**Figure 23.** FEA results for stress distributions on nine different interface regions of the fiber-matrix (RVE) under tensile load.

**Figure 24.** Comparison of stress distributions along the main y-axis of the fiber composite in microscale for the fiber diameters = 40 m and = 80 m.

**Figure 25.** Comparison of stress distributions along the main y-axis of the fiber composite in microscale for the fiber diameters = 40 m and = 80 m.

$$\left| \tau\_{\rm xy} \right|\_{\rm max} = 5.8 \times 10^{-10} \text{ N/} \mu \text{m}^2 \quad d = 40 \text{ } \mu \text{m} \tag{12}$$

$$\left. \tau\_{xy} \right|\_{\text{max}} = 3.5 \times 10^{-10} \text{ N/} \mu \text{m}^2 \quad d = 80 \text{ } \mu \text{m} \tag{13}$$

$$\left| \tau\_{yz} \right|\_{\text{max}} = 3.0 \times 10^{-9} \text{ N/} \mu \text{m}^2 \quad d = 40 \text{ } \mu \text{m} \tag{14}$$

$$\left| \tau\_{\wp z} \right|\_{\text{max}} = 0.5 \times 10^{-9} \text{ N/} \mu \text{m}^2 \quad d = 80 \text{ } \mu \text{m} \tag{15}$$

The stress-strain − curves were plotted in FE models for diameters = 40 m, = 70 m, and = 80 m. These are presented in **Figure 26**. From the slopes of these curves, the elastic constant values parallel to the fiber and loading direction 1 were obtained and results are listed in **Table 2** for both fiber-matrix and fiber-interface-matrix models. These models have provided an opinion to the resultant data for 1 . FE model for the short fiber composite without interface case gave the value as 1 = 1.058 GPa, and for the short fiber composite with interface case, it was 1 = 1.446 GPa.

Micromechanical Analysis of Polymer Fiber Composites under Tensile Loading by Finite Element Method http://dx.doi.org/10.5772/65002 107

**Figure 26.** Stress-strain curves of FE-RVE models for fiber diameters = 40 m, = 75 m and = 80 m.

**Figure 27.** Comparison of stress distributions obtained from the interface segment of the RVE-FE model ( = 75 m): (a) fiber with interface, (b) interface, (c) fiber without interface.

**Figure 27** presented the comparison of distribution for (a) fiber without interface, (b) interface, and (c) fiber with interface regions. As it has shown in the curve for = 75 m, fiber with interface and also interface region stresses were near to each other in magnitudes 5.7 × 10−7 N/m2. In case, fiber without interface showed the distribution around 5.2 × 10−7 N/m2. The difference between them was equal to 0.5 × 10−7 N/m2.

#### **4. Discussion and conclusion**

**Figure 25.** Comparison of stress distributions along the main y-axis of the fiber composite in microscale for the

10 2

10 2

9 2

9 2

The stress-strain − curves were plotted in FE models for diameters = 40 m, = 70 m, and = 80 m. These are presented in **Figure 26**. From the slopes of these curves, the elastic constant values parallel to the fiber and loading direction 1 were obtained and results are listed in **Table 2** for both fiber-matrix and fiber-interface-matrix models. These models have provided an opinion to the resultant data for 1 . FE model for the short fiber

= 1.446 GPa.

*xy* 5.8 10 N/ m 40 m *<sup>d</sup>* - = ´ m =m (12)

*xy* 3.5 10 N/ m 80 m *<sup>d</sup>* - = ´ m =m (13)

*yz* 3.0 10 N/ m 40 m *<sup>d</sup>* - = ´ m =m (14)

*yz* 0.5 10 N/ m 80 m *<sup>d</sup>* - = ´ m =m (15)

= 1.058 GPa, and for the short fiber

fiber diameters = 40 m and = 80 m.

106 Perusal of the Finite Element Method

max t

max t

max t

max t

composite without interface case gave the value as 1

composite with interface case, it was 1

The effectiveness of the imposed interface volume and its geometric properties on the homogeneously distributed stress components in the polymer type fiber composites have been shown in this study by using finite element method.

The results obtained from this study were summarized as below::


symmetric configuration in the center of the main axis = 0.1 × 10−4 (**Figure 28**). The developingdisplacementdistributioninthecompositematrixsectionlocalizedattheend portions of the matrix portion of the cylinder.


**9.** When three layers of the composite compared, the fiber material carried higher stresses than the interface and the interface material carried higher stresses than the matrix. This was displayed as ≥ int ≥ .

The results obtained from this study were summarized as below::

reduction in stress values.

108 Perusal of the Finite Element Method

0.046 and 0.061 GPa mutually.

be measured after loading stage.

portions of the matrix portion of the cylinder.

confirmed by using FEA. With less fiber content (*Vf*

matrix interface was found to be larger.

matrix section.

**1.** The developing stress concentrations at the fiber-interface and interface-matrix common surfaces showed that the increase in fiber content was inversely proportional to the

**2.** Fiber volume fraction range from 4.98 to 11.87 was examined. Although the generated number of elements was different in the FE modeling of fiber composite structures, the obtained results of analysis were converged to the definite values. In the examined fiber volume fractions, the numerically calculated elastic constant values changed in small degree. The distribution of all values obtained from FEM was calculated in terms of polynomials, so that all resultant values were followed the polynomial properties.

**3.** In the FE modeling, the thickness of the interface region was changed in proportional to the fiber diameter and then obtained resultant elastic constant remained in a stable range. The total changes for fiber-interface-matrix and fiber-matrix models were obtained as

**4.** The application of the mechanical load per node = 0.1 × 10−4 <sup>N</sup> gave the elastic

**5.** The maximum developing stress distribution was seen on the composite fiber section while the maximum developing *y*-displacement was developed on the composite

**6.** The generated mesh system was not symmetric in composite cylinder transverse cross sections.Thedevelopingdisplacementdistributiononthecompositefibersectionshowed symmetric configuration in the center of the main axis = 0.1 × 10−4 (**Figure 28**). The developingdisplacementdistributioninthecompositematrixsectionlocalizedattheend

**7.** In the analysis, the fiber in the matrix material was moved along the *y*-axis under tensile loading. The fiber material was deformed by the help of stress transmission from matrix to fiber. The role of the difference in Young's modulus and Poisson's ratio of the fiber and

**8.** In this study, it was determined that the fiber content of fiber/interface/matrix composition was significant in the stress distribution in all PPE/PP composite models and this was

%), stress concentration at the fiber-

matrix material on deformation analysis should be considered (**Figure 29**).

constants 1.39 and 1.18 GPa for fiber and matrix sections of the = 80 m microcomposite by Ansys code FE analysis. While the applied load affected the entire combined structure, some portions of the stress were transferred to the other sections of the composite body. Therefore, the original individual material properties of the composite material could not


**Figure 28.** Comparison of stress distributions obtained from the cross section of the RVE FE model for mechanical load ( = 80 m): (a) fiber without interface volume 1, (b) fiber without interface volume 2.

**Figure 29.** Comparison of displacement distributions (deformed/undeformed shapes) of the fiber obtained from the cross section of the RVE FE model ( = 80 m): (a) fiber without interface volume 1, (b) fiber without interface volume 2 (force loading / = 0.1 N).

Three main conclusions obtained from this research were evaluated for the values of the matrix/ fiber diameters and their effect on the equivalent elastic constant.

The first conclusion was presented by the comparison curve that plotted by the "linear trendline curve" representing the average differences between two FE models. The relative percentage change of the equivalent elastic constant value 1 in the two main models was decreased as the diameter increased proportionally (**Table 2** and **Figure 30**). Adding the interface with the small diameter fiber into the cylinder strengthens the composite structure.

The second conclusion was described in the fiber-matrix models. The diameter of the fiber increased from 40 to 55 m in an order. As a result of this application, the equivalent elastic constant of the composite cylinder 1 was increased from 1.054 to 1.081 GPa in proportion. The decrease in elastic modulus value 1 was obtained from 1.063 to 1.047 GPa in a nonlinear manner for the diameter ranges from 60 to 80 μm.

**Figure 30.** Comparison 1 between fiber-matrix and fiber-interface-matrix FE models.

The third conclusion was explained for the fiber-interface-matrix model. As the diameter increased from 40 to 65 μm, the equivalent elastic constant 1 of the microcomposite decreased from 1.472 to 1.415 GPa linearly. However, the decrease in elastic modulus value 1 between the diameters 70 and 80 μm was obtained as 1.428 and 1.411 GPa in a nonlinear manner.

These results explained us that within the limits of 4.98–9.37% fiber volume fraction, proportionally increased interface volume caused the linearly decreasing elastic modulus value 1. Within the same limits, the fiber-matrix microcomposite had a linearly increasing value. The interface volume worked as a brake in the composite system. The interface volume reduced the stresses passing from matrix to fiber, and this region worked as a damper in a springdamper mechanical system. For the fiber volume, fraction ranges from 1022 to 11.87%, the interface volume caused a nonlinear decreasing in the equivalent elastic modulus values, and similarly, the fiber-matrix microcomposite had a decreasing value as nonlinear.

In other words, the physical meaning of the fiber matrix interaction behaviors was dependent on the interface volume. Interface volume caused a general linearity between the dimensions of the fiber, matrix, and the calculated equivalent elastic constant value (**Table 1**).

As a future work,, the effects of dimensional properties of composite models obtained in this study can be introduced into semi-empirical model (Halpin-Tsai) based on elasticity equations by curve fitting.

## **Acknowledgements**

Thanks for the great supports of FIGES and BAP project under grant no. 06/2011-57. This study was performed in Department of Mechanical Engineering, Gazi University, Ankara, Turkey.

## **Nomenclature**

## **Symbols**

Three main conclusions obtained from this research were evaluated for the values of the matrix/

The first conclusion was presented by the comparison curve that plotted by the "linear trendline curve" representing the average differences between two FE models. The relative percentage change of the equivalent elastic constant value 1 in the two main models was decreased as the diameter increased proportionally (**Table 2** and **Figure 30**). Adding the interface with the small diameter fiber into the cylinder strengthens the composite structure. The second conclusion was described in the fiber-matrix models. The diameter of the fiber increased from 40 to 55 m in an order. As a result of this application, the equivalent elastic constant of the composite cylinder 1 was increased from 1.054 to 1.081 GPa in proportion. The decrease in elastic modulus value 1 was obtained from 1.063 to 1.047 GPa in a nonlinear

fiber diameters and their effect on the equivalent elastic constant.

**Figure 30.** Comparison 1 between fiber-matrix and fiber-interface-matrix FE models.

The third conclusion was explained for the fiber-interface-matrix model. As the diameter increased from 40 to 65 μm, the equivalent elastic constant 1 of the microcomposite decreased from 1.472 to 1.415 GPa linearly. However, the decrease in elastic modulus value 1 between the diameters 70 and 80 μm was obtained as 1.428 and 1.411 GPa in a nonlinear manner.

These results explained us that within the limits of 4.98–9.37% fiber volume fraction, proportionally increased interface volume caused the linearly decreasing elastic modulus value 1. Within the same limits, the fiber-matrix microcomposite had a linearly increasing value. The interface volume worked as a brake in the composite system. The interface volume reduced the stresses passing from matrix to fiber, and this region worked as a damper in a springdamper mechanical system. For the fiber volume, fraction ranges from 1022 to 11.87%, the interface volume caused a nonlinear decreasing in the equivalent elastic modulus values, and

similarly, the fiber-matrix microcomposite had a decreasing value as nonlinear.

manner for the diameter ranges from 60 to 80 μm.

%

110 Perusal of the Finite Element Method


,, displacements parallel to the , , directions in an order m , fiber and matrix volume fractions

## **Greek symbols**


## **Author details**

#### Ezgi Günay

Address all correspondence to: ezgigunay@gazi.edu.tr

Department of Mechanical Engineering, Gazi University, Maltepe, Ankara, Turkey

## **References**


[4] Termonia Y. Effect of strain rate on the mechanical properties of composites with a weak fibre/matrix interface. Journal of Materials Science. 1992;27:4878–4882.

,, displacements parallel to the , , directions in an order m

, fiber and matrix volume fractions

, Poisson's ratio of fiber and matrix

, -directional stress component N/m2

2 Von Mises stress component N/m2

,, displacements parallel to the , , directions in an order (rad).

Address all correspondence to: ezgigunay@gazi.edu.tr

of Materials Science. 1987;22:504–508.

Department of Mechanical Engineering, Gazi University, Maltepe, Ankara, Turkey

filled polymers. Journal of Materials Science. 1987;22:1733–1736.

reinforced materials. Journal of Materials Science. 1990;25:103–106.

[1] Termonia Y. Computer model for the elastic properties of short fibre and particulate

[2] Termonia Y. Theoretical study of the stress transfer in single fibre composites. Journal

[3] Termonia Y. Fibre coating as a means to compensate for poor adhesion in fiber-

12 major in-plane ratio

stress vector N/m2

reinforcing factor

strain vector

**Author details**

Ezgi Günay

**References**

**Greek symbols**

112 Perusal of the Finite Element Method


#### **Estimation of Shallow Water Flow Based on Kalman Filter FEM Estimation of Shallow Water Flow Based on Kalman Filter FEM**

Takahiko Kurahashi, Taichi Yoshiara and Yasuhide Kobayashi Takahiko Kurahashi, Taichi Yoshiara and Yasuhide Kobayashi

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64595

#### **Abstract**

In this chapter, we present numerical examples of an estimation of shallow water flow based on Kalman filter finite element method (Kalman filter FEM). Shallow water equations are adopted as the governing equations. The Galerkin method, using triangular elements, is employed to discretize the governing equation in space, and the selective lumping method is used to discretize time. We describe the influence on the numerical results of setting the observation points.

**Keywords:** Kalman filter FEM, shallow water equation, Galerkin method, selective lumping method

## **1. Introduction**

Conventionally, shallow water flow analysis based on the FEM has been carried out to investigate the marine environment, specifically coastal drift sand, storm surges, tsunamis, and so on [1, 2]. However, the computed results are rarely close to the observed values if the appropriate governing equation is not employed, the appropriate discretization technique is not applied to the governing equation, or the boundary conditions are not defined appropri‐ ately. This has prompted a great deal of inverse analysis in recent years, aiming to obtain flow fields that more closely approach the observed values. Boundary control analysis is one of these inverse analytic techniques, in which an unknown boundary condition is numerically determined by iterative computation such that the computed water elevation homes in on the target value or the observed value [3, 4]. Deterministic methods such as the adjoint variable or

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

sensitivity equation methods are commonly employed. On the other hand, the stochastic approach has also been adopted, in which estimation of the flow field is employed using a Kalman filter. Removal of the noise data from the observed values and the flow field estimation are also performed. Conventionally, flow fields are estimated using Kalman filter FEM, and this method is applied to the practical coastal model [5, 6]. However, the effect of boundary conditions on the estimated flow field has not been investigated. This prompted us to research and describe in detail in this chapter the formulation of Kalman filter FEM and numerical experiments for flow estimation when changing the positions of observation points [7].

## **2. Concept of state estimation**

In the Kalman filter, system and observation equations are employed. Eqs. (1) and (2) are sample equations:

$$
\phi^{n \ast 1} = a\phi^n + q^n \tag{1}
$$

Here, and *p* indicate the estimation value for the state variable and estimation error. From **Figure 1**, it is found that the estimation value is expressed as a summation of the true value and the estimation error, and the observation value is expressed by the summation of the true value and the observation error. Based on this concept, our formulation of the Kalman filter is

The shallow water equations, shown below, are employed as the governing equations.

*<sup>i</sup>* , 0 *<sup>i</sup> u g* & + = h

h

governing equation is written as Eq. (5) and is represented by Eq. (6),

þ D

f

1


[ ] [0] [ ] { } [ ] [0] [0] { } { } [0] [ ] [0] [0] [ ] [ ] { } { } [0] [0] [ ] { } [] [] []

*<sup>M</sup> g tS u u <sup>M</sup> v M M g tS v*

*n x n n n*

%

*n n*

ï ï <sup>î</sup> ê ú î þ ë û -D - ê ú ë û

<sup>1</sup> ˆ ˆ { } [ ]{ } *n n*

, 0 *i i*

gravitational acceleration, and water depth, respectively. The Galerkin method and the selective lumping method are applied to discretize the governing equation. The discretized

, *η*, *g*, and *h* indicate flow velocity component for the *x* and *y* directions, water elevation,

*x y*

é ù

%

], [*M*], and denote the time increment, the mean water depth, the matrix for

*<sup>M</sup> h tS h tS M*

ê ú - D é ù ì ü ê ú ï ï <sup>=</sup> ê ú - D í ý ê ú ï ï

 f

the pressure, the consistent mass matrix, and diagonal mass matrix, respectively. is represented by Eq. (7) by using the lumping parameter *e*(0 ≤ *e* ≤ 1). indicates the estimation

The system equation in the Kalman filter is obtained by adding the vector represented by the multiplication of the driving matrix [Γ] and the system noise vector {g} and is shown as Eq.

(3)

Estimation of Shallow Water Flow Based on Kalman Filter FEM

http://dx.doi.org/10.5772/64595

117

&+ = *hu* (4)

*y*

%

*A* <sup>+</sup> = (6)

[ ] (1 )[ ] [ ] *M e M eM* % =- + (7)

 h (5)

shown below.

where *ui*

1 1 1

+ + +

ì ü ï ï í ý

h

where Δ, ℎ, [*Si*

value.

**3. Discretization of the governing equation**

$$z^{n+1} = \phi^{n+1} + r^{n+1} \tag{2}$$

where *ϕ*, *z*, *a*, *q*, and *r* indicate the true value of the state variable, the observation value, the constant coefficient, the system, and the observation noise, respectively. *n* denotes the number of time steps. **Figure 1** shows relationship between the observation and the estimation values.

**Figure 1.** Diagram of the relationship between the observation and the estimation values.

Here, and *p* indicate the estimation value for the state variable and estimation error. From **Figure 1**, it is found that the estimation value is expressed as a summation of the true value and the estimation error, and the observation value is expressed by the summation of the true value and the observation error. Based on this concept, our formulation of the Kalman filter is shown below.

## **3. Discretization of the governing equation**

sensitivity equation methods are commonly employed. On the other hand, the stochastic approach has also been adopted, in which estimation of the flow field is employed using a Kalman filter. Removal of the noise data from the observed values and the flow field estimation are also performed. Conventionally, flow fields are estimated using Kalman filter FEM, and this method is applied to the practical coastal model [5, 6]. However, the effect of boundary conditions on the estimated flow field has not been investigated. This prompted us to research and describe in detail in this chapter the formulation of Kalman filter FEM and numerical experiments for flow estimation when changing the positions of observation points [7].

In the Kalman filter, system and observation equations are employed. Eqs. (1) and (2) are

*a q* <sup>+</sup> = + (1)

+ ++ = + *n nn z r* (2)

*n nn* 1

1 11 f

where *ϕ*, *z*, *a*, *q*, and *r* indicate the true value of the state variable, the observation value, the constant coefficient, the system, and the observation noise, respectively. *n* denotes the number of time steps. **Figure 1** shows relationship between the observation and the estimation values.

 f

f

**Figure 1.** Diagram of the relationship between the observation and the estimation values.

**2. Concept of state estimation**

sample equations:

116 Perusal of the Finite Element Method

The shallow water equations, shown below, are employed as the governing equations.

$$
\dot{\mu}\_l + \mathbf{g}\eta\_\gamma = \mathbf{0} \tag{3}
$$

$$
\dot{\eta} + h u\_{i,i} = 0 \tag{4}
$$

where *ui* , *η*, *g*, and *h* indicate flow velocity component for the *x* and *y* directions, water elevation, gravitational acceleration, and water depth, respectively. The Galerkin method and the selective lumping method are applied to discretize the governing equation. The discretized governing equation is written as Eq. (5) and is represented by Eq. (6),

$$
\begin{Bmatrix}
\{\boldsymbol{\upmu}^{n+1}\} \\
\{\boldsymbol{\upnu}^{n+1}\} \\
\{\boldsymbol{\upmu}^{n+1}\} \\
\end{Bmatrix} = \begin{bmatrix}
[\overline{M}] & [0] & [0] \\
[0] & [\overline{M}] & [0] \\
[0] & [0] & [\overline{M}] \\
\end{bmatrix}^{-1} \begin{bmatrix}
[\boldsymbol{\uptilde{M}}] & [0] & -\mathtt{g}\Delta\mathbb{I}[\boldsymbol{S}\_{x}] \\
[0] & [\boldsymbol{M}] & -\mathtt{g}\Delta\mathbb{I}[\boldsymbol{S}\_{y}] \\
\end{bmatrix} \begin{Bmatrix}
\{\boldsymbol{u}^{n}\} \\
\{\boldsymbol{v}^{n}\} \\
\{\boldsymbol{\upeta}^{n}\} \\
\end{Bmatrix} \tag{5}
$$

$$\{\hat{\phi}^{\mu \leftrightarrow 1}\} = [A] \{\hat{\phi}^{\mu}\} \tag{6}$$

where Δ, ℎ, [*Si* ], [*M*], and denote the time increment, the mean water depth, the matrix for the pressure, the consistent mass matrix, and diagonal mass matrix, respectively. is represented by Eq. (7) by using the lumping parameter *e*(0 ≤ *e* ≤ 1). indicates the estimation value.

$$\mathbb{E}\left[\tilde{M}\right] = (\mathbb{I} - e)\{\overline{M}\} + e\{M\} \tag{7}$$

The system equation in the Kalman filter is obtained by adding the vector represented by the multiplication of the driving matrix [Γ] and the system noise vector {g} and is shown as Eq. (8). Here, the vector {*ø*} indicates the vector of true value. The observation equation shown in Eq. (9) is also introduced. The vectors {*z*} and {*r*} and the matrix [*H*] denote the observation value and the observation noise vectors and the observation matrix.

$$\{\phi^{n+1}\} = \left[A\right]\{\phi^{n}\} + \left[\Gamma\right]\{q^{n}\} \tag{8}$$

$$\{z^{n^{n+1}}\} = [H]\{\phi^{n^{n+1}}\} + \{r^{n^{n+1}}\} \tag{9}$$

## **4. Derivation of computational equations in the Kalman filter FEM**

The process of modification of the estimation vector is referred to as "assimilation," and the estimation vector before and after assimilation is expressed by "(−)" and "(+)" marks, respectively. First of all, it is assumed that the estimated value after assimilation is expressed by the observation value and the estimated value before assimilation, and that the estimated value after assimilation, i.e., the optimal estimated vector, is expressed by Eq. (10).

$$\{\hat{\phi}\_{(\*)}^{n+1}\} = \left[K\_1^{n+1}\right] \{z^{n+1}\} + \left[K\_2^{n+1}\right] \{\hat{\phi}\_{(-)}^{n+1}\} \tag{10}$$

where 1 + 1 and 2 + 1 indicate arbitrary matrices. In addition, the optimal estimated vector (+) + 1 is expressed by true value + 1 and estimation error (+) + 1 as shown in Eq. (11).

$$\{\hat{\phi}^{n^{n+1}}\_{(\*)}\} = \{\phi^{n^{n+1}}\} + \{p^{n+1}\_{(\*)}\} \tag{11}$$

Consequently, Eq. (12) is obtained from Eqs. (9), (10), and (11).

$$\begin{aligned} \{\phi^{n+1}\} + \{p^{n+1}\_{(+)}\} &= \\ \{K^{n+1}\_1\} (\{H\} \{\phi^{n+1}\} + \{r^{n+1}\}) \\ + \{K^{n+1}\_2\} (\{\phi^{n+1}\} + \{p^{n+1}\_{(-)}\}) \end{aligned} \tag{12}$$

Here, we consider the expected value for Eq. (12) in time. In addition, the expected value for vectors + 1 , (+) + 1 and ( <sup>−</sup> ) + 1 in time is assumed to be zero, and Eq. (12) is represented as Eq. (13).

$$<\langle [K\_2^{n+1}] + [K\_1^{n+1}][H] - [I] \rangle \langle \langle \phi^{n+1} \rangle \rangle = \langle 0 \rangle \tag{13}$$

where [*I*] denotes the unit matrix, and + 1 indicates the expected value of + 1 in time. Because of + 1 <sup>≠</sup> <sup>0</sup> , the following relation equation is obtained (see Eq. (14)).

$$[K\_2^{s+l}] + [K\_1^{s+l}][H] - [I] = [0] \tag{14}$$

Substituting Eq. (14) for Eq. (10), Eq. (15) is obtained.

(8). Here, the vector {*ø*} indicates the vector of true value. The observation equation shown in Eq. (9) is also introduced. The vectors {*z*} and {*r*} and the matrix [*H*] denote the observation

*A q* <sup>+</sup> = +G (8)

+ ++ = + (9)

<sup>1</sup> { } [ ]{ } [ ]{ } *n nn*

1 11 { } [ ]{ } { } *n nn zH r* f

**4. Derivation of computational equations in the Kalman filter FEM**

value after assimilation, i.e., the optimal estimated vector, is expressed by Eq. (10).

1 11 1 1 () 1 2 () { } [ ]{ } [ <sup>ˆ</sup> ]{ <sup>ˆ</sup> } *n nn n n*

+ 1 is expressed by true value + 1 and estimation error (+)

\$ <sup>1</sup> 1 1 ( ) ( ) { }{ }{ } *<sup>n</sup> n n*

> 11 1 2 ( )


+

Here, we consider the expected value for Eq. (12) in time. In addition, the expected value for

[ ]([ ]{ } { }) [ ]({ } { })

f

*n nn nn n*

 f*<sup>p</sup>* <sup>+</sup> + +

1 1 ( ) 1 11

+ + + + ++ ++ +

+ +

*p KH r K p*

+ =

f

{ }{ }

*n n*

f

1

f

+ 1 and ( <sup>−</sup> )

Consequently, Eq. (12) is obtained from Eqs. (9), (10), and (11).

*Kz K* + ++ ++

 f

+ - = + (10)

<sup>+</sup> = + <sup>+</sup> (11)

+ 1 in time is assumed to be zero, and Eq. (12) is represented

+ 1 as shown in

(12)

+ 1 indicate arbitrary matrices. In addition, the optimal estimated

The process of modification of the estimation vector is referred to as "assimilation," and the estimation vector before and after assimilation is expressed by "(−)" and "(+)" marks, respectively. First of all, it is assumed that the estimated value after assimilation is expressed by the observation value and the estimated value before assimilation, and that the estimated

 f

value and the observation noise vectors and the observation matrix.

f

f

where 1

Eq. (11).

vectors

as Eq. (13).

+ 1 , (+)

vector (+)

+ 1 and 2

118 Perusal of the Finite Element Method

$$\{\hat{\phi}\_{(\!)}^{n+1}\} = \{\hat{\phi}\_{(\!\!)}^{n+1}\} + [K\_1^{n+1}](\{z^{n+1}\} - [H]\{\hat{\phi}\_{(\!\!)}^{n+1}\})\tag{15}$$

where 1 + 1 is referred to as the Kalman gain matrix.

Next, let us consider the Kalman gain matrix 1 + 1 so as to minimize the trace norm of the error covariance matrix after assimilation. The estimation vector is represented by Eq. (16).

$$\{p\_{(\*)}^{n+1}\} = \{\hat{\phi}\_{(\*)}^{n+1}\} - \{\phi^{n+1}\} \tag{16}$$

Eq. (17) is obtained by substituting Eq. (15) with Eq. (16), and substituting Eq. (9) with the obtained equation.

$$\begin{aligned} \{p\_{(\*)}^{n+1}\} &= \\ \{ [I] - [K\_1^{n+1}][H] \} \{p\_{(-)}^{n+1}\} + [K\_1^{n+1}] \{r^{n+1}\} \end{aligned} \tag{17}$$

Here, the error covariance matrix after assimilation (+) + 1 , i.e., the predicted error covariance matrix, is expressed by the expected value of the error covariance matrix as shown in Eq. (18).

$$\mathbb{E}\left[P\_{(\*)}^{u\*+1}\right] = \langle \{p\_{(\*)}^{u\*+1}\} \{p\_{(\*)}^{u\*+1}\}^T \rangle \tag{18}$$

Also, in calculating Eq. (18), it is assumed that the expected value of the covariance matrix of the observation error and the estimation error before assimilation is a zero vector (see Eq. (19)).

$$<\langle \{ r^{n+1} \} \{ p\_{(-)}^{n+1} \}^{\top} \rangle = [0] \tag{19}$$

Consequently, the predicted error covariance matrix can be expressed, such as in Eq. (20),

$$\begin{array}{l} [P\_{(\*)}^{n+1}] = \\ \mathbb{E}\left( [I] - [K\_1^{n+1}][H] \right) [P\_{(-)}^{n+1}] ([I] - [K\_1^{n+1}][H])^T \\ + [K\_1^{n+1}] [R^{n+1}] [K\_1^{n+1}]^T \end{array} \tag{20}$$

where + 1 is the observation error covariance matrix, and is represented by + 1 + 1 . Here, let us consider the Kalman gain matrix 1 + 1 so as to minimize the trace norm of the predicted error covariance matrix such as ∂ <sup>∂</sup> 1 + 1 tr (+) + 1 = 0. Conse‐ quently, we can obtain the equation for calculating the Kalman gain matrix, as shown in Eq. (21).

$$\begin{aligned} [K\_1^{\ast \ast 1}] &= \\ [P\_{(-)}^{\ast \ast 1}][H]^T ([H][P\_{(-)}^{\ast \ast 1}][H]^T + [R]^{\ast \ast 1})^{-1} \end{aligned} \tag{21}$$

In addition, Eq. (20) is represented by Eq. (22) taking Eq. (21) into account.

$$[P\_{(\*)}^{\ast \ast 1}] = [P\_{(-)}^{\ast \ast 1}] - [K\_1^{\ast \ast 1}][H][P\_{(-)}^{\ast \ast 1}] \tag{22}$$

Next, let us consider the computation of the estimated error covariance matrix before assimi‐ lation ( <sup>−</sup> ) + 1 , i.e., the estimated error covariance matrix. First of all, Eq. (6) is represented as Eq. (23) based on the consideration expressed in Eq. (11).

$$\{\phi^{n+1}\} + \{p\_{(-)}^{n+1}\} = [A](\{\phi^{n}\} + \{p\_{(-)}^{n}\})\tag{23}$$

Subtracting Eq. (8) from Eq. (23) gives us Eq. (24).

$$\{p\_{(-)}^{\ast + 1}\} = [A]\{p\_{(+)}^{\ast}\} - [\Gamma]\{q^{\ast}\} \tag{24}$$

Here, the estimated error covariance matrix ( <sup>−</sup> ) + 1 is also represented by the expected value of the error covariance matrix, as shown in Eq. (25).

$$\{P\_{(-)}^{n+1}\} = \langle \{p\_{(-)}^{n+1}\} \{p\_{(-)}^{n+1}\}^T \rangle \tag{25}$$

Considering the relation equations (+) <sup>=</sup> <sup>0</sup> and (+) <sup>=</sup> <sup>0</sup> in the calcula‐ tion in Eq. (25) allows Eq. (26) to be derived.

$$[I\_{(-)}^{n+1}] = [A][I\_{(+)}^{n+1}][A]^{\mathbb{T}} + [\Gamma][Q^\*][\Gamma]^{\mathbb{T}} \tag{26}$$

where is the system error covariance matrix and is represented by + 1 + 1 .

## **5. Computational algorithm based on the Kalman filter FEM**

If the matrices [*A*], [*Q*], and [*R*] are steady coefficient matrices, the computational algo‐ rithm can be divided into two parts: the computations of the Kalman gain matrix and the estimated value in time. The computational algorithm for the Kalman filter FEM is shown below.

#### **5.1. Computation of Kalman gain matrix**

1 1 ( ) { }{ } [0] *n nT r p* + +

1 ( )

=

+ +

[ ]

*P*

120 Perusal of the Finite Element Method

+

 + 1

(21).

*n*

11 1 1 1

*n n nT*

++ +

[ ][ ][ ]

trace norm of the predicted error covariance matrix such as ∂

( ) ( )

1 1

1

In addition, Eq. (20) is represented by Eq. (22) taking Eq. (21) into account.


1 1

=

+

[ ]

Eq. (23) based on the consideration expressed in Eq. (11).

f

Subtracting Eq. (8) from Eq. (23) gives us Eq. (24).

of the error covariance matrix, as shown in Eq. (25).

*K*

*n*

*KRK*

Consequently, the predicted error covariance matrix can be expressed, such as in Eq. (20),

11 1 1 ( ) 1

++ + -

*n n nT*

where + 1 is the observation error covariance matrix, and is represented by

quently, we can obtain the equation for calculating the Kalman gain matrix, as shown in Eq.

1 1 1 1

111 1 () () 1 ( ) [ ] [ ] [ ][ ][ ] *nnn n P P K HP* +++ +

( ) ( ) { } { } [ ]({ } { }) *n n nn*

( ) ( ) { } [ ]{ } [ ]{ } *n nn p Ap q* <sup>+</sup>

Next, let us consider the computation of the estimated error covariance matrix before assimi‐ lation ( <sup>−</sup> ) + 1 , i.e., the estimated error covariance matrix. First of all, Eq. (6) is represented as

> f

Here, the estimated error covariance matrix ( <sup>−</sup> ) + 1 is also represented by the expected value

+ + + -

[ ][ ] ([ ][ ][ ] [ ] )

*P H HP H R*

*n T n Tn*

([ ] [ ][ ])[ ]([ ] [ ][ ])

*I K HP I K H*


+ 1 . Here, let us consider the Kalman gain matrix 1


<sup>∂</sup> 1

+ - = - - (22)

*pA p* + + += + - - (23)


(20)

+ 1 so as to minimize the

+ 1 = 0. Conse‐

+ 1 tr (+)

<sup>+</sup> (21)


#### **5.2. Computation of estimated value in time**


## **6. Numerical example 1**

**Figures 2** and **3**, respectively, show the computational model and the position of observation points in the computation using the Kalman filter FEM. The estimation of shallow water flow was carried out based on the Kalman filter FEM. **Table 1** shows the computational conditions. In this study, we simulated the observed data from the shallow water flow analysis based on the FEM. The boundary condition for the water elevation is given by *η*(*t*) = sin (*t*) on the left‐ hand side of the open channel, and the artificial observed data is obtained by summation of the numerical result and observation noise, i.e., the white noise. In the computation of the Kalman filter FEM shown in this section, the inflow boundary condition is ignored, and the artificial observation data flow velocities *ux*, *uy*, and water elevation *η*, i.e., the computational result by the FEM, are given at the observation points.

**Figure 2.** Computational model.

**Figure 3.** Numerical examples.


**Table 1.** Computational conditions.

**6. Numerical example 1**

122 Perusal of the Finite Element Method

**Figure 2.** Computational model.

**Figure 3.** Numerical examples.

result by the FEM, are given at the observation points.

**Figures 2** and **3**, respectively, show the computational model and the position of observation points in the computation using the Kalman filter FEM. The estimation of shallow water flow was carried out based on the Kalman filter FEM. **Table 1** shows the computational conditions. In this study, we simulated the observed data from the shallow water flow analysis based on the FEM. The boundary condition for the water elevation is given by *η*(*t*) = sin (*t*) on the left‐ hand side of the open channel, and the artificial observed data is obtained by summation of the numerical result and observation noise, i.e., the white noise. In the computation of the Kalman filter FEM shown in this section, the inflow boundary condition is ignored, and the artificial observation data flow velocities *ux*, *uy*, and water elevation *η*, i.e., the computational

> The computational results are shown below. **Figures 4** and **5** show the convergence criterion expressed by the Frobenius norm. The equation for the Frobenius norm is shown in Step 5 of the flowchart in Section 4. It is seen that the convergence criterion monotonically decreases and converges. **Figures 6** and **7** show the time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case A and Case B, respectively. It is found that the observation noise can be removed in both cases. **Figures 8** and **9** show the distribution of water elevation on line *y* = 0.4 m in Case A and Case B, respectively. From **Figures 8** and **9**, it can be seen that if the observation point is not set at the upstream side in the channel, the distribution of the water elevation cannot be appropriately obtained. Conversely, from the result of Case A, it can be said that the distribu‐ tion of water elevation can be correctly obtained if the position of the observation point is appropriately given in the computational model. **Figures 10** and **11** show the distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s. It can be seen that water elevation on the upstream side in Case B is not appropriately obtained in comparison with the result of Case A. Therefore, from this numerical example, it can be said that it is necessary to set the observation points uniformly in the computational domain.

**Figure 4.** Convergence criterion in Case A.

**Figure 5.** Convergence criterion in Case B.

**Figure 6.** Time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case A.

**Figure 7.** Time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case B.

**Figure 8.** Distribution of water elevation on line *y* = 0.4 m in Case A.

**Figure 5.** Convergence criterion in Case B.

124 Perusal of the Finite Element Method

**Figure 6.** Time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case A.

**Figure 7.** Time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case B.

**Figure 9.** Distribution of water elevation on line *y* = 0.4 m in Case B.

**Figure 10.** Distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s in Case A.

**Figure 11.** Distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s in Case B.

## **7. Numerical example 2**

In this section, the numerical experiment is carried out with the inflow boundary conditions given in Case B, shown in the previous section. An image diagram of the numerical example is shown in **Figure 12**. The inflow boundary condition is given by *η*(*t*) = sin (*t*). The computa‐ tional conditions in this section are the same as in **Table 1**. As the computational result, **Figure 13** shows the convergence criterion expressed by the Frobenius norm. It can be seen that the convergence criterion decreases and converges monotonically. **Figure 14** shows the time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case C. It is found that the observation noise can be removed. **Figure 15** shows the distribution of water elevation on line *y* = 0.4 m in Case C. From **Figure 15**, it is found that the distribution of the water elevation can be appro‐ priately obtained in comparison with the result shown in **Figure 9**. In addition, **Figure 16** shows the distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s. It is seen that water elevation on the upstream side can be more accurately obtained than with the result for Case B. Therefore, from this numerical experiment, it is necessary to give the boundary conditions if the computation of the flow field estimation is carried out based on the Kalman filter FEM, and the observation point cannot be uniformly given.

**Figure 12.** Numerical example.

**Figure 13.** Convergence criterion in Case C.

**Figure 11.** Distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s in Case B.

and the observation point cannot be uniformly given.

In this section, the numerical experiment is carried out with the inflow boundary conditions given in Case B, shown in the previous section. An image diagram of the numerical example is shown in **Figure 12**. The inflow boundary condition is given by *η*(*t*) = sin (*t*). The computa‐ tional conditions in this section are the same as in **Table 1**. As the computational result, **Figure 13** shows the convergence criterion expressed by the Frobenius norm. It can be seen that the convergence criterion decreases and converges monotonically. **Figure 14** shows the time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case C. It is found that the observation noise can be removed. **Figure 15** shows the distribution of water elevation on line *y* = 0.4 m in Case C. From **Figure 15**, it is found that the distribution of the water elevation can be appro‐ priately obtained in comparison with the result shown in **Figure 9**. In addition, **Figure 16** shows the distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s. It is seen that water elevation on the upstream side can be more accurately obtained than with the result for Case B. Therefore, from this numerical experiment, it is necessary to give the boundary conditions if the computation of the flow field estimation is carried out based on the Kalman filter FEM,

**7. Numerical example 2**

126 Perusal of the Finite Element Method

**Figure 12.** Numerical example.

**Figure 14.** Time history of water elevation at (*x*, *y*) = (5.0 m, 0.4 m) in Case C.

**Figure 15.** Distribution of water elevation on line *y* = 0.4 m in Case C.

**Figure 16.** Distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s in Case C.

## **8. Comparison of estimation accuracy in Cases A, B, and C**

Based on the estimation result at (*x*, *y*) = (5.0 m, 0.4 m) shown in **Figures 6**, **7**, and **14**, the numerical accuracy is investigated by comparison of data between the true and the estimated values in each case. The *L*2 norm with respect to the difference between the true and the estimated values in each case is shown in **Table 2**.


**Table 2.** Comparison of *L*<sup>2</sup> norm.

A comparison of the results in Cases A and B shows that it is necessary to set the observation points uniformly in the computational domain to obtain a highly accurate estimation result. In addition, when comparing the results in Case B and Case C, it is found that the estimation accuracy increases if the inflow boundary condition is taken into account.

## **9. Conclusions**

In this study, flow field estimation analysis in an open channel was carried out based on the Kalman filter FEM. The shallow water equations were employed as the governing equations, and the Galerkin and the selective lumping methods were used to discretize the governing equation in space and time, respectively. In the numerical experiments, the observed value is created by adding white noise to the simulation result in the open channel model. The following conclusions were obtained from this study.


## **Nomenclature**

**Figure 16.** Distribution of flow velocity *u* and *v* and water elevation *η* at *T* = 2.0 s in Case C.

**8. Comparison of estimation accuracy in Cases A, B, and C**

estimated values in each case is shown in **Table 2**.

norm.

**Table 2.** Comparison of *L*<sup>2</sup>

128 Perusal of the Finite Element Method

Based on the estimation result at (*x*, *y*) = (5.0 m, 0.4 m) shown in **Figures 6**, **7**, and **14**, the numerical accuracy is investigated by comparison of data between the true and the estimated values in each case. The *L*2 norm with respect to the difference between the true and the

 **norm**

**Case Number of observation points Inflow boundary condition** *L***<sup>2</sup>**

Case A 10 Not given 1.899 Case B 5 (only downstream side) Not given 4.161 Case C 5 (only downstream side) Given 2.463

A comparison of the results in Cases A and B shows that it is necessary to set the observation points uniformly in the computational domain to obtain a highly accurate estimation result. In addition, when comparing the results in Case B and Case C, it is found that the estimation

accuracy increases if the inflow boundary condition is taken into account.



## **Acknowledgements**

This work was supported by Grants‐in‐Aid for Scientific Research (C) Grant Number 15K05786, and the contents of this work, i.e., formulation by FEM and the theory of the Kalman filter FEM, is based on seminars by Emeritus Professor Mutsuto Kawahara at Chuo University's Department of Civil Engineering. The computations were mainly carried out using the Fujitsu PRIMERGY CX400 computer facilities at Kyushu University's Research Institute for Informa‐ tion Technology. We wish to thank Emeritus Professor Mutsuto Kawahara and the staff at Kyushu University's Research Institute for Information Technology.

## **Author details**

Takahiko Kurahashi\* , Taichi Yoshiara and Yasuhide Kobayashi

\*Address all correspondence to: kurahashi@mech.nagaokaut.ac.jp

Department of Mechanical Engineering, Nagaoka University of Technology, Nagaoka, Japan

## **References**


[4] Kurahashi, T. and Kawahara, M., Flood control of urban stormwater conduit systems using the finite element methods. International Journal for Numerical Methods in Flu‐ ids, Vol. 52 (2005), pp. 1029–1057.

{*p*}: estimation error vector [*K*1]: Kalman gain matrix

130 Perusal of the Finite Element Method

**Acknowledgements**

**Author details**

Takahiko Kurahashi\*

**References**

[*P*]: estimated error covariance matrix [*Q*]: system error covariance matrix [*R*]: observation error covariance matrix

This work was supported by Grants‐in‐Aid for Scientific Research (C) Grant Number 15K05786, and the contents of this work, i.e., formulation by FEM and the theory of the Kalman filter FEM, is based on seminars by Emeritus Professor Mutsuto Kawahara at Chuo University's Department of Civil Engineering. The computations were mainly carried out using the Fujitsu PRIMERGY CX400 computer facilities at Kyushu University's Research Institute for Informa‐ tion Technology. We wish to thank Emeritus Professor Mutsuto Kawahara and the staff at

Kyushu University's Research Institute for Information Technology.

\*Address all correspondence to: kurahashi@mech.nagaokaut.ac.jp

matics with Applications, Vol. 16 (1988), pp. 139–152.

Scientific Computing, Vol. 46, 3 (2011), pp. 329–358.

Vol. 19 (1994), pp. 463–477.

, Taichi Yoshiara and Yasuhide Kobayashi

Department of Mechanical Engineering, Nagaoka University of Technology, Nagaoka, Japan

[1] Kawahara, M., Kodama, T. and Kinoshita, M., Finite element method for tsunami wave propagation analysis considering the open boundary condition. Computers & Mathe‐

[2] Tanaka, S., Bunya, S., Westerink, J.J., Dawson, C.N. and Luettich, R.A., Scalability of an unstructured grid continuous Galerkin‐based hurricane storm surge model. Journal of

[3] Kawahara, M. and Shimada, Y., Gradient method of optimal control applied to the operation of a dam water gate. International Journal for Numerical Methods in Fluids,


**Chapter 6**

**Provisional chapter**

**Improved Reduced Order Mechanical Model and Finite**

**Element Analysis of Three-Dimensional Deformations of**

**Improved Reduced Order Mechanical Model and Finite**

**Element Analysis of Three-Dimensional Deformations**

In this chapter, we analyse non-uniform bending of single-layer cell tissues—epithelia, surrounding organs throughout the body. Dimensionally reduced model is suggested, which is equivalent to membranes with bending stiffness: the total elastic energy of the tissue is a combination of stretching and bending energies. The energy, suggested in this chapter, is a piecewise function, the branches of which correspond to a specific

Under specific assumptions on a single cell, such as all cells are identical, elastic, homogeneous, have the shape of a hexagonal prism, are linked to each other by pure elastic junctions, and so on, tissues of different shapes are analysed using finite element discretization. The deformation of initially flat, cylindrical and spherical tissues is visualized to reproduce similar deformations in real epithelial tissues observed during

**Keywords:** morphogenesis, embryonic tissue, adherens junctions, relaxation, quasiconvex energy densities, isometric deformations, short maps, tissue mechanics, Γ-con-

Morphogenesis, the evolutionary process of shape and structure development of organisms or their parts, is driven by certain types of cell shape changes or, in other words, deformations.

and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**of Epithelial Tissues**

**Epithelial Tissues**

Sergey K. Ohanyan

**Abstract**

experiments.

**1. Introduction**

Sergey K. Ohanyan

http://dx.doi.org/10.5772/65027

Ara S. Avetisyan, Asatur Zh. Khurshudyan and

Ara S. Avetisyan, Asatur Zh. Khurshudyan and

Additional information is available at the end of the chapter

deformation regime: compression, pure bending and stretch.

vergence, lower semi-continuity, hexagonal elements

Additional information is available at the end of the chapter

#### **Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations of Epithelial Tissues Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations of Epithelial Tissues**

Ara S. Avetisyan, Asatur Zh. Khurshudyan and Sergey K. Ohanyan Ara S. Avetisyan, Asatur Zh. Khurshudyan and Sergey K. Ohanyan

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65027

#### **Abstract**

In this chapter, we analyse non-uniform bending of single-layer cell tissues—epithelia, surrounding organs throughout the body. Dimensionally reduced model is suggested, which is equivalent to membranes with bending stiffness: the total elastic energy of the tissue is a combination of stretching and bending energies. The energy, suggested in this chapter, is a piecewise function, the branches of which correspond to a specific deformation regime: compression, pure bending and stretch. tissue is a combination of stretching and bending energies. The energy, suggested in

Under specific assumptions on a single cell, such as all cells are identical, elastic, homogeneous, have the shape of a hexagonal prism, are linked to each other by pure elastic junctions, and so on, tissues of different shapes are analysed using finite element discretization. The deformation of initially flat, cylindrical and spherical tissues is visualized to reproduce similar deformations in real epithelial tissues observed during experiments. elastic junctions, and so on, tissues of different shapes are analysed using finite

**Keywords:** morphogenesis, embryonic tissue, adherens junctions, relaxation, quasiconvex energy densities, isometric deformations, short maps, tissue mechanics, Γ-convergence, lower semi-continuity, hexagonal elements

## **1. Introduction**

Morphogenesis, the evolutionary process of shape and structure development of organisms or their parts, is driven by certain types of cell shape changes or, in other words, deformations.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

One type of such deformations is the apical constriction—visible shrinkage of the apical side of cells, leading to bending of epithelia—cell sheets, which surround organs throughout the body. First occurring at early stages of embryogenesis, the apical constriction results initially flat epithelia to be bent obtaining three-dimensional form, which, depending on physiological context and morphogenetic stage, leads to different consequences. It has been first hypothesized in [1] that in various developmental systems the apical constriction may drive the bending of epithelia. Using bulky mechanical construction, the hypothesis of Rhumbler is first practically tested in [2]. The testing mechanism consists of 13 identical bars (cell walls) that are kept apart by means of stiff tubes connecting their centres (cell kernels), and are held in a row by rubber bands connecting their ends (cell membrane). In the first stage, the rubber bands at both sides are stretched equally and the mechanism is in straight equilibrium. Shortening the rubber bands at, for instance, the upper side uniformly in each segment (cell), the mechanism is bent on the side of the greater tension, as it could be expected. Thus, Rhumbler's hypothesis could take place.

A more illustrative, computer model for verifying Rhumbler's hypothesis is suggested in [3], where bars and bands are replaced by virtual cells. Prescribing certain mechanical properties to the cell membrane and cytoplasmic components, and assuming that as a result of deformations the volume of a single cell is preserved, which is implicitly done by Rhumbler and Lewis, a model of cuboidal epithelia folding is suggested, based on the local behaviour of individual cells. The model is demonstrated on the example of ventral furrow formation in *Drosophila*. A sequence of cells in the form of a cylindrical shell, representing the cross section of ventral furrow, is deformed such that apical constriction occurs in its lower row of cells. Increasing the applied stresses, different steps of the furrow formation are illustrated. **Figure 1** expresses how much Odell's model is close to the real microscopic picture [4].

**Figure 1.** Ventral furrow formation in *Drosophila* according to [3] (upper) and [4] (lower).

Much is known about the causes of the apical constriction, but some issues still remain unexplored [5–8]. The most studied causes include contraction of actin filaments—fibrous network localized at the cortex of the cell, by interacting with motor protein myosin—the most known converter of chemical energy into mechanical work. Under influence of myosin, actin fibres contract leading to shrinkage of the cell in the area of their localization [9, 10]. The contraction size and principal direction of the fibres, that is, the microscopical deformation of a single cell, mainly depend on the tissue type: actin-myosin network contraction may deform columnar (occurring, for instance, in digestive tract and female reproductive system) or cuboid (resp. in kidney tubules) cells into trapezoidal-, wedge- or bottle-shaped cells. The deformation size strongly depends also on the emplacement of deforming cells: cells with different placement are constricted differently, so that macroscopically one observes localized wrinkles (see **Figure 2**).

**Figure 2.** Apical constriction of ventral furrow at different times [10].

One type of such deformations is the apical constriction—visible shrinkage of the apical side of cells, leading to bending of epithelia—cell sheets, which surround organs throughout the body. First occurring at early stages of embryogenesis, the apical constriction results initially flat epithelia to be bent obtaining three-dimensional form, which, depending on physiological context and morphogenetic stage, leads to different consequences. It has been first hypothesized in [1] that in various developmental systems the apical constriction may drive the bending of epithelia. Using bulky mechanical construction, the hypothesis of Rhumbler is first practically tested in [2]. The testing mechanism consists of 13 identical bars (cell walls) that are kept apart by means of stiff tubes connecting their centres (cell kernels), and are held in a row by rubber bands connecting their ends (cell membrane). In the first stage, the rubber bands at both sides are stretched equally and the mechanism is in straight equilibrium. Shortening the rubber bands at, for instance, the upper side uniformly in each segment (cell), the mechanism is bent on the side of the greater tension, as it could be expected. Thus, Rhumbler's hypothesis

A more illustrative, computer model for verifying Rhumbler's hypothesis is suggested in [3], where bars and bands are replaced by virtual cells. Prescribing certain mechanical properties to the cell membrane and cytoplasmic components, and assuming that as a result of deformations the volume of a single cell is preserved, which is implicitly done by Rhumbler and Lewis, a model of cuboidal epithelia folding is suggested, based on the local behaviour of individual cells. The model is demonstrated on the example of ventral furrow formation in *Drosophila*. A sequence of cells in the form of a cylindrical shell, representing the cross section of ventral furrow, is deformed such that apical constriction occurs in its lower row of cells. Increasing the applied stresses, different steps of the furrow formation are illustrated. **Figure 1** expresses

how much Odell's model is close to the real microscopic picture [4].

**Figure 1.** Ventral furrow formation in *Drosophila* according to [3] (upper) and [4] (lower).

could take place.

136 Perusal of the Finite Element Method

Because of the mechanical nature of cell shape change, in particular, apical constriction, its theoretical study first of all must rely on mechanical principles and constitutive laws. In early mechanical models, the tissue is modelled as a continuum material, so that the position and behaviour of individual cells are unimportant, that is, only macroscopic deformation of the tissue is studied. Moreover, the height of the cells (viz. the thickness of the tissue) is supposed to be negligible with respect to other measures, such that the deformation of the tissue can be described in terms of its mid-surface. In such models, the actin network is not explicitly accounted and the contraction forces are modelled as a force term acting on the outer surface of the epithelium. The actin network is explicitly accounted in [11], where thin elastic shell model based on linear Cauchy relations is derived to describe apical constriction in initially non-flat epithelia. The model is tested to simulate apical constriction in initially flat, cylindrical and spherical tissues. Ventral furrow formation in *Drosophila* is simulated (see **Figure 3**), and it becomes evident from comparison of **Figures 1** and **3** that the model, being three-dimensional, is in good correspondence with that from [3]: **Figure 1** corresponds to cross section of cylindrical shell from **Figure 3**. However, it does not involve the behaviour of individual cells and describes only macroscopic deformation. For further introduction into other mechanical models, we refer to [11–26].

**Figure 3.** Ventral furrow formation in *Drosophila* according to [11].

The physical and geometrical characteristics of individual cells are taken into account by mechanical model suggested in [27] describing three-dimensional deformations of cell sheet. Both actin-myosin network contractile tensions and cell-cell adhesion stresses are involved to contribute in epithelial tissue bending. Prescribing specific mechanical characteristics to cytoplasmic components and to the kernel, the effective energy of a cell, which is assumed to be a hexagonal prism with constant volume, is expressed in terms of its basal length. As a result of simulations, it is particularly established that when adhesion stresses, distributed on lateral sides of the cell, are large enough, the effective energy has two local minima, that is, there are two equilibrium cell shapes. More particular, depending on position, cell shape may be discontinuously transited from squamous to columnar forms. **Figure 4** shows that there is a whole domain in the plane of lateral adhesion and actin-myosin tensions for which the effective energy has two local minima.

**Figure 4.** Epithelial cell aspect ratio as a bistable phenomenon [27]: is the contractile force and is the actin-myosin tension.

In mathematical terms, it means that the total energy of the tissue is not *lower semi-continuous*, which implies that there is no convergence of its minimizers, and, therefore, it becomes impossible to derive real deformation of the tissue [28, 29]. To be more illustrative, consider a membrane, that is, a tissue without bending stiffness, which is compressed by boundary stresses. The only way that the membrane can accommodate a compressed state is due to *wrinkling* [30]. Wrinkles may occur more and more finely, so the limiting deformation will be smooth, but evidently it will not minimize the membrane energy.

sional, is in good correspondence with that from [3]: **Figure 1** corresponds to cross section of cylindrical shell from **Figure 3**. However, it does not involve the behaviour of individual cells and describes only macroscopic deformation. For further introduction into other mechanical

The physical and geometrical characteristics of individual cells are taken into account by mechanical model suggested in [27] describing three-dimensional deformations of cell sheet. Both actin-myosin network contractile tensions and cell-cell adhesion stresses are involved to contribute in epithelial tissue bending. Prescribing specific mechanical characteristics to cytoplasmic components and to the kernel, the effective energy of a cell, which is assumed to be a hexagonal prism with constant volume, is expressed in terms of its basal length. As a result of simulations, it is particularly established that when adhesion stresses, distributed on lateral sides of the cell, are large enough, the effective energy has two local minima, that is, there are two equilibrium cell shapes. More particular, depending on position, cell shape may be discontinuously transited from squamous to columnar forms. **Figure 4** shows that there is a whole domain in the plane of lateral adhesion and actin-myosin tensions for which the effective

is the contractile force and is the actin-myo-

models, we refer to [11–26].

138 Perusal of the Finite Element Method

energy has two local minima.

sin tension.

**Figure 3.** Ventral furrow formation in *Drosophila* according to [11].

**Figure 4.** Epithelial cell aspect ratio as a bistable phenomenon [27]:

To overcome such difficulties, the membrane energy density is usually substituted by its *quasiconvex envelope* [28], which ensures the lower semi-continuity of its total energy. In calculus of variations, there are several ways to construct or even compute convex envelopes for functionals, depending on first- and second-order derivatives [28, 31–36].

Another possibility for incorporating the lack of lower semi-continuity of membrane energy functionals is accounting the bending stiffness of the membrane and adding to the energy the bending contribution [37]. So, it is established in [33, 34] that the total energy of a membrane with some bending stiffness and thickness ℎ, that is, the functional

$$E\left[\mathbf{r}\right] = hE\_m\left[\mathbf{r}\right] + h^3E\_b\left[\mathbf{r}\right] \tag{1}$$

of deformation consisting of membrane () and bending () contributions is lower semicontinuous for arbitrary as far as is lower semi-continuous. Explicit forms for and as well as sufficient conditions on material constants for which (and therefore ) is lower semi-continuous are derived ibid. Such materials are called *stable*.

However, the most rigorous and general approach seems to be the derivation of low-order energies from general three-dimensional non-linear elasticity by means of -convergence. The concept was originally developed by De Giorgi, see [29]. It turns out that the -limit of threedimensional elastic energy functional, when the thickness of the body goes to zero, is always lower semi-continuous and since most commonly used two-dimensional energy functionals, that is, membrane energy, pure bending (Kirchhoff) energy and higher-order terms (von Kármán–like energies), are already derived as such limits (see Section 3), it becomes possible to analyze real deformations of particular tissues.

By suggested improvements, it is supposed to get rid of disadvantages of previous models and take into account the fact that the total energy of the tissue can have two local minima. Thus, we are intended


**•** to use the quasi-convexification of the total energy, so the real three-dimensional deformations of the epithelial tissues can be identified as minimizers of the total energy by a gradient flow technique.

The former allows making simplified toolboxes for analysing three-dimensional deformations of epithelial tissues. The rough diagram of the improved model looks like in **Figure 5**, so that the characteristics of each individual cell are important, as well as the actin belt is explicitly involved in the model.

**Figure 5.** Diagrams of a single-cell layer: cross section (left) and 3D shape (right).

The chapter is organized as follows: In Section 2, some preliminary definitions and main notations are brought to make the chapter independent of outer sources. In Section 3, known results from -convergent energies are shortly described. In Section 4, the quasi-convex envelope of membrane energy densities of general form is derived explicitly and sufficient conditions for their convexity are derived. Moreover, a specific energy is suggested for different deformation regimes that fully satisfies the needs of basic mechanical theory suggestion. Finally, in Section 5 main results of finite element analysis of tissues of particular initial shapes (flat, cylindrical, spherical) are discussed. Overall conclusion completes the body of the chapter.

## **2. Preliminaries**

We begin with the definition of concepts used throughout the paper, which and much more on this topic can be found in [28, 29].

There are different concepts of convexity in higher dimensions, such as *quasi-, poly-, rank-one* and *separately convexity*. Here, we use all but poly-convexity, so it is of no use to enlarge the chapter by its definition.

**Definition 1** (quasi-convexity). *A function is said to be quasi-convex if*

Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations... http://dx.doi.org/10.5772/65027 141

$$W(\mathbf{G}) \le \frac{1}{\mu(\Omega)} \int\_{\Omega} W(\mathbf{G} + \nabla \mathbf{r}\_0(\mathbf{x})) d\mathbf{x},\tag{2}$$

*for every bounded open set* Ω, *for every and for every*

**Definition 2** (rank-one convexity). *A function is said to be rank-one convex if*

$$W(\mathbf{G} + \theta \mathbf{F}) \le (1 - \theta)W(\mathbf{G}) + \theta W(\mathbf{G} + \mathbf{F}),\tag{3}$$

*for every* [0, 1], *with* rank = 1.

**•** to use the quasi-convexification of the total energy, so the real three-dimensional deformations of the epithelial tissues can be identified as minimizers of the total energy by a gradient

The former allows making simplified toolboxes for analysing three-dimensional deformations of epithelial tissues. The rough diagram of the improved model looks like in **Figure 5**, so that the characteristics of each individual cell are important, as well as the actin belt is explicitly

The chapter is organized as follows: In Section 2, some preliminary definitions and main notations are brought to make the chapter independent of outer sources. In Section 3, known results from -convergent energies are shortly described. In Section 4, the quasi-convex envelope of membrane energy densities of general form is derived explicitly and sufficient conditions for their convexity are derived. Moreover, a specific energy is suggested for different deformation regimes that fully satisfies the needs of basic mechanical theory suggestion. Finally, in Section 5 main results of finite element analysis of tissues of particular initial shapes (flat, cylindrical, spherical) are discussed. Overall conclusion completes the body of the

We begin with the definition of concepts used throughout the paper, which and much more

There are different concepts of convexity in higher dimensions, such as *quasi-, poly-, rank-one* and *separately convexity*. Here, we use all but poly-convexity, so it is of no use to enlarge the

**Definition 1** (quasi-convexity). *A function is said to be quasi-convex if*

**Figure 5.** Diagrams of a single-cell layer: cross section (left) and 3D shape (right).

flow technique.

140 Perusal of the Finite Element Method

involved in the model.

chapter.

**2. Preliminaries**

chapter by its definition.

on this topic can be found in [28, 29].

**Definition 3** (separately convexity). *A function is said to be separately convex if*

$$W\left(\mathbf{x}\_{l},\ldots,\mathbf{x}\_{l-1},\mathbf{x}\_{l},\mathbf{x}\_{l+1},\ldots,\mathbf{x}\_{N}\right)\tag{4}$$

*is convex for every* = 1, …, *and for every fixed*

In other words, a function is separately convex if it is convex on every line parallel to a coordinate axis.

In general, convexity implies quasi-convexity, which implies rank-one convexity, and finally rank-one convexity implies separately convexity.

The quasi-convexity notion is generalized by Meyers [38] for functionals depending on higherorder derivatives. We refer to [32, 35, 36] for practical use of the definition.

**Definition 4** (*k*-quasi convexity). *A function is said to be quasiconvex if*

$$W(\mathbf{G}) \le \frac{1}{\mu(\Omega)} \int\_{\Omega} W(\mathbf{G} + \nabla^k \mathbf{r}\_0(\mathbf{x})) d\mathbf{x},\tag{5}$$

*for every open-bounded set* Ω, *for every and for every*

In particular, 2-quasi-convex functions, which we use here, are the restriction of quasi-convex functions to symmetric matrices [39] (in the case of -quasi convexity, see [40]).

We widely use the quasi-convex envelope of () or *apparent energy density*, (), which can be defined in terms of deformations of the form () = <sup>+</sup> 0() with constant, and 0()=0 on the boundary ∂Ω [41, 42]:

$$\mathcal{W}\_q(\mathbf{G}) = \frac{1}{\mu(\Omega)} \inf\_{\mathbf{r}\_0|\_{\Omega} = 0} \int\_{\Omega} W(\mathbf{G} + \nabla \mathbf{r}\_0) d\mathbf{x}.\tag{6}$$

It is evident that for any and when = 0, = 0 as well.

Deformation is always denoted by with open-bounded We introduce the first and second fundamental forms of a deformation

$$\mathbf{I} = \left(\nabla \mathbf{r}\right)^{\mathsf{T}} \nabla \mathbf{r}, \quad \mathbf{II} = \left\{ \frac{\partial^{2} \mathbf{r}}{\partial \mathbf{x}\_{i} \partial \mathbf{x}\_{j}} \cdot \mathbf{n} \right\}\_{i,j=1,2},\tag{7}$$

$$=\frac{1}{\left|\frac{\partial \mathbf{r}}{\partial x\_1} \times \frac{\partial \mathbf{r}}{\partial x\_2}\right|} \frac{\partial \mathbf{r}}{\partial x\_1} \times \frac{\partial \mathbf{r}}{\partial x\_2}$$

respectively, in which is the unit of the outer normal to Ω. The first fundamental form quantitatively characterizes the stretch or compression of the membrane, while the second fundamental form characterizes its curvature.

**Definition 5**. *Energy density function* is called frame indifferent if for all rotations 3 <sup>∈</sup> SO(3)

$$W\left(\mathbf{R}\_{\natural}\mathbf{G}\right) = W\left(\mathbf{G}\right). \tag{8}$$

*If for all* 2 <sup>∈</sup> SO(2),

$$W\left(\mathbf{GR}\_2\right) = W\left(\mathbf{G}\right),\tag{9}$$

*then it is called isotropic*.

In other words, frame indifferent energy densities are invariant against three-dimensional rotations before a deformation is applied and that isotropic energy densities are invariant against two-dimensional rotations after a deformation is applied.

**Definition 6** (short (-short) deformation). *A deformation is called short* (*-short) if* ∇ <sup>≤</sup> Id2×2 (≤ Id2×2) *a.e. in* <sup>Ω</sup>.

**Definition 7** (isometric deformation). *A deformation is called isometric if* ∇ = Id2×2 *a.e. on* Ω.

Short deformations result compressive stresses, while isometric deformations do not allow stretching or compression. The case Id2×2 corresponds to tensile or, in general, noncompressive stresses. It is known that for non-compressive stresses, the membrane energy density function is already quasi-convex [41, 42].

## **3. -convergence and -limits of three-dimensional non-linear elastic energy**

Rigorous derivation of two-dimensional energy functionals for thin bodies from threedimensional functionals is of particular interest. There are several ways for dimension reduction, such as asymptotic expansion, Γ-convergence technique, and so on (for general survey, see [43]). In [44], a hierarchy of plate models is derived as Γ-limit of three-dimensional elastic energy functional when the thickness of the body ℎ 0. It turns out that the scaling of external force in ℎ plays an important role for Γ-limit derivation. Suppose a body occupies the domain Ω × <sup>−</sup> <sup>ℎ</sup> 2, ℎ <sup>2</sup> , and the applied forces active on the outer surface of the body satisfy

$$\frac{1}{h^{\alpha}}\mathbf{f}^{(h)} \rightharpoonup \mathbf{f} \quad \text{in} \quad L^{2}\left(\Omega, \mathbb{R}^{3}\right), \tag{10}$$

and

<sup>0</sup> =0 <sup>0</sup> <sup>1</sup> ( )= inf ( ). ( ) *W Wd <sup>q</sup>*

Deformation is always denoted by with open-bounded We introduce the

respectively, in which is the unit of the outer normal to Ω. The first fundamental form quantitatively characterizes the stretch or compression of the membrane,

**Definition 5**. *Energy density function* is called frame indifferent if for all rotations

In other words, frame indifferent energy densities are invariant against three-dimensional rotations before a deformation is applied and that isotropic energy densities are invariant

**Definition 6** (short (-short) deformation). *A deformation is called short* (*-short) if*

**Definition 7** (isometric deformation). *A deformation is called isometric if* ∇ = Id2×2

Short deformations result compressive stresses, while isometric deformations do not allow stretching or compression. The case Id2×2 corresponds to tensile or, in general, noncompressive stresses. It is known that for non-compressive stresses, the membrane energy

**G G r** *x* (6)

, =1,2

**<sup>r</sup> I r r II n** (7)

*W W* (**RG G** <sup>3</sup> ) = . ( ) (8)

*W W* (**GR G** <sup>2</sup> ) = , ( ) (9)

*i j i j x x*

ï ï ¶ ¶ î þ

 ¶W <sup>W</sup> + Ñ <sup>W</sup> ò **<sup>r</sup>**

It is evident that for any and when = 0, = 0 as well.

( ) <sup>2</sup>

= ,= , *<sup>T</sup>*

ì ü ï ï ¶ Ñ Ñ í ý <sup>×</sup>

m

first and second fundamental forms of a deformation

while the second fundamental form characterizes its curvature.

against two-dimensional rotations after a deformation is applied.

3 <sup>∈</sup> SO(3)

*a.e. on* Ω.

*If for all* 2 <sup>∈</sup> SO(2),

142 Perusal of the Finite Element Method

*then it is called isotropic*.

∇ <sup>≤</sup> Id2×2 (≤ Id2×2) *a.e. in* <sup>Ω</sup>.

density function is already quasi-convex [41, 42].

$$E^{h}\left[\mathbf{r}\right] = \frac{1}{h}E\left[\mathbf{r}\right] = \int\_{\Omega \times \left(-\frac{1}{2}, \frac{1}{2}\right)} W\left(\nabla\_{h} \mathbf{r}\right) d\mathbf{x} \tag{11}$$

denotes the rescaled elastic energy of the body, with

$$\nabla\_{\mathbf{A}}\mathbf{r} = \left(\frac{\partial\mathbf{r}\_{1}}{\partial\mathbf{x}\_{1}}, \frac{\partial\mathbf{r}\_{2}}{\partial\mathbf{x}\_{2}}, \frac{1}{h}\frac{\partial\mathbf{r}\_{3}}{\partial\mathbf{x}\_{3}}\right). \tag{12}$$

The total energy of the body will be

$$E^{\hbar}\left[\mathbf{r}\right] = E^{\hbar}\left[\mathbf{r}\right] - \int\_{\Omega \times \left(-\frac{1}{2}, \frac{1}{2}\right)} \mathbf{f}^{(\hbar)}\left(\mathbf{x}\right) \cdot \mathbf{r} \, d\mathbf{x}.\tag{13}$$

The Γ-limit

$$\left[\Gamma - \lim\_{h \to 0} \frac{1}{h^{\beta}} E^{\hbar} \left[\mathbf{r}\right] \right] \tag{14}$$

for 0 will provide constitutive mechanical models to analyse deformed three-dimensional shape of initially flat thin bodies.

We combine the results of [44–48] in Theorem 1.

**Theorem 1**. *Let the stored energy density function is frame indifferent and satisfy*

$$W = 0 \quad \text{on} \quad \text{SO(3)},\tag{15}$$

$$W(\mathbf{F}) \ge c \operatorname{dist}^2(\mathbf{F}, \operatorname{SO}(\mathfrak{I})), c \ge 0,\tag{16}$$

*and in a neighbourhood of* SO(3), 2.

*Then*

*(i) (Membrane theory [45]). Suppose* = = 0 *and*

$$W\left(\mathbf{F}\right) \le C\left(1 + \left|\mathbf{F}\right|^p\right), \ W\left(\mathbf{F}\right) \ge c\left|\mathbf{F}\right|^p - C, \ p \ge 2, c \ge 0. \tag{17}$$

*Then is a* -*minimizing sequence, that is, if*

$$\limsup\_{h \to 0} \frac{1}{h^{\beta}} \left[ \mathbb{E}^{\mathbb{A}} \left[ \mathbf{r} \right] - \inf \, \mathbb{E}^{\mathbb{A}} \right] = \mathbf{0},\tag{18}$$

*then (for a subsequence). The limiting deformation is independent of* 3 *and minimizes*

$$\mathbb{E}\_{o}^{0}\Big[\mathbf{r}\Big] = \int\_{\Omega} \Big[\boldsymbol{W}\_{q}\left(\nabla\mathbf{r}\right) - \mathbf{f} \cdot \mathbf{r}\Big] d\mathbf{x} \tag{19}$$

*among all* . *For derivation of from given see Section 4 subsequently*.

*(ii) (Constrained membrane theory [46]). Suppose that* 0 < <sup>&</sup>lt; <sup>5</sup> <sup>3</sup> *and* <sup>=</sup> . *Then and every* -*minimizing sequence* (ℎ) *has a subsequence with in* . *The limit is independent of* 3 *and minimizes*

$$E\_{\alpha}^{0}\left[\mathbf{r}\right] = \int\_{\Omega} \left[-\mathbf{f} \cdot \mathbf{r}\right] d\mathbf{x} \tag{20}$$

*among all short deformations* .

*(iii) (Non-linear bending theory [47]). Suppose* = = 2. *Then* inf <sup>ℎ</sup> <sup>≤</sup> ℎ2 *and if* (ℎ) *is a minimizing sequence then there is strong convergence* (ℎ) *in (for a subsequence). The* Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations... http://dx.doi.org/10.5772/65027 145

*limiting deformation is isometric, independent of* 3*, belongs to and minimizes*

$$\mathbb{E}\_2^0[\mathbf{r}] = \int\_{\Omega} \left[ \frac{1}{24} \mathcal{Q}\_2(\mathbf{II}) - \mathbf{f} \cdot \mathbf{r} \right] d\mathbf{x} \tag{21}$$

*among all isometric deformations , which belong to The non-linear strain satisfies*

$$\frac{1}{h} \Big[ \left( \nabla \mathbf{r}^{(h)} \right)^{T} \nabla \mathbf{r}^{(h)} - \text{Id} \right] \to \mathbf{x}\_{\mathfrak{z}} \Big( \mathbf{II} + \text{sym} \, a\_{\text{min}} \otimes e\_{\mathfrak{z}} \Big),\tag{22}$$

*where* 2 sym <sup>=</sup> <sup>+</sup> *,* min *is the solution of*

**Theorem 1**. *Let the stored energy density function is frame indifferent and satisfy*

( ) (1 , ) ( ) , 2, > 0. *p p*

[ ]

[ ] ( ) <sup>0</sup> E = <sup>0</sup> *W d <sup>q</sup>* <sup>W</sup>

[ ] [ ] <sup>0</sup> E = *d*

a

*(iii) (Non-linear bending theory [47]). Suppose* = = 2. *Then* inf

*among all* . *For derivation of from given see Section 4 subsequently*.

*then (for a subsequence). The limiting deformation is*

<sup>1</sup> limsup E inf E = 0, *h h*

*and in a neighbourhood of* SO(3), 2.

144 Perusal of the Finite Element Method

*independent of* 3 *and minimizes*

*every* -*minimizing sequence*

*The limit is independent of* 3 *and minimizes*

*among all short deformations* .

*minimizing sequence then there is strong convergence*

*(i) (Membrane theory [45]). Suppose* = = 0 *and*

*Then is a* -*minimizing sequence, that is, if*

0

®

*(ii) (Constrained membrane theory [46]). Suppose that* 0 < <sup>&</sup>lt; <sup>5</sup>

*<sup>h</sup> h*b

*Then*

*W* = 0 on SO(3), (15)

é ù - ë û **<sup>r</sup>** (18)

é ù Ñ -× ò ë û **r r fr** *<sup>x</sup>* (19)

(ℎ) *has a subsequence with in* .

<sup>W</sup> - × ò **r fr** *<sup>x</sup>* (20)

<sup>3</sup> *and* <sup>=</sup> . *Then and*

<sup>ℎ</sup> <sup>≤</sup> ℎ2 *and if*

(ℎ) *in (for a subsequence). The*

(ℎ) *is a* -

( ) ( ) <sup>2</sup> *Wc c* **F F** ³ dist ,SO(3) , > 0, (16)

*W C W c Cp c* **F F FF** £+ ³ - ³ (17)

$$\underline{Q}\_{2}\left(\mathbf{G}\right) = \min\_{a \in \mathbb{R}^{3}} \underline{Q}\_{3}\left(\mathbf{G} + a \otimes e\_{3} + e\_{3} \otimes a\right), \quad \underline{Q}\_{3}\left(\mathbf{G}\right) = \frac{\hat{\sigma}^{2}W}{\hat{\sigma}\mathbf{G}^{2}}\bigg|\_{a \cdot \omega = 0}.\tag{23}$$

*In all cases, there is convergence of energy, that is*,

$$\lim\_{h \to 0} \frac{1}{h^{\theta}} \mathbb{E}^{h} \left[ \mathbf{r}^{(h)} \right] = \lim\_{h \to 0} \frac{1}{h^{\theta}} \inf \, \mathbf{E}^{h} = \mathbb{E}\_{a}^{0} \left[ \overline{\mathbf{r}} \right] = \min \, \mathbf{E}\_{a}^{0}. \tag{24}$$

For > 0, the limiting deformation is not only isometric but is close to a rigid motion. For that reason in [44], for deformation the deformation

$$\tilde{\mathbf{r}}^{(h)} = \left(\mathbf{R}^{(h)}\right)^{T}\mathbf{r}^{(h)} - \mathbf{c}^{(h)} \tag{25}$$

with (ℎ) <sup>∈</sup> SO(3) and is introduced and the averaged in-plane and out-of-plane displacements

$$\mathbf{U}^{(h)}\left(\mathbf{x}\_{1},\mathbf{x}\_{2}\right) = \int\_{-\frac{1}{2}}^{\frac{1}{2}} \left[ \begin{pmatrix} \tilde{\mathbf{r}}\_{1}^{(h)} \\ \tilde{\mathbf{r}}\_{2}^{(h)} \end{pmatrix} \begin{pmatrix} \mathbf{x}\_{1},\mathbf{x}\_{2},\mathbf{x}\_{3} \end{pmatrix} - \begin{pmatrix} \mathbf{x}\_{1} \\ \mathbf{x}\_{2} \end{pmatrix} \right] d\mathbf{x}\_{3}, \quad \mathbf{V}^{(h)}\left(\mathbf{x}\_{1},\mathbf{x}\_{2}\right) = \int\_{-\frac{1}{2}}^{\frac{1}{2}} \tilde{\mathbf{r}}\_{3}^{(h)} d\mathbf{x}\_{3} \tag{26}$$

are rescaled by

$$\mathbf{u}^{(h)} = \frac{1}{h^{\mathcal{V}}} \mathbf{U}^{(h)}, \quad \mathbf{v}^{(h)} = \frac{1}{h^{\mathcal{S}}} \mathbf{V}^{(h)},\tag{27}$$

respectively.

Then, we have

**Theorem 2**. *(iv) (Linearized isometry constraint). Suppose* 2 < < 3 *and set*  = 2 2*,* = 2( 2)*,* 2 <sup>=</sup> *. If* 2 < <sup>&</sup>lt; <sup>5</sup> 2*, suppose in addition that* <sup>Ω</sup> *is simply connected. Then,* ℎ <sup>≤</sup> inf <sup>ℎ</sup> <sup>≤</sup> <sup>0</sup>*. If*  (ℎ) *is a -minimizing sequence then there exist constant* (ℎ) <sup>∈</sup> SO(3) *and c(h)* ∈ ℝ*3 such that*

$$
\mathbf{R}^{(h)} \to \overline{\mathbf{R}},\tag{28}
$$

$$\nabla\_h \tilde{\mathbf{r}} \to \mathrm{Id} \quad \text{in} \quad L^2 \left( \Omega \times \left( -\frac{1}{2}, \frac{1}{2} \right); \mathbb{R}^{3 \times 3} \right), \tag{29}$$

$$\mathbf{u}^{(h)} \to \overline{\mathbf{u}}, \quad \mathbf{v}^{(h)} \to \overline{\mathbf{v}}, \quad \text{in} \quad W^{1,2} \left( \Omega; \mathbb{R}^2 \right), \quad \overline{\mathbf{v}} \in W^{2,2} \left( \Omega; \mathbb{R}^2 \right), \tag{30}$$

$$
\nabla \overline{\mathbf{u}} + \left(\nabla \overline{\mathbf{u}}\right)^{\mathsf{T}} + \nabla \overline{\mathbf{v}} \otimes \nabla \overline{\mathbf{v}} = \mathbf{0}.\tag{31}
$$

*Moreover, the pair* , *minimizes the functional*

$$E\_{\rm liv}^{\rm vK}\left(\mathbf{v}, \mathbf{R}\right) = \int\_{\Omega} \left[\frac{1}{24} Q\_2\left(\nabla^2 \mathbf{v}\right) - \mathbf{R}\_{33} \mathbf{f}\_3 \cdot \mathbf{v}\right] d\mathbf{x},\tag{32}$$

*subject to* det∇2 = 0.

*(v) (von Kármán theory). Suppose that*  = 3 *and set*  = 4*,*  = 2*,*  = 1*. Then and for a subsequence of a -minimizing sequence, (1) and (2) hold and the limit triple* ,, *minimizes von Kármán functional*

$$\begin{split} \mathbf{E}^{\mathcal{K}} \left( \mathbf{u}, \mathbf{v}, \mathbf{R} \right) \quad &= \frac{1}{2} \int\_{\Omega} \Big[ Q\_{2} \Big( \frac{1}{2} \Big[ \nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{T} + \nabla \mathbf{v} \otimes \nabla \mathbf{v} \Big] \Big) + \frac{1}{24} Q\_{2} \Big( \nabla^{2} \mathbf{v} \Big) \Big] d\mathbf{x} - \\ &\quad - \mathbf{R}\_{33} \Big[ \mathbf{f}\_{3} \cdot \mathbf{v} \, d\mathbf{x}. \end{split} \tag{33}$$

*(vi) (Linearized von Kármán theory). Suppose*  > 3 *and set*  = 2 2*,*  = 1 *and* = 2. *Then* ℎ <sup>≤</sup> inf ε<sup>ℎ</sup> <sup>≤</sup> <sup>0</sup> *and for a sequence of a -minimizing sequence. (2) holds with*  = 0 *and the pair* , *minimizes the linearized von Kármán functional*

$$\varepsilon^{\rm vK}\left(\mathbf{v},\mathbf{R}\right) = \int\_{\Omega} \left[\frac{1}{24}\mathcal{Q}\_2\left(\nabla^2\mathbf{v}\right) - \mathbf{R}\_{33}\mathbf{f}\_3 \cdot \mathbf{v}\right] d\mathbf{x}.\tag{34}$$

*Moreover, the non-linear strain satisfies*

() () () () 1 1 = ,= , *h hh h h h*

**Theorem 2**. *(iv) (Linearized isometry constraint). Suppose* 2 < < 3 *and set*  = 2 2*,*

( ) = 0. *<sup>T</sup>*

2 33 3

é ù Ñ- × ê ú

*(v) (von Kármán theory). Suppose that*  = 3 *and set*  = 4*,*  = 2*,*  = 1*. Then and for a subsequence of a -minimizing sequence, (1) and (2) hold and the limit triple* ,, *minimizes*

2 2

*Q Q d*

é ù æ ö é ù Ñ + Ñ +Ñ ÄÑ + Ñ - ê ú ç ÷ ë û ë û è ø

( ) ( ) <sup>2</sup>

<sup>1</sup> E ,= , <sup>24</sup>

*lin Q d* <sup>W</sup>

( ) ( ) ( ) <sup>2</sup>

**uvR u u vv v**

1 1 <sup>1</sup> E ,, = 2 2 <sup>24</sup>

.

*x*

*d*

 d

**u Uv V** (27)

2*, suppose in addition that* <sup>Ω</sup> *is simply connected. Then,*

( ) , **R R** *<sup>h</sup>* ® (28)

Ñ + Ñ +Ñ ÄÑ **u u vv** (31)

ë û ò **v R v Rf v** *<sup>x</sup>* (32)

*x*

(29)

(30)

(33)

(ℎ) *is a -minimizing sequence then there exist constant* (ℎ) <sup>∈</sup> SO(3) *and c(h)*

g

respectively.

146 Perusal of the Finite Element Method

Then, we have

ℎ <sup>≤</sup> inf

 *such that*

*subject to* det∇2 = 0.

*von Kármán functional*

∈ ℝ*3*

= 2( 2)*,* 2 <sup>=</sup> *. If* 2 < <sup>&</sup>lt; <sup>5</sup>

<sup>ℎ</sup> <sup>≤</sup> <sup>0</sup>*. If* 

*Moreover, the pair* , *minimizes the functional*

*vK*

33 3

ò


W

**R fv**

*vK T*

W

ò

$$\frac{1}{h^{1-\alpha}} \left( \left[ \left( \nabla\_h \mathbf{r}^{(h)} \right)^T \nabla\_h \mathbf{r}^{(h)} \right]^{\frac{1}{2}} - \text{Id} \right) \to \text{sym} \, \mathbf{G}. \tag{35}$$

*In all cases, we have convergence of the rescaled energy to the minimum of the limit functional* . *Moreover, for* <sup>3</sup> ≡0 *we have* 33 = 1 *or* 1.

**Remark 1. (***i) Practically, Theorems 1 and 2 establish -convergence and give the -limits of threedimensional non-linear energy functional in the range*  0,<sup>5</sup> <sup>3</sup> <sup>∪</sup> 2,∞ *. The range*  <sup>5</sup> 3,2 *remains unexplored*.

*(ii) Even in the case* <sup>=</sup> <sup>5</sup> <sup>3</sup>, *it is still possible to find a sequence* (ℎ) *for which the functional is bounded when* ℎ 0. *The limiting deformations are composed of finitely many affine isometries separated by sharp folds and are called origami deformations [46]. It is proved ibid that the class of origami deformations is the closure of the class of short deformations with respect to the uniform convergence*.

*(iii) In particular, when*  = = 2*, for isotropic homogeneous bodies we obtain*

$$\mathcal{Q}\_z(\mathbf{II}) = 2\mu \left| \mathbf{II} \right|^2 + \frac{\lambda \cdot 2\mu}{\lambda + 2\mu} \text{tr}^2 \mathbf{II},\tag{36}$$

*in which and are Lamé constants, coinciding with plate energy derived by Kirchhoff much earlier in [49]*.

## **4. Relaxation of epithelial elastic energy by Pipkin's procedure and by adding the bending contribution**

As was mentioned in Section 1, there are two ways to relax the epithelial elastic energy. The first way is to relax the stretching energy for deformation regimes corresponding to compressive stresses. For that reason, Pipkin's procedure [41] seems to be the most common tool.

Suppose that the stretching energy density function = ∇ is frame indifferent, isotropic and positive except the cases

$$
\nabla \mathbf{r} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix} \text{and } \nabla \mathbf{r} = \begin{pmatrix} \mathcal{S} & 0 \\ 0 & \mathcal{S} \\ 0 & 0 \end{pmatrix}, \quad 1 \le \mathcal{S} \in \mathbb{R}, \tag{37}
$$

where it turns to zero, that is, attains its minimum. On elastic energies with two minima, see [50]. This assumption is motivated by bistability of cell shapes during apical constriction (see **Figure 4**).

We restrict attention to deformations with

$$
\nabla \mathbf{r} = \begin{pmatrix} \sigma\_1 & 0 \\ 0 & \sigma\_2 \\ 0 & 0 \end{pmatrix}, \tag{38}
$$

and denote ∇ : = (1,2). Here <sup>=</sup> , 1; 2 , are the principal stretches, the squares of which are the eigenvalues of the first fundamental form. Therefore

$$\mathcal{A}\_{\mathbf{r},2}\left(\nabla\mathbf{r}\right) = \sqrt{\frac{1}{2}\left[\operatorname{tr}\left(\mathbf{I}\left(\nabla\mathbf{r}\right)\right) \pm \sqrt{\left\|\mathbf{I}\left(\nabla\mathbf{r}\right)\right\|^2 - 2\det\left(\mathbf{I}\left(\nabla\mathbf{r}\right)\right)}\right]}\,\tag{39}$$

In view of frame indifference of , and thence, , we conclude that ( ± 1, ± 2) = (1,2).

From quasi-convexity of , in particular, follows its rank-one convexity, therefore since the matrix

Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations... http://dx.doi.org/10.5772/65027 149

$$\mathbf{A} = \begin{pmatrix} \delta\_{11} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}, \quad \delta\_{11} \neq \mathbf{0}, \tag{40}$$

is rank one, we have

**4. Relaxation of epithelial elastic energy by Pipkin's procedure and by**

As was mentioned in Section 1, there are two ways to relax the epithelial elastic energy. The first way is to relax the stretching energy for deformation regimes corresponding to compressive stresses. For that reason, Pipkin's procedure [41] seems to be the most common tool.

Suppose that the stretching energy density function = ∇ is frame indifferent, isotropic

= 0 1 and = 0 , 1 < R,

d

where it turns to zero, that is, attains its minimum. On elastic energies with two minima, see [50]. This assumption is motivated by bistability of cell shapes during apical constriction (see

1

( ) ( ( )) ( ) ( ( )) <sup>2</sup>

é ù Ñ Ñ± Ñ - Ñ ê ú

= tr 2det .

In view of frame indifference of , and thence, , we conclude that

From quasi-convexity of , in particular, follows its rank-one convexity, therefore since the

ë û **r Ir Ir Ir** (39)

æ ö ç ÷ <sup>Ñ</sup> ç ÷ ç ÷ è ø

s

squares of which are the eigenvalues of the first fundamental form. Therefore

2 0 =0 , 0 0

s

d

 d

**r r** (37)

**r** (38)

, 1; 2 , are the principal stretches, the

1 0 0

æö æ ö ç÷ ç ÷ ÑÑ Î

00 00

èø è ø

**adding the bending contribution**

We restrict attention to deformations with

and denote ∇ : = (1,2). Here <sup>=</sup>

1

2

1,2

l

( ± 1, ± 2) = (1,2).

matrix

and positive except the cases

148 Perusal of the Finite Element Method

**Figure 4**).

therefore is convex with respect to 1. Repeating the argument when

$$\mathbf{A} = \begin{pmatrix} 0 & 0 \\ 0 & \delta\_{22} \\ 0 & 0 \end{pmatrix}, \quad \delta\_{22} \neq 0,\tag{41}$$

we see that (1,2) is separately convex.

Under assumptions made, we have (1,2)=0 if 1 <sup>=</sup> 2 = 1 and 1 <sup>=</sup> 2 <sup>=</sup> . Since > 1 and <sup>0</sup> ≤ are convex along lines parallel to 1 = 0 and 2 = 0, therefore = 0 on edges connecting the vertexes ±, ± in the 12 plane. Repeating the argument, we conclude that = 0 inside the square Ω0 <sup>=</sup> 1,2 ; 1 ≤ , 2 ≤ . So, for -short maps <sup>≡</sup> <sup>0</sup>.

Since (1,2) is even and convex in 2, it will attain its minimum with respect to 2 only when 2 = 0. Denote by 0(1) = min 2 (1,2) when 1 <sup>&</sup>gt; . Suppose that 0(1) is the largest value of 2 for which (1,0) = 0(1), 1 <sup>&</sup>gt; . The convexity of in 2 implies that (1,2) = 0(1) for 2 ≤ 0(1), 1 <sup>&</sup>gt; . In the same way, using the symmetry of in its arguments, we will arrive at (1,2) = 0(2) for 1 ≤ 0(2), 2 <sup>&</sup>gt; . Evidently, for the rest of deformation regimes, is convex.

If we denote by Ω1 <sup>=</sup> 1,2 ; 1 <sup>&</sup>gt; , 2 ≤ 0(1) and Ω2 <sup>=</sup> 1,2 ; 1 ≤ 0(2), 2 <sup>&</sup>gt; , the ranges of principal stretches, corresponding to uniaxial tensions in directions of those stretches, respectively [51], we finally arrive at

$$
\tilde{\boldsymbol{W}}\_{q}(\boldsymbol{\lambda}\_{1}, \boldsymbol{\lambda}\_{2}) = \begin{cases} \boldsymbol{0}, & \text{in } \Omega\_{0}, \\ \tilde{\boldsymbol{W}}^{0}(\boldsymbol{\lambda}\_{1}), & \text{in } \Omega\_{1}, \\ \tilde{\boldsymbol{W}}^{0}(\boldsymbol{\lambda}\_{2}), & \text{in } \Omega\_{2}, \\ \tilde{\boldsymbol{W}}(\boldsymbol{\lambda}\_{1}, \boldsymbol{\lambda}\_{2}), & \text{else}. \end{cases} \tag{42}
$$

Besides opportunity of explicit relaxation, in [41] the derivation of criteria for convexity and quasi-convexity is described. In view of rank-one convexity, satisfies Legendre-Hadamard inequality from which it follows

$$
\tilde{W}\_q^1 \ge 0, \quad \tilde{W}\_q^2 \ge 0, \quad \tilde{W}\_q^{11} \ge 0, \quad \tilde{W}\_q^{22} \ge 0,\tag{43}
$$

$$G \ge 0, \quad \sqrt{\tilde{W}\_q^{11} \tilde{W}\_q^{22}} - \tilde{W}\_q^{12} \ge H - G, \quad \sqrt{\tilde{W}\_q^{11} \tilde{W}\_q^{22}} + \tilde{W}\_q^{12} \ge -H - G. \tag{44}$$

For convexity of in corresponding domain instead of the last two equalities, the stronger conditions

$$\left(\tilde{W}\_q^{11}\tilde{W}\_q^{22} - \left(\tilde{W}\_q^{12}\right)^2\right) \ge 0, G \succeq \left| H \right|, \tag{45}$$

must be satisfied. Above

$$
\tilde{W}\_q^{'} = \frac{\partial \tilde{W}\_q}{\partial \lambda\_j}, \quad \tilde{W}\_q^{'k} = \frac{\partial^2 \tilde{W}\_q}{\partial \lambda\_j \partial \lambda\_k}, \quad G = \frac{\lambda\_l \tilde{W}\_q^{1} - \lambda\_z \tilde{W}\_q^{2}}{\lambda\_1^2 - \lambda\_2^2}, \quad H = \frac{\lambda\_z \tilde{W}\_q^{1} - \lambda\_l \tilde{W}\_q^{2}}{\lambda\_1^2 - \lambda\_2^2}. \tag{46}
$$

The other way is the adding of pure bending contribution to the elastic energy of the tissue. Since actin-myosin network contraction leads to compressive or non-stretching stresses, we have to incorporate the elastic energy mainly for such deformations. According to Section 3 for non-stretching stresses, we have

**i.** (short deformations and compressive stresses)

$$\Gamma - \lim\_{h \to 0} \frac{1}{h^{\beta}} E^{(h)} = 0 \text{ in the range } \beta \in \left[0, \frac{\mathfrak{F}}{\mathfrak{Z}}\right), \tag{47}$$

**ii.** (isometric deformations, neither stretch nor compressive stresses)

$$
\Gamma - \lim\_{h \to 0} \frac{1}{h^2} E^{(h)} = E\_b. \tag{48}
$$

The deformation regime Id corresponding to tensile stresses leads to , which is already lower semi-continuous. In the case of uniaxial tension, there is no bending and the energy expression will look like the middle rows of (42). Thence, the total energy might be given as follows:

$$\mathbf{E}\left[\mathbf{r}\right] = -\int\_{\Omega} \mathbf{f} \cdot \mathbf{r} \, d\mathbf{x} + \frac{1}{2} \int\_{\Omega} \tilde{W}^{0} \left(\lambda\_{i}(\nabla \mathbf{r})\right) d\mathbf{x}, \text{ for unit-axial tension in the direction of } \lambda\_{i},$$

$$\mathbf{E}\left[\mathbf{r}\right] = -\int\_{\Omega} \mathbf{f} \cdot \mathbf{r} \, d\mathbf{x} + \frac{1}{2} \int\_{\Omega} \tilde{W}^{0} \left(\lambda\_{z}(\nabla \mathbf{r})\right) d\mathbf{x}, \text{ for unit-axial tension in the direction of } \lambda\_{z}, \tag{49}$$

$$\begin{aligned} \frac{1}{24} \int\_{\Omega} Q\_{z}(\mathbf{H}) \, d\mathbf{x}, \text{ I = Id,} \\ \frac{1}{2} \int\_{\Omega} W\_{\neq}(\nabla \mathbf{r}) \, d\mathbf{x}, \text{ otherwise.} \end{aligned} \tag{40}$$

We assume that the tissue is homogeneous and isotropic, so according to Remark 1

Besides opportunity of explicit relaxation, in [41] the derivation of criteria for convexity and quasi-convexity is described. In view of rank-one convexity, satisfies Legendre-Hadamard

*qq q q* ³³ ³ ³ (43)

*qq q* - ³³ (45)

 ll

> l l

*qq q* + ³- - (44)

<sup>1</sup> <sup>2</sup> <sup>11</sup> <sup>22</sup> 0, 0, 0, 0, *WWW W* %% % %

11 22 12 11 22 12 0, , . *G WW W H G WW W H G* ³ %% % *qq q* - ³- %% %

> ( ) <sup>2</sup> 11 22 12 0, | |, *WW W G H* %% %

= , = ,= , = . *<sup>j</sup> <sup>q</sup> jk q q q qq*

¶¶ - - ¶ ¶¶ - -

*W W WW WW*

 l l

The other way is the adding of pure bending contribution to the elastic energy of the tissue. Since actin-myosin network contraction leads to compressive or non-stretching stresses, we have to incorporate the elastic energy mainly for such deformations. According to Section 3

> 1 5 lim = 0 in the range 0, , <sup>3</sup>

> > ( )

The deformation regime Id corresponding to tensile stresses leads to , which is already lower semi-continuous. In the case of uniaxial tension, there is no bending and the energy expression will look like the middle rows of (42). Thence, the total energy might be given

<sup>2</sup> <sup>0</sup> 1 lim = . *<sup>h</sup> <sup>b</sup> <sup>h</sup> E E*

® *h*

ll

For convexity of in corresponding domain instead of the last two equalities, the stronger

2 1 2 12

% % % % %% % % (46)

b

é ö G - <sup>Î</sup> <sup>ê</sup> <sup>÷</sup> ë ø (47)

G - (48)

12 21 2 2 2 2 1 2 1 2

inequality from which it follows

150 Perusal of the Finite Element Method

conditions

as follows:

must be satisfied. Above

*q q*

l

for non-stretching stresses, we have

*j j k*

**i.** (short deformations and compressive stresses)

0

®

*h*

 ll

*WW G H*

( )

*E h*b

*h*

**ii.** (isometric deformations, neither stretch nor compressive stresses)

$$\mathcal{Q}\_z(\mathbf{II}) = 2\mu \left| \mathbf{II} \right|^2 + \frac{\lambda \cdot 2\mu}{\lambda + 2\mu} \text{tr}^2 \mathbf{II}. \tag{50}$$

Furthermore, the membrane energy density function we take in the following form [52, 53]:

$$W\left(\nabla \mathbf{r}\right) = \gamma\_1 \left(I\_1 - 3\right) + \gamma\_2 \left(I\_1 - 3\right)^2,\tag{51}$$

which is valid for large deformations of incompressible hyper-elastic membranes. Above, <sup>1</sup> = tr is the first strain invariant, 1 > 0 and 2 > 0 are material parameters, evaluated experimentally in [53]. For a neo–Hookean material 21 <sup>=</sup> and 22 <sup>=</sup> , in which is the shear modulus and is the bulk modulus. and 0 are determined according to the procedure described above. For that reason, <sup>1</sup> must be represented in terms of the principal stretches 1 and 2 as follows:

$$I\_1(\boldsymbol{\lambda}\_1, \boldsymbol{\lambda}\_2) = \boldsymbol{\lambda}\_1^2 + \boldsymbol{\lambda}\_2^2 + \boldsymbol{\lambda}\_3^2 = \boldsymbol{\lambda}\_1^2 + \boldsymbol{\lambda}\_2^2 + \frac{1}{\boldsymbol{\lambda}\_1^2 \boldsymbol{\lambda}\_2^2},\tag{52}$$

in which the incompressibility condition 123 = 1 is taken into account.

For any fixed 1 > 1 (uniaxial tension in the first principal direction), the corresponding 1,2 attains its minimum with respect to 2 at the maximal positive root of

$$\frac{\partial \tilde{W}}{\partial \lambda\_2} = 2\lambda\_2 \left( 1 - \frac{1}{\left(\lambda\_1^2 + \lambda\_2^2\right)^2} \right) \left[ \gamma\_1 + 2\gamma\_2 \left( I\_1 - 3\right) \right] = 0. \tag{53}$$

Since in the uniaxial tension regime 1 <sup>2</sup> <sup>+</sup> 2 <sup>2</sup> > 1, we finally get

$$
\mathcal{N}^0(\mathcal{A}\_1) = \frac{1}{\sqrt{\mathcal{A}\_1}}.\tag{54}
$$

Then,

$$W^0(\mathcal{J}\_\mathfrak{l}) = \frac{1}{\mathcal{J}\_\mathfrak{l}^2} (\mathcal{J}\_\mathfrak{l}^3 - 3\mathcal{J}\_\mathfrak{l} + 2) \left[ \mathcal{Y}\_\mathfrak{l}\mathcal{J}\_\mathfrak{l} + \mathcal{Y}\_2(\mathcal{J}\_\mathfrak{l}^3 - 3\mathcal{J}\_\mathfrak{l} + 2) \right]. \tag{55}$$

The correspondent tension is defined by

$$\pi\_1(\boldsymbol{\lambda}\_t) = \frac{\partial W^0(\boldsymbol{\lambda}\_t)}{\partial \boldsymbol{\lambda}\_t} = 2 \frac{\boldsymbol{\lambda}\_1^3 - 1}{\boldsymbol{\lambda}\_1^3} [\boldsymbol{\gamma}\_1 \boldsymbol{\lambda}\_t + 2 \boldsymbol{\gamma}\_2 (\boldsymbol{\lambda}\_t^3 - 3 \boldsymbol{\lambda}\_1 + 2)]. \tag{56}$$

Since is obviously symmetric in its arguments, in the case of uniaxial tension in the second principal direction, we would have

$$W^0(\mathcal{J}\_2) = \frac{1}{\lambda\_2^2} (\mathcal{\lambda}\_2^3 - 3\mathcal{\lambda}\_2 + 2)[\mathcal{\chi}\_1\mathcal{\lambda}\_2 + \mathcal{\chi}\_2(\mathcal{\lambda}\_2^3 - 3\mathcal{\lambda}\_2 + 2)],\tag{57}$$

and

$$\pi\_2(\boldsymbol{\lambda}\_2) = \frac{\partial W^0(\boldsymbol{\lambda}\_2)}{\partial \boldsymbol{\lambda}\_2} = 2 \frac{\boldsymbol{\lambda}\_2^3 - 1}{\boldsymbol{\lambda}\_2^3} [\boldsymbol{\gamma}\_1 \boldsymbol{\lambda}\_2 + 2 \boldsymbol{\gamma}\_2 (\boldsymbol{\lambda}\_2^3 - 3 \boldsymbol{\lambda}\_2 + 2)].\tag{58}$$

It is evident, that when we substitute 1 = 1 in the first case and 2 = 1 in the second case, we arrive at undeformed configuration of the membrane, so the relaxed energy is fully consistent.

Forces driving tissue deformation are strongly localized and in general are compressive. Force must possess those properties.

## **5. Finite element analysis of three-dimensional shape of some characteristic deformations of tissues during morphogenesis**

In this section, we summarize main results of finite element analysis of a single layer tissue model, the elastic energy of which is given by

$$E\left[\mathbf{r}\right] = hE\_{\text{w}}\left[\mathbf{r}\right] + h^{\text{3}}E\_{b}\left[\mathbf{r}\right] \tag{59}$$

in which and are defined in Eqs. (49)–(51). Since the discretization procedure is standard, we omit the details and bring only the main results.

In **Figure 6**, we bring the model of a single-cell (element) and cell-cell junction (in red). All structures (plate and shell) considered in this section entirely consist of such cell groups. In all tissues considered below, the height of a cell ℎ=4 and ℎ = 200, in which is its side and is the diameter of actin fibres. The ratio of Young's moduli of a cell and actin fibres is act = 3cell, and of actin fibres and cell-cell links act = 1.5link.

We consider

Since in the uniaxial tension regime 1

152 Perusal of the Finite Element Method

Then,

and

<sup>2</sup> <sup>+</sup> 2

0 1

0 3 3 1 1 1 11 2 1 1 2

 l

0 3

¶ -

l

l

2

l

 l

l

1 1

 l

0 3 3 2 2 2 12 2 2 2 2

 l

0 3

¶ -

 l

> l

l

l

<sup>1</sup> *<sup>W</sup>* ( ) ( 3 2)[ ( 3 2)],

 gl g l

2 2 3 2 2 3 12 2 2 2 2 2

gl

It is evident, that when we substitute 1 = 1 in the first case and 2 = 1 in the second case, we arrive at undeformed configuration of the membrane, so the relaxed energy is fully consistent.

Forces driving tissue deformation are strongly localized and in general are compressive. Force

 g l

() 1 ( ) 2 [ 2 ( 3 2)]. *<sup>W</sup>*

= = + -+

 l

<sup>1</sup> *<sup>W</sup>* ( ) ( 3 2) [ ( 3 2)].

 gl g l

1 1 3 1 1 3 11 2 1 1

gl

Since is obviously symmetric in its arguments, in the case of uniaxial tension in the second

 g l

() 1 ( ) 2 [ 2 ( 3 2)]. *<sup>W</sup>*

= = + -+

1

l

 l

l

The correspondent tension is defined by

t l

principal direction, we would have

t l

must possess those properties.

l l() . l

<sup>2</sup> > 1, we finally get

<sup>=</sup> (54)

 l

= -+ + -+ (55)

 l

 l

> l

¶ (58)

= -+ + -+ (57)

¶ (56)

1 1


**Figure 6.** Single-cell model (left) and adherens junction model (right): diagonals of the top hexagon imitate actin fibres, red areas between cells imitate junction bonds.

**Figure 7.** Apical constriction of initially flat tissue. It consists from 5050 elements or cells and has 529,245 DOFs.

**Figure 8.** von Mises stress distribution in deformed configurations.

**Figure 9.** Apical constriction of initially cylindrical tissue. It consists from 5975 elements or cells and has 621,375 DOFs.

**Figure 10.** von Mises stress distribution in deformed configurations.

Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations... http://dx.doi.org/10.5772/65027 155

**Figure 11.** Apical constriction of initially spherical tissue. It consists from 3126 elements or cells and has 325,245 DOFs.

Elements of the middle part of the rectangular tissue are compressed in apical sides to imitate apical constriction in cells. Increasing the compressing stresses, the tissue is bent and a blastershaped pattern is formed as shown in **Figure 7** (right). The quantitative picture of the stresses arising in the tissue is drawn in **Figure 8**.

**Figure 12.** von Mises stress distribution in deformed configurations.

Next, we consider a cylindrical shell to imitate ventral furrow (generally all tubular patterns) formation. Elements of the top part of the cylinder are constrained in the apical sides by compressing the links standing for apical fibres (see **Figure 9**). Increasing the compressing stresses, the ventral furrow formation is simulated similar to stages presented in **Figure 1**. The quantitative picture of the stresses arising in the tissue is drawn in **Figure 10**.

Finally, a spherical shell is simulated. Cells at the top of a semi-sphere are constrained in apical sides and depending on values of compressing stresses various stages of blastopore formation in archenteron can be described (see **Figure 11** (right)). The quantitative picture of the stresses arising in the tissue is drawn in **Figure 12**.

## **6. Conclusions**

**Figure 7.** Apical constriction of initially flat tissue. It consists from 5050 elements or cells and has 529,245 DOFs.

**Figure 9.** Apical constriction of initially cylindrical tissue. It consists from 5975 elements or cells and has 621,375 DOFs.

**Figure 8.** von Mises stress distribution in deformed configurations.

154 Perusal of the Finite Element Method

**Figure 10.** von Mises stress distribution in deformed configurations.

Analysis based on lower semi-continuous energy functionals reveals real three-dimensional deformations of soft epithelial tissues. Having different forms for different deformation regimes, such as compressive stresses (short deformations), uniaxial tensions collinear to principal directions of the first fundamental form, no stretching or pure bending stresses (isometric deformations) and stretching stresses (large deformations), the resulting total energy is lower semi-continuous, so the existence of its minimizers, that is, real deformations, is ensured. Particular energy density functions are chosen and the explicit form of the total energy functional is obtained, thereby the discretization is made easy.

On the basis of obtained energy functional, a three-dimensional discretized model of epithelial tissues undergoing combined stretching and bending deformations is constructed. Discretization elements correspond to single cells forming the tissue. Actin fibres and cell-cell adhesion links, mainly contributing on the tissue energy, are explicitly embedded in elements. Deformations characteristic to specific embryonic tissues (ventral furrow, neutral tube, neurosphere) observed earlier are described quantitatively increasing contractile stresses in fibres.

## **Dedication**

We thankfully dedicate the chapter to the blessed memory of our good fellow and colleague, a candidate of physical and mathematical sciences, Hamlet V. Hovhannisyan (1956–2016), who unexpectedly died before he could realize his best scientific ideas.

## **Acknowledgements**

The theoretical part of the chapter is investigated under the guidance of Doctor, Professor Benedikt Wirth, Institute for Computational and Applied Mathematics, University of Münster, whom we are heartily thankful. The work of As. Kh. and S. O. was made possible in part by a research grant from the Armenian National Science and Education Fund (ANSEF) based in New York, NY, USA.

## **Nomenclature**



## **Author details**

principal directions of the first fundamental form, no stretching or pure bending stresses (isometric deformations) and stretching stresses (large deformations), the resulting total energy is lower semi-continuous, so the existence of its minimizers, that is, real deformations, is ensured. Particular energy density functions are chosen and the explicit form of the total

On the basis of obtained energy functional, a three-dimensional discretized model of epithelial tissues undergoing combined stretching and bending deformations is constructed. Discretization elements correspond to single cells forming the tissue. Actin fibres and cell-cell adhesion links, mainly contributing on the tissue energy, are explicitly embedded in elements. Deformations characteristic to specific embryonic tissues (ventral furrow, neutral tube, neurosphere)

We thankfully dedicate the chapter to the blessed memory of our good fellow and colleague, a candidate of physical and mathematical sciences, Hamlet V. Hovhannisyan (1956–2016), who

The theoretical part of the chapter is investigated under the guidance of Doctor, Professor Benedikt Wirth, Institute for Computational and Applied Mathematics, University of Münster, whom we are heartily thankful. The work of As. Kh. and S. O. was made possible in part by a research grant from the Armenian National Science and Education Fund (ANSEF) based in

observed earlier are described quantitatively increasing contractile stresses in fibres.

energy functional is obtained, thereby the discretization is made easy.

unexpectedly died before he could realize his best scientific ideas.

**Dedication**

156 Perusal of the Finite Element Method

**Acknowledgements**

New York, NY, USA.

**Nomenclature**

⋅ Inner (dot) product , Lamé coefficients

1, 2 Principal stretches

ℎ Rescaled three-dimensional total energy

(Ω) Measure of bounded open set Ω

∇ Gradient (nabla) operator

Ara S. Avetisyan, Asatur Zh. Khurshudyan\* and Sergey K. Ohanyan

\*Address all correspondence to: khurshudyan@mechins.sci.am

Department of Dynamics of Deformable Systems and Connected Fields, Institute of Mechanics, National Academy of Sciences of Armenia, Yerevan, Armenia

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#### **Simulating Contact Instability in Soft Thin Films through Finite Element Techniques Simulating Contact Instability in Soft Thin Films through Finite Element Techniques**

Jayati Sarkar, Hemalatha Annepu and Jayati Sarkar, Hemalatha Annepu and Satish Kumar Mishra

Satish Kumar Mishra

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65357

## **Abstract**

When a thin film of soft elastic material comes in contact with an external surface, contact instability triggered by interaction forces, such as van der Waals, engenders topologi‐ cally functionalized surfaces. Innumerable technological applications such as adhesives; microelecromechanical systems (MEMS), and nanoelectromechanical systems (NEMS) demand understanding of the physics behind the mechanical contact, relationship between the morphologies, and detachment forces in such films. Indentation tests are important experimental approach toward this; there also exist many simulation procedures to model the mechanical contact. Both atomistic level and analytical continuum simulations are computationally expensive and are restricted by the domain geometries that can be handled by them. Polymeric films also particularly demonstrate a rich variety of nonlinear behavior that cannot be adequately captured by the aforementioned methods. In this chapter we show how finite element techniques can be utilized in crack opening and in contact‐instability problems.

**Keywords:** ABAQUS, thin films, patterned substrate, VDLOAD, contact instability

## **1. Introduction**

In nature, there are several examples such as lotus leaves, peacock's feathers, butterfly wings, gecko feet, and moths' eyes where meso‐nano‐scaled physical patterns on the surface impart characteristic properties such as hydrophobicity, structural color, adhesiveness, and optical properties, which are absent otherwise, to the surfaces. To engineer such surfaces artificially and to fabricate such features, there are mainly two most commonly used approaches: one a

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

top‐down approach and the other a bottom‐up approach. The top‐down approach consists of the various lithographic techniques such as electron beam lithography, dip‐pen lithography, embossing lithography, laser lithography [1], laser photoablation [2], ion implantation [3], micromachining [4], etc. The bottom‐up approach, on the other hand, consists of self‐assembly of molecules to produce bigger building blocks to a growing structure as in chemical vapor deposition [5]. While the self‐assembly procedure involves numerous steps, lithographic methods such as dip‐pen lithography, which are used for creating patterns on hard silicon surfaces, are time‐consuming, involve great cost and may not at all be suitable to create patterns on soft surfaces such as polymers. Self‐organization methods, on the other hand, are for bulk production of desired structures involving lower cost and therefore are more potentially applicablefabricationtechniquesonsoft surfaces.Inself‐organizationtechnique,materials such as polymeric films, which have highly tunable mechanical properties, can be used to obtain a desired mesoscopic structure due to the influence of internal or external forces. The mechanical and interaction properties of these films can be controlled by the amount of cross‐linker added during the synthesis step itself. The instabilities during self‐organization can be harnessed to engineer desired periodic arrangements of sub‐micron features in technological applications such as in flexible electronics [6], optoelectronics [7], functional coatings [8], sensors [9], microfluidic devices, microelecromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) [10], pressure‐sensitive adhesives [11], high‐efficiency light‐emitting diodes, thin‐filmtransistors,etc.Thus,thesethinfilmsaremodel‐mesoscopicsystemsforunderstanding physicalphenomenasuchas tunableadhesion,adhesiondebonding,patternformation,wetting dewetting and friction at soft interfaces, etc.

When a thin elastic film (shear modulus of <1 MPa) is brought in contact proximity of an external contactor, it forms miniaturized surface patterns because of a competition between the destabilizing interaction forces between the film and the contactor and the restoring elastic forces present in the bulk of the elastic film. The wavelengths of instabilities scale as 4 h in the case of peeling geometry [12–14] (whereas glass slide rests on a rigidly bonded film in cantilever configuration) and application of normal force at the end of the glass slide leads to well‐defined one‐dimensional (1D) finger‐like patterns at the crack opening and 3 h in the case of adhesive geometry [15–26]. In the adhesive‐contact geometry, the free surface of the film in contact proximity with an external contactor spontaneously roughens to form a two‐dimen‐ sional (2D) labyrinthic pattern when the stiffness of the attractive forces due to van der Waals (VDW) interactions (arising because of the contactor) exceeds the elastic stiffness of the film. This pattern remained unchanged in experiments either when the upper glass slide was silanized with a chemisorbed layer of fluorinated monochlorodecyl silane [15] or when the interactions were switched to electrostatic ones [21, 24, 25]. In recent past, much effort has been devoted to understand how further miniaturization of length scales and hence enhanced functional properties can be made technologically feasible in these already short wavelength exhibiting adhesive soft elastic thin films in contact geometry. From a fabrication point of view, the smallest patterns realized till date at such soft interfaces are ~ 0.1 h and are obtained with elastic bilayers when the surface energies of the films are very low and the top film is considered to be much thinner and of very small shear modulus compared to the bottom film [27, 28]. But pattern formation in elastic bilayers through squeezing modes of instabilities has the disad‐ vantage of delaminating at the film-film interface. To alleviate this problem, one has found that films cast on patterned substrates can give rise to surface patterns, which are at length scales that are about an order of magnitude less than that found in films cast on flat substrates [29–31].

Apart from direct experiments, these systems are well analyzed both theoretically and numerically. In both cases, atomistic scale models can be considered but are computationally expensive and are restricted by the domain geometries that can be handled by them. For elastic films, linear stability analysis is generally carried out to understand the critical length scales at the incipience of instability by perturbing Navier's equations with bifurcation modes and obtaining the nontrivial solutions [16, 17]. In nonlinear analysis [18–20], on the other hand, it is possible to find the surface morphology by trying to find the equilibrium displacements that minimize the total energy, which is composed of the interaction, and the elastic energies. Finding the optimized Fourier coefficients, which define the top surface, with the help of conjugate-gradient method, does this. However, for elastic films cast on generally patterned substrates the obtained equations show highly nonlinear behavior, which cannot be solved by the simplistic analytical methods or even by the energy-minimizing schemes developed for the base case. On top of these, polymeric films particularly demonstrate a rich variety of nonlinear behavior ranging from viscoelasticity to plasticity that cannot be adequately captured by the aforementioned methods. Finite element analysis (FEA) methods can be used to alleviate these problems. FEA methods have been used extensively in recent years to study crack opening, crack propagation [32–34], delamination [35], film instability over soft substrates, and bifurcation analysis of elastic contact instability [36–39].

In this chapter we show how to implement cohesive-zone modeling (CZM) for simple shear and normal modes of crack opening in the finite element software and how to simulate contact instability and adhesion debonding in soft thin films cast on complex geometries.

## **2. Cohesive-zone modeling**

top‐down approach and the other a bottom‐up approach. The top‐down approach consists of the various lithographic techniques such as electron beam lithography, dip‐pen lithography, embossing lithography, laser lithography [1], laser photoablation [2], ion implantation [3], micromachining [4], etc. The bottom‐up approach, on the other hand, consists of self‐assembly of molecules to produce bigger building blocks to a growing structure as in chemical vapor deposition [5]. While the self‐assembly procedure involves numerous steps, lithographic methods such as dip‐pen lithography, which are used for creating patterns on hard silicon surfaces, are time‐consuming, involve great cost and may not at all be suitable to create patterns on soft surfaces such as polymers. Self‐organization methods, on the other hand, are for bulk production of desired structures involving lower cost and therefore are more potentially applicablefabricationtechniquesonsoft surfaces.Inself‐organizationtechnique,materials such as polymeric films, which have highly tunable mechanical properties, can be used to obtain a desired mesoscopic structure due to the influence of internal or external forces. The mechanical and interaction properties of these films can be controlled by the amount of cross‐linker added during the synthesis step itself. The instabilities during self‐organization can be harnessed to engineer desired periodic arrangements of sub‐micron features in technological applications such as in flexible electronics [6], optoelectronics [7], functional coatings [8], sensors [9], microfluidic devices, microelecromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) [10], pressure‐sensitive adhesives [11], high‐efficiency light‐emitting diodes, thin‐filmtransistors,etc.Thus,thesethinfilmsaremodel‐mesoscopicsystemsforunderstanding physicalphenomenasuchas tunableadhesion,adhesiondebonding,patternformation,wetting

When a thin elastic film (shear modulus of <1 MPa) is brought in contact proximity of an external contactor, it forms miniaturized surface patterns because of a competition between the destabilizing interaction forces between the film and the contactor and the restoring elastic forces present in the bulk of the elastic film. The wavelengths of instabilities scale as 4 h in the case of peeling geometry [12–14] (whereas glass slide rests on a rigidly bonded film in cantilever configuration) and application of normal force at the end of the glass slide leads to well‐defined one‐dimensional (1D) finger‐like patterns at the crack opening and 3 h in the case of adhesive geometry [15–26]. In the adhesive‐contact geometry, the free surface of the film in contact proximity with an external contactor spontaneously roughens to form a two‐dimen‐ sional (2D) labyrinthic pattern when the stiffness of the attractive forces due to van der Waals (VDW) interactions (arising because of the contactor) exceeds the elastic stiffness of the film. This pattern remained unchanged in experiments either when the upper glass slide was silanized with a chemisorbed layer of fluorinated monochlorodecyl silane [15] or when the interactions were switched to electrostatic ones [21, 24, 25]. In recent past, much effort has been devoted to understand how further miniaturization of length scales and hence enhanced functional properties can be made technologically feasible in these already short wavelength exhibiting adhesive soft elastic thin films in contact geometry. From a fabrication point of view, the smallest patterns realized till date at such soft interfaces are ~ 0.1 h and are obtained with elastic bilayers when the surface energies of the films are very low and the top film is considered to be much thinner and of very small shear modulus compared to the bottom film [27, 28]. But pattern formation in elastic bilayers through squeezing modes of instabilities has the disad‐

dewetting and friction at soft interfaces, etc.

164 Perusal of the Finite Element Method

Most of the failures in engineering structures are the result of cracks that are preexisting (during production) or the cracks that are formed in service. Continuum mechanics calculations fail to handle such cracks and these are handled by fracture mechanics, which primarily studies resistance of a material to fracture. When a crack is formed, stress is usually not transmitted between the two "virtual" surfaces that form the following crack formation. We can consider a cohesive zone, of zero thickness, to bridge these two virtual surfaces and it can be suggested that fracture happens when material strength in this zone decays. The basic premise for cohesive-zone model is that infinite stresses (stress singularity) at crack tip are unrealistic. Dugdale [40] and Barenblatt [41] developed the first models to address this shortcoming and model brittle fracture. Their models divide the crack tip into stress-free region and cohesively stressed regions. More than a decade later, Hillerborg [42] was the first one to use finite element analysis for cohesive-zone modeling of brittle fracture. Tvergaard's [43–44] modification to Dugdale and Barenblatt's model ensured that continuum elements remain intact in the cohesive model. It is their first decohesion model where traction is dependent on both normal and tangential separation, whereas in Barenblatt's model, it is dependent on the distance from the crack tip. They do this by defining the cohesive elements between the continuum elements such that damage initiation is characterized by opening of the cohesive elements (and governed by some material law) and at failure these elements lose their stiffness, resulting in the continuum elements disjointing from each other. This model is also called the cohesive‐zone model or damage zone model (DZM). The initiation and evolution of the damage are defined according to material‐softening law/cohesive law/bridging law/traction‐ separation law (TSL). TSL can be defined based on the properties of the cohesive zone. Traction‐ separation laws are not universal, and these are not physical laws to characterize fracture at bonded interfaces. Rather, they are highly material dependent. We can define the TSLs in many ways but all separation laws have one thing in common: they all suggest that the traction initially increases with separation and beyond a critical separation distance it diminishes to zero (material softening) at which point these cohesive surfaces detach from each other and a real crack tip is defined.

For implementing TSL in CZM, two constitutive relations are considered: one for the contin‐ uum or the "bulk behavior" of the blocks and another for the gap between the surfaces that is defined by the traction‐displacement relationships. If the cohesive zone is a thin adhesive layer bonding the blocks, we can define the TSL as the traction at this interface with respect to the separation at the interface. In the modeling by commercial finite element software ABAQUS, the cohesive zone is represented by a single layer of elements that connect the two blocks/ surfaces and is constrained by one of the blocks as can be seen from **Figure 1**. Each element on the surface of one block is "tied" with its corresponding neighboring element in the other block through a series of "inelastic springs" which govern the interaction between these two surfaces. This tie constraint prevents the propagation of singular modes in deformation that can arise due to lack of membrane stiffness. The debonding/crack‐opening process is akin to the failure of these springs.

**Figure 1.** Meshing with tie constraints for cohesive elements in ABAQUS.

#### **2.1. Cohesive zone implementation in ABAQUS with an example problem**

To show the implementation of cohesive zone model with traction‐separation law in ABAQUS, we have provided an example problem [43] below. In this, we study the propagation of a crack in a material block. A debonding model is taken from Tvergaard's work [43] and is described in brief below. This model does not consist of a debonding potential, but is designed to address the effect of fiber debonding from a metal‐matrix composite on the tensile properties (as this requires one to consider both normal and tangential separations in the debonding). For this, first, a function *F*(*λ*) was chosen such that

$$F\left(\mathcal{X}\right) = \frac{27}{4}\sigma\_{\text{max}}\left(1 - 2\mathcal{X} + \mathcal{X}^2\right) \text{ and} \\ 0 \le \mathcal{X} \le 1 \tag{1}$$

The nondimensional parameter *λ* is defined as

both normal and tangential separation, whereas in Barenblatt's model, it is dependent on the distance from the crack tip. They do this by defining the cohesive elements between the continuum elements such that damage initiation is characterized by opening of the cohesive elements (and governed by some material law) and at failure these elements lose their stiffness, resulting in the continuum elements disjointing from each other. This model is also called the cohesive‐zone model or damage zone model (DZM). The initiation and evolution of the damage are defined according to material‐softening law/cohesive law/bridging law/traction‐ separation law (TSL). TSL can be defined based on the properties of the cohesive zone. Traction‐ separation laws are not universal, and these are not physical laws to characterize fracture at bonded interfaces. Rather, they are highly material dependent. We can define the TSLs in many ways but all separation laws have one thing in common: they all suggest that the traction initially increases with separation and beyond a critical separation distance it diminishes to zero (material softening) at which point these cohesive surfaces detach from each other and a

For implementing TSL in CZM, two constitutive relations are considered: one for the contin‐ uum or the "bulk behavior" of the blocks and another for the gap between the surfaces that is defined by the traction‐displacement relationships. If the cohesive zone is a thin adhesive layer bonding the blocks, we can define the TSL as the traction at this interface with respect to the separation at the interface. In the modeling by commercial finite element software ABAQUS, the cohesive zone is represented by a single layer of elements that connect the two blocks/ surfaces and is constrained by one of the blocks as can be seen from **Figure 1**. Each element on the surface of one block is "tied" with its corresponding neighboring element in the other block through a series of "inelastic springs" which govern the interaction between these two surfaces. This tie constraint prevents the propagation of singular modes in deformation that can arise due to lack of membrane stiffness. The debonding/crack‐opening process is akin to the failure

real crack tip is defined.

166 Perusal of the Finite Element Method

of these springs.

**Figure 1.** Meshing with tie constraints for cohesive elements in ABAQUS.

**2.1. Cohesive zone implementation in ABAQUS with an example problem**

To show the implementation of cohesive zone model with traction‐separation law in ABAQUS, we have provided an example problem [43] below. In this, we study the propagation of a crack

$$\mathcal{A} = \sqrt{\left(\frac{\mu\_n}{\mathcal{S}\_n}\right)^2 + \left(\frac{\mu\_t}{\mathcal{S}\_t}\right)^2} \tag{2}$$

For a monotonically increasing value of the tractions are <sup>=</sup> and <sup>=</sup> . Here, the normal and tangential tractions are and respectively, normal and tangential displace‐ ments are and respectively, δ and δ are the maximum separation distances in normal and traction modes, respectively. Here, we have considered that the two virtual surfaces that form after a crack growth are of unit dimension. For solving this, we have developed our own code using MATLAB and ABAQUS. MATLAB is excellent to call external commands to run the PYTHON scripts for generating the input files for ABAQUS.

ABAQUS has vast libraries of constitutive material models but this library is neither a complete one nor a completely flexible one. However, fortunately, user‐defined constitutive material models can be implemented in ABAQUS through either UMAT or VUMAT. UMAT is an implicit time integration method (used with ABAQUS/Standard) and it requires the material stiffness matrix as input for forming the consistent Jacobian (*C*), which is also called algorithmic tangential stiffness or ATS, of the nonlinear equations. For linearly elastic materials, it is defined as

$$C = \frac{\partial \Delta \sigma}{\partial \Delta \varepsilon} \tag{3}$$

where is the increment in Cauchy stress and is the increment in strain.

VUMAT, on the other hand, uses explicit time integration and hence does not require Jacobian matrix. It is implemented in ABAQUS/Explicit and is easier to write. In the case of cohesive zone modeling, we can include element failure and progressive damage failure in the VUMAT subroutine. The VUMAT subroutine consists of the strain decomposition of the total strain into elastic (computed from a linear elastic or hyperelastic constitutive model) and plastic compo‐ nents and flow rule to define the material's plastic behavior. A sample of the input file (filename.inp) and VUMAT subroutine (vumat.f) are annotated and provided at the end of this section. These are run in ABAQUS through the following command:

abaqus job=*filename.inp* user=*vumat.f*

There are a minimum of five parameters that have to be prescribed for the CZM. They are as follows:


Cohesive strength (related to yield stress) and cohesive energy signify the maximum resistance to fracture and dissipation following material separation, respectively. For the cohesive zone modeling with traction‐separation law, we need to specify the initial elastic stiffness (*K*).

**Figure 2.** Cohesive zone model (CZM) in ABAQUS. Loads and boundary conditions for the case of purely normal trac‐ tion (T*<sup>t</sup>* = 0).

**Figure 3.** Normal traction (*Tn*) versus separation (*un*) obtained after CZM in ABAQUS.

We cannot use the exact value of the material's Young's modulus (*E*) for this as this stiffness here refers to a "penalty stiffness," that is, we are free to choose a value that is necessary to prevent interpenetration of the surfaces and for the convergence of our model. One rule of thumb is to take a value of *K* that is equal to *E/h* (*h* is the cohesive layer thickness). Hence, *K* can be taken to be a very large value as the cohesive elements are considered to be a layer of zero thickness. Also, for simplification, identical values of *K* are taken for the crack‐opening mode (mode I) and crack‐shearing modes (mode II). The chosen damage initiation and damage evolution criteria affect how these two modes of crack opening interact with each other.

The variables we output (stress, strain, etc.) depend on the kind of problems being modeled by the cohesive elements. This is indicated in the .inp file itself in the definition of the section properties by specifying the response type required. The stress components are as follows:

11—Direct membrane stress

nents and flow rule to define the material's plastic behavior. A sample of the input file (filename.inp) and VUMAT subroutine (vumat.f) are annotated and provided at the end of

There are a minimum of five parameters that have to be prescribed for the CZM. They are as

Cohesive strength (related to yield stress) and cohesive energy signify the maximum resistance to fracture and dissipation following material separation, respectively. For the cohesive zone modeling with traction‐separation law, we need to specify the initial elastic stiffness (*K*).

**Figure 2.** Cohesive zone model (CZM) in ABAQUS. Loads and boundary conditions for the case of purely normal trac‐

We cannot use the exact value of the material's Young's modulus (*E*) for this as this stiffness here refers to a "penalty stiffness," that is, we are free to choose a value that is necessary to prevent interpenetration of the surfaces and for the convergence of our model. One rule of

, the peak values of the traction‐separation curve).

, usually the separation value corresponding to the cohesive

this section. These are run in ABAQUS through the following command:

**2.** Cohesive energy (area under the traction‐separation curve).

**Figure 3.** Normal traction (*Tn*) versus separation (*un*) obtained after CZM in ABAQUS.

abaqus job=*filename.inp* user=*vumat.f*

168 Perusal of the Finite Element Method

**1.** Cohesive strength (*Tn* and *Tt*

**3.** Characteristic length (*un* and *ut*

strength).

follows:

tion (T*<sup>t</sup>* = 0).


**Figure 4.** Cohesive zone model (CZM) in ABAQUS. Loads and boundary conditions shown for the case of purely shear traction (T*n*= 0).

**Figure 5.** Tangential traction (*Tt* ) versus shear separation (*ut* ) obtained from CZM in ABAQUS.

During a purely normal crack opening ( = 0), according to our TSL, the normal traction initially increases with the normal separation , reaches a maximum value max and reduces to 0 when <sup>=</sup> that is, when the final separation occurs (see **Figures 2** and **3**). Similarly, during a purely shear crack opening ( = 0), the shear traction initially increases with tangential separation , reaches a maximum value max and then at <sup>=</sup> the final separation occurs (see **Figures 4** and **5**). The negative values in **Figure 5** indicate compressive stresses.

### **2.2. Writing ABAQUS .inp file and VUMAT subroutine**

## *2.2.1. Snippet of input file*

Snippet of input file (filename.inp) containing the geometry, meshing, material parameters, and output variables

\*Heading

```
** Job name: cohesive‐zone Model name: Model‐1
```

```
** Generated by: Abaqus/CAE 6.9‐1
```

```
*Preprint, echo=NO, model=NO, history=NO, contact=NO
```
\*\* PARTS

```
*Part, name=Part‐1
```
\*End Part

\*\*

```
** ASSEMBLY
```
\*\*

```
*Assembly, name=Assembly
```
\*\*

```
*Instance, name=Part‐1‐1, part=Part‐1 ***(two unit blocks/surfaces, each with 4 nodes)
```
\*Node

⋮

```
*Element, type=CPE4R***(4 noded continuum plane-strain elements for the blocks)
```
⋮

```
*Element, type=COH2D4 ***(4 noded cohesive elements for the cohesive zone)
```
⋮

\*Nset, nset=\_PickedSet*<sup>i</sup>* , internal, generate

⋮

```
*Elset, elset=_PickedSeti
                       , internal
⋮
** Section: Section‐1
*Solid Section, elset=_PickedSeti
                               , material=steel
,
** Section: Section‐2
*Cohesive Section, elset=_PickedSeti
                                     , material=coh, response=TRACTION SEPARATION
***(specifying response as TRACTION SEPARATION to implement TSL)
*End Instance
**
⋮
*Surface, type=ELEMENT, name=_PickedSurf21, internal
__PickedSurf21_S3, S3
*End Assembly
*Amplitude, name=Amp‐1 ***(time dependent loading conditions)
0., 0., 0.5, 0.5, 1., 1.
t=0 amp_at_t=0 t=0.5 amp_at_t=0.5 t=1 amp_at_t=1
**
** MATERIALS
**
*Material, name=coh ***(cohesive zone parameters)
*User material, Constants = 5 ***(user defined subroutine with 5 parameters)
200.0,0.25,0.25,1.0,0.5 * * *(, , 
                                     ,  .   these are read through props( ) matrix in vumat.f)
*Density
5.4e‐7
*Depvar
7, ***(number of solution-dependent variables SDVs)
*Initial conditions, type=solution
Part‐1‐1._PickedSet4,0.0,0.0,0,0.0,0.0,0.0,0.0 ***(initial values of the 7 SDVs)
*Material, name=material‐1 ***(material properties of the continuum surfaces)
```
to 0 when <sup>=</sup> that is, when the final separation occurs (see **Figures 2** and **3**). Similarly,

separation occurs (see **Figures 4** and **5**). The negative values in **Figure 5** indicate compressive

Snippet of input file (filename.inp) containing the geometry, meshing, material parameters,

\*Instance, name=Part‐1‐1, part=Part‐1 *\*\*\*(two unit blocks/surfaces, each with 4 nodes)*

\*Element, type=CPE4R*\*\*\*(4 noded continuum plane-strain elements for the blocks)*

\*Element, type=COH2D4 *\*\*\*(4 noded cohesive elements for the cohesive zone)*

, internal, generate

, reaches a maximum value max and then at <sup>=</sup>

initially increases with

the final

during a purely shear crack opening ( = 0), the shear traction

**2.2. Writing ABAQUS .inp file and VUMAT subroutine**

\*\* Job name: cohesive‐zone Model name: Model‐1

\*Preprint, echo=NO, model=NO, history=NO, contact=NO

\*\* Generated by: Abaqus/CAE 6.9‐1

tangential separation

170 Perusal of the Finite Element Method

*2.2.1. Snippet of input file*

and output variables

stresses.

\*Heading

\*\* PARTS

\*End Part

\*\* ASSEMBLY

\*Assembly, name=Assembly

\*Nset, nset=\_PickedSet*<sup>i</sup>*

\*\*

\*\*

\*\*

⋮

⋮

⋮

⋮

\*Node

\*Part, name=Part‐1

```
*Density
7.8e‐06,
*Elastic
2.5e+8, 0.25***(Young's modulus E and Poisson's ratio of Material-1)
** ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐*
** STEP: Step‐1
**
*Step, name=Step‐1
*Dynamic, Explicit
, 1.
*Bulk Viscosity
0.06, 1.2
** Mass Scaling: Semi‐Automatic
** Whole Model
*Fixed Mass Scaling, factor=50.
**
** BOUNDARY CONDITIONS
**
** Name: BC‐2 Type: Displacement/Rotation
*Boundary
_PickedSet17, 2, 2
** Name: BC‐3 Type: Displacement/Rotation
*Boundary
_PickedSet18, 1, 1
** Name: BC‐4 Type: Displacement/Rotation
*Boundary, amplitude=Amp‐1
_PickedSet19, 2, 2, 1.
**
** LOADS
**
```
\*\* Name: Load‐1 Type: Pressure

\*Dsload, amplitude=Amp‐3

\_PickedSurf21, P,50. *\*\*\*(Surface-2 of and permitted to have maximum displacement 0.5 in xdirection i.e. =0.5)*

\*\*

\*Density 7.8e‐06, \*Elastic

172 Perusal of the Finite Element Method

\*\* STEP: Step‐1

\*Bulk Viscosity

\*\* Whole Model

\*\* Mass Scaling: Semi‐Automatic

\*Fixed Mass Scaling, factor=50.

\*\* BOUNDARY CONDITIONS

\*\* Name: BC‐2 Type: Displacement/Rotation

\*\* Name: BC‐3 Type: Displacement/Rotation

\*\* Name: BC‐4 Type: Displacement/Rotation

\*Boundary, amplitude=Amp‐1

\*Step, name=Step‐1 \*Dynamic, Explicit

\*\*

, 1.

\*\*

\*\*

\*Boundary

\*Boundary

\*\*

\*\*

\*\* LOADS

\_PickedSet17, 2, 2

\_PickedSet18, 1, 1

\_PickedSet19, 2, 2, 1.

0.06, 1.2

2.5e+8, 0.25*\*\*\*(Young's modulus E and Poisson's ratio of Material-1)*

\*\* ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐\*

\*\* OUTPUT REQUESTS

\*\*

\*Restart, write, number interval=1, time marks=NO

\*\*

\*\* FIELD OUTPUT: F‐Output‐2

\*\*

\*Output, field,op=new,number interval=50

\*Node Output

RF, U *\*\*\*(output variables at the nodes: reaction forces and displacements)*

\*Element Output, directions=YES

S, SDV *\*\*\*(output variables over the elements: stresses and SDVs which are defined in the .f file of VUMAT)*

\*\*

\*\* FIELD OUTPUT: F‐Output‐1

\*\*

\*Output, field, variable=PRESELECT

\*\*

\*\* HISTORY OUTPUT: H‐Output‐1

\*\*

\*Output, history, variable=PRESELECT

\*End Step

*2.2.2. Snippet of vumat.f subroutine*

The headers of this subroutine are common to every VUMAT and can be obtained from ABAQUS documentation. We have listed below only portions of the user‐defined calculations for the properties.

*! Reading material properties specified in abaqus input file through props() array* = props(1) *! max nominal stress at which damage is initiated* = props(2) *! max normal displacement at which the material fails and Tn=0* = props(3) *! max tangential displacement at which the material fails and Tt=0*

 $a = \text{props(4)}$   $\left[ \text{pparameter in } T\_t = a \frac{u\_t}{\delta\_t} F(\lambda) \right]$ 

mu = props(5)

do ii = 1,nblock

*! Reading the old state variables or SDVs*.

= stateOld(ii,1) *! normal displacement*

= stateOld(ii,2) *!tangential displacement*

counter = stateOld(ii,3)

 = stateOld(ii,4)*!tangential traction*

= stateOld(ii,5)*!normal traction*

*! Reading the strain Increments*.

*! In VUMAT shear components are stored as tensor components and not as engineering components*.

*! so* Δ  *will be twice of tensorial shear strain*.

```
Δ = strainInc(ii,1)
```
Δ = 2.0\*strainInc(ii,2)

*!Depending on pure normal traction or pure shear traction or mixed mode of debonding, calculate the following:*

$$\begin{array}{rcl} \mathcal{A} & = & \cdots & \cdots \\\\ T\_{\kappa, \text{new}} & = & \frac{\mathcal{U}\_{\kappa}}{\delta\_{\kappa}} F(\mathcal{A}) \end{array}$$

$$T\_{\iota, \text{new}} \quad = \quad \alpha \frac{\mathcal{U}\_{\iota}}{\delta\_{\iota}} F(\mathcal{A})$$

Simulating Contact Instability in Soft Thin Films through Finite Element Techniques http://dx.doi.org/10.5772/65357 175

$$\begin{split} \boldsymbol{\sigma}\_{11} &= \left(\bigvee\_{\boldsymbol{\beta}\boldsymbol{\lambda}\_{n}} \right) \Bigg( \boldsymbol{F}\left(\boldsymbol{\lambda}\right) + \boldsymbol{u}\_{n} \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{u}\_{n}} \Bigg); \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{u}\_{n}} = \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{\lambda}} \frac{\partial \boldsymbol{\lambda}}{\partial \boldsymbol{u}\_{n}} \\ \boldsymbol{\sigma}\_{12} &= \left(\bigvee\_{\boldsymbol{\beta}\boldsymbol{\lambda}\_{n}} \right) \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{u}\_{i}}; \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{u}\_{i}} = \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{\lambda}} \frac{\partial \boldsymbol{\lambda}}{\partial \boldsymbol{u}\_{i}} \\ \boldsymbol{\sigma}\_{13} &= \boldsymbol{\alpha} \Big(\boldsymbol{u}\_{\boldsymbol{\beta}} \Big/ \boldsymbol{F}\left(\boldsymbol{\lambda}\right) \Bigg) \\ \boldsymbol{\sigma}\_{14} &= \left(\boldsymbol{\alpha} \Big{\boldsymbol{\beta}}\_{1}\right) \Big(\boldsymbol{F}\left(\boldsymbol{\lambda}\right) + \boldsymbol{u}\_{i} \frac{\partial \boldsymbol{F}\left(\boldsymbol{\lambda}\right)}{\partial \boldsymbol{u}\_{i}} \Bigg) \end{split}$$

*!Increment in stress*

*! Reading material properties specified in abaqus input file through props() array*

= props(2) *! max normal displacement at which the material fails and Tn=0*

= props(3) *! max tangential displacement at which the material fails and Tt=0*

*! In VUMAT shear components are stored as tensor components and not as engineering*

*!Depending on pure normal traction or pure shear traction or mixed mode of debonding,*

,new

*<sup>u</sup> T F*

=

=

*<sup>u</sup> T F*

=

*n*

l

*t*

,new

( )

l

*n*

LLL

*n t*

d

a

*t*

d

( )

 l

= props(1) *! max nominal stress at which damage is initiated*

 

*components*.

Δ = strainInc(ii,1)

= 2.0\*strainInc(ii,2)

*calculate the following:*

*! so* Δ

Δ

mu = props(5)

do ii = 1,nblock

counter = stateOld(ii,3)

= props(4) *!parameter in*  <sup>=</sup>

174 Perusal of the Finite Element Method

*! Reading the old state variables or SDVs*.

= stateOld(ii,1) *! normal displacement*

= stateOld(ii,4)*!tangential traction*

 *will be twice of tensorial shear strain*.

= stateOld(ii,5)*!normal traction*

*! Reading the strain Increments*.

= stateOld(ii,2) *!tangential displacement*

$$\begin{aligned} \Delta T\_{\boldsymbol{u}} &= (\sigma\_{11} \, ^\ast \Delta \boldsymbol{u}\_{\boldsymbol{n}}) + (\sigma\_{12} \, ^\ast \Delta \boldsymbol{u}\_{\boldsymbol{\iota}}), \\ \Delta T\_{\boldsymbol{\iota}} &= (\sigma\_{21} \, ^\ast \Delta \boldsymbol{u}\_{\boldsymbol{n}}) + (\sigma\_{22} \, ^\ast \Delta \boldsymbol{u}\_{\boldsymbol{\iota}}). \end{aligned}$$

*! stress update*

```
stressNew(ii,1) = stressOld(ii,1)+ 
stressNew(ii,2) = stressOld(ii,2)+ 
! state variable update
stateNew(ii,1) = stateOld(ii,1) + 
stateNew(ii,2) = stateOld(ii,2) + 
stateNew(ii,3) = stateOld(ii,3) + 1
stateNew(ii,4) = 
stateNew(ii,5) = 
enddo
return
end
```
## **3. Elastic contact instability through FEM simulations**

**Figure 6A** and **B** shows the schematic setup of a soft incompressible initially stress‐free elastic film of shear modulus <1 MPa, mean thickness *h*, and lateral length *L*. It is cast over a rough substrate. To demonstrate the effect of roughness here, we have chosen a substrate with a uniform sawtooth profile characterized by its amplitude ℎ and wavelength = / where is the number of substrate troughs or crests. In the asymptotic case when ℎ 0 or / ∞, we get the classical case of a smooth substrate, though in real systems it is almost impossible to come across a substrate that is atomistically smooth and almost all surfaces have inherent roughness. Again in real situations mechanical/thermal noises and vibrations almost always disturb the film surface at the air interface. To incorporate them for numerical simulations, the film surface is perturbed by sinusoidal disturbances having different frequencies and random amplitudes of the order of 10−3 nm. The mean distance from the free surface of the film and the contactor is termed the gap distance and is represented by the symbol *d* (refer to **Figure 6A**). In simulations, the contactor, as in experiments, is brought near the film surface from a finite distance at an approach velocity ( <sup>1</sup> /) maintained below the critical distance where the instabilities first initiate (adhesion) for around ~15 s. This is necessary to allow the film to have sufficient time to arrange itself under the interplay of imposed force and elasticity of the film. Once the patterns get mature, the debonding phase is started. In the debonding phase, the contactor is pulled off at different velocities (debonding: ). The contactor positions at different times can be visualized from **Figure 6B**. The thickness of the film considered for the simulations is 500 nm and length *L* of the film considered is 8ℎ. The *β* is varied from 0.1 to 0.7 and *n* is varied from 1 to 12 (for *L = 8h*) in the simulations. Though in reality the film is three‐ dimensional (3D) as shown in **Figure 6A**, for ease of computation the simulations are carried out considering a plane‐strain model (as because of normal traction from the contactor the displacements can be considered to be confined in the plane of the paper) and a 2D geometry as shown in **Figure 6B** has been mostly considered. To understand the morphology in greater detail, 3D simulations are also carried out in certain cases with the 3D geometry.

**Figure 6.** Schematic diagram of an elastic thin film case on a (A) 3D sawtooth and (B) 2D sawtooth‐patterned substrate and in contact proximity to an external contactor. The position of the contactor at different time is also shown.

If one considers an infinitesimal control volume inside the elastic film, a force balance will give rise to the following governing thin‐film (Navier's) equation:

$$
\vec{\nabla} \cdot \overleftarrow{\sigma} + \vec{f} = 0 \tag{4}
$$

where, = + ∇ + ∇ is the stress developed inside the film, is the displacement vector, *p* is the pressure across the film, and **I** is the identity matrix. The second term, , represents the body force (per unit volume) that the film experiences because of the contactor. The equation as can be seen is a partial differential equation in the unknown and does not yield an analytical solution in the case when the substrate is patterned or when the film material displays geometric or material nonlinearity. A numerical solution can, however, be obtained when the differential equation is discretized by considering a trial/shape function for the displacement and converted into a set of algebraic equations. In the Galerkin finite element formulation, the residual obtained because of the trial function can now be minimized in a weighted fashion according to

$$\int\_{\Omega} \left(\vec{\nabla} \cdot \vec{\sigma} + \vec{f}\right) \phi d\Omega = 0 \tag{5}$$

where Ω is the discretized domain inside the film and Φ is the shape function. The desired discretization and subsequent simulations to realize the film deformation and morphology due to influence of the external contactor are carried out with the help of the commercial finite element software ABAQUS

#### **3.1. Geometry and mesh creation**

is the number of substrate troughs or crests. In the asymptotic case when ℎ 0 or / ∞, we get the classical case of a smooth substrate, though in real systems it is almost impossible to come across a substrate that is atomistically smooth and almost all surfaces have inherent roughness. Again in real situations mechanical/thermal noises and vibrations almost always disturb the film surface at the air interface. To incorporate them for numerical simulations, the film surface is perturbed by sinusoidal disturbances having different frequencies and random amplitudes of the order of 10−3 nm. The mean distance from the free surface of the film and the contactor is termed the gap distance and is represented by the symbol *d* (refer to **Figure 6A**). In simulations, the contactor, as in experiments, is brought near the film surface from a finite distance at an approach velocity ( <sup>1</sup> /) maintained below the critical distance where the instabilities first initiate (adhesion) for around ~15 s. This is necessary to allow the film to have sufficient time to arrange itself under the interplay of imposed force and elasticity of the film. Once the patterns get mature, the debonding phase is started. In the debonding phase, the contactor is pulled off at different velocities (debonding: ). The contactor positions at different times can be visualized from **Figure 6B**. The thickness of the film considered for the simulations is 500 nm and length *L* of the film considered is 8ℎ. The *β* is varied from 0.1 to 0.7 and *n* is varied from 1 to 12 (for *L = 8h*) in the simulations. Though in reality the film is three‐ dimensional (3D) as shown in **Figure 6A**, for ease of computation the simulations are carried out considering a plane‐strain model (as because of normal traction from the contactor the displacements can be considered to be confined in the plane of the paper) and a 2D geometry as shown in **Figure 6B** has been mostly considered. To understand the morphology in greater

176 Perusal of the Finite Element Method

detail, 3D simulations are also carried out in certain cases with the 3D geometry.

**Figure 6.** Schematic diagram of an elastic thin film case on a (A) 3D sawtooth and (B) 2D sawtooth‐patterned substrate and in contact proximity to an external contactor. The position of the contactor at different time is also shown.

If one considers an infinitesimal control volume inside the elastic film, a force balance will give

*<sup>f</sup>* <sup>0</sup> <sup>r</sup> <sup>r</sup> (4)

Ñ× + = s

rise to the following governing thin‐film (Navier's) equation:

To mimic the physical geometry and to simulate several cases of substrates (having varying frequency and amplitudes) and to incorporate different perturbation conditions at the filmcontactor interface, the graphical user interface in the software proves to be difficult. Python scripting was used instead to generate the multitude of geometries. For the film's lateral edges and the substrate boundary, straight lines are used, while for achieving the perturbed top surface cubic splines are fitted to get a continuous line out of the random sinusoidal fluctuations at the film-contactor interface. Meshing forms an integral part of the preprocessing steps, which helps in discretizing the geometry. The accuracy that can be obtained from any finite element analysis is directly related to the finite element mesh that is used. Generally, for the large deformations in 2D, eight node elements are used but those are computationally intensive. In 2D as discussed earlier for the present scenario one can consider a plain-strain condition, therefore, continuum plain-strain 4 node-reduced integration elements (CPE4R) are used in 2D. For 3D geometry, continuum three-dimensional eight-node-reduced integration (C3D8R) elements are used. To circumvent the problem of artificial stiffness due to shear locking that can arise due to four node elements, a large number of elements are used while maintaining the aspect ratio of the elements close to 1, which also helps in avoiding problems associated with skewness. Further, for the large deformation problem, reduced integration elements are shown to be successfully implemented without impacting the results and reduce the computational load. On the top of the film, the force is much higher and decreases drastically with depth in the film due to the highly nonlinear dependence of force on distance. Thus, it becomes necessary to have high density of mesh toward the top of the film. The geometry of the film in case of a smooth substrate is seeded with a bias ratio to have smaller elements toward the top as shown in **Figure 7A** and **B** (for 2D and 3D geometries, respectively). For sawtooth profile, the geometry is divided into two blocks. The top block has similar seeding as in case of smooth substrate and the bottom is meshed as shown in **Figure 7C** and **D** (for 2D and 3D geometries, respectively). Since the force applied depends on the amplitude of substrate roughness as well, the meshed geometry is tested with the grid independence test for every case and the finalized meshes are then chosen.

**Figure 7.** The meshing done inside the film geometry for (A)&(B) a planar substrate, (C)&(D) for a sawtooth substrate, respectively. Figures on the left and right represent meshing in 2D and 3D respectively.

## **3.2. Boundary conditions**

To imitate the rigid boundary conditions of the base in the simulations, the bottom edge of the film is pinned to the substrate and movement toward either direction is restricted with the help of an Encastre boundary condition. To incorporate periodic boundary conditions on the lateral sides of the film, x‐symmetric boundary conditions are applied.

#### **3.3. Force calculations**

As discussed earlier, so far the failure at the interfaces was captured via a cohesive zone model. In the model, the contactor (a rigid body) and the elastic film were considered to be in intimate contact separated only by a very thin layer of cohesive zone. In the cohesive zone, a traction‐ separation law (inbuilt in ABAQUS) is defined in such a way that the traction initially increases with separation and beyond a critical separation distance it diminishes to zero (material softening) at which point these cohesive surfaces detach from each other. The energy release rate at debonding is given by the area under the curve. The problem with the method is except for the energy all the other parameters are not well characterized and the traction‐separation law is not based on any physical law. Also, while the model is good for peeling/debonding experiments and almost always considers the initial stage as one where the rigid contactor and the film is in intimate contact, it fails to capture the essential morphologies at critical separation distance.

geometry of the film in case of a smooth substrate is seeded with a bias ratio to have smaller elements toward the top as shown in **Figure 7A** and **B** (for 2D and 3D geometries, respectively). For sawtooth profile, the geometry is divided into two blocks. The top block has similar seeding as in case of smooth substrate and the bottom is meshed as shown in **Figure 7C** and **D** (for 2D and 3D geometries, respectively). Since the force applied depends on the amplitude of substrate roughness as well, the meshed geometry is tested with the grid independence test

**Figure 7.** The meshing done inside the film geometry for (A)&(B) a planar substrate, (C)&(D) for a sawtooth substrate,

To imitate the rigid boundary conditions of the base in the simulations, the bottom edge of the film is pinned to the substrate and movement toward either direction is restricted with the help of an Encastre boundary condition. To incorporate periodic boundary conditions on the

As discussed earlier, so far the failure at the interfaces was captured via a cohesive zone model. In the model, the contactor (a rigid body) and the elastic film were considered to be in intimate contact separated only by a very thin layer of cohesive zone. In the cohesive zone, a traction‐ separation law (inbuilt in ABAQUS) is defined in such a way that the traction initially increases with separation and beyond a critical separation distance it diminishes to zero (material softening) at which point these cohesive surfaces detach from each other. The energy release rate at debonding is given by the area under the curve. The problem with the method is except

respectively. Figures on the left and right represent meshing in 2D and 3D respectively.

lateral sides of the film, x‐symmetric boundary conditions are applied.

**3.2. Boundary conditions**

178 Perusal of the Finite Element Method

**3.3. Force calculations**

for every case and the finalized meshes are then chosen.

Thus, to capture both the adhesion and debonding of an elastic film due to the interaction between the film and the contactor, a body‐force approach has been considered, where each node in the film experiences a varying magnitude body force depending on its relative position with the contactor. The most generic interaction between two different materials when it comes in contact proximity is given by van der Waals type of potential. The attractive potential of VDW prevalent between two atoms or molecules separated at a distance *r* can be described as [45]

$$\Psi(r) = -M \,/ \, r^6 \tag{6}$$

where *M* is the coefficient of attraction. However, a molecule in the film experiences attraction from all the molecules in the contactor. Thus, if one considers a ring‐like element in the contactor as shown in **Figure 8A**, the total number of atoms/molecules present in the ring will be 2 12 where denotes the number density of atoms/molecules of the contactor. Hence, the total effective potential that a molecule in the film will experience due to a contactor that is semi‐infinitely thick is laterally unbounded and is situated at a surface‐to‐surface separation distance of *D* which is given by

$$\text{cov}(D) = -2\pi \rho\_c M \int\_{\mathbf{x}\_2 - D}^{\mathbf{x}\_2 - \epsilon} d\mathbf{x}\_2 \int\_{\mathbf{x}\_1 - 0}^{\mathbf{x}\_1 - \epsilon} \frac{\mathbf{x}\_1 d\mathbf{x}\_1}{\left(\sqrt{\mathbf{x}\_1^2 + \mathbf{x}\_2^2}\right)^6} = -\frac{\pi M \rho\_c}{6D^3} \tag{7}$$

If the single molecule of the film is now replaced by a solid body of unit volume having number density of atoms/molecules , the body force that the elastic film will experience is given by

$$f\left(D\right) = -\rho\_f \frac{dw}{dD} = -\frac{\pi M \,\rho\_f \rho\_c}{2D^4} = -\frac{A}{2\pi D^4} \tag{8}$$

where *A* is known as the Hamaker constant and has the value of 10−19 J for the present simulations. In the simulations, the above body‐force term is modified and a born‐repulsion term is also added to avoid extremely high forces at total contact such that

$$f\left(D\right) = -\frac{A}{2\pi D^4} + \frac{9B}{D^{10}}\tag{9}$$

where <sup>=</sup> 6/18 Jm6 (*le* is an equilibrium distance and has a value of 0.138 nm for the present case). The nature of the body force can be seen in **Figure 8B**.

**Figure 8.** (A) Schematic diagram of the calculation of force between a contactor surface and a molecule inside the elas‐ tic film. (B) It shows the van der Waals body force present between the contacting surface and any nodal point in the mesh inside the film geometry.

The contactor in the simulations is virtual in nature, and its presence is felt only through the body‐force term *f*. Since the contactor position keeps on changing with time (refer to **Figure 6B**) so also does the body force. Hence, the gap distance between a particular node and the surface of the contactor keeps on changing with time as per the following:

During adhesion: <sup>=</sup> initial − <sup>×</sup> − 2 for <sup>0</sup> <sup>≤</sup> <sup>t</sup> ≤ 1 where <sup>1</sup> = ( + 1)/ During resting: <sup>=</sup> initial <sup>−</sup> + 1 − 2 for <sup>1</sup> <sup>≤</sup> <sup>t</sup> ≤ 2 where 2( + 1)/ + 15 During debonding phase:

> ( initial ( )) ( 2 2 ) ( ) <sup>2</sup> 1 For t t *D d v tt ut c d* x= - + + ´- - £ (10)

In the above set of equations, initial denotes the initial gap distance, 2 the normal position of the particular node at time *t*, the critical distance at which the film begins to experience substantial force due to the contactor and starts deforming. Its value can be roughly estimated analytically from linear stability analysis on sinusoidal substrates according to [30]

$$\frac{hA}{2\pi\mu d\_c^4} = \frac{6.22}{\left(1-\beta\right)}\tag{11}$$

In VDLOAD subroutine of ABAQUS, the body force for a particular node is thus calculated according to

Simulating Contact Instability in Soft Thin Films through Finite Element Techniques http://dx.doi.org/10.5772/65357 181

$$f\left(\xi\right) = -\frac{A}{2\pi\xi^4} + \frac{9B}{\xi^{10}}\tag{12}$$

The morphological evolution of the film due to the presence of the contactor is thus dependant on the dynamic contactor position and thus is a time‐dependent problem. The solution is obtained by marching forward in time using an explicit scheme such that the body force calculated at a particular time *t* is given by

$$\int f\left(t\right) = f\left(t - \Delta t\right) + \frac{\partial \tilde{\xi}}{\partial t} \times \frac{\partial f}{\partial \tilde{\xi}}\bigg|\_{\left(t - \Delta t\right)}\tag{13}$$

#### **3.4. Structure of VDLOAD subroutine**

The headers of this subroutine are common to every VDLOAD and can be obtained from ABAQUS documentation. We have listed below only portions of the user‐defined load subroutine

**\*\*\* Declare new variables used (if any)**

**\*\*\* Declare the values of contactor approach velocity, retracting velocity, van der Waals potential/force parameters, initial separation distance, cutoff distance**

do k = 1, nblock

where <sup>=</sup>

180 Perusal of the Finite Element Method

mesh inside the film geometry.

During debonding phase:

according to

case). The nature of the body force can be seen in **Figure 8B**.

6/18 Jm6 (*le* is an equilibrium distance and has a value of 0.138 nm for the present

**Figure 8.** (A) Schematic diagram of the calculation of force between a contactor surface and a molecule inside the elas‐ tic film. (B) It shows the van der Waals body force present between the contacting surface and any nodal point in the

The contactor in the simulations is virtual in nature, and its presence is felt only through the body‐force term *f*. Since the contactor position keeps on changing with time (refer to **Figure 6B**) so also does the body force. Hence, the gap distance between a particular node

<sup>1</sup> = ( + 1)/

2( + 1)/ + 15

<sup>1</sup> <sup>≤</sup> <sup>t</sup> ≤ 2 where

*<sup>d</sup>* <sup>=</sup> - (11)

= - + + ´- - £ (10)

and the surface of the contactor keeps on changing with time as per the following:

( initial ( )) ( 2 2 ) ( ) <sup>2</sup> 1 For t t *D d v tt ut c d*

analytically from linear stability analysis on sinusoidal substrates according to [30]

2 1 *<sup>c</sup> hA* pm

In the above set of equations, initial denotes the initial gap distance, 2 the normal position of the particular node at time *t*, the critical distance at which the film begins to experience substantial force due to the contactor and starts deforming. Its value can be roughly estimated

> ( ) <sup>4</sup> 6.22

In VDLOAD subroutine of ABAQUS, the body force for a particular node is thus calculated

 b

During adhesion: <sup>=</sup> initial − <sup>×</sup> − 2 for <sup>0</sup> <sup>≤</sup> <sup>t</sup> ≤ 1 where

During resting: <sup>=</sup> initial <sup>−</sup> + 1 − 2 for

x

u2= curCoords (k,2) **\*\*\* curCoords(k,1) and curCoords(k,2) represent the updated coordinates in** *x***1 and** *x***2 direction respectively)**

**\*\* Calculate gap distance according to Eq. 10**

**\*\*\* Calculate forces according to Eq. 12**

**\*\* Calculate force update according to Eq. 13 and input the magnitude in** the array value (*k*), which is the prescribed **ABAQUS declared array to contain forces**

end do

The FE results displaying the effect of the substrate roughness on the morphologies formed, the work of adhesion exhibited, are discussed next in "Results" section.

#### **3.5. Results**

Linear stability analysis performed with the help of a single Fourier mode [16, 17] and energy‐ minimizing nonlinear analysis [18–20] has already been able to identify the nontrivial bifur‐ cation modes for which instability is possible. For critical conditions at which instability first initiates, it has been observed that the critical wave length λ 2.96 h is at par with the exper‐ imental observations. In FEM to obtain the critical wave mode, the geometry is created with the help of a series of cosine waves having random amplitudes and frequencies in the vicinity of a mean value *j*. When the particular geometries are simulated using VDLOAD, only in the adhesive branch, the displacements at the corner‐most node, which has the highest amplitude (because of the cosine nature of the waves), is tracked. The displacement of the node thus obtained at any generic time near critical separation distance is plotted against the average wave modes in **Figure 9A**. It can be seen that the curve has a bell shape indicating that near a particular mean surface wave mode the deformation of the surface is highest. This marks the dominant wave mode, whose value matches with those from Linear stability analysis (LSA) and experiments involving flat substrates.

**Figure 9.** (A) The displacement of the top‐most corner node of the film for different perturbed film surfaces at a partic‐ ular time near critical. The dominant wave mode is ~3 for flat substrate. (B) The dominant wave mode and the wave‐ length for different substrate amplitude averaged over the substrate wavelength. With increased roughness in substrate miniaturization of the surface patterns occurs.

**Figure 10.** The trajectory of the corner‐most point with time.

To study whether the substrate roughness has any influence on the emerging surface patterns, similar studies have been conducted for sawtooth‐profiled substrates having different substrate parameters of and *n*. The results reveal that though the substrate wavelengths have negligible effect, the substrate amplitudes have a profound influence on the emerging wavelengths. The surface wave mode/wave lengths are found to be an increasing/decreasing function of the substrate amplitude averaged over the different substrate wave modes considered, indicating that the patterned substrates do engender miniaturized length scales as can be seen from **Figure 9B**.

of a mean value *j*. When the particular geometries are simulated using VDLOAD, only in the adhesive branch, the displacements at the corner‐most node, which has the highest amplitude (because of the cosine nature of the waves), is tracked. The displacement of the node thus obtained at any generic time near critical separation distance is plotted against the average wave modes in **Figure 9A**. It can be seen that the curve has a bell shape indicating that near a particular mean surface wave mode the deformation of the surface is highest. This marks the dominant wave mode, whose value matches with those from Linear stability analysis (LSA)

**Figure 9.** (A) The displacement of the top‐most corner node of the film for different perturbed film surfaces at a partic‐ ular time near critical. The dominant wave mode is ~3 for flat substrate. (B) The dominant wave mode and the wave‐ length for different substrate amplitude averaged over the substrate wavelength. With increased roughness in

To study whether the substrate roughness has any influence on the emerging surface patterns, similar studies have been conducted for sawtooth‐profiled substrates having different

and experiments involving flat substrates.

182 Perusal of the Finite Element Method

substrate miniaturization of the surface patterns occurs.

**Figure 10.** The trajectory of the corner‐most point with time.

The displacement profile of the corner node at different time is shown in **Figure 10**. It can be seen that at initial time *t* = 0 since the initial gap is much greater than the critical value, the film surface remains planar with zero displacement. Near critical distance (near <sup>1</sup>), the dis‐ placements start to have nonzero values and at just greater than <sup>1</sup> when the contactor is in resting position the film comes in total contact. The node continues to be in contact with the contactor even in the debonding phase when the contactor is pulled up and it gets de‐ tached only at snap‐off distance dsnap. The trajectory of the corner node arising due to dif‐ ferent debonding velocity is found to be different in the debonding phase. It can be seen that the snap‐off also occurs at different times. Though the snap‐off time t snap for the lower velocity is found to be highest, the snap‐off distance dsnap = vd × tsnap is in reverse order 30, 16, 5, 10, 6.6 nm showing that in case of lower debonding velocity the detachment oc‐ curs at the lowest snap‐off distance.

**Figure 11.** (A1‐E1) Morphologies at different contactor positions. (A2‐E2) Stresses in the film at different contactor po‐ sitions for a film cast on a sawtooth‐profiled substrate of amplitude = 0.1 and = 4.

In ABAQUS, not only a single node but the whole evolution of the film can be traced throughout adhesion‐debonding. In **Figure 11**, the labels A1‐E1 indicate the morphologies and A2‐E2 the stress profiles that develop across a film cast on a sawtooth‐profiled substrate (of amplitude = 0.1 and = 4) for the different contactor positions. In the inset, a contactor displacement graph is provided for easy understanding of the positions. At position A1 where the contactor is just near critical distance the surface is seen to begin destabilizing where the structures formed have very less asperities. At stage B1, the film surface comes in physical contact with the contactor and the top surface takes the shape of columns bridging the gap with intervening cavities (which form the dispersed phase). At C1, the end of rest‐ ing phase, the columnar structures mature to form continuous ridges, which upon with‐ drawal to D1 shrinks in size due to peeling from the edges. Continuous peeling ultimately gives rise to isolated columns (which now forming the dispersed phase) as shown in E1 leading to snap‐off. Thus, from adhesion to debonding, the morphologies at the surface go through a phase inversion. The corresponding stress profiles can be seen from **Figure 11** A2‐ E2. It can be seen that the stresses lie concentrated at the column edges. Thus, more number or greater width of columns (B2 and C2) are seen to have higher stresses at the edges which are responsible for subsequent peeling and stress release as seen in **Figure 11** D2 and E2.

**Figure 12.** Contour plots of the work of adhesion (J/m2 ) for different substrate parameters for a film cast on a rough (sawtooth‐profiled) substrate. *(A) For debonding velocity* = 0.125 /. *(B) For debonding velocity*, = 0.25 /.

The sum of the stresses across the total top surface area is a signature of the forces re‐ quired by the contactor to pull the film up to a certain pull‐off distance. The area under the force displacement curve from the beginning of debonding phase to snap‐off yields the work of adhesion, which is required to separate the two surfaces of the film and the con‐ tactor. **Figure 12** reveals the work of adhesion exhibited by films cast on substrates having various degrees of roughness. It can be seen vividly that while the effect of the wavelength of the substrate is negligible, the higher the substrate amplitude, more is the substrate roughness more is the work of adhesion and better is the performance of the elastic film as an adherent. The work of adhesion is also seen to be a function of the debonding velocity. For a higher debonding velocity (**Figure 12B**), it can be seen that the work of adhesion is also high (compared to **Figure 12A**). The fact is intuitive considering in band‐aids when we pull at higher velocity higher adhesiveness is felt compared to that felt at lower velocities [46].

## **4. Conclusions**

In ABAQUS, not only a single node but the whole evolution of the film can be traced throughout adhesion‐debonding. In **Figure 11**, the labels A1‐E1 indicate the morphologies and A2‐E2 the stress profiles that develop across a film cast on a sawtooth‐profiled substrate (of amplitude = 0.1 and = 4) for the different contactor positions. In the inset, a contactor displacement graph is provided for easy understanding of the positions. At position A1 where the contactor is just near critical distance the surface is seen to begin destabilizing where the structures formed have very less asperities. At stage B1, the film surface comes in physical contact with the contactor and the top surface takes the shape of columns bridging the gap with intervening cavities (which form the dispersed phase). At C1, the end of rest‐ ing phase, the columnar structures mature to form continuous ridges, which upon with‐ drawal to D1 shrinks in size due to peeling from the edges. Continuous peeling ultimately gives rise to isolated columns (which now forming the dispersed phase) as shown in E1 leading to snap‐off. Thus, from adhesion to debonding, the morphologies at the surface go through a phase inversion. The corresponding stress profiles can be seen from **Figure 11** A2‐ E2. It can be seen that the stresses lie concentrated at the column edges. Thus, more number or greater width of columns (B2 and C2) are seen to have higher stresses at the edges which are responsible for subsequent peeling and stress release as seen in **Figure 11** D2 and E2.

(sawtooth‐profiled) substrate. *(A) For debonding velocity* = 0.125 /. *(B) For debonding velocity*,

The sum of the stresses across the total top surface area is a signature of the forces re‐ quired by the contactor to pull the film up to a certain pull‐off distance. The area under the force displacement curve from the beginning of debonding phase to snap‐off yields the work of adhesion, which is required to separate the two surfaces of the film and the con‐ tactor. **Figure 12** reveals the work of adhesion exhibited by films cast on substrates having various degrees of roughness. It can be seen vividly that while the effect of the wavelength of the substrate is negligible, the higher the substrate amplitude, more is the substrate roughness more is the work of adhesion and better is the performance of the elastic film as an adherent. The work of adhesion is also seen to be a function of the debonding velocity.

) for different substrate parameters for a film cast on a rough

**Figure 12.** Contour plots of the work of adhesion (J/m2

= 0.25 /.

184 Perusal of the Finite Element Method

Opening of crack surfaces through traction‐separation laws in cohesive‐zone modeling is demonstrated for cohesive blocks. In the case of adhesion‐debonding of elastic films, simula‐ tions are performed for topologically rough substrates to demonstrate the ability of the method to resolve complex geometries, which these analytical methods are incapable of. At incipience of instability when the film surface is perturbed by random fluctuations at the vicinity of prescribed wavelengths, the displacements made by those contacting surface nodes are found highest near the dominant wavelength. These dominant perturbations grow into columnar structures at contact. A thorough FE study reveals that film cast on rough substrate is ideal for fabrication of features/columns at miniaturized length scales and exhibits high work of adhesion. Thus, these films can act as better adhesives and can produce enhanced functional surfaces.

#### **Nomenclature**



*j* Mean wave mode of surface

## **Acknowledgements**

*L* Lateral length

*λs* Substrate wavelength *n* Substrate wave mode *d* Film‐contactor gap distance

186 Perusal of the Finite Element Method

*va* Approach velocity *vd* Debonding velocity Stress tensor in the film

 Displacement vector *P* Pressure across the film **I** The identity matrix

Φ The shape function

Body force (per unit volume)

*x1*‐*x2* Cartesian coordinate system *Ψ* Attractive van der Waals potential

*M* The coefficient of attraction

*w* Modified interaction potential

*A* The Hamaker constant *B* Born‐repulsion coefficient *le* An equilibrium distance

Dintial The initial gap distance

*t2* Adhesive phase end time

*j* Mean wave mode of surface

*dc* Critical gap distance *u2* Normal displacement λ Surface wavelength

*t2* Resting phase end time of contactor

*t* Time

Ω The discretized domain inside the film

*r* Separation distance between two atoms or molecules

*ρc* The number density of atoms/molecules of the contactor

*ρf* The number density of atoms/molecules of the film

*ξ* Gap distance between a particular node and the surface of the contactor

*D* Surface‐to‐surface separation distance of D

*β* Nondimensional substrate amplitude

Funding from the Department of Science & Technology, New Delhi, India, is highly acknowl‐ edged.

## **Author details**

Jayati Sarkar1\*, Hemalatha Annepu2 and Satish Kumar Mishra3

\*Address all correspondence to: jayati@chemical.iitd.ac.in

1 Department of Chemical Engineering, Indian Institute of Technology Delhi, New Delhi, India

2 Institute of Advanced Simulations‐2, Forschungszentrum Gmbh, Jülich, Germany

3 Department of Chemical Engineering, Mississippi State University, MS, USA

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## **The Role of Finite Element Analysis in Studying Potential Failure of Mandibular Reconstruction Methods The Role of Finite Element Analysis in Studying Potential Failure of Mandibular Reconstruction Methods**

Raymond C.W. Wong, John S.P. Loh and I. Islam Raymond C.W. Wong, John S.P. Loh and I. Islam Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64890

#### **Abstract**

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terials for Regenerative Medicine, Springer, 2012.

Defects of the mandible occur after trauma or resection after infection or tumours. There have been many methods espoused, but many methods can fail especially if the biomechanics of the mandible is not considered fully. As the only moveable, loadbearing bone of the skull, the mandible is subject to loads and stresses unique to it due to its shape, location and function. This chapter reviews the basic knowledge of the mandible necessary to perform finite element analysis, the challenges and then reviews several studies that have been done. The authors' personal research is detailed to illustrate how finite element analysis can be used to look at potential failure of a new method for mandibular reconstruction and implant evaluation.

**Keywords:** mandible reconstruction, finite element analysis, potential failure

## **1. Introduction**

The lower jaw or mandible is the only load bearing, moveable bone of the skull. Defects in the lower jaw or mandible can happen as a result of trauma, infection or after a resection for pathology, which can be benign or malignant. The field of medicine or surgery has never been able to satisfactorily reconstruct a mandible after the original tissue has been gone. An unrepaired defect in the mandible, depending on the location of the defect would lead to (1) constriction of the remaining tissue around the defect leading to a malocclusion due to the pull of the muscles, (2) collapse of the arches leading to inability to eat and function and (3) difficulty swallowing or even breathing especially when lying flat due to loss of attachment of the tongue, sometimes leading to aspiration or asphyxiation.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **2. Methods for mandibular reconstruction**

There have been many methods advocated for reconstruction of the mandible, most of which have never fully considered the biomechanical forces acting on the mandible both in the short and long term. These methods have been "tried", most times with catastrophic results. Among the methods for mandibular reconstruction advocated are:


The current gold standard is still the vascularized free flap, which needs a long operation time and harvest of tissue from another part of the body [1, 2].

This brings the question of "what is the ideal method of mandibular reconstruction?" to the fore.

## **2.1. Ideal method**

The ideal method of mandibular reconstruction would:


No such ideal method exists.

#### **2.2. Challenges to mandibular reconstruction**

The challenges to replacing a section of the mandible lie in its form and function as well as its unique location in the body. Any hardware (plates and screws) used to fix the mandible would undergo unique stresses not seen in other parts of the body. There is a non-axial load in that the long axis of the teeth is about perpendicular to the long axis of the mandible. The overlying tissue in the mouth is thin; any break in the soft tissue could lead to exposure of the hardware to the bacteria and saliva in the mouth, leading to the formation of biofilms. Ingestion of hot and cold food and liquids can cause expansion and contraction of the hardware, which is dissimilar to that of the underlying bone.

## **3. Mandibular biomechanics**

**2. Methods for mandibular reconstruction**

the methods for mandibular reconstruction advocated are:

and harvest of tissue from another part of the body [1, 2].

The ideal method of mandibular reconstruction would:

**5)** Not need to take tissue from another part of the body

**8)** Allow the patient to eat and function early or immediately

**7)** Not need prolonged hospitalization and recovery

**2.2. Challenges to mandibular reconstruction**

**1)** Reconstruct the missing soft tissue and bone

**6)** Newer methods like the endoprosthesis or alloplastic replacement

**3)** Mandibular reconstruction/bridging plate

**4)** Cancellous bone in titanium mesh

**7)** Tissue engineered bone scaffold

**2)** Allow replacement of teeth **3)** Not need a long surgery

No such ideal method exists.

**1)** Soft tissue flap

192 Perusal of the Finite Element Method

**2)** Autogenous bone blocks

**5)** Vascularized free flap

fore.

**2.1. Ideal method**

**6)** Cost effective

There have been many methods advocated for reconstruction of the mandible, most of which have never fully considered the biomechanical forces acting on the mandible both in the short and long term. These methods have been "tried", most times with catastrophic results. Among

The current gold standard is still the vascularized free flap, which needs a long operation time

This brings the question of "what is the ideal method of mandibular reconstruction?" to the

**4)** Easy to learn without needing extensive training and skills that are hard to learn

**9)** Be able to withstand the forces of biting and chewing permanently or for a long time

The challenges to replacing a section of the mandible lie in its form and function as well as its unique location in the body. Any hardware (plates and screws) used to fix the mandible would The biomechanics of the mandible has not been very well studied. Mandibular biomechanics can be studied with finite element analysis, strain gauges or photoelastic models. Finite element analysis allows study of the forces throughout an entire structure but is limited to static forces. An understanding of the anatomy and biomechanics of the intact mandible is necessary prior to looking at how to conduct a finite element analysis of the mandible.

## **3.1. Anatomy of the mandible**

The mandible is a U-shaped bone, connected at the temporomandibular joints at both ends to the skull. It consists of a corpus (body), the symphysis, which connects both right and left at the midline, the alveolar bone, which supports the teeth, the ramus with the condylar as well as the coronoid processes (**Figure 1**).

**Figure 1.** Schema showing the gross anatomy of the mandible.

The teeth are connected to the alveolar bone by periodontal ligaments, which act as a gomphosis (allowing minute movement). These ligaments are inserted into the bone on one end and into the cementum of the teeth.

The muscular attachments can be divided into three groups:


The outer surface of the mandible consists of dense cortical bone, the thickness of which varies. In certain areas, there is only cortical bone throughout. The alveolar processes and in the middle of the mandibular body and part of the ramus consist of cancellous bone (bone marrow). There is a nerve coursing through the mandible in a canal, which usually does not play a role in terms of biomechanics.

The attachment of the mandible to the skull consists of the attachments of the muscles of mastication and the temporomandibular joint. The temporomandibular joint consists of two joint spaces, the superior and inferior joint spaces, surrounded by a capsule consisting of elastic collagen fibres and divided by the articular cartilage, which is a fibrocartilage. The movement of the joint consists of two distinct movements, which are Phase I (rotation) about a hinge for the first 20 mm followed by Phase 2 (translation), which is affected mainly by the action of the lateral pterygoid muscle pulling the entire condyle to the front and out of the glenoid fossa onto the part of the zygomatic process of the temporal bone. The forward movement of the condyle is limited somewhat by a protrusion called the articular eminence.

## **3.2. Biomechanics of the intact mandible**

The mandible functions as a Class III lever. During function, there is a zone of tension on the alveolar part of the mandible and a zone of compression on the lower border. Studies by Meyer [3, 4] showed bone deformation in the mandibular condyle region, with tensile stress along the anterior ramus as well as along the sigmoid notch area and compressive stress along the posterior ramus border. This suggested that there is a tendency of the mandible to straighten during function. This somewhat simplistic model holds true when there is bilateral and equal function and bite forces (**Figure 2**).

Upon contraction of the muscles of mastication, the mandible is bent in a sagittal plane; this is produced by the vertical component of the muscle forces, the joint reaction forces and the reaction forces from chewing motions. During asymmetrical loading (biting on one side), the largest shear forces happen between the bite force and the muscle force on the working side (the biting side) and between the muscle force and joint force on the balancing side (non-biting side). This produces then a converse load distribution with a zone of tension on the lower border and a zone of compression on the alveolar portion in the working side and vice versa on the balancing side. This means that during incisal biting (biting on the front teeth), there is an equal amount of sagittal bending on both sides but a different deformation on the working and balancing sides during molar biting [5–7].

**Figure 2.** Zones of tension and compression in the mandible.

The teeth are connected to the alveolar bone by periodontal ligaments, which act as a gomphosis (allowing minute movement). These ligaments are inserted into the bone on one end

**1)** Muscles of mastication—medial pterygoid, lateral pterygoid, masseter and temporalis; only the lateral pterygoid muscle assist in opening the mandible, the rest closes the

**2)** Suprahyoid muscles that assist in some opening of the mouth and swallowing—mylo-

**3)** Muscles of facial expression that insert into the mandible—buccinators, depressor anguli

The outer surface of the mandible consists of dense cortical bone, the thickness of which varies. In certain areas, there is only cortical bone throughout. The alveolar processes and in the middle of the mandibular body and part of the ramus consist of cancellous bone (bone marrow). There is a nerve coursing through the mandible in a canal, which usually does not

The attachment of the mandible to the skull consists of the attachments of the muscles of mastication and the temporomandibular joint. The temporomandibular joint consists of two joint spaces, the superior and inferior joint spaces, surrounded by a capsule consisting of elastic collagen fibres and divided by the articular cartilage, which is a fibrocartilage. The movement of the joint consists of two distinct movements, which are Phase I (rotation) about a hinge for the first 20 mm followed by Phase 2 (translation), which is affected mainly by the action of the lateral pterygoid muscle pulling the entire condyle to the front and out of the glenoid fossa onto the part of the zygomatic process of the temporal bone. The forward movement of the

The mandible functions as a Class III lever. During function, there is a zone of tension on the alveolar part of the mandible and a zone of compression on the lower border. Studies by Meyer [3, 4] showed bone deformation in the mandibular condyle region, with tensile stress along the anterior ramus as well as along the sigmoid notch area and compressive stress along the posterior ramus border. This suggested that there is a tendency of the mandible to straighten during function. This somewhat simplistic model holds true when there is bilateral and equal

Upon contraction of the muscles of mastication, the mandible is bent in a sagittal plane; this is produced by the vertical component of the muscle forces, the joint reaction forces and the reaction forces from chewing motions. During asymmetrical loading (biting on one side), the largest shear forces happen between the bite force and the muscle force on the working side (the biting side) and between the muscle force and joint force on the balancing side (non-biting side). This produces then a converse load distribution with a zone of tension on the lower

condyle is limited somewhat by a protrusion called the articular eminence.

and into the cementum of the teeth.

play a role in terms of biomechanics.

**3.2. Biomechanics of the intact mandible**

function and bite forces (**Figure 2**).

mandible.

194 Perusal of the Finite Element Method

oris, mentalis.

The muscular attachments can be divided into three groups:

hyoid, hyoglossus, genioglossus, digastric muscles.

There is also a tendency for narrowing of the mandibular arch from parasagittal and transverse deformation upon clenching and incisal biting. This is caused by bilateral torsion of both mandibular bodies and bending at the symphyseal region leading to compression at the superior margin of the symphysis and tension at the inferior margin.

Hylander showed that the mandibular symphysis undergoes three distinct patterns of stress and deformation, that is, corporal rotation (relative outward rotation of both halves of the mandible), medial convergence (change in mandibular width during function) and dorsoventral shear (movement of both mandibular halves relative to one another in the vertical plane) [8] (**Figure 3**).

There is also some difference in the deformation between the outer (buccal) surface of the mandible and the inner (lingual) surface. Lateral transverse bending occurs and the bending moment increases from back to front during the late power stroke of biting/clenching. The maximum magnitude of the bending occurs near the symphysis. This bending produces compressive stresses at the buccal cortex and tensile stress at the lingual cortex. The deformation has been calculated to be as large as 0.6 mm in a simulated molar bite of 526N using finite element analysis [6]. The mandible deformed in a helical pattern upwards and towards the working side, with regions of high tensile stress (15–25 MPa) from the coronoid process and ramus towards the lingual side of the symphysis. The highest value of compressive stress (15– 25 MPa) was at the bite point and bilateral sigmoid notches, at the working side angle and in an area from the posterior surface of the balancing side ramus running to the lower border of the body till the symphysis. This then runs up to the buccal side from the inferior until the bite point. Overall, the shear stresses were larger on the working side with the exception of the balancing side condyle (peak shear stress of 25 MPa). In a nutshell, this means that the mandible changes in dimensions during function as a result of its shape and muscle pull with the greatest change in dimension and deformation in the midline.

What does all this information mean and what is the practical application? It means that application of any hardware to the mandible must take into account these forces and change in dimension. This has led to the creation of Champy's ideal lines of internal fixation for fixation of a mandibular fracture [9] (**Figure 4**). Placement of bone plates and screws in the area between the zones of tension and zones of compression will tend neutralize the forces and stabilize the bony fragments enough for healing. This however uses the principle of cross bracing and load sharing, that is, using inter-fragmentary bone friction to help stabilize the bony segments.

**Figure 3.** (A) Tendency for mandible to straighten as well as undergo torsion. (B) Forces acting about the midline of the mandible. CR – corporal rotation; MC – medial convergence; DVS – dorso-ventral shear.

In comminuted fractures or in a segmental mandibular defect, the principle of load sharing cannot be applied. We are then dependent on using load bearing bone plates, which means the material strength of the plate is the only thing keeping the mandibular segments together. The method of fixation of the bone plate to the bone also plays a factor.

**Figure 4.** Champy's lines for ideal internal fixation – placement of bone plates along these lines will stabilize the bone fragments enough for healing of fractures.

## **4. Finite element model of the mandible**

A very accurate finite element model is practically impossible to create due to the complex anatomy of the mandible. A true to accurate model that takes everything into account would need several supercomputers. Assumptions will have to be made to simplify the model and reduce the need for computing power.

Several authors have created finite element models of the mandible, each with different levels of complexity [10, 11]. Following are the steps taken to create an accurate finite element model [12].

## **4.1. Creation of a 3D mesh**

point. Overall, the shear stresses were larger on the working side with the exception of the balancing side condyle (peak shear stress of 25 MPa). In a nutshell, this means that the mandible changes in dimensions during function as a result of its shape and muscle pull with the greatest

What does all this information mean and what is the practical application? It means that application of any hardware to the mandible must take into account these forces and change in dimension. This has led to the creation of Champy's ideal lines of internal fixation for fixation of a mandibular fracture [9] (**Figure 4**). Placement of bone plates and screws in the area between the zones of tension and zones of compression will tend neutralize the forces and stabilize the bony fragments enough for healing. This however uses the principle of cross bracing and load sharing, that is, using inter-fragmentary bone friction to help stabilize the bony segments.

**Figure 3.** (A) Tendency for mandible to straighten as well as undergo torsion. (B) Forces acting about the midline of the

mandible. CR – corporal rotation; MC – medial convergence; DVS – dorso-ventral shear.

change in dimension and deformation in the midline.

196 Perusal of the Finite Element Method

The information about the external shaped of the mandible is needed to be able to create a mesh to input into the FEA software. Several possible methods that have been used to get the geometry of the mandible:


Once the 3D geometry has been obtained, it is subdivided into a finite, large number of geometrically simplified elements, connected together at the nodes. The mesh is a contiguous collection of these simple-shaped elements. Most FE software have automated mesh generation features, creating relatively dense meshes, which can be refined in different regions.

## **4.2. Input of material properties and boundary conditions**

The material properties of the elements, namely the elastic modulus and Poisson's ratio must be defined. Since the mandible consists of cortical and cancellous bone together with the teeth, several assumptions need to be made to simplify the model further. It is well known that bone is anisotropic in different dimensions. For purposes of simplifying the calculations, most authors have tended to assume that bone is isotropic and that the mandible is purely cortical bone. Any teeth present, which in real life would have periodontal ligaments and allows minute movements would tend to be assumed to be ankylosed to the bone, that is, fused to the bone. The teeth are composed of enamel on the outer surface of the crown followed by dentine and the pulp (a hollow cavity which contains the nerve fibres and blood supply). The outer surface of the root is covered by cementum. All these tissues have different material properties. The values for the material properties are have already been determined by studies. Some errors abound when it comes to the values of the cancellous bone. Most studies have tended to remove a block of cancellous bone and then subject the block to mechanical testing to ascertain its material properties. There is evidence to suggest that this is not entirely accurate. Misch et al. (1999) showed that the presence of cortical bone increases the elastic modulus of cancellous bone [14]. When the cortical bone was present, the elastic modulus ranged from 24.9 to 240 MPa (mean 96.2 MPa). When cancellous bone only was tested, the elastic modulus reduced dramatically (3.5–125.6 MPa). This means that the values from the literature for cancellous bone are not accurate.

Boundary conditions are important to prevent movement of the individual units so that the model can be loaded and deformed as a rigid structure, allowing computations to be performed. It can be divided into essential boundary conditions (displacement constraints to anchor the model and the non-essential boundary conditions or loading conditions, which are the forces to be applied to the model). Decisions will need to be made also about the insertion of the muscle forces and the force of each muscle. Some muscles like the masseter have three distinct types of fibres with different vectors. Most of these values are already in the literature.

## **4.3. Solution**

**1)** Digital creation of 3D model—not accurate representation of the anatomical detail. The question then arises: "At what level of intricacy would there be detriment to the level of

**2)** Conversion of digitized slices or sections of a human mandible into a whole 3D structure

**3)** Computer tomographic (CT) scans of a human mandible or mandibular equivalent and conversion of the radiographic images in Digitised Communication in Medicine (DICOM) format into a 3D structure in Standard Tessellation Language (STL). This information is then meshed with any number of software programs. This is the most popular method. Some authors have also used the cone beam CT, which is another way to get the CT images with a cone beam instead of a fan beam. This tends to produce many sharp triangles, which

Once the 3D geometry has been obtained, it is subdivided into a finite, large number of geometrically simplified elements, connected together at the nodes. The mesh is a contiguous collection of these simple-shaped elements. Most FE software have automated mesh generation features, creating relatively dense meshes, which can be refined in different regions.

The material properties of the elements, namely the elastic modulus and Poisson's ratio must be defined. Since the mandible consists of cortical and cancellous bone together with the teeth, several assumptions need to be made to simplify the model further. It is well known that bone is anisotropic in different dimensions. For purposes of simplifying the calculations, most authors have tended to assume that bone is isotropic and that the mandible is purely cortical bone. Any teeth present, which in real life would have periodontal ligaments and allows minute movements would tend to be assumed to be ankylosed to the bone, that is, fused to the bone. The teeth are composed of enamel on the outer surface of the crown followed by dentine and the pulp (a hollow cavity which contains the nerve fibres and blood supply). The outer surface of the root is covered by cementum. All these tissues have different material properties. The values for the material properties are have already been determined by studies. Some errors abound when it comes to the values of the cancellous bone. Most studies have tended to remove a block of cancellous bone and then subject the block to mechanical testing to ascertain its material properties. There is evidence to suggest that this is not entirely accurate. Misch et al. (1999) showed that the presence of cortical bone increases the elastic modulus of cancellous bone [14]. When the cortical bone was present, the elastic modulus ranged from 24.9 to 240 MPa (mean 96.2 MPa). When cancellous bone only was tested, the elastic modulus reduced dramatically (3.5–125.6 MPa). This means that the values from the literature for

Boundary conditions are important to prevent movement of the individual units so that the model can be loaded and deformed as a rigid structure, allowing computations to be performed. It can be divided into essential boundary conditions (displacement constraints to anchor the model and the non-essential boundary conditions or loading conditions, which are

accuracy?" Is it necessary to recreate all the intricacies (concavities, canals)?

need to be simplified prior to mesh creation.

**4.2. Input of material properties and boundary conditions**

cancellous bone are not accurate.

[13].

198 Perusal of the Finite Element Method

With all the information, the completed model is then solved to obtain the displacements and the resulting stress and strains. In biomechanical models, what is most often sought is information on the stresses and strains, as the force is usually known.

The external forces {F} and the mechanical properties/geometry {K} are used to solve the nodal displacements {D}. With the nodal displacements known, the displacement fields are then interpolated from the nodal values using standard interpolating polynomial functions. The strain distribution is the differentiation of the displacement field yields and the stress distribution is then determined mathematically.

## **4.4. Validation and interpretation**

Validation can be performed by evaluating the precision and accuracy of the model. Precision, defined as how close the model's results are to the exact solution to the biomechanical model, can be ascertained by conducting a convergence test where meshes of different refinements are created and the strains/stresses at specific locations are compared. Most reported studies have tended to use precision studies as a measure of validation as it is difficult to affix strain gauges to the human subject to test for accuracy for ethical and practical purposes.

## **5. Finite element analysis of the various mandibular reconstruction methods**

There have been very few finite element analysis conducted on reconstructed mandibles, mainly due to this being a field that is not very well understood by the people who operate in this area. In the field of orthopaedics, the biomechanics of the limbs has been well studied and there are numerous studies using finite element analysis. This section reviews the few studies that have been conducted followed by the authors' own studies. By necessity, it is impractical to go into very much detail for each study. We will concentrate on two studies in detail at the end of this section.

## **5.1. Marginal resection of the mandible**

Marginal resection of the mandible is performed for tumours that affect the alveolar process of the mandible but does not extend to the mandibular basal bone. One of the complications that can occur is fracture of the remaining portion of the mandible, usually from the corners of the resection where areas of stress concentration occur (**Figure 5**).

**Figure 5.** Marginal resection of the alveolar portion of the mandible, leaving the basal bone intact.

Wittkampf et al. [13] conducted studies on prevention of mandible fracture after a marginal resection of the mandible. The authors sectioned a human mandible, photographed the slices and digitized it. Marginal resections of different radii were placed in digitally and the areas of maximum stress concentrations in the corner of the resections were compared. It was found that an enlarge radius of resection at the corners offered the best resistance to minimize fracture after resection.

#### **5.2. Reconstruction plates for bridging mandibular defects**

Reconstruction plates alone are sometime used to bridge a defect to maintain the space and contours of the jaw in patients in poor health or with advanced tumours. A patient might not be fit for a long surgery or it might not be worthwhile to subject a patient to a long surgery if the prognosis is poor. Sometimes also the reconstruction plate is placed together with bone grafts, which is then subject to other considerations in terms of biomechanics (**Figure 6**).

The most common complications in using a reconstruction plate to bridge a defect are plate fracture, sometimes after a few years, loosening of the screws and exposure of the plate (dehiscence) either in the mouth or through the skin.

For pure mandibular bridging plates alone, the size and location of the defect and whether the defect crosses the midline (symphysis region) plays a large role in terms of complications [11, 15]. Recall the earlier section where it was shown that the midline is subject to many different forces of tension, compression and torsion with changes in dimensions. The masticatory loads on the plates cause vertical discrepancies that can lead to bone resorption and screw loosening. Arden et al. [16] reported that defects larger than 5 cm of bone length are associated with a high complication rate as high as 81% when plates alone are used to repair lateral defects.

Martola et al. [17] hypothesized that residual stresses from bending a stiff plate (sometimes repeatedly) to adapt to the jaw contour can be a main reason for plate fracture. This makes sense from a material science point of view, in that repeated bending of a metal results in work hardening of the metal, sometime with creation of small micro-cracks which may affect the mean stress in fatigue loading.

**Figure 5.** Marginal resection of the alveolar portion of the mandible, leaving the basal bone intact.

**5.2. Reconstruction plates for bridging mandibular defects**

(dehiscence) either in the mouth or through the skin.

after resection.

200 Perusal of the Finite Element Method

Wittkampf et al. [13] conducted studies on prevention of mandible fracture after a marginal resection of the mandible. The authors sectioned a human mandible, photographed the slices and digitized it. Marginal resections of different radii were placed in digitally and the areas of maximum stress concentrations in the corner of the resections were compared. It was found that an enlarge radius of resection at the corners offered the best resistance to minimize fracture

Reconstruction plates alone are sometime used to bridge a defect to maintain the space and contours of the jaw in patients in poor health or with advanced tumours. A patient might not be fit for a long surgery or it might not be worthwhile to subject a patient to a long surgery if the prognosis is poor. Sometimes also the reconstruction plate is placed together with bone grafts, which is then subject to other considerations in terms of biomechanics (**Figure 6**).

The most common complications in using a reconstruction plate to bridge a defect are plate fracture, sometimes after a few years, loosening of the screws and exposure of the plate

For pure mandibular bridging plates alone, the size and location of the defect and whether the defect crosses the midline (symphysis region) plays a large role in terms of complications [11, 15]. Recall the earlier section where it was shown that the midline is subject to many different

**Figure 6.** (A) Reconstruction plate bridging a defect in the anterior. (B) Fracture of the reconstruction plate, a common complication. Indicated by arrow.

Kimura et al. [18] investigated the most suitable method in dispersing stresses around the screws in plate fixation to the remnant mandible after a resection of the mandible. The authors took CT scans of dry human mandibles and created eight digital edentulous (no teeth) mandible models. Defects were created on the models in the front (midline) and lateral areas and the plates were drawn onto the defects with different screw configurations. A dental implant was drawn into the opposing (contralateral) side of the defect. The material properties (based on reported data) of all the components in the model were defined: cancellous bone, cortical bone and titanium. A maximum bite force of 300 N in a vertical load pattern was chosen and used. In the analysis, the stresses were concentrated around the implant, the screw closest to the defect on both sides (crucial screw) and the plate on the non-loaded side. Defects in the midline (central defects) placed greater maximum stress on the screws. If three screws were placed for lateral defects, there was greater stress on the crucial screws, which could have been due to bowing of the plate.

A partially dentate mandible of a cadaveric male was used to create a model with defect in the front and back. The authors (Schuller-Gotzburg et al, [19]) studied the effects and change in bone stress after caudal and buccal placement of placing a bridging plate. In this study, bridging plate alone in a conventional placement, bone grafts fixed with small miniplates to the remnant mandible and also placement of a bridging plate in a caudal and then buccal location for the reconstruction. The load was 50 N on the right mandibular second premolar. The conclusion was that there would be a better biomechanical advantage and lesser stress with the plate in a caudal position.

Knoll et al. [20] investigated replacement of angle defects with a standard 2.7 mm reconstruction plate with a linear screw configuration. An edentulous finite element mandible model was created; a defect placed in the right angle with a virtual bone plate placed and the model was loaded with 135 N of force in the front. The model showed that the stresses were far in excess of the material strength of titanium and cortical bone. This result showed that there was a high possibility of plate fracture, bone loss and screw loosening. The recommendation was to redesign the plate to allow screw placement in a triangular or square configuration to further maximize the interface between bone and plate.

## **5.3. Vascularized free flaps**

Vascularized free flaps are flaps of soft tissue and bone, harvested from another part of the body with its own arterial blood supply and venous drainage. The flaps are then placed into the part of the body that needs it, connected to the local blood supply and then fixed to the surrounding bone and soft tissue. For defects of the mandible, the two most common flaps used are the fibula and iliac crest free flaps. This is still the gold standard after a resection.

Tie et al. [21] constructed finite element models to study which flaps, the fibula or iliac crest free flaps would be best biomechanically to replace a segment of the mandible. The authors scanned the mandible, iliac crest and fibula of a healthy 30-year old volunteer and used the images to create a finite element model of the mandible. Defects were created in the anterior and lateral mandible; the outlines of the defects outlined and extracted using the software and the volume of the fibula and iliac crest made to fit the defects. The stress distribution for the iliac crest was found to be similar to that of the intact mandible. The fibula reconstruction, however, had greater stresses (compressive and tensile) at the grafted bone with the maximum stress at the interface between native and grafted bone. The increased binding or interface between the iliac crest and native bone contributed towards better transmission of the bite forces. Their conclusion is that for smaller defects, the iliac crest would be better due to the above findings. As the fibula has greater length, it would be more suitable for larger defects.

## **5.4. Endoprosthesis/customized implant**

Kimura et al. [18] investigated the most suitable method in dispersing stresses around the screws in plate fixation to the remnant mandible after a resection of the mandible. The authors took CT scans of dry human mandibles and created eight digital edentulous (no teeth) mandible models. Defects were created on the models in the front (midline) and lateral areas and the plates were drawn onto the defects with different screw configurations. A dental implant was drawn into the opposing (contralateral) side of the defect. The material properties (based on reported data) of all the components in the model were defined: cancellous bone, cortical bone and titanium. A maximum bite force of 300 N in a vertical load pattern was chosen and used. In the analysis, the stresses were concentrated around the implant, the screw closest to the defect on both sides (crucial screw) and the plate on the non-loaded side. Defects in the midline (central defects) placed greater maximum stress on the screws. If three screws were placed for lateral defects, there was greater stress on the crucial screws, which could have been

A partially dentate mandible of a cadaveric male was used to create a model with defect in the front and back. The authors (Schuller-Gotzburg et al, [19]) studied the effects and change in bone stress after caudal and buccal placement of placing a bridging plate. In this study, bridging plate alone in a conventional placement, bone grafts fixed with small miniplates to the remnant mandible and also placement of a bridging plate in a caudal and then buccal location for the reconstruction. The load was 50 N on the right mandibular second premolar. The conclusion was that there would be a better biomechanical advantage and lesser stress with the plate in a

Knoll et al. [20] investigated replacement of angle defects with a standard 2.7 mm reconstruction plate with a linear screw configuration. An edentulous finite element mandible model was created; a defect placed in the right angle with a virtual bone plate placed and the model was loaded with 135 N of force in the front. The model showed that the stresses were far in excess of the material strength of titanium and cortical bone. This result showed that there was a high possibility of plate fracture, bone loss and screw loosening. The recommendation was to redesign the plate to allow screw placement in a triangular or square configuration to further

Vascularized free flaps are flaps of soft tissue and bone, harvested from another part of the body with its own arterial blood supply and venous drainage. The flaps are then placed into the part of the body that needs it, connected to the local blood supply and then fixed to the surrounding bone and soft tissue. For defects of the mandible, the two most common flaps used are the fibula and iliac crest free flaps. This is still the gold standard after a resection. Tie et al. [21] constructed finite element models to study which flaps, the fibula or iliac crest free flaps would be best biomechanically to replace a segment of the mandible. The authors scanned the mandible, iliac crest and fibula of a healthy 30-year old volunteer and used the images to create a finite element model of the mandible. Defects were created in the anterior and lateral mandible; the outlines of the defects outlined and extracted using the software and the volume of the fibula and iliac crest made to fit the defects. The stress distribution for the

due to bowing of the plate.

202 Perusal of the Finite Element Method

maximize the interface between bone and plate.

**5.3. Vascularized free flaps**

caudal position.

An endoprosthesis is an implant, mostly titanium that is placed to replace a defect in the line of the remaining bone stumps. It is usually attached to the bone stumps with a stem, which is usually cemented or press fit. This concept has been used in orthopaedics in the long bones with great success for decades [22–25].

The concept of using an endoprosthesis for mandibular replacement was introduced by Tideman, initially as proof of concept and with several animal studies [26–28]. The decision was made to look into the mandibular endoprosthesis as a modular format rather than customized. The reasons for this were as follows:


The modular endoprosthesis has been used, again with great success in the field of orthopaedics and musculoskeletal surgery.

The difference between the long limbs and the mandible has already been discussed previously. There is also the addition of the curvature of the mandible in the anterior region, which varies between individuals, making it difficult for a stock endoprosthesis.

Nevertheless, the animal experiments, which largely were conducted in the monkey model yielded interesting results. There were two designs: (1) the mandibular body replacement and (2) the condyle replacement. The condyle replacement had no problems with loosening and infection. The body replacement design had persistent problems with loosening between the module connections, causing infection and loosening. The cemented stems had no problems; however, a decision was made to investigate the biomechanical forces that acted on the entire reconstruction for the body replacement design [29].

The design of the endoprosthesis was changed as follows [30]:


This new design was then tested for the following aims prior to any animal or human testing:


The new endoprosthesis design was made to fit the dimensions of a human mandible and underwent mechanical testing in a jig, mounted on a synthetic mandible, which had similar elastic properties to cortical bone. The methods of mechanical testing depend on the question asked. It can range from simple three-point bending test, compressive and tensile strength to complex tests for load to failure or fatigue loading. The dimensions of the assembled endoprosthesis were 18-mm stem length with 4-mm diameter and body dimensions of 15-mm length, 16-mm high and 8.5-mm thick (**Figure 7**).

**Figure 7.** The endoprosthesis consisted of two screwed stems, which connected to each other in a dovetail. This was locked together by a central screw, which is inserted from the top.

Static testing revealed a tendency for the screw stem to pull out of the substrate of the synthetic mandible. Cyclic testing was then performed for up to 500,000 cycles. This revealed a tendency for the endoprosthesis to fail with fracture or bending at the superior surface of the stem but with no loosening of the module connection, which had plagued the earlier animal experiments.

**1)** The stem was changed from cemented to screw retained, which was to be screwed into

**2)** The module connection was changed to a male and female part in a dovetail fashion, which was connected by a screw. A slight movement of 0.1 mm was designed as s tress breaker

This new design was then tested for the following aims prior to any animal or human testing: **1)** Experimental evaluation of the new design to look at fatigue performance and failure patterns by mechanical testing. The entire setup was also investigated with a finite element analysis to see if the model correctly predicted the site of failure in the mechanical testing.

**2)** Finite element analysis of the new design in a simulated human mandible under certain conditions. The defect was then to be made bigger and the stem length was to be changed in length to see what happens to the stress distribution to better predict the ability of the reconstruction to withstand failure (if the stress in any part is more than the material

The new endoprosthesis design was made to fit the dimensions of a human mandible and underwent mechanical testing in a jig, mounted on a synthetic mandible, which had similar elastic properties to cortical bone. The methods of mechanical testing depend on the question asked. It can range from simple three-point bending test, compressive and tensile strength to complex tests for load to failure or fatigue loading. The dimensions of the assembled endoprosthesis were 18-mm stem length with 4-mm diameter and body dimensions of 15-mm

**Figure 7.** The endoprosthesis consisted of two screwed stems, which connected to each other in a dovetail. This was

Static testing revealed a tendency for the screw stem to pull out of the substrate of the synthetic mandible. Cyclic testing was then performed for up to 500,000 cycles. This revealed a tendency

the marrow part of the mandible.

204 Perusal of the Finite Element Method

between the connection of the male and female part.

strength of the material) and also the location of failure.

length, 16-mm high and 8.5-mm thick (**Figure 7**).

locked together by a central screw, which is inserted from the top.

The line drawings of the endoprosthesis from the manufacturer were imported into Abaqus v6.10 (Simulia, Dassault Systemes, France). The stems were modelled as smooth cylinders to simplify calculations. Rectangular cuboids were modelled as the synthetic mandibles and bores made for stem insertion. The stems of the endoprosthesis were perfectly tied to the bores of the holes as well as the central connection screw. The cuboids were assigned the elastic properties of cortical bone and meshed with linear tetrahedral elements. It was assumed that the bone was isotropic. A bolt load of 10 N was applied to simulate tightening of the central screw. A downward force of 150 N (calculated at 80% of average static load to failure of 185 N, this was the maximum force used for the fatigue testing) was loaded on to one end of the reconstruction while the other end was given fixed boundary conditions. The load was kept constant to identify peak stresses, which lead to fatigue failure (**Figure 8**).

**Figure 8.** (A) Rendering of endoprosthesis mounted in bone blocks. (B) Von Mises stress distribution.

The finite element analysis of the setup showed areas of high stresses accumulating in the screw hole of the connection screw as well as on the superior surface of the stems. The von Mises stress recorded a maximum of 188.838 MPa, which is way below the strength of titanium alloy at 897 MPa. This corresponds well with the results of the fatigue testing with had failure of crack lines in the same areas. The conclusion of this bench top experiment was that although the forces are way below the material strength of titanium, micro-cracks as well as areas of stress concentrations from the indentation of the screw threads can lead to eventual failure over a long time of loading at a much lower load level. The connection problem of the modules seemed to have been solved.

A human-sized mandible synthetic was scanned with a cone beam CT scan to get the geometrical information of the mandible [31]. A synthetic mandible was used due to biohazard concerns. Due to the nature of the cone beam CT, using the DICOM information without alteration tended to produce a mesh with a lot of sharp, irregular and thin triangles. This was re-meshed with 3-Matics (Materialise, Belgium) into linear tetrahedrons. The mesh was dense enough to justify the use of linear elements to save on computational resources. The teeth were also removed digitally as it does not contribute structurally to the mandible.

The endoprosthesis was modelled as previously described above (termed Case I) and then the stem length was shortened (Case II) followed by Case III where the length of the body was doubled to 30 mm (**Figure 9**).

**Figure 9.** Endoprosthesis in mandible model.

A standardized defect as well as boreholes was created to fit the dimensions of Case I, II and III on the right side of the mandible using Abaqus. The model was assumed to be made up of only cortical bone, the bone was assumed to be isotropic and the values and vectors of the muscle pull as well as the joint reaction forces were taken from the literature. The boreholes and stems were assumed to be bonded just like the previous FEM study of the experimental setup. Case I was used as the model for the standard endoprosthesis design, Case II and III were used as models for looking at the effects of a decrease in stem length as well as an increase in defect length, respectively. A 300 N load was applied directly in the incisor region. Several studies from the literature have supported our assumptions to be relatively accurate while saving on computational resources. As a form of validation, a convergence check was conducted on a mandible with quadratic elements, which gave a finer mesh. The differences were less than 10%.

The analysis was conducted and divided into three separate parts:


The finite element analysis of the setup showed areas of high stresses accumulating in the screw hole of the connection screw as well as on the superior surface of the stems. The von Mises stress recorded a maximum of 188.838 MPa, which is way below the strength of titanium alloy at 897 MPa. This corresponds well with the results of the fatigue testing with had failure of crack lines in the same areas. The conclusion of this bench top experiment was that although the forces are way below the material strength of titanium, micro-cracks as well as areas of stress concentrations from the indentation of the screw threads can lead to eventual failure over a long time of loading at a much lower load level. The connection problem of the modules

A human-sized mandible synthetic was scanned with a cone beam CT scan to get the geometrical information of the mandible [31]. A synthetic mandible was used due to biohazard concerns. Due to the nature of the cone beam CT, using the DICOM information without alteration tended to produce a mesh with a lot of sharp, irregular and thin triangles. This was re-meshed with 3-Matics (Materialise, Belgium) into linear tetrahedrons. The mesh was dense enough to justify the use of linear elements to save on computational resources. The teeth were

The endoprosthesis was modelled as previously described above (termed Case I) and then the stem length was shortened (Case II) followed by Case III where the length of the body was

A standardized defect as well as boreholes was created to fit the dimensions of Case I, II and III on the right side of the mandible using Abaqus. The model was assumed to be made up of only cortical bone, the bone was assumed to be isotropic and the values and vectors of the muscle pull as well as the joint reaction forces were taken from the literature. The boreholes and stems were assumed to be bonded just like the previous FEM study of the experimental

also removed digitally as it does not contribute structurally to the mandible.

seemed to have been solved.

206 Perusal of the Finite Element Method

doubled to 30 mm (**Figure 9**).

**Figure 9.** Endoprosthesis in mandible model.


Under the prescribed loading conditions, the intact mandible bent upwards almost equally on both sides. As this model was not totally symmetrical, there were minute differences on both sides. With a defect in place, reconstructed with an endoprosthesis, the mandible became less stiff, causing the left intact side to arch less than the right reconstructed side. This led to the mandible shifting to the left by 0.354 mm. There was a tendency for the endoprosthesis to bend outwards. There was little difference between Case I and II, which led to the conclusion that a case could be put forward for a shorter stem to reduce the amount of hardware needed. A longer/larger defect, as in Case III led to a tendency for separation at the module connection, leading to a possibility for eventual loosening of the connection screw in the long term. Although there was a tendency for separation at the module connection, the stresses in the connection screw were all below 100 MPA, below the material strength of titanium. The stress distribution for Case I and II was similar, but with slightly larger magnitudes in Case II. In Case III, however, since it was a larger defect, the endoprosthesis tended to bend more at the stem, although, again, the stresses were below the strength of titanium at 898 MPa (**Figures 10**–**12**).

#### *(2) Stress in the mandible*

With regard to the stresses within the mandible, areas of stress concentration were in the left condyle (due to unequal deflection of the mandible from torsional stress), lower border of the mandibular body close to the abutment with the endoprosthesis and top edge of the hole in the stumps. There tended to be a pull out tendency of the stem from the holes, as experienced in the experimental setup. This was worse in Case II and III, with the peak stress exceeding the material strength of cortical bone at 85 MPa. The condyles were restricted from moving in this model, while in real life, there would certainly be movements that could dissipate stress; thus there is some mitigation factor in vivo. Whether this would be true could only be answered in animal experimental models.

## *(3) Deflection of the mandible*

Deflection of the mandible was measured with respect to an *x*-*y*-*z* axis in three dimensions at points of interest. Since the boundary conditions were applied to the incisors and condyles, this caused any deflection to show up in the body of the mandible. The greatest displacement for Case I was 0.638 mm, for Case II 0.8 mm and III 0.608 mm.

The conclusions drawn from the study were that the modular endoprosthesis in its current dimensions should be adequate for small defects. Altering the stem dimensions by shortening it showed a slight increase in magnitude but no significant alteration to the stress distribution. A larger defect, however, would be more difficult to reconstruct and more studies needed to be done.

This work was followed by Pinheiro and Alves [32] who performed a finite element analysis for an endoprosthesis that was not modular and comprised of a solid component, which was customized. This was a feasibility study in which the authors removed the screw stem component and designed a stem that looks to be press fit. The endoprosthesis performed well based on the finite element study. There was no tendency for separation of the module due to the entire prosthesis being a solid framework and the stress distribution as well as the displacement field were very similar to that of the intact mandible and yet did not exceed the material strength of titanium.

**Figure 10.** Case I Von Mises stress distribution.

The Role of Finite Element Analysis in Studying Potential Failure of Mandibular Reconstruction Methods http://dx.doi.org/10.5772/64890 209

**Figure 11.** Case II Von Mises stress distribution. The stem length has been halved.

*(3) Deflection of the mandible*

208 Perusal of the Finite Element Method

material strength of titanium.

**Figure 10.** Case I Von Mises stress distribution.

be done.

Deflection of the mandible was measured with respect to an *x*-*y*-*z* axis in three dimensions at points of interest. Since the boundary conditions were applied to the incisors and condyles, this caused any deflection to show up in the body of the mandible. The greatest displacement

The conclusions drawn from the study were that the modular endoprosthesis in its current dimensions should be adequate for small defects. Altering the stem dimensions by shortening it showed a slight increase in magnitude but no significant alteration to the stress distribution. A larger defect, however, would be more difficult to reconstruct and more studies needed to

This work was followed by Pinheiro and Alves [32] who performed a finite element analysis for an endoprosthesis that was not modular and comprised of a solid component, which was customized. This was a feasibility study in which the authors removed the screw stem component and designed a stem that looks to be press fit. The endoprosthesis performed well based on the finite element study. There was no tendency for separation of the module due to the entire prosthesis being a solid framework and the stress distribution as well as the displacement field were very similar to that of the intact mandible and yet did not exceed the

for Case I was 0.638 mm, for Case II 0.8 mm and III 0.608 mm.

**Figure 12.** Case III Von Mises stress distribution. The defect length has been doubled.

## **6. Conclusion**

Finite element analysis is a good method to analyse solid mechanics. It has been used extensively in studying the forces in the long limbs as well as the bone plates and the pattern of bone resorption and formation. In the field of head and neck surgery, there are much fewer studies. A number of studies have been recently published as surgeons notice problems with their methods of reconstruction. We have looked at some of the studies and although not exhaustive, it should serve as an illustration to how it is used to look at potential failure of mandibular reconstruction.

## **Author details**

Raymond C.W. Wong\* , John S.P. Loh and I. Islam

\*Address all correspondence to: raginwcw@yahoo.com

Discipline of Oral and Maxillofacial Surgery, National University Hospital, National University of Singapore, Singapore

## **References**


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**6. Conclusion**

210 Perusal of the Finite Element Method

reconstruction.

**Author details**

**References**

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123–136.

Raymond C.W. Wong\*

University of Singapore, Singapore

Finite element analysis is a good method to analyse solid mechanics. It has been used extensively in studying the forces in the long limbs as well as the bone plates and the pattern of bone resorption and formation. In the field of head and neck surgery, there are much fewer studies. A number of studies have been recently published as surgeons notice problems with their methods of reconstruction. We have looked at some of the studies and although not exhaustive, it should serve as an illustration to how it is used to look at potential failure of mandibular

, John S.P. Loh and I. Islam

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clenching. J. Dent. Res. 1994; 73: 56–66.

Discipline of Oral and Maxillofacial Surgery, National University Hospital, National

[1] Wong RC, Tideman H, Kin L, Merkx MA: Biomechanics of mandibular reconstruction:

[2] Goh BT, Lee S, Tideman H, Stoelinga PJ: Mandibular reconstruction in adults: a review.

[3] Meyer C, Kahn JL, Boutemi P, Wilk A: Photoelastic analysis of bone deformation in the region of the mandibular condyle during mastication. J. Craniomaxillofac. Surg. 2002;

[4] Meyer C, Sehir L, Boutemi P: Experimental evaluation of three osteosynthesis devices used for stabilizing condylar fractures of the mandible. J. Craniomaxillofac. Surg. 2006;

[5] Korioth TW, Hannam AG: Deformation of the human mandible during simulated tooth

[6] Korioth TW, Romilly DP, Hannam AG: Three dimensional finite element stress analysis

[7] Van Eijden TMGJ: Biomechanics of the mandible. Crit. Rev. Oral Biol. Med. 2000; 11:

of the dentate human mandible. Am. J. Phy. Anthropol. 1992; 88: 69–96.

\*Address all correspondence to: raginwcw@yahoo.com


## **Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis**

Yutaka Inaba, Hiroyuki Ike, Masatoshi Oba and Tomoyuki Saito Yutaka Inaba, Hiroyuki Ike, Masatoshi Oba and Tomoyuki Saito

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65031

#### **Abstract**

[22] Callaghan JJ, Albright JC, Goetz DD, Olejniczak JP, Johnston RC: Charnley total hip arthroplasty with cement: minimum twenty-five year follow-up. J. Bone Joint Surg. Am.

[23] Charnley J: The reaction of bone to self-curing acrylic cement: a long-term histological

[24] Henshaw R, Malawer M: Musculoskeletal Cancer Surgery: Treatment of Sarcomas and Allied Diseases: A Review of Endoprosthetic Reconstruction During Limb-sparing

[25] Huiskes R: Biomechanics of artificial joint fixation. Basic Orthopaedic Biomechanics. In: Van C Mowe and Wilson Hayes (Eds.). Raven Press Ltd, New York 1991; 375–437.

[26] Tideman H, Lee S: The TL endoprosthesis for mandibular reconstruction: a metallic yet

[27] Lee S, Goh BT, Tideman H, Stoelinga PJW: Modular endoprosthesis for mandibular reconstruction: a preliminary animal study. Int. J. Oral Maxillofac. Surg. 2008; 37: 935–

[28] Lee S, Goh BT, Tideman H, Stoelinga PJW, Jansen J: Modular endoprosthesis for mandibular reconstruction: a clinical, micro-computed, tomographic and histologic evaluation in 8 macaca fascicularis. Int. J. Oral Maxillofac. Surg. 2009; 38: 40–47. [29] Wong RC, Lee S, Tideman S, Merkx MAW, Jansen J, Liao K: Effect of replacement of mandibular defects with a modular endoprosthesis on bone mineral density in a

[30] Wong RC, Tideman H, Merkx MAW, Jansen J, Goh SM: The mandibular modular endoprosthesis for body replacement. Part 1: mechanical testing of the reconstruction.

[31] Wong RC, Tideman H, Merkx, MAW, Jansen J, Goh SM: The modular endoprosthesis for mandibular body replacement. Part 2: finite element analysis of the reconstruction.

[32] Pinheiro M, Alves JL: The feasibility of a custom-made endoprosthesis in mandibular reconstruction: implant design and finite element analysis. J. Craniomaxillofac. Surg.

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212 Perusal of the Finite Element Method

942.

We compared equivalent stress and strain energy density (SED) to bone mineral density (BMD) in the femur after total hip arthroplasty (THA) using subject-specific finite element analysis (FEA). Equivalent stress and BMD were maintained in the distal femur after THA, whereas both decreased in the proximal femur. A significant correlation was observed between the rates of changes in BMD and equivalent stress before and after THA. Therefore, FEA can predict adaptive bone remodeling after mechanical loading changes. Additionally, we evaluated the effects of two different types of stem geometries (Zweymüller-type stem and fit-and-fill-type stem) on load distribution and BMD using the same method. Equivalent stress and BMD in the medial side of the proximal femur were significantly lower with the Zweymüller-type stem than with the fit-and-fill-type stem. Therefore, FEA can assess the effects of stem geometry on bone remodeling after THA. Moreover, we evaluated the effects of bone geometry on load distribution and BMD after THA. Equivalent stress in the medial side of the proximal femur was significantly lower in the stovepipe model implanted with large tapered wedge-type stems than in the champagne flute and intermediate models, and there was a significant loss of BMD in the stovepipe model. Therefore, a large tapered wedge-type stem and stovepipe femur may be associated with significant proximal BMD loss.

**Keywords:** finite element analysis, dual-energy X-ray absorptiometry, bone remodeling, hip, arthroplasty

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

Periprosthetic bone loss, i.e., a decrease in bone mineral density (BMD) around an implant, is a major concern following total hip arthroplasty (THA) [1, 2]. Severe periprosthetic bone loss may contribute to aseptic loosening of the prosthesis or the risk of periprosthetic fracture [3] and presents serious problems if revision surgery becomes necessary [4]. Several studies have reported that BMD decreases following THA, particularly in the proximal femur [5, 6]. Most changes in BMD occur during the first postoperative year, and subsequently, BMD is maintained [7–9]. This phenomenon is considered to be an adaptive remodeling response of the bone structure to alter its stress environment owing to the presence of an implant. Therefore, mechanical stress may be associated with the patterns of BMD changes. In particular, in cementless THA, an optimal fit of the stem into the metaphyseal region of the femur has been reported to lead to a reduction in periprosthetic bone loss [10].

Wolff [11] proposed that trabecular bone in the proximal femur adapts to external mechanical loads. This structural change is caused by bone remodeling, which couples bone resorption by osteoclasts and bone formation by osteoblasts [12]. It is generally accepted that bone tissue can detect and respond to its mechanical environment, but the exact mechanical signal that drives remodeling remains controversial [13]. Various biomechanical parameters have been investigated to determine the remodeling stimulus. Huiskes et al. [14] examined whether strain energy density (SED) controls bone remodeling. Carter et al. [15] examined stress, and Stulpner et al. [16] examined equivalent strain as signals for remodeling. Adachi et al. [17] proposed that the driving force of remodeling is the local nonuniformity of the scalar function of stress. The existence of various theories indicates that the mechanism of bone remodeling is not well understood.

Subject-specific finite element (FE) models based on geometry and mechanical properties derived from computed tomography (CT) scans have been described in several studies [18– 20]. Furthermore, relationships between results of subject-specific finite element analysis (FEA) and bone density have been investigated previously. Cody et al. [21] performed subject-specific FEA and bone density measurements for predicting the fracture load of the femur. Tawara et al. [22] evaluated vertebral strength using subject-specific FEA and BMD. We previously examined the usefulness of subject-specific FEA, which was validated by BMD changes, after THA. We found that equivalent stress through subject-specific FEA correlated with BMD changes before and after THA [23]. The distribution of mechanical loads within the femur after THA might be influenced by the stem geometry and the femur structure. Our previous results suggest that subject-specific FEA could be used to evaluate the effects of stem and femoral geometry on adaptive bone remodeling after THA [24].

This chapter initially provides general information on subject-specific FEA and then presents a brief overview of our work regarding adaptive bone remodeling after THA.

## **2. Finite element analysis**

**1. Introduction**

214 Perusal of the Finite Element Method

loss [10].

understood.

Periprosthetic bone loss, i.e., a decrease in bone mineral density (BMD) around an implant, is a major concern following total hip arthroplasty (THA) [1, 2]. Severe periprosthetic bone loss may contribute to aseptic loosening of the prosthesis or the risk of periprosthetic fracture [3] and presents serious problems if revision surgery becomes necessary [4]. Several studies have reported that BMD decreases following THA, particularly in the proximal femur [5, 6]. Most changes in BMD occur during the first postoperative year, and subsequently, BMD is maintained [7–9]. This phenomenon is considered to be an adaptive remodeling response of the bone structure to alter its stress environment owing to the presence of an implant. Therefore, mechanical stress may be associated with the patterns of BMD changes. In particular, in cementless THA, an optimal fit of the stem into the metaphyseal region of the femur has been reported to lead to a reduction in periprosthetic bone

Wolff [11] proposed that trabecular bone in the proximal femur adapts to external mechanical loads. This structural change is caused by bone remodeling, which couples bone resorption by osteoclasts and bone formation by osteoblasts [12]. It is generally accepted that bone tissue can detect and respond to its mechanical environment, but the exact mechanical signal that drives remodeling remains controversial [13]. Various biomechanical parameters have been investigated to determine the remodeling stimulus. Huiskes et al. [14] examined whether strain energy density (SED) controls bone remodeling. Carter et al. [15] examined stress, and Stulpner et al. [16] examined equivalent strain as signals for remodeling. Adachi et al. [17] proposed that the driving force of remodeling is the local nonuniformity of the scalar function of stress. The existence of various theories indicates that the mechanism of bone remodeling is not well

Subject-specific finite element (FE) models based on geometry and mechanical properties derived from computed tomography (CT) scans have been described in several studies [18– 20]. Furthermore, relationships between results of subject-specific finite element analysis (FEA) and bone density have been investigated previously. Cody et al. [21] performed subject-specific FEA and bone density measurements for predicting the fracture load of the femur. Tawara et al. [22] evaluated vertebral strength using subject-specific FEA and BMD. We previously examined the usefulness of subject-specific FEA, which was validated by BMD changes, after THA. We found that equivalent stress through subject-specific FEA correlated with BMD changes before and after THA [23]. The distribution of mechanical loads within the femur after THA might be influenced by the stem geometry and the femur structure. Our previous results suggest that subject-specific FEA could be used to evaluate the effects of stem and femoral

This chapter initially provides general information on subject-specific FEA and then presents a brief overview of our work regarding adaptive bone remodeling after THA.

geometry on adaptive bone remodeling after THA [24].

## **2.1. Finite element analysis and bone mineral density**

It is difficult to assess the stress and strain distributions throughout the entire bone using simplified mathematical models or implanted prostheses or through experiments with cadaveric tissue. An alternative approach to analyze bone mechanics is the FE method, which can accommodate large intersubject variations in bone geometry and material properties [25]. When the ability to perform three-dimensional FE contact analysis is available, subject-specific geometry should be analyzed. This provides the opportunity to examine normal populations of individuals and populations with specific injuries or other pathologies. Ultimately, patientspecific modeling might be used in surgical planning and large-scale studies of treatment efficacy [26]. As mentioned above, we investigated the relationships between the FEA results and BMD. Several previous studies have described the application of FEA for investigating bone-remodeling patterns around implants. However, most of these were not subject-specific models. Weinans et al. [27] reported that subject-specific FE models might be useful for explaining variations in bone adaptation responsiveness among different subjects.

**Figure 1.** Diagrams of (a) collarless VerSys Fiber Metal MidCoat, (b) SL-PLUS, and (c) Accolade TMZF stems. (a) Collarless VerSys Fiber Metal MidCoat (Zimmer Inc., Warsaw, IN) is a fit-and-fill-type stem composed of titanium alloy with fiber-metal porous coating at the proximal region. (b) SL-PLUS (Smith & Nephew Inc., Memphis, TN) is a Zweymüller-type stem composed of titanium alloy with a dual taper, rectangular, and trochanteric wing design. (c) Accolade TMZF stem (Stryker Orthopaedics, Mahwah, NJ) is a cementless tapered wedge-type stem composed of beta titanium alloy with a proximal hydroxyapatite-coated porous surface and a distal smooth-finished surface.

We investigated 24 patients (16 women and 8 men) who underwent primary cementless THA using the same fit-and-fill type of prosthesis (collarless VerSys Fiber Metal MidCoat, Zimmer Inc., Warsaw, IN) (**Figure 1**). Collarless VerSys Fiber Metal MidCoat is a stem composed of titanium alloy with a fiber-metal porous coating at the proximal region. Patients were excluded from the study if they were receiving systemic estrogen, vitamin D, or bisphosphonate for osteoporosis. The mean patient age at surgery was 63 years (range, 44–82 years). After surgery, all patients were allowed to use a wheelchair, with full weight bearing, on postoperative day 1, and gait exercise was performed as soon as possible [23]. BMD was assessed with dualenergy X-ray absorptiometry (DEXA) using the Hologic Discovery system (Hologic Inc., Waltham, MA). Baseline DEXA measurements were performed 1 week after surgery as reference, and subsequent measurements were performed at 3, 6, and 12 months postoperatively. Patients were placed in the supine position, and the affected leg was maintained in 20 ° internal rotation to minimize measurement errors. DEXA measurements were performed using the "array prosthetic" mode, which excludes metal from analysis of the regions of interest (ROIs). ROIs were defined according to Gruen's system [28]. The pixel size was 1.1 × 0.56 mm.

**Figure 2.** Finite element mesh of the femur. Three-dimensional FE models are constructed using 1- to 4-mm tetrahedral elements for the trabecular and inner cortical bone. The outer surface of the cortical bone is represented with three nodal-point shell elements having a thickness of 0.3 mm.

CT scans of the femurs of all patients were performed pre- and postoperatively (1 week after surgery). The scanner settings were approximately 140 kV and 300 mA, with a slice thickness of 2 mm, pixel resolution of 512 × 512, and voxel size of 0.70 × 0.70 × 2 mm. Three-dimensional FE models of the femur and stem were constructed according to pre- and postoperative CT data using the Mechanical Finder ver. 6.0 software (Research Center of Computational Mechanics Inc., Tokyo, Japan). This software creates FE models after considering individual bone shape and density distribution [29]. We used a global threshold algorithm for determining the bone area with a threshold of 200 mg/cm3 and with manual correction if necessary. We applied a closing algorithm in addition to the global threshold algorithm for determining the bone area when the density was extremely low [29]. In addition, the global threshold algorithm was applied to determine the stem area with a threshold of 2,000 Hounsfield units. We used 1 to 4 mm tetrahedral elements to construct three-dimensional FE models for the trabecular and inner cortical bone. The outer surface of the cortical bone was represented with three nodalpoint shell elements having a thickness of 0.3 mm (**Figure 2**). The FE models of the femur consisted of approximately 600,000 elements in addition to 200,000 elements for the stem. Bone density was determined according to CT density values, using a calibration equation (**Table 1**). Additionally, the elastic modulus of the bone was determined according to bone density values, using the equations proposed by Keyak et al. [18, 30] and Keller [31] (**Table 1**). Poisson's ratio of the bone was assumed to be 0.40. The stem was an isotropic titanium alloy, and it had an elastic modulus of 109.0 GPa and a Poisson's ratio of 0.28. The models assumed a completely bonded interface between the stem and bone. The models were restrained at the distal end of the femoral shaft, and loads were applied to the femoral head and greater trochanter. We applied a load of 2400 N to the surface of the femoral head at an angle of 15° to the femoral axis and applied a load of 1200 N to the greater trochanter at an angle of 20° [32, 33]. Linear FEA was performed. Equivalent stress and SED were analyzed in ROIs 1–7 and were compared with the DEXA data.


$$= 0.0 \text{ (H.U.} \le -1\text{)}$$

We investigated 24 patients (16 women and 8 men) who underwent primary cementless THA using the same fit-and-fill type of prosthesis (collarless VerSys Fiber Metal MidCoat, Zimmer Inc., Warsaw, IN) (**Figure 1**). Collarless VerSys Fiber Metal MidCoat is a stem composed of titanium alloy with a fiber-metal porous coating at the proximal region. Patients were excluded from the study if they were receiving systemic estrogen, vitamin D, or bisphosphonate for osteoporosis. The mean patient age at surgery was 63 years (range, 44–82 years). After surgery, all patients were allowed to use a wheelchair, with full weight bearing, on postoperative day 1, and gait exercise was performed as soon as possible [23]. BMD was assessed with dualenergy X-ray absorptiometry (DEXA) using the Hologic Discovery system (Hologic Inc., Waltham, MA). Baseline DEXA measurements were performed 1 week after surgery as reference, and subsequent measurements were performed at 3, 6, and 12 months postoperatively. Patients were placed in the supine position, and the affected leg was maintained in 20 ° internal rotation to minimize measurement errors. DEXA measurements were performed using the "array prosthetic" mode, which excludes metal from analysis of the regions of interest (ROIs). ROIs were defined according to Gruen's system [28]. The pixel size was 1.1 ×

**Figure 2.** Finite element mesh of the femur. Three-dimensional FE models are constructed using 1- to 4-mm tetrahedral elements for the trabecular and inner cortical bone. The outer surface of the cortical bone is represented with three no-

CT scans of the femurs of all patients were performed pre- and postoperatively (1 week after surgery). The scanner settings were approximately 140 kV and 300 mA, with a slice thickness

dal-point shell elements having a thickness of 0.3 mm.

0.56 mm.

216 Perusal of the Finite Element Method

H.U. represents CT density values in Hounsfield Units. The elastic modulus of the bone is determined according to computed tomography density values, using the equations proposed by Keyak et al. [18, 30] and Keller [31].

**Table 1.** Equations used for calculating the elastic modulus of the bone.

Changes in the BMD of the femur during the first postoperative year are shown in **Figure 3**. BMD was maintained in ROIs 3, 4, 5, and 6, whereas it significantly decreased in ROIs 1, 2, and 7 by 19, 11, and 27%, respectively, 12 months after THA. The changes in BMD noted in this study were similar to those reported in other studies [34, 35].

**Figure 3.** The median percentage change in bone mineral density (BMD) after surgery with the fit-and-fill-type stem. The periprosthetic BMDs at 3, 6, and 12 months after total hip arthroplasty (THA) are expressed as percentage changes from the baseline BMD measured at 1 week after surgery. BMD is maintained 12 months after THA in regions of interest (ROIs) 3, 4, 5, and 6, whereas it has significantly decreased by 19, 11, and 27% in ROIs 1, 2, and 7, respectively, 12 months after THA (\**p*< 0.05).

The pre- and postoperative equivalent stress and SED are shown in **Figure 4**. The equivalent stress and SED were maintained in ROIs 3, 4, and 5, whereas both decreased in ROIs 1, 2, 6, and 7 after THA (**Figure 5**). The lowest values of equivalent stress, SED, and relative BMD were observed in ROI 7 (*p* < 0.05). There was a relatively strong correlation between equivalent stress and BMD changes (*R* = 0.426, *p* < 0.01), whereas there was a weak correlation between SED and BMD changes (*R* = 0.183, *p* = 0.053) (**Figure 6**).

**Figure 4.** Distributions of equivalent stress and strain energy density (SED) in the femur. Equivalent stress and SED are maintained in the distal femur after total hip arthroplasty, whereas both decreased in the proximal femur.

Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis http://dx.doi.org/10.5772/65031 219

**Figure 3.** The median percentage change in bone mineral density (BMD) after surgery with the fit-and-fill-type stem. The periprosthetic BMDs at 3, 6, and 12 months after total hip arthroplasty (THA) are expressed as percentage changes from the baseline BMD measured at 1 week after surgery. BMD is maintained 12 months after THA in regions of interest (ROIs) 3, 4, 5, and 6, whereas it has significantly decreased by 19, 11, and 27% in ROIs 1, 2, and 7, respectively,

The pre- and postoperative equivalent stress and SED are shown in **Figure 4**. The equivalent stress and SED were maintained in ROIs 3, 4, and 5, whereas both decreased in ROIs 1, 2, 6, and 7 after THA (**Figure 5**). The lowest values of equivalent stress, SED, and relative BMD were observed in ROI 7 (*p* < 0.05). There was a relatively strong correlation between equivalent stress and BMD changes (*R* = 0.426, *p* < 0.01), whereas there was a weak correlation between SED and

**Figure 4.** Distributions of equivalent stress and strain energy density (SED) in the femur. Equivalent stress and SED are

maintained in the distal femur after total hip arthroplasty, whereas both decreased in the proximal femur.

12 months after THA (\**p*< 0.05).

218 Perusal of the Finite Element Method

BMD changes (*R* = 0.183, *p* = 0.053) (**Figure 6**).

**Figure 5.** Strain energy density (SED), equivalent stress, and bone mineral density (BMD) 12 months after total hip arthroplasty. In ROI 7, equivalent stress, SED, and relative BMD show the lowest values of all regions examined.

**Figure 6.** Scatter plots of the relationships (a) between bone mineral density (BMD) and equivalent stress and (b) between BMD and strain energy density (SED). The periprosthetic BMD at 12 months after total hip arthroplasty (THA) is expressed as the percentage change from the baseline BMD measured at 1 week after surgery. Postoperative equivalent stress and SED are expressed as percentage changes from the preoperative values. (a) Equivalent stress has a relatively strong correlation with the changes in BMD (*R* = 0.426, *p* < 0.01). (b) SED has a weak correlation with the changes in BMD (*R* = 0.183, *p* = 0.053).

Periprosthetic BMD loss may be related to changes in the mechanical loading environment. The purpose of this study was to investigate the relationships between periprosthetic BMD loss and changes in the mechanical loading environment after cementless THA according to subject-specific FEA. This subject-specific FEA exhibited a significant correlation between BMD changes and equivalent stress. After THA, the stress load is transferred to the distal femur through the stem rather than to the periprosthetic bone owing to the high elastic modulus of the stem [36, 37]. Therefore, equivalent stress in the proximal femur may decrease after stem implantation. These results are consistent with those of previous studies [20, 33]. We investigated the effect of a porous-coated titanium-alloy fit-and-fill-type stem on bone remodeling after THA. Although this implant is designed to achieve proximal fixation, equivalent stress and BMD decreased postoperatively at the proximal femur. These results indicate that load transmission to the proximal femur may not be sufficient for maintaining BMD.

The present study demonstrated that equivalent stress correlates with BMD changes in the femur 12 months after THA. In this study, CT scans were performed from all patients pre- and postoperatively, and subject-specific FE models were constructed to investigate bone heterogeneity by using the equations proposed by Keyak et al. [18, 30] and Keller [31]. Our results suggest that the rate of changes in equivalent stress could be an important factor for BMD changes.

The strain-adaptive bone-remodeling theory has been used to simulate BMD changes after THA; Huiskes et al. [14] reported that the SED distribution indicates the regions where resorption and apposition are expected. However, other studies have reported that bone remodeling can be better predicted with stress parameters than with strain parameters. Simulation results and clinical data have shown that there is a positive correlation between average equivalent stress and BMD [38, 39]. Therefore, controversy still exists with regard to whether equivalent stress or SED is an appropriate parameter for evaluating bone remodeling. Our results suggest that a decrease in equivalent stress caused a decrease in BMD in the proximal femur after THA.

#### **2.2. Effects of stem geometry**

A number of factors, such as geometry, roughness and coating of the stem, technique of preparation, and bone quality, influence initial stability or primary fixation. Femoral stems with various geometries are currently in use. The implant shape has been shown to determine cortical contact and initial stability [40]. Our recent study showed that differences in stem design could affect the postoperative BMD loss of the proximal femur [41]. We studied the effects of stem geometry on BMD changes and equivalent stress in the femur after THA.

We investigated 20 patients who underwent primary cementless THA for osteoarthritis or osteonecrosis of the hip. Of these patients, 10 underwent THA with a Zweymüller-type stem (SL-PLUS; Smith & Nephew Inc.), and the other 10 underwent THA with a fit-and-fill-type stem (collarless VerSys Fiber Metal MidCoat; Zimmer Inc.) (**Figure 1**). The mean age of the patients at the time of THA was 63 years (range, 54–72 years) in the Zweymüller-type stem group and 64 years (range, 57–69 years) in the fit-and-fill-type stem group. There was no significant difference in age between the two groups. In the Zweymüller-type stem group, all patients were diagnosed with osteoarthritis, while in the fit-and-fill-type stem group, eight patients were diagnosed with osteoarthritis, and two were diagnosed with osteonecrosis. We compared equivalent stress and BMD between the two groups.

In the preoperative models, no significant differences in equivalent stress were observed between the two groups. In the postoperative models, maximum equivalent stress was observed in ROI 4, and minimum equivalent stress was observed in ROI 1 in both groups. In ROI 4, equivalent stress was significantly higher in the Zweymüller-type stem group than in the fit-and-fill-type stem group (*p* < 0.05). On the other hand, in ROI 7, equivalent stress was significantly higher in the fit-and-fill-type stem group than in the Zweymüller-type stem group (*p* < 0.05) (**Figure 7**).

subject-specific FEA. This subject-specific FEA exhibited a significant correlation between BMD changes and equivalent stress. After THA, the stress load is transferred to the distal femur through the stem rather than to the periprosthetic bone owing to the high elastic modulus of the stem [36, 37]. Therefore, equivalent stress in the proximal femur may decrease after stem implantation. These results are consistent with those of previous studies [20, 33]. We investigated the effect of a porous-coated titanium-alloy fit-and-fill-type stem on bone remodeling after THA. Although this implant is designed to achieve proximal fixation, equivalent stress and BMD decreased postoperatively at the proximal femur. These results indicate that load

The present study demonstrated that equivalent stress correlates with BMD changes in the femur 12 months after THA. In this study, CT scans were performed from all patients pre- and postoperatively, and subject-specific FE models were constructed to investigate bone heterogeneity by using the equations proposed by Keyak et al. [18, 30] and Keller [31]. Our results suggest that the rate of changes in equivalent stress could be an important factor for BMD

The strain-adaptive bone-remodeling theory has been used to simulate BMD changes after THA; Huiskes et al. [14] reported that the SED distribution indicates the regions where resorption and apposition are expected. However, other studies have reported that bone remodeling can be better predicted with stress parameters than with strain parameters. Simulation results and clinical data have shown that there is a positive correlation between average equivalent stress and BMD [38, 39]. Therefore, controversy still exists with regard to whether equivalent stress or SED is an appropriate parameter for evaluating bone remodeling. Our results suggest that a decrease in equivalent stress caused a decrease in BMD in the

A number of factors, such as geometry, roughness and coating of the stem, technique of preparation, and bone quality, influence initial stability or primary fixation. Femoral stems with various geometries are currently in use. The implant shape has been shown to determine cortical contact and initial stability [40]. Our recent study showed that differences in stem design could affect the postoperative BMD loss of the proximal femur [41]. We studied the effects of stem geometry on BMD changes and equivalent stress in the femur after THA.

We investigated 20 patients who underwent primary cementless THA for osteoarthritis or osteonecrosis of the hip. Of these patients, 10 underwent THA with a Zweymüller-type stem (SL-PLUS; Smith & Nephew Inc.), and the other 10 underwent THA with a fit-and-fill-type stem (collarless VerSys Fiber Metal MidCoat; Zimmer Inc.) (**Figure 1**). The mean age of the patients at the time of THA was 63 years (range, 54–72 years) in the Zweymüller-type stem group and 64 years (range, 57–69 years) in the fit-and-fill-type stem group. There was no significant difference in age between the two groups. In the Zweymüller-type stem group, all patients were diagnosed with osteoarthritis, while in the fit-and-fill-type stem group, eight patients were diagnosed with osteoarthritis, and two were diagnosed with osteonecrosis. We

compared equivalent stress and BMD between the two groups.

transmission to the proximal femur may not be sufficient for maintaining BMD.

changes.

proximal femur after THA.

220 Perusal of the Finite Element Method

**2.2. Effects of stem geometry**

**Figure 7.** Comparison of equivalent stress between the fit-and-fill-type stem and Zweymüller-type stem groups. The equivalent stress in ROI 4 is higher, and the stress in ROI 7 is lower in the Zweymüller-type stem group than in the fitand-fill-type stem group \**p*< 0.05.

Changes in the BMD of the femur during the first postoperative year are shown in **Figure 8**. In both the Zweymüller-type stem and fit-and-fill-type stem groups, BMD decreased in ROIs 2, 6, and 7, whereas BMD was maintained in ROIs 3, 4, and 5. In ROIs 6 and 7, BMD was significantly lower in the Zweymüller-type stem group than in the fit-and-fill-type stem group (*p* < 0.05). In ROI 1, the fit-and-fill-type stem group showed a continuous decrease in BMD for 12 months after surgery, while the Zweymüller-type stem group showed a decrease in BMD up to 6 months after surgery and then showed an increase 12 months after surgery. In ROI 4, BMD was significantly lower in the fit-and-fill-type stem group than in the Zweymüller-type stem group at 3 months after surgery (*p* < 0.05); however, there were no differences in BMD between the two groups at 6 months or 12 months after surgery [24].

Stress at the proximal femur was lower, and stress at the distal femur was higher in the Zweymüller-type stem group than in the fit-and-fill-type stem group. BMD at the proximal femur was significantly lower in the Zweymüller-type stem group than in the fit-and-fill-type stem group (*p* < 0.05). The FEA results and postoperative BMD changes in the femur suggest that implantation of the Zweymüller-type stem leads to insufficient load transmission to the proximal femur and may be associated with proximal BMD loss. These findings indicate that subject-specific FEA can assess the effects of stem geometry on adaptive bone remodeling after THA.

**Figure 8.** Median percentage change in bone mineral density (BMD) after surgery in the fit-and-fill-type stem and Zweymüller-type stem groups. The periprosthetic BMDs at 3, 6, and 12 months after surgery are expressed as percentage changes from the baseline BMD measured at 1 week after surgery (\**p*< 0.05).

#### **2.3. Effects of bone geometry**

Noble et al. [42] demonstrated the femoral anatomy variations in the mediolateral and anteroposterior dimensions and the need for multiple stem designs to achieve a close fit. Close adaptation of the prosthesis to the bone geometry is required to achieve optimal primary stability and secondary biologic fixation [43]. We studied the effects of bone geometry on BMD changes and equivalent stress in the femur after THA.

We investigated 20 patients who underwent THA using the Accolade TMZF stem (Stryker Orthopaedics) after reviewing the preoperative anterior-posterior hip radiographs of 90 consecutive patients who had undergone THA using the Accolade TMZF stem (**Figure 1**). The width of the proximal femoral canal on the operated side was measured from preoperative anterior-posterior hip radiographs, and Noble's canal flare index (CFI) was calculated for each patient [42]. Based on Noble's classification, the femurs were divided into the following three groups of femoral canal shapes: champagne flute (CFI ≥ 4.7), intermediate (CFI 3–4.7), and stovepipe (CFI < 3). Among the 20 study femurs, 7 had a champagne flute canal shape, 5 had a stovepipe shape, and 8 had an intermediate canal shape. The 8 femurs with an intermediate canal shape were randomly selected from among the remaining 78 hips with an intermediate canal shape to balance the groups for analysis.

that implantation of the Zweymüller-type stem leads to insufficient load transmission to the proximal femur and may be associated with proximal BMD loss. These findings indicate that subject-specific FEA can assess the effects of stem geometry on adaptive bone remodeling after

**Figure 8.** Median percentage change in bone mineral density (BMD) after surgery in the fit-and-fill-type stem and Zweymüller-type stem groups. The periprosthetic BMDs at 3, 6, and 12 months after surgery are expressed as percent-

Noble et al. [42] demonstrated the femoral anatomy variations in the mediolateral and anteroposterior dimensions and the need for multiple stem designs to achieve a close fit. Close adaptation of the prosthesis to the bone geometry is required to achieve optimal primary stability and secondary biologic fixation [43]. We studied the effects of bone geometry on BMD

We investigated 20 patients who underwent THA using the Accolade TMZF stem (Stryker Orthopaedics) after reviewing the preoperative anterior-posterior hip radiographs of 90 consecutive patients who had undergone THA using the Accolade TMZF stem (**Figure 1**). The width of the proximal femoral canal on the operated side was measured from preoperative anterior-posterior hip radiographs, and Noble's canal flare index (CFI) was calculated for each patient [42]. Based on Noble's classification, the femurs were divided into the following three

age changes from the baseline BMD measured at 1 week after surgery (\**p*< 0.05).

changes and equivalent stress in the femur after THA.

**2.3. Effects of bone geometry**

THA.

222 Perusal of the Finite Element Method

The mean percentage changes in equivalent stress in each ROI between the pre- and postoperative femur models were calculated from the FEA results, and they are plotted in **Figure 9**. Between-group differences in the pattern of postoperative changes in equivalent stress were identified, and specific differences were observed in ROIs 6 (*p* = 0.01) and 7 (*p* = 0.02). Post hoc analysis indicated that the decrease in equivalent stress in ROI 6 was significantly greater in the stovepipe group (−63.8%) than in the champagne flute group (−38.8%) and intermediate group (−47.2%) (*p* = 0.01). Additionally, the decrease in equivalent stress in ROI 7 was significantly greater in the stovepipe group (−70.9%) than in the champagne flute group (−53.7%) and intermediate group (−51.4%) (*p* = 0.03).

**Figure 9.** Comparison of the relative percentage changes in equivalent stress between the pre- and postoperative femur models for the three femoral canal shape groups. Postoperative relative percentage changes in equivalent stress in each region of interest compared to that of the preoperative models. Dots in the box plot show the outliers (\**p* = 0.02, \*\**p* = 0.01).

BMD changes in the femurs during the first postoperative year are shown in **Figure 10**. Significant between-group differences in BMD were evident in ROIs 6 (*p* = 0.01) and 7 (*p* = 0.04). In ROI 6, a marked decrease in BMD was identified only in the stovepipe group during the first postoperative year; the ROI 6 BMD showed a significantly higher decrease in the stovepipe group (−14.3%) than in the champagne flute group (−1.4%) and intermediate group (+3.6%) (*p* = 0.01). A postoperative decrease in ROI 7 BMD was evident in all the groups during the first postoperative year (champagne flute group, −23.1%; intermediate group, −22.7%; and stovepipe group, −36.5%). Additionally, the ROI 7 BMD showed a significantly higher decrease in the stovepipe group than in the other two groups (*p* = 0.045). BMDs in ROIs 1 and 2 were relatively lower in the stovepipe group than in the champagne flute and stovepipe groups during the first postoperative year; however, the differences were not significant.

**Figure 10.** Comparison of bone mineral density (BMD) in the three femoral canal shape groups. The periprosthetic BMDs at 3, 6, and 12 months after surgery are expressed as percentage changes from the baseline BMD measured at 1 week after surgery (\**p* = 0.04, \*\**p* = 0.01).

The percentage changes in equivalent stress in each ROI were positively correlated with the postoperative BMD changes (**Figure 11**). A significant correlation was observed between equivalent stress and BMD changes (*p* < 0.01), and the adjusted *R*<sup>2</sup> value showed that the percentage changes in equivalent stress (adjusted *R*<sup>2</sup> = 0.79) could predict the postoperative BMD changes in our simulation model.

Evaluation of Adaptive Bone Remodeling after Total Hip Arthroplasty Using Finite Element Analysis http://dx.doi.org/10.5772/65031 225

first postoperative year; the ROI 6 BMD showed a significantly higher decrease in the stovepipe group (−14.3%) than in the champagne flute group (−1.4%) and intermediate group (+3.6%) (*p* = 0.01). A postoperative decrease in ROI 7 BMD was evident in all the groups during the first postoperative year (champagne flute group, −23.1%; intermediate group, −22.7%; and stovepipe group, −36.5%). Additionally, the ROI 7 BMD showed a significantly higher decrease in the stovepipe group than in the other two groups (*p* = 0.045). BMDs in ROIs 1 and 2 were relatively lower in the stovepipe group than in the champagne flute and stovepipe groups

**Figure 10.** Comparison of bone mineral density (BMD) in the three femoral canal shape groups. The periprosthetic BMDs at 3, 6, and 12 months after surgery are expressed as percentage changes from the baseline BMD measured at

The percentage changes in equivalent stress in each ROI were positively correlated with the postoperative BMD changes (**Figure 11**). A significant correlation was observed between

value showed that the

= 0.79) could predict the postoperative

equivalent stress and BMD changes (*p* < 0.01), and the adjusted *R*<sup>2</sup>

percentage changes in equivalent stress (adjusted *R*<sup>2</sup>

1 week after surgery (\**p* = 0.04, \*\**p* = 0.01).

224 Perusal of the Finite Element Method

BMD changes in our simulation model.

during the first postoperative year; however, the differences were not significant.

**Figure 11.** Scatter plot illustrating the relationship between the percentage changes in equivalent stress and postoperative bone mineral density (BMD). The 21 points plotted in this scatter plot display the average relative percentage changes in equivalent stress (finite element analysis results) and BMD for the three femoral canal shape groups. A significant correlation is observed between equivalent stress and BMD changes (adjusted *R*<sup>2</sup> = 0.79, *p* < 0.01).

The FEA results and postoperative BMD changes of the femur suggest that the combination of a large, tapered wedge-type stem and stovepipe femur may be associated with proximal BMD loss. Our simulation-based outcomes predicted a stronger stress-shielding effect in the stovepipe femur group than in the other groups. These results indicate that subject-specific FEA can assess the effects of bone geometry on adaptive bone remodeling after THA.

The present studies have limitations. First, the interface condition between the stem and the femur in the FEA model was assumed to be fully bonded. Consideration of the surface properties of the implant and degree of stem fixation would be useful for constructing realistic models. Second, the maximum length of the tetrahedral elements (4 mm) was relatively large, and therefore the elements might straddle the boundary between the cortical and cancellous area in the FE models. Therefore, the elastic modulus of the elements straddling the boundary would contain some errors.

## **3. Summary**

We focused on equivalent stress as a mechanical parameter associated with adaptive bone remodeling and found that equivalent stress changes were positively correlated with postoperative BMD changes in the femur. BMD and equivalent stress were maintained in the distal femur and decreased in the proximal femur after THA. Thus, the proposed subject-specific FEA is useful for evaluating adaptive bone remodeling after THA and could predict BMD changes after THA. Furthermore, subject-specific FEA can be used to evaluate the effects of stem geometry and bone geometry on bone remodeling and can help in the understanding of the altered mechanical environment after THA.

## **Abbreviations**

BMD, bone mineral density CFI, canal flare index CT, computed tomography DEXA, dual-energy X-ray absorptiometry FEA, finite element analysis ROIs, regions of interest SED, strain energy density THA, total hip arthroplasty

## **Author details**

Yutaka Inaba\* , Hiroyuki Ike, Masatoshi Oba and Tomoyuki Saito

\*Address all correspondence to: yute0131@med.yokohama-cu.ac.jp

Department of Orthopaedic Surgery, Yokohama City University, Yokohama, Japan

## **References**

[1] Malchau H, Herberts P, Ahnfelt L. Prognosis of total hip replacement in Sweden. Follow-up of 92,675 operations performed 1978-1990. Acta Orthop Scand. 1993; 64:497– 506.

[2] Dorr LD, Lewonowski K, Lucero M, Harris M, Wan Z. Failure mechanisms of anatomic porous replacement I cementless total hip replacement. Clin Orthop Relat Res. 1997; 334:157–167.

**3. Summary**

226 Perusal of the Finite Element Method

**Abbreviations**

CFI, canal flare index

BMD, bone mineral density

CT, computed tomography

FEA, finite element analysis

ROIs, regions of interest SED, strain energy density THA, total hip arthroplasty

**Author details**

Yutaka Inaba\*

**References**

506.

DEXA, dual-energy X-ray absorptiometry

the altered mechanical environment after THA.

We focused on equivalent stress as a mechanical parameter associated with adaptive bone remodeling and found that equivalent stress changes were positively correlated with postoperative BMD changes in the femur. BMD and equivalent stress were maintained in the distal femur and decreased in the proximal femur after THA. Thus, the proposed subject-specific FEA is useful for evaluating adaptive bone remodeling after THA and could predict BMD changes after THA. Furthermore, subject-specific FEA can be used to evaluate the effects of stem geometry and bone geometry on bone remodeling and can help in the understanding of

, Hiroyuki Ike, Masatoshi Oba and Tomoyuki Saito

Department of Orthopaedic Surgery, Yokohama City University, Yokohama, Japan

[1] Malchau H, Herberts P, Ahnfelt L. Prognosis of total hip replacement in Sweden. Follow-up of 92,675 operations performed 1978-1990. Acta Orthop Scand. 1993; 64:497–

\*Address all correspondence to: yute0131@med.yokohama-cu.ac.jp


[28] Gruen TA, McNeice GM, Amstutz HC. "Modes of failure" of cemented stem-type femoral components: a radiographic analysis of loosening. Clin Orthop Relat Res. 1979; 141:17–27.

[15] Carter DR, Van Der Meulen MC, Beaupre GS. Mechanical factors in bone growth and

[16] Stulpner MA, Reddy BD, Starke GR, Spirakis A. A three-dimensional finite analysis of adaptive remodelling in the proximal femur. J Biomech. 1997; 30:1063–1066.

[17] Adachi T, Tomita Y, Sakaue H, Tanaka M. Simulation of trabecular surface remodeling

[18] Keyak JH, Rossi SA, Jones KA, Skinner HB. Prediction of femoral fracture load using

mapping strategy influences the accuracy of CT-based finite element models of bones: an evaluation against experimental measurements. Med Eng Phys. 2007;

[19] Taddei F, Schileo E, Helgason B, Cristofolini L, Viceconti M. The material

[20] Pettersen SH, Wik TS, Skallerud B. Subject specific finite element analysis of stress shielding around a cementless femoral stem. Clin Biomech (Bristol, Avon).

[21] Cody DD, Gross GJ, Hou FJ, Spencer HJ, Goldstein SA, Fyhrie DP. Femoral strength is better predicted by finite element models than QCT and DXA. J

for osteoporotic vertebrae. J Mech Behav Biomed Mater. 2010; 3:31–40.

[22] Tawara D, Sakamoto J, Murakami H, Kawahara N, Oda J, Tomita K. Mechanical evaluation by patient-specific finite element analyses demonstrates therapeutic effects

[23] Ike H, Inaba Y, Kobayashi N, Hirata Y, Yukizawa Y, Aoki C, Choe H, Saito T. Comparison between mechanical stress and bone mineral density in the femur after total hip arthroplasty by using subject-specific finite element analyses. Comput Methods

[24] Hirata Y, Inaba Y, Kobayashi N, Ike H, Fujimaki H, Saito T. Comparison of mechanical stress and change in bone mineral density between two types of femoral implant using

[25] Anderson AE, Peters CL, Tuttle BD, Weiss JA. Subject-specific finite element model of the pelvis: development, validation and sensitivity studies. J Biomech Eng. 2005;

[26] Ateshian GA, Henak CR, Weiss JA. Toward patient-specific articular contact mechanics.

[27] Weinans H, Sumner DR, Igloria R, Natarajan RN. Sensitivity of periprosthetic stressshielding to load and the bone density-modulus relationship in subject-specific finite

based on local stress uniformity. Jpn Soc Mech Eng. 1997; 40:782–792.

automated finite element modeling. J Biomech. 1998; 31:125–133.

development. Bone. 1996; 18:5S–10S.

29:973–979.

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element models. J Biomech. 2000; 33:809–817.

finite element analysis. J Arthroplasty. 2013; 28:1731–1735.


#### **Finite Elements Method in Implant Prosthetics Finite Elements Method in Implant Prosthetics**

Iulia Roateşi Iulia Roateşi

[40] Khanuja HS, Vakil JJ, Goddard MS, Mont MA. Cementless femoral fixation in total hip

[41] Inaba Y, Kobayashi N, Oba M, Ike H, Kubota S, Saito T. Difference in postoperative periprosthetic bone mineral density changes between 3 major designs of uncemented stems: a 3-year follow-up study. J Arthroplasty. DOI: 10.1016/j.arth.2016.02.009. [42] Noble PC, Alexander JW, Lindahl LJ, Yew DT, Granberry WM, Tullos HS. The anatomic basis of femoral component design. Clin Orthop Relat Res. 1988; 235:148–165.

[43] Massin P, Geais L, Astoin E, Simondi M, Lavaste F. The anatomic basis for the concept of lateralized femoral stems: a frontal plane radiographic study of the proximal femur.

arthroplasty. J Bone Joint Surg Am. 2011; 93:500–509.

J Arthroplasty. 2000; 15:93–101.

230 Perusal of the Finite Element Method

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64969

### **Abstract**

This chapter is devoted to the study of behavior of functional loadings for implant prosthetics rehabilitation by finite elements method (FEM). It presents a numerical calculation of stress, displacement, and strain in implant and surrounding bone, which is used to assess risk factors from a biomechanical point. The masticatory forces are simulated by axial and/or non-axial loads, and they are responsible for the biomechanical response of the bone-tissue-implant-crown system. This chapter represents an analysis of this response in view of highlighting the factors involved in implant stability and success. The safety factor for different loading cases is calculated as well. A good agreement with other study results and clinical studies is obtained.

**Keywords:** finite elements methods, implant prosthodontics, biomechanics, stress distribution, numerical solutions

## **1. Introduction**

This chapter presents a study of the biomechanical behavior of dental implant prosthodontics in a mandible under masticatory loads using numerical models made by the finite elements method (FEM). This method is widely used in the study of dental implants and has proven very useful in analyzing the distribution of strains and stress in the entire bone tissue, implant, and crown structure [1].

These studies are equally relevant in achieving optimal design of the implant, in osseointegration study, in oral rehabilitation study by developing various loading and orientation versions of the implant, and so on [2].

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Numerical analysis of implants is useful for doctors in a deeper and thorough understanding of the characteristics of the state of stress and strain in the bone structure to ensure success of implants [3].

FEM is a numerical method developed by engineers and substantiated by mathematicians to find approximate solutions to complex structure problems. This method is also successfully used in medicine for biological structures and, in particular, in dentistry.

The method consists of dividing (mesh) the problem structure domain in smaller areas (finite element) connected by nodes, called mesh; finding the solution to these subdomains, finite elements (FEs), considering both loads (forces, pressures, etc.) applied to the structure boundary, as well as the boundary conditions and finally assembling these elementary solutions for finding global solutions.

FEM is considered by many specialists as one of the most common numerical methods for structural analysis. Regardless of their type and complexity, FEM is used in industry and science, taking increasingly greater scale in medicine and, certainly, its usefulness being quite rigorously outlined in dentistry and all of its specialties [4].

The large variety of types of finite elements, of loads that can be taken into consideration, of material modes of behavior (linear or non-linear), of the application regime (static, elastic, or dynamic), of the types of analysis (structural response, modal analysis, fatigue, fracture, optimization, etc.) gives the method and, therefore, the user the possibility to carry out an accurate calculation for any problem, with results as close as possible to reality [5].

FEM possibilities are closely linked to the performance of programs and computers. Currently, software products of analysis by FEM are made, which, under increasingly high hard performance, enables the possibility that the size or difficulty of the problem to be solved to no longer constitutes an insurmountable obstacle.

Correctly solving the structure analysis problems by FEM is based primarily on creating a numerical model as suitable as possible; this model should be based on the correct understanding of the problem to be solved and equally on the knowledge of the theoretical foundations of the method.

The great versatility and efficiency of FEM determined its utilization in areas as highly diverse as mechanical structures, fluids, thermal processes, and recently in areas of medicine such as orthopedics, stress flow, dental medicine, and so on.

Therefore, FEM is currently one of the most powerful tools for investigating many phenomena of the most complex and highly diverse areas.

The FEM uses as a starting point an integral model of the phenomenon under consideration [6, 7]. It applies separately for a series of small parts of a continuous structure obtained by the meshing process, known as finite elements, connected to each other at points called nodes [8]. These finite elements must be designed so that their ensemble reconstructs as closely as possible the actual analyzed structure. In principle, these connections must be designed as to allow a numerical convergence to the exact solution when the structure is discretized in finite elements with dimensions increasingly reduced.

## **1.1. Finite elements analysis steps**

Numerical analysis of implants is useful for doctors in a deeper and thorough understanding of the characteristics of the state of stress and strain in the bone structure to ensure success of

FEM is a numerical method developed by engineers and substantiated by mathematicians to find approximate solutions to complex structure problems. This method is also successfully

The method consists of dividing (mesh) the problem structure domain in smaller areas (finite element) connected by nodes, called mesh; finding the solution to these subdomains, finite elements (FEs), considering both loads (forces, pressures, etc.) applied to the structure boundary, as well as the boundary conditions and finally assembling these elementary

FEM is considered by many specialists as one of the most common numerical methods for structural analysis. Regardless of their type and complexity, FEM is used in industry and science, taking increasingly greater scale in medicine and, certainly, its usefulness being quite

The large variety of types of finite elements, of loads that can be taken into consideration, of material modes of behavior (linear or non-linear), of the application regime (static, elastic, or dynamic), of the types of analysis (structural response, modal analysis, fatigue, fracture, optimization, etc.) gives the method and, therefore, the user the possibility to carry out an

FEM possibilities are closely linked to the performance of programs and computers. Currently, software products of analysis by FEM are made, which, under increasingly high hard performance, enables the possibility that the size or difficulty of the problem to be solved to no

Correctly solving the structure analysis problems by FEM is based primarily on creating a numerical model as suitable as possible; this model should be based on the correct understanding of the problem to be solved and equally on the knowledge of the theoretical foun-

The great versatility and efficiency of FEM determined its utilization in areas as highly diverse as mechanical structures, fluids, thermal processes, and recently in areas of medicine such as

Therefore, FEM is currently one of the most powerful tools for investigating many phenomena

The FEM uses as a starting point an integral model of the phenomenon under consideration [6, 7]. It applies separately for a series of small parts of a continuous structure obtained by the meshing process, known as finite elements, connected to each other at points called nodes [8]. These finite elements must be designed so that their ensemble reconstructs as closely as possible the actual analyzed structure. In principle, these connections must be designed as to allow a numerical convergence to the exact solution when the structure is discretized in finite

accurate calculation for any problem, with results as close as possible to reality [5].

used in medicine for biological structures and, in particular, in dentistry.

rigorously outlined in dentistry and all of its specialties [4].

longer constitutes an insurmountable obstacle.

orthopedics, stress flow, dental medicine, and so on.

of the most complex and highly diverse areas.

elements with dimensions increasingly reduced.

dations of the method.

implants [3].

232 Perusal of the Finite Element Method

solutions for finding global solutions.

Finite elements analysis of a structure should take the following steps [6–8]:

## *1. Dividing analysis area in finite elements (mesh)*

Meshing a structure means that it is subdivided in a seldom number of finite elements or in a numerical integration point mesh, interconnected in their exterior nodes. During this operation, the types of finite elements to be used will be chosen and their distribution among the meshed area is established, resulting in their number, size, and form.

Meshing is done by computer. The finite element type is defined by several characteristics, such as dimension (one-, two-, three-dimensional), the number of the element's nodes, the associated approximation functions, and others. Choosing the finite element is of great importance for the necessary of internal memory, for the required calculation effort of the computer, and for results quality.

In case of two-dimensional structures, for modeling, triangular or quadrilateral finite elements can be used; the triangular elements ensure better opportunities in terms of contours geometry approximation, while quadrilateral elements accurately reproduce the distribution of stress. It is appropriate to use the elements as close as possible of the equilateral triangle or square. The use of very obtuse triangles or rectangle angles is not recommended, with too elongated elements. Similarly, in three-dimensional structures, tetrahedral or parallelepipedic types of finite elements can be used.

The starting point for the mathematical construction of various finite elements methods is represented by respecting the following principles:


#### **1.2. Effect of meshing on the numerical results**

Calculation results (displacements strain and stress), which are obtained by FEM, are dependent of the meshing choosing. For this reason, there are situations, especially in the case of complicated geometries when the problem addressed with these methods should be investigated in several meshing variants and subsequently sorting out the results [8].

On the other hand, the effect of errors increases with the number of elements used. Numerical errors are due to truncation, rounding, and of input data errors.

To study the influence of meshing, the most common method is to halve the mesh and compare it, and if the results are negligible, the analysis is considered acceptable. To effectively achieve more accurate results by successive refining of the finite elements mesh, the following criteria must be considered:

**•** Each previous mesh should be reflected in the new one;


Nevertheless, it must be noted that from a certain number of finite elements, results can no longer be improved by increasing their number, and changing the type of finite element used becomes compulsory.

The model obtained by meshing the structure into finite elements must meet the following requirements:

– to represent with sufficient fidelity the actual behavior of the structure;

– to allow easy generation of results (displacements, strains, stress);

– to not involve an excessively high labor for input data preparation or results processing and hence a very long work time for the computer and a great part of its memory.

Some of these requirements are becoming less restrictive following the improvement of software and technical performance of computers.

It should be emphasized that computer programs are capable of performing automatic meshing, being able to perform even an automatic refinement of meshing in areas where there are large gradients of stresses (strains).

### *2. Establishment of finite element equations (elemental equations)*

Material behavior within a finite element is described by finite element equations called elemental equations. These form a system of equations of the element. Basic equations can be derived directly on a variational way, through, for example, residues method (Galerkin) or by the energy balance method.

#### *3. Assembling the elemental equations in the equations system of the structure*

The whole structure behavior is modeled by assembling systems of finite elements equations in the system of structure equations, which physically means that the structure balance is conditioned by the balance of finite elements. The assembly requires that in the common nodes of the elements, the unknown function or functions to have the same value.

#### *4. Implementation of boundary conditions and solving the system of the structure equations*

The equations system obtained from implementing appropriate boundary conditions suitable for the problem under consideration is solved by one of the common procedures, for example, by Gauss elimination or by Cholesky method, and so on, obtaining function values in nodes. These are called primary or first-order unknowns.

#### *5. Performing additional calculations to determine the secondary unknowns*

In some problems, after finding the primary unknowns, the analysis closes. In other problems, however, knowing only the primary unknown is not sufficient, the analysis must proceed with the determination of the secondary unknowns. These are higher-order derivatives of the primary unknowns. Thus, for example, in dental implants analysis, primary unknowns are the nodal displacements. With their help, in this step, secondary unknowns are determined, represented by strains and stress [6, 8].

## **2. Material and method**

**•** Each point of the structure must always be found in a finite element;

– to represent with sufficient fidelity the actual behavior of the structure;

hence a very long work time for the computer and a great part of its memory.

– to allow easy generation of results (displacements, strains, stress);

software and technical performance of computers.

These are called primary or first-order unknowns.

*2. Establishment of finite element equations (elemental equations)*

*3. Assembling the elemental equations in the equations system of the structure*

of the elements, the unknown function or functions to have the same value.

*4. Implementation of boundary conditions and solving the system of the structure equations*

are large gradients of stresses (strains).

the energy balance method.

one element to another.

234 Perusal of the Finite Element Method

becomes compulsory.

requirements:

**•** The approximation function (type of element) should remain the same when passing from

Nevertheless, it must be noted that from a certain number of finite elements, results can no longer be improved by increasing their number, and changing the type of finite element used

The model obtained by meshing the structure into finite elements must meet the following

– to not involve an excessively high labor for input data preparation or results processing and

Some of these requirements are becoming less restrictive following the improvement of

It should be emphasized that computer programs are capable of performing automatic meshing, being able to perform even an automatic refinement of meshing in areas where there

Material behavior within a finite element is described by finite element equations called elemental equations. These form a system of equations of the element. Basic equations can be derived directly on a variational way, through, for example, residues method (Galerkin) or by

The whole structure behavior is modeled by assembling systems of finite elements equations in the system of structure equations, which physically means that the structure balance is conditioned by the balance of finite elements. The assembly requires that in the common nodes

The equations system obtained from implementing appropriate boundary conditions suitable for the problem under consideration is solved by one of the common procedures, for example, by Gauss elimination or by Cholesky method, and so on, obtaining function values in nodes.

## **2.1. The use of FEM in dental implants studies**

FEM is very useful in the study of dental implants because it determines stress, strain, and displacement both in the implant, crown, as well as in the bone tissue. This calculation plays a crucial role both in the optimal design of implants [9] and in determining the factors that control the whole process of post-implant osseointegration [3].

In recent years, studies that are based on the use of FEM in oral implantology became very numerous and are dedicated either to oral rehabilitation or to analysis of the state of stress in and around the implant in the peri-implant bone (cortical and trabecular bone). We can mention works that use two-dimensional (2D) representations of the implant and mandible or maxilla and consider a geometry of the implant-bone system and axial-symmetric loads [10] or numerous three-dimensional (3D) studies [11], which model the bone geometry or loads application in a proper way. On the other hand, there are studies that consider a more realistic modeling of mastication, namely cyclical dynamic loads directed in an occlusal angle [12].

The importance of FEM numerical analysis in the study of dental implants involves several aspects. The method is equally useful both for clinicians, by investigating alternative treatments, and for dental implants producers [13]. They can change the macro-design according to clinical benefits. The purpose of improving the design and use of dental implants is represented by the bone absorption as reduced as possible in the region around the implant, certain micro-displacement of the blunt, a better distribution of the loading on the implant structures [14]. All these properties are often related to the biomechanical behavior and should be investigated not only in clinical trials but also in FEM studies [2].

A new design of dental implants and materials must be subjected to thorough investigation and compared to traditional structures. FEM analysis allows the comparison of old and new treatment modelings, taking into account the limitations and deeper understanding of the applicability areas [15].

FEM studies are not generally designed for clinicians without the help of engineers. Typically, clinicians have limited knowledge of modeling and simulation using computers and therefore they need the help of engineers. Therefore, collaboration between a clinician and an engineer in this field is very useful. The clinician should devote too much time to acquire adequate knowledge on techniques of computational for the preparation and development of a numerical model implementation. However, the clinician must manage the study and provide necessary directions and indications concerning the dental implant parts, bone physiology, masticatory forces, the implant bonds with the bone, generally, the dental implantology matter. Also, the clinician should make every effort to maintain contact with the engineer to achieve effective assessment and adaptation of the numerical model [16].

The number of elements and nodes can be increased to achieve a more detailed modeling. But this can be time consuming, which could hinder the development of the model and complicate the computational calculations. Therefore, the engineer must clearly understand the purpose of the study, the boundaries, the limitations that can be applied to the model, and the number of elements can be increased only in the area of interest, therefore in the vicinity of the implant.

The difficulties in finite element modeling in dental implants study are linked on one hand to the assignment of material properties for biological materials: cortical bone, trabecular bone, gingiva; on modeling of loads that act on the bone-tissue-implant structure [17], and also on the actual faithful realization of the geometric model of the implant, which can be extremely laborious in the case of accurate modeling [11, 18].

As previously mentioned, 3D models are widely used. Given the nature of applied loads, unbalance of loads, and of geometric structures involved as well as their complexity, 2D models are unsatisfactory, 3D models being rather preferred [13, 19]. 2D models [20] cannot simulate the behavior of real structures as realistic as 3D models, so, the latest research focuses on 3D models [3]. The present paper continues the modern trends of recent years of using detailed 3D models that are much closer to reality both in geometric modeling accuracy loads application and in boundary conditions.

This is the case of the present study, which consists of the calculation and analysis of the state of stress and strain and displacement of the implant and its interaction with the mandibular bone.

In implants cases, the most critical area is that in which there is maximum stress concentration [9]. This is the neck of the implant and the surrounding area, namely the cervical edge (marginal bone) [18]. Therefore, this area should be kept as intact as possible in order to maintain a structural and functional bone-implant interface [21].

## **2.2. Numerical analysis program**

Modeling and simulation of dental implants behavior are made using Solid Works program regarding the geometric model development to simulate the implant and bone-tissue structure and Cosmos program for numerical solving of the obtained scheme [22].

Cosmos program is one of the best professional programs of FEM analysis of the continuum mechanics. This program was used in the version of its integration into the Solid Works product —one of the most powerful media of structure modeling—thus ensuring an efficient and accessible interface, both during data preprocessing and during the post-processing of obtained results using graphs, isocurves, isosurfaces to represent the values of the studied entities fields, and so on.

Since implementing the geometry model of dental implant, crown, and bone tissue requires special preprocessing resources, SolidWorks software was used. The geometric model made with this program was exported and used for the Cosmos program calculations.

The present study analyzes the insertion of a dental implant in a section of the mandible, focusing primarily on highlighting and predicting areas of stress concentration both in the trabecular bone and in the cortical bone for different clinical situations and loads caused by mastication. These areas are the most vulnerable areas, where potential failures, fractures, or damage structure may occur.

## **2.3. The geometric model of the dental implant**

## *2.3.1. General considerations*

knowledge on techniques of computational for the preparation and development of a numerical model implementation. However, the clinician must manage the study and provide necessary directions and indications concerning the dental implant parts, bone physiology, masticatory forces, the implant bonds with the bone, generally, the dental implantology matter. Also, the clinician should make every effort to maintain contact with the engineer to achieve

The number of elements and nodes can be increased to achieve a more detailed modeling. But this can be time consuming, which could hinder the development of the model and complicate the computational calculations. Therefore, the engineer must clearly understand the purpose of the study, the boundaries, the limitations that can be applied to the model, and the number of elements can be increased only in the area of interest, therefore in the vicinity of the implant.

The difficulties in finite element modeling in dental implants study are linked on one hand to the assignment of material properties for biological materials: cortical bone, trabecular bone, gingiva; on modeling of loads that act on the bone-tissue-implant structure [17], and also on the actual faithful realization of the geometric model of the implant, which can be extremely

As previously mentioned, 3D models are widely used. Given the nature of applied loads, unbalance of loads, and of geometric structures involved as well as their complexity, 2D models are unsatisfactory, 3D models being rather preferred [13, 19]. 2D models [20] cannot simulate the behavior of real structures as realistic as 3D models, so, the latest research focuses on 3D models [3]. The present paper continues the modern trends of recent years of using detailed 3D models that are much closer to reality both in geometric modeling accuracy loads appli-

This is the case of the present study, which consists of the calculation and analysis of the state of stress and strain and displacement of the implant and its interaction with the mandibular

In implants cases, the most critical area is that in which there is maximum stress concentration [9]. This is the neck of the implant and the surrounding area, namely the cervical edge (marginal bone) [18]. Therefore, this area should be kept as intact as possible in order to

Modeling and simulation of dental implants behavior are made using Solid Works program regarding the geometric model development to simulate the implant and bone-tissue structure

Cosmos program is one of the best professional programs of FEM analysis of the continuum mechanics. This program was used in the version of its integration into the Solid Works product —one of the most powerful media of structure modeling—thus ensuring an efficient and accessible interface, both during data preprocessing and during the post-processing of

maintain a structural and functional bone-implant interface [21].

and Cosmos program for numerical solving of the obtained scheme [22].

effective assessment and adaptation of the numerical model [16].

laborious in the case of accurate modeling [11, 18].

cation and in boundary conditions.

236 Perusal of the Finite Element Method

**2.2. Numerical analysis program**

bone.

The first step of finite element modeling is to provide implant-crown-bone-tissue structure by creating a geometric model. The next step is choosing materials for structural components [23, 24]. Their properties are available in SolidWorks materials library and studies based on experimental determinations [19, 25].

In our case, the analysis model is static and materials are considered elastic. We indicate the material constants for the different materials involved in the modeled structure, such as Young elasticity module (*E*), Poisson coefficient (*ν*), the density, tensile strength, and yield strength. The third step is the application of restrictions related to boundary conditions and then applying the loads to simulate the mastication forces in the considered problem.

Both the geometric model and the complete finite elements model of the implant components and of the bone tissue were performed with great accuracy, considering the construction and functional details (radius of fillet, fillets, releases, and contact) for the results to be as close to reality as possible. All of this can be seen in **Figures 1**–**12**.

## *2.3.2. Geometric model components*

The case of a structure consisting of a 13.5-mm long implant, with a diameter of 3.8 mm with two threaded zones (fine step and big step) placed in a section of the mandible with extension of about 20 mm from the implant axis as shown in **Figures 1(a)** and **1(b)**, was considered [26].

The geometric model consists on one hand of the biological material, mandibular part, which comprises the cancellous bone, cortical bone, and gingiva tissue, and on the other hand of the implant, prosthetic blunt, and crown. The implant is usually made of titanium alloy and crown of ceramic [27]. In the studied Denti implant case, the blunt is made of magnesium alloy (**Figure 1b**).

**Figure 1.** Geometric model: (a) overview and (b) section.

The geometric model of the bone tissue may also be considered in a simpler geometrical shape, for example, in the form of a cylinder, which comprises a layer of trabecular bone lined both in the upper and in the lower sides by a thinner layer of cortical bone, as shown in **Figure 2(a)**. The geometric model of the bone tissue could be simplified by considering a geometric shape in the form of a cylinder, but which contains a layer of trabecular bone bounded only at the top by a thinner layer of cortical bone, as shown in **Figure 2(b)**, which represents bone constituents who contact the implant.

**Figure 2.** Geometric model, bone tissue, simplified bone versions, with the implant and the crown inserted: (a) trabecular bone placed between two cortical bone disks (marked in red); (b) trabecular bone superior bounded by a disk of cortical bone (marked in red).

For structures shown in **Figures 1(a,b)** and **2(a,b)**, various calculations by applying appropriate boundary conditions and some axial and lateral loads between 100 and 400 N have been carried out, which are simulating the masticatory forces [28].

Calculations revealed no significant differences either in values obtained for stress, strains, and displacements, or in their localization which is why it can be considered for subsequent calculations the simpler geometry of the bone tissue as shown in **Figure 2(a,b)**. As the time of creating the geometric model and the running time is much lower in simplified geometry, considering a simplified geometry model is useful for quick guidance calculations, which allow running many cases, with variation of the different factors taken into account, for example, different values, localizations, and positions of the loading forces (simulating masticatory forces), types of material with different values of the material constants, and so on.

For problems that require highly accurate calculations in areas of interest, it is preferable to chose a geometry as close to reality as possible.

The analyzed structure components are implants, crown, and mandible portion. All these components were created by a computer and were used in the finite element calculation [26, 29].

### *2.3.2.1. Implant*

**Figure 1.** Geometric model: (a) overview and (b) section.

238 Perusal of the Finite Element Method

constituents who contact the implant.

cortical bone (marked in red).

out, which are simulating the masticatory forces [28].

The geometric model of the bone tissue may also be considered in a simpler geometrical shape, for example, in the form of a cylinder, which comprises a layer of trabecular bone lined both in the upper and in the lower sides by a thinner layer of cortical bone, as shown in **Figure 2(a)**. The geometric model of the bone tissue could be simplified by considering a geometric shape in the form of a cylinder, but which contains a layer of trabecular bone bounded only at the top by a thinner layer of cortical bone, as shown in **Figure 2(b)**, which represents bone

**Figure 2.** Geometric model, bone tissue, simplified bone versions, with the implant and the crown inserted: (a) trabecular bone placed between two cortical bone disks (marked in red); (b) trabecular bone superior bounded by a disk of

For structures shown in **Figures 1(a,b)** and **2(a,b)**, various calculations by applying appropriate boundary conditions and some axial and lateral loads between 100 and 400 N have been carried

Calculations revealed no significant differences either in values obtained for stress, strains, and displacements, or in their localization which is why it can be considered for subsequent calculations the simpler geometry of the bone tissue as shown in **Figure 2(a,b)**. As the time of creating the geometric model and the running time is much lower in simplified geometry, considering a simplified geometry model is useful for quick guidance calculations, which allow running many cases, with variation of the different factors taken into account, for example, different values, localizations, and positions of the loading forces (simulating masticatory

forces), types of material with different values of the material constants, and so on.

It is cone-shaped with two threaded zones (see **Figure 3**).

**Figure 3.** The geometric model of the implant.

The interior of the implant (see **Figure 4**) allows installation without a threaded zone of the intermediate part (blunt) (see **Figure 5**) and at the top has a hexagonal bore for mounting with an Allen key in the mandible. The thread of the inner part is used to assemble the implant with the prosthetic blunt by a titanium screw M2 (see **Figure 6**).

**Figure 4.** Longitudinal section through the geometric model of the implant where implant system parts can be observed.

**Figure 5.** Geometric model of abutment.

**Figure 6.** Geometric model of screw.

#### *2.3.2.2. Crown*

This component is made from ceramic and has been modeled by a geometric shape as close to the real one as possible (**Figure 7**).

**Figure 7.** Geometric model of crown in section.

## *2.3.2.3. Bone tissue*

**Figure 5.** Geometric model of abutment.

240 Perusal of the Finite Element Method

**Figure 6.** Geometric model of screw.

the real one as possible (**Figure 7**).

**Figure 7.** Geometric model of crown in section.

This component is made from ceramic and has been modeled by a geometric shape as close to

*2.3.2.2. Crown*

This is the portion of the mandible around the implant at a distance from it which does not influence the stress and strain of the studied ensemble.

Adoption of the mandible portion size was based on the Saint-Venant's principle (the new distribution of forces produces appreciable differences in stress in the area of application, but remain with no effect or insignificant effect at large distances from the place of forces application) and by preliminary numerical simulations which confirmed the correct choice of size [30].

Modeling the mandible portion took into account the different structure of the bone portion (cortical and trabecular) by assigning suitable material properties in respective areas (see **Figure 8**) [17].

**Figure 8.** Geometric model of bone as a piece of mandible, in section: cancellous bone portion bordered by the upper and lower thinner layers of cortical bone.

For example, in the case of simplified modeling of the bone tissue in a cylindrical shape, the cortical bone is shown in **Figure 9** and the trabecular bone is shown in **Figure 10**.

**Figure 9.** The geometric model of the cortical bone component in the simplified version in the form of a cylinder. The implant thread with a small step can be seen.

**Figure 10.** Geometric model of cancellous bone component in the form of a simplified cylinder version. The implant thread with a large step can be seen.

For the same example, the whole geometric model, consisting of the trabecular bone side, cortical bone, implant, and crown, is shown in **Figure 11**, and a complete section in **Figure 12**.

**Figure 11.** The geometric model of the assembly, the visible portion in the image is represented by compact bone (a), trabecular bone (b), and crown (c).

**Figure 12.** The longitudinal section of the system, composed of bone portions represented by compact bone, trabecular bone, the implant (d) fixed along the two portions of bone, the clamping screw (e), blunt (f), and crown.

All components were modeled respecting all elements and technical details (threads, taperings, releases, etc.).

The assembly was designed to transmit the mastication force from the crown to blunt and then to the upper part of the implant; mastication force transmission mechanism does not intervene in any threaded part [31].

## **2.4. The dental implant finite elements model**

### *2.4.1. The finite elements*

**Figure 10.** Geometric model of cancellous bone component in the form of a simplified cylinder version. The implant

For the same example, the whole geometric model, consisting of the trabecular bone side, cortical bone, implant, and crown, is shown in **Figure 11**, and a complete section in **Figure 12**.

**Figure 11.** The geometric model of the assembly, the visible portion in the image is represented by compact bone (a),

**Figure 12.** The longitudinal section of the system, composed of bone portions represented by compact bone, trabecular

All components were modeled respecting all elements and technical details (threads, taper-

The assembly was designed to transmit the mastication force from the crown to blunt and then to the upper part of the implant; mastication force transmission mechanism does not intervene

bone, the implant (d) fixed along the two portions of bone, the clamping screw (e), blunt (f), and crown.

thread with a large step can be seen.

242 Perusal of the Finite Element Method

trabecular bone (b), and crown (c).

ings, releases, etc.).

in any threaded part [31].

3D finite elements model used for the study of an implant in a mandible portion was built using SolidWorks software and uses tetrahedral elements both in the implant and in the bone. In the following, we represent one meshing option that easily allows viewing of the considered details.

The FE model is made of several parts corresponding to implant components, as shown in **Figures 13** (FE model of the implant) and **14** (FE model of implant-crown assembly) [26].

**Figure 13.** FE model of implant and abutment.

**Figure 14.** FE model of implant-crown assembly.

**Figures 15** and **16** represent a complete FE model of the implant and mandibular portion, respectively, complete FE model for the implant and mandibular cylindrical portion.

**Figure 15.** Complete FE model of crown, implant, and mandible portion.

**Figure 16.** Simplified FE model (cylindrical mandible portion, implant, crown).

Usually, when a problem is to be modeled, first, simple models are created, beginning with the geometry for a quick development, calculation, and running, considering different scenarios (e.g., loading size, localization of loading application, etc.). Further, more complicated models are created, in order to bring it as close as possible to reality. The main difference between the two models is the easy way to construct, to handle, to run for a much shorter time, and so on, the simpler one. But the complete model offers more accurate information, especially the localizations of critical zones, despite the difficulty of its construction and a longer running time.

The features of present numerical studies obtained on the two models, full (portion of mandible) and simplified (cylindrical), are presented in **Table 1**.


**Table 1.** Mesh information.

The large number of finite elements relative to the size of the implant is explained by the existence of fine threads in the structural elements. Meshing fineness appeared as a need for a closer-to-reality modeling of constructive forms of great finesse, as threads, releases, and so on.

#### *2.4.2. Material models and their characteristics*

In this paragraph, the material types and the characteristics of the material are presented for each component, namely the bone tissue (composed of trabecular bone and cortical bone), implant, and crown.


**Table 2.** Data on the used materials.

**Figure 16.** Simplified FE model (cylindrical mandible portion, implant, crown).

mandible) and simplified (cylindrical), are presented in **Table 1**.

**Information about the mesh Full model Simplified model**

Element size 0.4 mm 0.4 mm Tolerance 0.02 mm 0.02 mm Number of elements 216904 165382 Number of nodes 318030 243874 Meshing time (hh; mm; ss): 00:07:44 00:01:18

time.

244 Perusal of the Finite Element Method

**Table 1.** Mesh information.

implant, and crown.

*2.4.2. Material models and their characteristics*

on.

Usually, when a problem is to be modeled, first, simple models are created, beginning with the geometry for a quick development, calculation, and running, considering different scenarios (e.g., loading size, localization of loading application, etc.). Further, more complicated models are created, in order to bring it as close as possible to reality. The main difference between the two models is the easy way to construct, to handle, to run for a much shorter time, and so on, the simpler one. But the complete model offers more accurate information, especially the localizations of critical zones, despite the difficulty of its construction and a longer running

The features of present numerical studies obtained on the two models, full (portion of

The large number of finite elements relative to the size of the implant is explained by the existence of fine threads in the structural elements. Meshing fineness appeared as a need for a closer-to-reality modeling of constructive forms of great finesse, as threads, releases, and so

In this paragraph, the material types and the characteristics of the material are presented for each component, namely the bone tissue (composed of trabecular bone and cortical bone),


**Table 3.** Material constants used for ceramics.


**Table 4.** Material constants used for magnesium alloy.

These data are provided in [1, 32, 33] and by the technical presentation of implants used in the calculation. They were used as input to perform the numerical calculations.

For an easier presentation, data are summarized in tables as follows: type of material for each component, mass, and volume are shown in **Table 2**, and in **Tables 3**–**7** are represented characteristics of material that is used, respectively, for crown, for intermediate piece of implant, implant, trabecular bone, and cortical bone.


**Table 5.** Material constants used for titanium alloy.


**Table 6.** Material constants used for trabecular bone.


**Table 7.** Material constants used for cortical bone (mandible portion).

## *2.4.3. Contact modeling*

For an easier presentation, data are summarized in tables as follows: type of material for each component, mass, and volume are shown in **Table 2**, and in **Tables 3**–**7** are represented characteristics of material that is used, respectively, for crown, for intermediate piece of

Constant name Value Units Elastic module 1.048 × 10+011 N/m2

Shear modulus 4.1024 × 10+010 N/m2 Density 4428.8 kg/m3 Tensile strength 8.2737 × 10+008 N/m2 Yield strength 1.05 × 10+009 N/m2 Thermal expansion coefficient 9 × 10−006 /Kelvin Thermal conductivity 6.7 W/(m.K) Specific heat 586.04 J/(kg.K)

Constant name Value Units Elastic module 1.8 × 10+008 N/m2

Density 1500 kg/m3 Tensile strength 2 × 10+007 N/m2 Yield strength 1.8 × 10+007 N/m2

Constant name Value Units Elastic module 1.8 × 10+010 N/m2

Density 2000 kg/m3 Tensile strength 1.5 × 10+008 N/m2 Yield strength 1.3 × 10+008 N/m2

0.85

implant, implant, trabecular bone, and cortical bone.

Poisson coefficient 0.31

**Titanium alloy (implant and the screw)**

246 Perusal of the Finite Element Method

Hardening factor

(0.0–1.0; 0.0=isotropic; 1.0=kinematic)

**Trabecular bone (mandible portion)**

**Cortical bone (mandible portion)**

**Table 5.** Material constants used for titanium alloy.

**Table 6.** Material constants used for trabecular bone.

Poisson coefficient 0.3

Poisson Coefficient 0.25

**Table 7.** Material constants used for cortical bone (mandible portion).

In this analysis, we considered a model as close to real conditions as possible. In this regard, we paid special attention to the modeling of contact.

Contact zones are the threaded portions of the implant, mandibular segment, and screw. Contact phenomenon in these areas is modeled by special finite elements—contact elements, according to the actual behavior [33].

The contact model was used between adjacent surfaces of the threaded zones without allowing them for interpenetrating. In this way, up to seven pairs of contact surfaces were defined.

This contact modeling corresponds to a complete osseointegration, which corresponds to a period of implant analysis of more than 6 months after its insertion.

## **2.5. Boundary conditions**

In general, structural analysis, boundary conditions are set in displacements and/or forces in those regions where these entities of the structure are considered to be known [6, 8].

These regions are considered restricted to remain fixed (if they have null displacements and/or rotations) during the numerical simulation or they may have non-zero values for specified displacements and/or rotations.

Non-zero displacement restrictions should be placed on boundaries of model for maintaining the balance of the solution.

Also, restrictions should be placed in nodes that are away from the region of interest, in our case, in the vicinity of the implant. This is in order to prevent overlapping of the stress or strain field associated with the reaction forces to the bone-implant interface.

The models created with FEM presented in this study (**Figure 17**, complete model; **Figure 18**, cylindrical section of a simplified model), the lateral faces of bone tissue and the lower ones, were considered without moving, namely fixed (displacements, nodes are blocked on those faces in all directions).

**Figure 17.** Boundary conditions for FE model (green arrows).

**Figure 18.** Boundary conditions for the simplified FE model (green arrows).

## **2.6. Load application**

Loosing marginal bone in the peri-implantation region may be a result of excessive occlusal forces [34]. To determine and understand this correlation, intensive research, including engineering principles, biomechanical relationships of living tissues, and mechanical properties of bone around the implant, is necessary.

In this context, the loads setting in an FE model is an important part of the study. Each component of the model is contributing to the final analysis after loading. In other words, from beginning to end, all FE analysis steps contribute to the ability to extrapolate masticatory forces in the area around the implant and prosthesis.

Masticatory forces can be forces of compression, traction, bending, and shear. Compressive forces try to push materials into each other, the traction forces pull them apart, separate entities and shear forces cause slides. Most detrimental forces that can increase stress around the boneimplant interface and prosthesis are traction and shear. These forces tend to damage the integrity of the material and cause stress concentration [2, 34].

In general, the prosthesis-implant ensemble adapts to compressive forces. Under effective mastication, repetitive model of cyclical forces transmitted via dental implant causes load of the peri-implant bone. This generates a stress around the ridge and prosthesis. However, random cyclic forces of mastication are not easily simulated. Therefore, most FE studies use axial and/or non-axial forces [35].

Non-axial loadings generate a distinctive stress particularly in the cortical bone ridge. So, for a realistic simulation it is necessary to use a combination of vertical and/or oblique (axial and non-axial) forces. As mentioned, oblique loads are more destructive for the peri-implant bone region and clinically disruptive for the prosthesis.

Masticatory force size can be variable depending on age, sex, and para-functional habits, and can vary from anterior to posterior [36]. Loadings simulating mastication forces generate stress concentrations that must be evaluated.

The loadings applied in the present study were provided by own experimental determinations (maximum values of masticatory forces women/men) and by information analysis from other studies [37–39]. In this study, we used both axial and non-axial loads. More precisely, more sets of static loads were considered, namely axial loads, non-axial (**Figure 19**); axial and nonaxial (**Figure 20**), applied to the crown.

**Figure 19.** Non-axial load application (arrows indicate the application direction).

**Figure 18.** Boundary conditions for the simplified FE model (green arrows).

ties of bone around the implant, is necessary.

in the area around the implant and prosthesis.

axial and/or non-axial forces [35].

concentrations that must be evaluated.

region and clinically disruptive for the prosthesis.

integrity of the material and cause stress concentration [2, 34].

Loosing marginal bone in the peri-implantation region may be a result of excessive occlusal forces [34]. To determine and understand this correlation, intensive research, including engineering principles, biomechanical relationships of living tissues, and mechanical proper-

In this context, the loads setting in an FE model is an important part of the study. Each component of the model is contributing to the final analysis after loading. In other words, from beginning to end, all FE analysis steps contribute to the ability to extrapolate masticatory forces

Masticatory forces can be forces of compression, traction, bending, and shear. Compressive forces try to push materials into each other, the traction forces pull them apart, separate entities and shear forces cause slides. Most detrimental forces that can increase stress around the boneimplant interface and prosthesis are traction and shear. These forces tend to damage the

In general, the prosthesis-implant ensemble adapts to compressive forces. Under effective mastication, repetitive model of cyclical forces transmitted via dental implant causes load of the peri-implant bone. This generates a stress around the ridge and prosthesis. However, random cyclic forces of mastication are not easily simulated. Therefore, most FE studies use

Non-axial loadings generate a distinctive stress particularly in the cortical bone ridge. So, for a realistic simulation it is necessary to use a combination of vertical and/or oblique (axial and non-axial) forces. As mentioned, oblique loads are more destructive for the peri-implant bone

Masticatory force size can be variable depending on age, sex, and para-functional habits, and can vary from anterior to posterior [36]. Loadings simulating mastication forces generate stress

The loadings applied in the present study were provided by own experimental determinations (maximum values of masticatory forces women/men) and by information analysis from other studies [37–39]. In this study, we used both axial and non-axial loads. More precisely, more

**2.6. Load application**

248 Perusal of the Finite Element Method

**Figure 20.** Axial and non-axial load application (arrows indicate the application direction).

## **3. Results of FEM analysis of some clinical situations of implant prosthodontics rehabilitations**

## **3.1. Analysis of states of stress, strain, and displacement in the implant and in the bone tissue**

Under the action of mastication forces, stress concentration increases in the prosthesis and bone. The stress is a representation of internal forces which are acting within a deformable body; it is a measure of the average force per unit of surface within the internal body where forces act. These internal forces occur in response to the application of external loadings on the considered body. Internal resistance after applying external loads cannot be measured. Therefore, a simple method is to measure the forces applied to the cross section.

The FEM analysis of dental implants usually calculated von Mises stress (equivalent stress), a scalar quantity (number) characterizing stress size. This is very important in formulating the criteria of damage, plasticity, strength, and so on. This analysis is used to assess the effect of load forces on the region in the vicinity of dental implant or crown [34].

There is the convention that specifies that positive values of stress mean traction stress, while negative values of stress mean compressive stress.

FEM numerical calculation results are shown as diagrams in which different colors signify areas of physical quantities considered of equal value, the minimum value areas to areas with maximum values. In **Figures 21**–**23**, the red-colored areas represent critical areas and bluecolored areas represent areas with the least loading.

**Figure 21.** von Mises stress field in the longitudinal plane section for axial force.


**Figure 22.** Displacement field in the longitudinal plane section for axial force.

**Figure 23.** Equivalent strain field in the longitudinal plane section for axial force.

In our analysis, loads that simulate axial and non-axial type masticatory forces were considered, with a range between 20 and 140 N.

In **Figures 21**–**23**, the most favorable areas are those with minimum values of stress, strain, and displacement, while areas with the greatest damage, high risk are characterized by higher values.

We will present the most important results regarding the distribution of the following entities of the study: von Mises stress field, displacement field, strain field, the safety factor (SF) for *axial loadings, non-axial loadings, and axial and non-axial loading simultaneously applied*.

## *3.1.1. Axial loadings*

FEM numerical calculation results are shown as diagrams in which different colors signify areas of physical quantities considered of equal value, the minimum value areas to areas with maximum values. In **Figures 21**–**23**, the red-colored areas represent critical areas and blue-

colored areas represent areas with the least loading.

250 Perusal of the Finite Element Method

**Figure 21.** von Mises stress field in the longitudinal plane section for axial force.

**Figure 22.** Displacement field in the longitudinal plane section for axial force.

**Figure 23.** Equivalent strain field in the longitudinal plane section for axial force.

When axial loadings are applied, the results concerning the stress field distribution for the entire structure are shown in **Figure 21**, the displacement field in **Figure 22**, and the strain field in **Figure 23**.

For example, **Figure 21** confirms the correctness of dimensions adopted for mandibular bone by the fact that stress in marginal areas of it is practically null.

**Figures 21**–**23** show fields symmetry due to the application of axial loads and they were obtained for a load value of 100 N.

We note that the case of axial loads is an ideal case, in reality this case is combined with nonaxial loads.

## *3.1.2. Non-axial loadings*

For non-axial loadings, we present the results for the von Mises stress field (**Figure 24**), displacement field (**Figure 25**), and strain field (**Figure 26**).

**Figure 24.** Von Mises stress field in longitudinal plane section for non-axial loadings.

**Figure 25.** Displacement field in longitudinal plane section.

**Figure 26.** Strain field in longitudinal plane section for non-axial loadings for non-axial loadings.

**Figures 24**–**26** confirm instead the asymmetry for stress, strain, and displacement fields due to the application of non-axial forces. Results from the figures presented here are obtained for a non-axial load of 120 N and represent the distribution of von Mises field stress, displacements, and strains of the entire bone-implant-crown system.

The distribution of these quantities—stress, displacements, and strains—is more suggestive if we consider their distribution separately only for bone tissue or for implant.

**Figure 27.** Von Mises stress in the bone tissue.

Thus, we present the results for the *bone tissue*: the von Mises stress distribution in **Figure 27**, displacements distribution in **Figure 28**; *for implant*: the von Mises stress distribution in **Figure 29** and von Mises stress distribution of the implant in the longitudinal section in **Figure 30**.

**Figure 28.** Displacements in the bone tissue.

**Figure 25.** Displacement field in longitudinal plane section.

252 Perusal of the Finite Element Method

**Figure 26.** Strain field in longitudinal plane section for non-axial loadings for non-axial loadings.

we consider their distribution separately only for bone tissue or for implant.

and strains of the entire bone-implant-crown system.

**Figure 27.** Von Mises stress in the bone tissue.

**Figures 24**–**26** confirm instead the asymmetry for stress, strain, and displacement fields due to the application of non-axial forces. Results from the figures presented here are obtained for a non-axial load of 120 N and represent the distribution of von Mises field stress, displacements,

The distribution of these quantities—stress, displacements, and strains—is more suggestive if

**Figure 29.** Von Mises stress in the implant.

**Figure 30.** Von Mises stress in the longitudinal plane section of the implant.

From the results shown in **Figures 27** and **28**, it is observed that to obtain large displacements and stresses, which may become critical in the upper cortical bone of the implant neck, it is necessary to apply non-axial forces in the opposite direction.

### *3.1.3. Axial and non-axial loadings applied simultaneously*

A more realistic simulation of masticatory forces can be achieved by the simultaneous applications of axial (120 N) and non-axial loadings (20 N) (**Figure 31**). This case of application of forces is closer to the actual situation [26].

**Figure 31.** The simultaneous application of axial and non-axial forces.

In this case, for instance, the displacement distribution for the whole bone-implant-crown system is presented in **Figure 32**, while **Figure 33** shows the distribution of displacements only in the bone tissue.

**Figure 32.** Displacements in the case of application of axial and non-axial loads in the whole system.

**Figure 33.** Displacements in the case of application of axial and non-axial loads in bone tissue.

The comparison of these results with those obtained in the previous paragraph is very useful, as they allow an analysis concerning the localizations on which the forces are acting on. It appears that, for example, the maximum von Mises stress if only non-axial loads in Section 3.1.2 is 5.197 × 108 Pa, unlike the calculations in this paragraph, the maximum value of von Mises stress is 1.288 × 108 Pa. This fact is due to modeling of the masticatory forces which are applied over a larger area, case in which there is lower concentration (case of Section 3.1.3). If the application forces surface is smaller, then the maximum stress concentration is higher (case of Section 3.1.2).

The stress is directly proportional to the force and inversely proportional to the surface area on which the force is applied. It is important to determine the surface area on which the loads are applied, as that fact influences considerably the calculation of maximum stress magnitude. For example, the area of the occlusal surface on which the rehabilitation is carried out is less than 4 mm, so that the amount of stress in much rehabilitation is in the range of MPa [40].

## **3.2. Safety factor determination**

*3.1.3. Axial and non-axial loadings applied simultaneously*

**Figure 31.** The simultaneous application of axial and non-axial forces.

in the bone tissue.

of forces is closer to the actual situation [26].

254 Perusal of the Finite Element Method

A more realistic simulation of masticatory forces can be achieved by the simultaneous applications of axial (120 N) and non-axial loadings (20 N) (**Figure 31**). This case of application

In this case, for instance, the displacement distribution for the whole bone-implant-crown system is presented in **Figure 32**, while **Figure 33** shows the distribution of displacements only

**Figure 32.** Displacements in the case of application of axial and non-axial loads in the whole system.

**Figure 33.** Displacements in the case of application of axial and non-axial loads in bone tissue.

The CosmosWorks program has a section destined for determining the distribution of the factor of safety (FOS) or safety factor as a ratio between allowable stress limit values and actual stress values obtained by FEM calculation [22]. If SF < 1, the stress level is higher than the material limit and the structure is likely to fail [22].

Images are suggestive and may indicate the need for constructive appropriate solutions to eliminate dangerous areas that have a low safety factor.

**Figures 34** and **35** present safety factor distributions for implant determined by the von Mises method and Tresca, respectively, in the case of axial loading, and in **Figures 36** and **37**, the safety factor distributions for implant, determined by the von Mises and Tresca method in the case of non-axial loads, respectively.

**Figure 34.** Safety factor distributions in the longitudinal section, von Mises criterion, axial load.

**Figure 35.** Safety factor distributions in the longitudinal section, Tresca criterion, axial load.

**Figure 36.** SF distribution in longitudinal section, the von Mises criterion, non-axial loads.

**Figure 37.** SF distribution in longitudinal section, the criterion Tresca, non-axial loads.

From these figures, the obtained results show that the most strained zone is the contact area of the crown and fine thread parts from the cortical bone area (red-colored areas in **Figures 34** and **35**). Between the two alternative calculations, there are small differences, both in terms of factor value and in terms of its distribution.

A suggestive description of the SF distribution can be achieved by considering the bone tissue only (**Figure 38**).

**Figure 38.** SF distribution in the bone tissue, Tresca criteria for non-axial loads.

From the obtained results of **Figures 36, 37** and **38**, asymmetry is observed due to the non-axial forces application, and also an area with very low safety factor in cervical bone area adjacent to the implant neck, on opposite application of oblique force.

## **4. Discussions**

**Figure 35.** Safety factor distributions in the longitudinal section, Tresca criterion, axial load.

256 Perusal of the Finite Element Method

**Figure 36.** SF distribution in longitudinal section, the von Mises criterion, non-axial loads.

**Figure 37.** SF distribution in longitudinal section, the criterion Tresca, non-axial loads.

3D FEM analysis is very efficient for assessing the biomechanical behavior of a structure made of bone, implant, and crown under various loading conditions. In the past four decades, numerous studies have shown that FEM applied in dentistry is a very successful method used to investigate critical issues related to stress distribution [41]. Using detailed 3D models may be extremely useful in understanding the critical issues related to choosing the rehabilitation and application of procedures.

Results of a FEM analysis cannot be implemented directly in clinical situations, but it can design a model so as to simulate a real situation as well as possible [42]. Some limitations of the study using FEM are, however, the adoption of simplifications and assumptions.

FEM analysis should be interpreted with care. In most cases, the numerical studies of oral implantology use isotropic material, and not orthotropic, or anisotropic, as would be plausible [43].

On the other hand, the model by FEM is a static situation at a moment of loading application and not an actual clinical situation. In reality, structure loading is rather dynamic and cyclical. Materials used in various branches of dentistry are supposed isotropic, homogeneous, and elastic, and so remain after loadings, which do not reflect the real situation [43, 44]. For example, periodontal ligament has non-linear mechanical properties and bone tissue is heterogeneous. The numerical results obtained should be more precise and rigorous if the material would be considered anisotropic and inhomogeneous, but they would lead to complex mathematical calculations on one hand, but more difficult, it would require more complicated laboratory experiments to determine material constants, especially for materials of living tissue type [45].

The results obtained by FEM use von Mises plasticity criterion, which is used in engineering rather for ductile material, for which the compression stress is equal to the traction stress as steel and aluminum. By contrast, brittle materials such as ceramics, cements, or composites resins had a compressive strength value significantly greater than the tensile strength [7].

The structure response is different under the action of asymmetric loads. When the structure is loaded compressively, no significant displacements occur due to higher compressive strength. The situation is different when loading is asymmetrical and when traction stress occurs. When a further lateral load is applied, traction stress is generated in areas with higher values than when vertical loads are applied in the same areas [43].

An increase in load does not cause a change in the stress distribution, only an increase in values. A loading that a structure is subjected can cause micro-cracks in certain parts of the structure, but not an immediate rupture [36].

Most dental breaks of used materials are caused by traction stress. A prevention of this phenomenon can be avoided by adjusting the occlusal surfaces. Masticatory forces vary between 11 and 150 N, while the maximum value of force is 200 N anterior, 350 N posterior, and 1000 N in bruxism [46, 47].

New developments in computer technology and modeling techniques make FEM a reliable and accurate approach in dealing with biomechanical applications.

In clinical practice, it may be considered that as these applications are being made by computer, assumptions and critical limitations can clearly affect the application of these results on a real scenario. Another aspect of the analysis by FEM is overemphasis results, due to the simplification applied to simulation models.

Although usually an advanced computing technology is used to obtain numerical results, there are many factors that affect clinical features such as macro- and micro-design, material properties, loading conditions, and boundary conditions.

Consequently, correlating the obtained FEM results with preclinical and clinical studies of long duration can help to validate results obtained by numerical simulations.

Note that several versions of meshing were created, to a fineness of it that does not lead to errors in the solution 3.5% higher than similar studies conducted by other authors [3].

Loading simulations must be as realistic as possible, containing simultaneous axial forces and non-axial [26, 29].

Results from our FEM analysis are consistent with data from the literature for similar models [48, 49]. Regarding optimizing dental implants, the following can be mentioned [50, 51]:

– Placing the implant should be done with caution so this would be as much as possible in the cortical bone area, as strain and stress values in the trabecular bone zone is very small, they may cause atrophy in the surrounding area.

– The neck of the implant should be long enough so that it departs from the soft tissue and any implant adherence has no effect on the mucous membrane. Any inflammation of the soft tissue and/or marginal bone resorption can jeopardize the stability of the implant.

– The neck of the implants should be quite robust, since the maximum stress concentrations occur in the neck of the implant. If the implant is not quite strong in this region, it can affect the integrity of the implant.

– There is a high risk of overload in the mesial and distal areas. This should be considered in patients where there is a narrow ridge of bone.

– Since loading does not necessarily mean it has overcome bone resistance, continuous loading more likely causes fatigue damage (micro-cracks in bone, marginal bone resorption). From a mechanical point of view, the presence of bone defects seems disadvantageous due to lack of bone support. Instead, the peri-implant bone stresses and strains are not only depending on the in vivo loading but also determined by bone quality (mechanical properties of bone) and amount (thickness of the cortical bone, density of the trabecular bone), periodontal state, oral hygiene, and other factors that may play an important role in marginal bone remodeling.

## **5. Conclusions**

Materials used in various branches of dentistry are supposed isotropic, homogeneous, and elastic, and so remain after loadings, which do not reflect the real situation [43, 44]. For example, periodontal ligament has non-linear mechanical properties and bone tissue is heterogeneous. The numerical results obtained should be more precise and rigorous if the material would be considered anisotropic and inhomogeneous, but they would lead to complex mathematical calculations on one hand, but more difficult, it would require more complicated laboratory experiments to determine material constants, especially for materials

The results obtained by FEM use von Mises plasticity criterion, which is used in engineering rather for ductile material, for which the compression stress is equal to the traction stress as steel and aluminum. By contrast, brittle materials such as ceramics, cements, or composites resins had a compressive strength value significantly greater than the tensile strength [7].

The structure response is different under the action of asymmetric loads. When the structure is loaded compressively, no significant displacements occur due to higher compressive strength. The situation is different when loading is asymmetrical and when traction stress occurs. When a further lateral load is applied, traction stress is generated in areas with higher

An increase in load does not cause a change in the stress distribution, only an increase in values. A loading that a structure is subjected can cause micro-cracks in certain parts of the structure,

Most dental breaks of used materials are caused by traction stress. A prevention of this phenomenon can be avoided by adjusting the occlusal surfaces. Masticatory forces vary between 11 and 150 N, while the maximum value of force is 200 N anterior, 350 N posterior,

New developments in computer technology and modeling techniques make FEM a reliable

In clinical practice, it may be considered that as these applications are being made by computer, assumptions and critical limitations can clearly affect the application of these results on a real scenario. Another aspect of the analysis by FEM is overemphasis results, due to the simplifi-

Although usually an advanced computing technology is used to obtain numerical results, there are many factors that affect clinical features such as macro- and micro-design, material

Consequently, correlating the obtained FEM results with preclinical and clinical studies of long

Note that several versions of meshing were created, to a fineness of it that does not lead to errors in the solution 3.5% higher than similar studies conducted by other authors [3].

Loading simulations must be as realistic as possible, containing simultaneous axial forces and

values than when vertical loads are applied in the same areas [43].

and accurate approach in dealing with biomechanical applications.

properties, loading conditions, and boundary conditions.

duration can help to validate results obtained by numerical simulations.

of living tissue type [45].

258 Perusal of the Finite Element Method

but not an immediate rupture [36].

and 1000 N in bruxism [46, 47].

cation applied to simulation models.

non-axial [26, 29].

The present study constitutes a FEM calculation of stress, displacement, and strain in implant and surrounding bone, which is used to assess risk factors from a biomechanical point.

FEM studies have certain advantages over clinical studies, preclinical, and in vitro. First, patients are in no way affected by the application of new materials and new treatment that have not been previously tested. In the biomedical field, FEM is an important tool because it avoids the need for a traditional specimen, instead using a mathematical model that eliminates the need for a large number of teeth. FEM analysis helps in preparing the design, indicates the right materials, or a combination therefore to be used in various load conditions to reduce material consumption and/or failure of the clinical practice.

FEM analyses are useful for clinicians, although they require additional time to work, but they are a useful tool in predicting the implant choosing: type, shape, size. We believe that in the future, it will develop further, being available and accessible to a large number of doctors.

FEM simulations can be extended in numerous directions, such as parametric studies for the key factors involved in the analysis of the success/failure of the dental implant treatments; the analysis of the implant prostheses: bridges on the implants, connected crowns, and individual crowns on the implants, in order to formulate exclusion criteria regarding the success of one of the therapeutic alternatives; comparative analysis of the various models of implants in terms of their stability and highlighting the factors involved in it.

## **Acknowledgements**

I would like to express my gratitude to Prof. Dr. E. Avram for his support for the geometrical model development and the FEM calculation. I also like to thank Prof. Dr. V. Năstăsescu for his high professionalism and expertise on FEM use.

## **Author details**

Iulia Roateşi

Address all correspondence to: iulia.roatesi@gmail.com

Faculty of Dental Medicine, Titu Maiorescu University, Bucharest, Romania

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I would like to express my gratitude to Prof. Dr. E. Avram for his support for the geometrical model development and the FEM calculation. I also like to thank Prof. Dr. V. Năstăsescu for

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Faculty of Dental Medicine, Titu Maiorescu University, Bucharest, Romania

**Acknowledgements**

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**Author details**

Iulia Roateşi

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264 Perusal of the Finite Element Method

## *Edited by Radostina Petrova*

The finite element method (FEM) is a numerical technique for finding approximate solutions to different numerical problems. The practical applications of FEM are known as finite element analysis (FEA). FEA is a good choice for analyzing problems over complicated domains. The first three chapters of this book contribute to the development of new FE techniques by examining a few key hurdles of the FEM and proposing techniques to mitigate them. The next four chapters focus on the close connection between the development of a new technique and its implementation. Current state-of-the-art software packages for FEA allow the construction, refinement, and optimization of entire designs before manufacturing. This is convincingly demonstrated in the last three chapters of the book with examples from the field of biomechanical engineering.

This book presents a current research by highlighting the vitality and potential of the finite elements for the future development of more efficient numerical techniques, new areas of application, and FEA's important role in practical engineering.

Perusal of the Finite Element Method

Perusal of the

Finite Element Method

*Edited by Radostina Petrova*

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