Preface

The finite element method (FEM) is a widely used technique for numerical simulations in many areas of science and engineering. The method has gained increased popularity over the years for the solution of complex mathematical problems. It is now a powerful and popular numerical method for solving partial differential equations, with flexibility in dealing with complex geometric domains and various boundary conditions. Although the method has been extensively used in the field of structural mechanics, it has also been successfully applied to solve several other types of engineering problems, such as heat conduction, fluid dynamics, seepage flow, and electric and magnetic fields. In particular, FEM has been successfully applied to fluid-structure interaction, thermomechanical, thermochemical, and thermo-chemo-mechanical problems, biomechanics, biomedical engineering, piezoelectricity, ferroelectricity, electromagnetics, and more.

An important advantage of FEM, and the main reason for its popularity among academics and industrial developers, is the ability to handle mathematical problems on domains with arbitrary geometry. An attractive feature is the ability to generate solutions to problems governed by linear and nonlinear differential equations. Moreover, FEM enjoys a firm theoretical foundation that is mostly free of ad hoc schemes and heuristic numerical approximations, thereby inspiring confidence in the physical relevance of the solution.

This book provides several applications of FEM for solving real-world problems. It is a useful resource for students in science and engineering, researchers with diverse educational background, practicing scientists and engineers, computational scientists, and applied mathematicians.

Chapter 1 introduces the method for several one-dimensional and two-dimensional model problems. The remaining chapters consider applications of FEM to several problems. These applications include fluid problems, magnetostatic and magnetodynamic problems, stress predictions of early-age concrete members, application on cell migration, dentistry, nanotechnology research, ship composite-based on aluminum, and nonlinear solid mechanics. The emphasis of the text is on the simulation of several physical phenomena of FEM, but many mathematical and numerical aspects to important problems are also given.

Chapter 1 provides a summary of FEM. Since the remaining chapters of this textbook are based on FEM, we present it in the first chapter as a general method for approximating solutions of ordinary differential equations (ODEs) and partial differential equations (PDEs). To be more specific, we use simple one-dimensional and two-dimensional model problems to introduce FEM.

Chapter 2 studies the pulsatile flow of blood with different physiological pressure conditions and altered gravity. It summarizes the investigation on the effects of hypertension in comparison with normal blood pressure on normal and stenosed carotid artery bifurcation. In addition, it discusses the effects of

altered gravity during the change of posture from sleeping to standing under normal blood pressure conditions.

Chapter 3 applies FEM to solve magnetostatic and magnetodynamic problems.

Chapter 4 presents a two-dimensional finite difference scheme for thermal analysis of a concrete element. FEM is then used to calculate the thermal stresses in the concrete. The analysis results are compared with measurements of actual concrete elements. The combined approach can be a simple and useful tool for analyzing temperatures and thermal stresses in early-age concrete elements.

Chapter 5 presents a mathematical model for convective heat and mass transfer of two immiscible fluids in a vertical channel of variable width with thermo-diffusion, diffusion-thermal effects. The governing boundary layer equations generated for momentum, angular momentum, energy, and species concentration are solved with appropriate boundary conditions using Galerkin FEM. The effects of the pertinent parameters are studied in detail. Furthermore, the chapter analyzes the rate of heat transfer, mass transfer, and shear stress near both walls.

Chapter 6 introduces a newly developed finite element cellular model to simulate collective cell migration and explore the effects of mechanical feedback between cells and between cells and substrate. The viscoelastic model represents one cell with many triangular elements. Intercellular adhesions between cells are represented as linear springs. Furthermore, the chapter includes a mechano-chemical feedback loop between cell-substrate mechanics and cell migration. The results reproduce a set of experimental observations of patterns of collective cell migration during epithelial wound healing. In addition, the chapter demonstrates that cellsubstrate-determined mechanics play an important role in regulating persistent and oriented collective cell migration. It also illustrates that our finite element cellular model can be applied to study a number of tissue-related problems regarding cellular dynamic changes at the subcellular level.

Chapter 7 describes FEM coupled with Monte Carlo analysis as a methodology for quantification of a particularly important nuclear parameter that is primarily influenced by thermal and mechanical phenomena present in nuclear reactors. FEM described in this chapter is used to evaluate the reactivity coefficient associated with the thermal expansion-driven spacing of the assemblies along with the much more complicated reactivity coefficient associated with the thermal expansion and mechanical interaction of the fuel assembly hexagonal ducts.

Chapter 8 presents a brief application of FEM in dentistry. It provides an overview of several methods.

Chapter 9 presents rolling resistance estimation in the design process of passenger car radial tire by using FEM to digitally simulate the tire. The simulation firstly computes the deformation of several alternative designs of tires under certain loading and then calculates the value of deformation force in each tire component during deformation. The total force of deformation is considered as energy loss or hysteresis loss resulting in tire rolling resistance. The experiment was carried out on three different tire designs: two grooves, three grooves, and four grooves. The four-groove tire design gave the smallest rolling resistance coefficient. Finally, the simulation was continued to compare different crown radii of the tires and the results show that the largest crown radius generates the lowest rolling resistance.

**V**

is highlighted.

argument.

circuit.

Chapter 10 elaborates the applications of FEM in varied applications of nanotechnology including carbon nanotubes (CNTs), nanobeams, nanorods, nanobiomaterials,

graphene-coated materials, nanosensors, nanotips, and curved nanobeams.

ANSYS software to determine the distribution of stress.

Chapter 11 explains the use of alternative materials for ship building, namely aluminum-based composite material, which is an aluminum alloy AlSi10Mg (b) ship-building material based on the European Nation (EN) Aluminum Casting (AC)-43,100, with silicon carbide reinforcement. Composite ship models use

Chapter 12 proposes a symbolic mathematical approach for the rapid early-phase development of finite elements. The algebraic manipulator adopted is MATLAB and the applicative context is the analysis of hyperelastic solids or structures under the hypothesis of finite deformation kinematics. The work has been finalized through the production, in an object-oriented programming style, of three MATLAB classes implementing a truss element, tetrahedral element, and plane element. The approach proposed, starting with the mathematical formulation and finishing with the code implementation, is described and its effectiveness, in terms of minimization of the gap between the theoretical formulation and its actual implementation,

Chapter 13 investigates the error estimation of numerical approximation to a class of semi-linear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method, and the space discretization uses FEM for which the meshes are allowed to change in time. The key idea in the analysis is to adapt the elliptic reconstruction technique enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order of convergence for Lipschitz and non-Lipschitz nonlinearities. This chapter also addresses some challenges dealing with the nonlinear term by employing a continuation

Chapter 14 presents a finite element magnetic method for magnetorheological-based actuators. We consider several discussions such as necessary magnetostatic using free software finite element magnetic method; design consideration for the magnetic circuit of the device and case studies of several types of simulation in magnetorheological material-based devices. During the design process, magnetostatic simulation using the finite element magnetic method is carried out to make a better magnetic

We thank all the authors who contributed to this book with their studies that provide accessible and excellent explanations of the applications of FEM. This work would not have been possible without our excellent contributors. Finally, we express our thanks to Author Service Manager Mr. Josip Knapić and the staff at IntechOpen for

**Mahboub Baccouch**

Omaha, Nebraska, USA

Department of Mathematics, University of Nebraska at Omaha,

their invaluable support and editorial assistance.

Chapter 10 elaborates the applications of FEM in varied applications of nanotechnology including carbon nanotubes (CNTs), nanobeams, nanorods, nanobiomaterials, graphene-coated materials, nanosensors, nanotips, and curved nanobeams.

Chapter 11 explains the use of alternative materials for ship building, namely aluminum-based composite material, which is an aluminum alloy AlSi10Mg (b) ship-building material based on the European Nation (EN) Aluminum Casting (AC)-43,100, with silicon carbide reinforcement. Composite ship models use ANSYS software to determine the distribution of stress.

Chapter 12 proposes a symbolic mathematical approach for the rapid early-phase development of finite elements. The algebraic manipulator adopted is MATLAB and the applicative context is the analysis of hyperelastic solids or structures under the hypothesis of finite deformation kinematics. The work has been finalized through the production, in an object-oriented programming style, of three MATLAB classes implementing a truss element, tetrahedral element, and plane element. The approach proposed, starting with the mathematical formulation and finishing with the code implementation, is described and its effectiveness, in terms of minimization of the gap between the theoretical formulation and its actual implementation, is highlighted.

Chapter 13 investigates the error estimation of numerical approximation to a class of semi-linear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method, and the space discretization uses FEM for which the meshes are allowed to change in time. The key idea in the analysis is to adapt the elliptic reconstruction technique enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order of convergence for Lipschitz and non-Lipschitz nonlinearities. This chapter also addresses some challenges dealing with the nonlinear term by employing a continuation argument.

Chapter 14 presents a finite element magnetic method for magnetorheological-based actuators. We consider several discussions such as necessary magnetostatic using free software finite element magnetic method; design consideration for the magnetic circuit of the device and case studies of several types of simulation in magnetorheological material-based devices. During the design process, magnetostatic simulation using the finite element magnetic method is carried out to make a better magnetic circuit.

We thank all the authors who contributed to this book with their studies that provide accessible and excellent explanations of the applications of FEM. This work would not have been possible without our excellent contributors. Finally, we express our thanks to Author Service Manager Mr. Josip Knapić and the staff at IntechOpen for their invaluable support and editorial assistance.

> **Mahboub Baccouch** Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska, USA
