1. Introduction

In the analysis of complex structural problems, it is often challenging to formulate and apply exact closed-form solutions, as the realistic nature of such engineering systems exhibits varying complexities, high gradients and strong irregularities, e.g., suddenly varying loading conditions, contrasting material composition or geometric variations. Based on the existing

© 2018 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.

© 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 eproduction in any medium, provided the original work is properly cited.

mathematical tools available, such systems may require certain assumptions and generalisations to be implemented in order to simplify the model, which may lead to inability to correctly describe the properties and behaviour of the system under described conditions. However, the preferred approach is to find an approximate numerical solution, whilst retaining these complexities as accurately as possible, to better describe and predict the behaviour of such systems. This has given rise to numerical methods such as the classical Finite Element Method which employs polynomial interpolating functions to obtain approximate solutions for various engineering problems. Although this numerical analysis technique has grown in popularity, its use to tackle problems with regions of the solution domain where the gradient of the field variables are expected to vary suddenly or fast, bring on difficulties in the analysis of a complex system [1]. In order to improve on the accuracy and better represent the system's behaviour, higher order polynomial interpolating functions or finer meshes may be employed and this in turn significantly increases the computational costs; which is undesirable. Moreover, the resolution of the elements can only be analysed to a specific scale once the orders of the governing polynomial functions have been selected. Subsequently, overcoming these challenges has been the driving force in the formulation of other numerical approximation techniques such as the Wavelet Finite Element Method [1–6].

compact support lead to a reduction in computational costs since fewer elements are required to achieve acceptable levels of accuracy [4, 5]. Due to the adaptability of wavelets, different wavelet families are being developed and customised for specific problems. However, it must be noted that when selecting a particular wavelet basis function for WFEM, key requirements, such as compatibility, completeness and convergence, must be satisfied and should allow for

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

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51

The Daubechies wavelet based finite element was first introduced to solve a 1D and 2D second order Neumann problem via the formulation of a tensor product finite element [2]. The Daubechies wavelet Galerkin finite element was then used to analyse the bending of plates and beams [12] giving rise to the formulation of a wavelet based beam finite element [6] and two dimensional Daubechies wavelet plate finite element [13] for static analysis. The Daubechies wavelet base finite element stiffness matrices and load vectors were presented by Chen et al. at multiresolution scale j = 0 [14] and different multiresolution scales [4]. The Daubechies plate finite element was developed by Diaz et al. for the static analysis of plates based on Mindlin-Reissner plate theory [15], where shear deformation is taken into consideration through the thickness of the plate, and compared it with Kirchhoff plate theory formulations [16]. This wavelet family has also been used in the analysis of many other structural problems, including formulation of the Rayleigh-Euler and Rayleigh-Timoshenko beam elements [17], the wavelet based spectral finite element to study elastic wave propagation in 1-D connected waveguides [18] and also to investigate the thermal stress distribution along the vertical direction of the tank wall [19]. Overall, the wavelet family performed decently in providing accurate solutions for the various structural analysis problems tackled. However, the Daubechies wavelet lacks an explicit expression for the wavelet and scaling functions and possesses unusual smoothness characteristics, particularly for lower orders, making it challenging to evaluate the numerical integrals necessary for the formulation of the element matrices and load vectors. The evaluation of the connection coefficients is therefore necessary

In a bid to overcome the limitations presented by the Daubechies wavelet, further research has been carried out to identify other potential wavelet families that can be implemented in WFEM. Basic spline functions were initially used as interpolating functions for the free vibration analysis of frame structures [3]. Chui and Quak [20] constructed the semi-orthogonal Bspline Wavelet on the Interval, which has the desirable properties of multiresolution, compact support, explicit expressions, smoothness and symmetry. The BSWI was employed to construct the wavelet based C<sup>0</sup> type plane elastomechanics element and Mindlin plate element [21] as well as truncated conical shell wavelet finite elements [22]. Xiang et al. [5] significantly contributed to the use of BSWI in WFEM by constructing the axial rod, beam (Timoshenko and Euler Bernoulli) and spatial bar WFEs with a multiresolution lifting scheme. Furthermore, this research was extended to the static and dynamic analysis of plates based on Kirchhoff plate theory using BSWI based wavelet finite elements [23, 24]. Xiang et al. [25] were able to illustrate that the shear-locking phenomenon of a rotating Rayleigh-Timoshenko shaft was significantly eliminated when the BSWI based WFEs were employed. Majority of the problems examined by this point were of static analysis and this led Musuva and Mares [26] to develop and implement the Daubechies and BSWI homogenous beam WFEs for the analysis of dynamic response and moving load problems. The vibration and dynamic response analysis

the easy implementation and treatment of boundary conditions.

for the formulation of these element matrices and vectors.

The initial development of wavelet analysis came from separate efforts that led to the foundation of modern wavelet theory. Grossman and Morlet [7] used wavelet analysis as a tool for signal analysis of seismic data and are credited with the introduction of the term and methodology of wavelets as it is known today. Ingrid Daubechies is recognised for her major breakthrough and contribution by constructing a family of orthonormal wavelet with compact support known as the Daubechies wavelets [8]. Wavelet analysis was used mainly by mathematicians as a decomposition tool for data functions and operators and its application has vastly grown in various disciplines at an exponential rate e.g., medicine [9], finance [10] and astronomy [11]. Likewise, the range of wavelet families and bases available for selection has also increased and this is credited to the properties of wavelets that allow it to be tailored to suite numerous avenues for design manipulation to meet the necessary and specific requirements for its application. The properties of different wavelet families vary, and therefore the decision on which family is the 'most adequate', is paramount to its application. Nevertheless, the more general aspects of wavelets formulations make it an important and convenient tool for mathematical manipulation allowing for the decomposition of a function into a set of coefficients that are dependent on scale and location. The 'two-scale' relation gives rise to one of the most key features of wavelet theory, multiresolution analysis (MRA), which allows for the convenient transformation of wavelet basis functions between different resolution scales [8]. Furthermore, the compact support property of wavelets ensures that the wavelet basis functions are finitely bound (non-zero over a finite range). The vanishing moments of wavelets allow the basic functions of wavelets to represent polynomials and other complex functions.

These desirable properties of wavelets have led to the use of wavelet basis functions as interpolating functions, in contrast to conventional polynomial functions as used in classical FEM, in the formulation of the wavelet based finite element method. For example, MRA permits for specific WFEs to be selected and analysed locally at finer scales without altering the initial system model, thus improving the accuracy of the solution, particularly in areas with high gradients or singularities present. Furthermore, rapid convergence of the method and compact support lead to a reduction in computational costs since fewer elements are required to achieve acceptable levels of accuracy [4, 5]. Due to the adaptability of wavelets, different wavelet families are being developed and customised for specific problems. However, it must be noted that when selecting a particular wavelet basis function for WFEM, key requirements, such as compatibility, completeness and convergence, must be satisfied and should allow for the easy implementation and treatment of boundary conditions.

mathematical tools available, such systems may require certain assumptions and generalisations to be implemented in order to simplify the model, which may lead to inability to correctly describe the properties and behaviour of the system under described conditions. However, the preferred approach is to find an approximate numerical solution, whilst retaining these complexities as accurately as possible, to better describe and predict the behaviour of such systems. This has given rise to numerical methods such as the classical Finite Element Method which employs polynomial interpolating functions to obtain approximate solutions for various engineering problems. Although this numerical analysis technique has grown in popularity, its use to tackle problems with regions of the solution domain where the gradient of the field variables are expected to vary suddenly or fast, bring on difficulties in the analysis of a complex system [1]. In order to improve on the accuracy and better represent the system's behaviour, higher order polynomial interpolating functions or finer meshes may be employed and this in turn significantly increases the computational costs; which is undesirable. Moreover, the resolution of the elements can only be analysed to a specific scale once the orders of the governing polynomial functions have been selected. Subsequently, overcoming these challenges has been the driving force in the formulation of other numerical approximation techniques such as the

50 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

The initial development of wavelet analysis came from separate efforts that led to the foundation of modern wavelet theory. Grossman and Morlet [7] used wavelet analysis as a tool for signal analysis of seismic data and are credited with the introduction of the term and methodology of wavelets as it is known today. Ingrid Daubechies is recognised for her major breakthrough and contribution by constructing a family of orthonormal wavelet with compact support known as the Daubechies wavelets [8]. Wavelet analysis was used mainly by mathematicians as a decomposition tool for data functions and operators and its application has vastly grown in various disciplines at an exponential rate e.g., medicine [9], finance [10] and astronomy [11]. Likewise, the range of wavelet families and bases available for selection has also increased and this is credited to the properties of wavelets that allow it to be tailored to suite numerous avenues for design manipulation to meet the necessary and specific requirements for its application. The properties of different wavelet families vary, and therefore the decision on which family is the 'most adequate', is paramount to its application. Nevertheless, the more general aspects of wavelets formulations make it an important and convenient tool for mathematical manipulation allowing for the decomposition of a function into a set of coefficients that are dependent on scale and location. The 'two-scale' relation gives rise to one of the most key features of wavelet theory, multiresolution analysis (MRA), which allows for the convenient transformation of wavelet basis functions between different resolution scales [8]. Furthermore, the compact support property of wavelets ensures that the wavelet basis functions are finitely bound (non-zero over a finite range). The vanishing moments of wavelets allow the basic functions of wavelets to represent polynomials and other complex functions. These desirable properties of wavelets have led to the use of wavelet basis functions as interpolating functions, in contrast to conventional polynomial functions as used in classical FEM, in the formulation of the wavelet based finite element method. For example, MRA permits for specific WFEs to be selected and analysed locally at finer scales without altering the initial system model, thus improving the accuracy of the solution, particularly in areas with high gradients or singularities present. Furthermore, rapid convergence of the method and

Wavelet Finite Element Method [1–6].

The Daubechies wavelet based finite element was first introduced to solve a 1D and 2D second order Neumann problem via the formulation of a tensor product finite element [2]. The Daubechies wavelet Galerkin finite element was then used to analyse the bending of plates and beams [12] giving rise to the formulation of a wavelet based beam finite element [6] and two dimensional Daubechies wavelet plate finite element [13] for static analysis. The Daubechies wavelet base finite element stiffness matrices and load vectors were presented by Chen et al. at multiresolution scale j = 0 [14] and different multiresolution scales [4]. The Daubechies plate finite element was developed by Diaz et al. for the static analysis of plates based on Mindlin-Reissner plate theory [15], where shear deformation is taken into consideration through the thickness of the plate, and compared it with Kirchhoff plate theory formulations [16]. This wavelet family has also been used in the analysis of many other structural problems, including formulation of the Rayleigh-Euler and Rayleigh-Timoshenko beam elements [17], the wavelet based spectral finite element to study elastic wave propagation in 1-D connected waveguides [18] and also to investigate the thermal stress distribution along the vertical direction of the tank wall [19]. Overall, the wavelet family performed decently in providing accurate solutions for the various structural analysis problems tackled. However, the Daubechies wavelet lacks an explicit expression for the wavelet and scaling functions and possesses unusual smoothness characteristics, particularly for lower orders, making it challenging to evaluate the numerical integrals necessary for the formulation of the element matrices and load vectors. The evaluation of the connection coefficients is therefore necessary for the formulation of these element matrices and vectors.

In a bid to overcome the limitations presented by the Daubechies wavelet, further research has been carried out to identify other potential wavelet families that can be implemented in WFEM. Basic spline functions were initially used as interpolating functions for the free vibration analysis of frame structures [3]. Chui and Quak [20] constructed the semi-orthogonal Bspline Wavelet on the Interval, which has the desirable properties of multiresolution, compact support, explicit expressions, smoothness and symmetry. The BSWI was employed to construct the wavelet based C<sup>0</sup> type plane elastomechanics element and Mindlin plate element [21] as well as truncated conical shell wavelet finite elements [22]. Xiang et al. [5] significantly contributed to the use of BSWI in WFEM by constructing the axial rod, beam (Timoshenko and Euler Bernoulli) and spatial bar WFEs with a multiresolution lifting scheme. Furthermore, this research was extended to the static and dynamic analysis of plates based on Kirchhoff plate theory using BSWI based wavelet finite elements [23, 24]. Xiang et al. [25] were able to illustrate that the shear-locking phenomenon of a rotating Rayleigh-Timoshenko shaft was significantly eliminated when the BSWI based WFEs were employed. Majority of the problems examined by this point were of static analysis and this led Musuva and Mares [26] to develop and implement the Daubechies and BSWI homogenous beam WFEs for the analysis of dynamic response and moving load problems. The vibration and dynamic response analysis was carried out for frame structures using the two wavelet families [27] and the WFEM was compared with an analytical wavelet approach using coiflets for the analysis of vehicle-bridge interaction for fast moving loads [28]. Furthermore, the Daubechies and BSWI wavelets were used to construct a functionally graded beam wavelet finite element under various moving load conditions [29, 30].

f <sup>2</sup>ð Þ¼ x fð Þ 2x ∀x

f <sup>n</sup>ð Þ¼ x f xð Þ � n

The orthogonal complement subspace Wj of Vj contains the additional 'detail' for subspace Vjþ<sup>1</sup> i.e., Vjþ<sup>1</sup> ¼ V<sup>0</sup> ⊕ W<sup>0</sup> ⊕ W<sup>1</sup> ⊕ W2⋯ ⊕ Wj. The union of the subspaces Vj leads to the space

functions correspond to the subspaces Vj and Wj respectively. The difference between current subspace Vj and subsequent subspace Vjþ<sup>1</sup> is represented by the wavelet space Wj which becomes automatically orthogonal to all other Wj for k < j due to the inclusion in and orthogonality to Vj. For the fundamental space V0, the scaling function fð Þx and its translates fð Þ x � k produce an orthonormal basis for V0. The orthonormal basis for the next space V<sup>1</sup> is

Provided Eq. (6) and the above mentioned properties are satisfied, the wavelet orthonormal

functions within these subspaces inherit the scale and shift invariance properties from the scaling function subspaces Vj and are orthonormal [8]. The projections of a function f ∈ L<sup>2</sup>

<sup>k</sup> are coefficients in the subspaces Vj and Wj respectively. Thus, if all the

at scale j in the subspaces Vj and Wj, defined as Pjf and Qjf respectively, are expressed as:

Pjf <sup>¼</sup> <sup>X</sup> k a j kfj <sup>k</sup>ð Þx

Qjf <sup>¼</sup> <sup>X</sup> k b j kψj <sup>k</sup>ð Þx

conditions described above are met, then the scaling and wavelet functions satisfy [8] ð∞ �∞

ð∞ �∞ fð Þx dx 6¼ 0

ψð Þx dx ¼ 0

Daubechies wavelets are compact supported orthonormal wavelets developed by Ingrid Daubechies and for order L, the scaling and wavelet functions are described by the 'two-scale'

<sup>p</sup> <sup>f</sup>ð Þ <sup>2</sup><sup>x</sup> � <sup>k</sup> . Thus, the orthonormal basis of Vj is defined as:

ð Þ <sup>R</sup> from the condition in Eq. (2) [36]. The scaling <sup>f</sup>ð Þ<sup>x</sup> <sup>∈</sup>L<sup>2</sup>

fj <sup>k</sup>ð Þ¼ x 2 j <sup>2</sup>f 2<sup>j</sup>

ψj <sup>k</sup>ð Þ¼ x 2 j <sup>2</sup>ψ 2<sup>j</sup>

The orthogonal subspaces Wj result from the decomposition of L<sup>2</sup>

L2

where a j <sup>k</sup> and b j

relation [8]:

2.1. Daubechies wavelet

the rescaled function ffiffiffi

basis for subspace Wj at scale j is

2

<sup>f</sup> <sup>∈</sup> Vj <sup>⇔</sup> <sup>f</sup> <sup>2</sup> <sup>∈</sup> Vjþ<sup>1</sup> <sup>j</sup><sup>∈</sup> <sup>Z</sup> (4)

Multiscale Wavelet Finite Element Analysis in Structural Dynamics

<sup>f</sup> <sup>∈</sup>V<sup>0</sup> <sup>⇔</sup> <sup>f</sup> <sup>n</sup> <sup>∈</sup> <sup>V</sup><sup>0</sup> <sup>n</sup>∈<sup>Z</sup> (5)

<sup>x</sup> � <sup>k</sup> � � <sup>k</sup><sup>∈</sup> <sup>Z</sup> (6)

<sup>x</sup> � <sup>k</sup> � � <sup>k</sup> <sup>∈</sup><sup>Z</sup> (7)

ð Þ <sup>R</sup> and wavelet <sup>ψ</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

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ð Þ R and subsequently the

ð Þ R

53

ð Þ R

(8)

(9)

Other different wavelet families have been selected and employed in the formulation of the WFEM to solve a wide variety of structural analysis problems and research in this field is still ongoing. The trigonometric Hermite wavelet, which can be explicitly expressed, was used to construct beam [31] and thin plate WFEs [32] for static and free vibration analysis. The Hermite Cubic Spline Wavelet on the Interval (HCSWI), polynomial wavelets [33] and the second generation wavelets [34] are other wavelet based approaches that have been introduced and researched on. A more comprehensive synthesis and summary of wavelet based numerical methods for various engineering problems is presented in [35].

A generalised Wavelet based Finite Element Method framework is presented based on the BSWI and Daubechies wavelet families to derive rod and beam WFEs for homogenous and functionally graded materials for static and dynamic structural problems. A brief introduction of wavelet analysis is described in Section 2, with emphasis given to the Daubechies wavelets, BSWI, multiresolution and connection coefficients formulations. In Section 3, the wavelet based finite elements for a rod, Euler Bernoulli homogeneous beam and transversely varying functionally graded beam are presented. The evaluation of the element matrices and various load vectors, including the WFEM moving load formulation, are presented. A comparison on the performance of the Daubechies and BSWI WFEMs are highlighted via numerical examples for a variety of static and dynamic structural problems in Section 4 followed by conclusions.
