Open access peer-reviewed chapter

# Multiscale Wavelet Finite Element Analysis in Structural Dynamics

By Mutinda Musuva and Cristinel Mares

Submitted: May 10th 2017Reviewed: October 23rd 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71882

## Abstract

Over the recent past, various numerical analysis techniques have been formulated and used to obtain approximate solutions for numerous engineering problems to aid predict the behaviour of systems accurately and efficiently. One such approach is the Wavelet Finite Element Method (WFEM) which involves combining the classical Finite Element Method (FEM) with wavelet analysis. The key desirable properties exhibited by some wavelet families, such as compact support, multiresolution analysis (MRA), smoothness, vanishing moments and the ‘two-scale’ relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) of a rod and beam are formulated using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet scaling functions as interpolating functions due to their desirable properties, thus making it possible to alter the local scale of the WFE without changing the initial model mesh. Specific benchmark cases are presented to exhibit and compare the performance of the WFEM with FEM in static, dynamic, eigenvalue and moving load transient response analysis for homogenous systems and functionally graded materials, where the material properties continuously vary spatially with respect to the constituent materials.

### Keywords

• multiresolution
• wavelets
• wavelet finite element (WFE)
• eigenvalue analysis

## 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 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 . 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, 2, 3, 4, 5, 6].

The initial development of wavelet analysis came from separate efforts that led to the foundation of modern wavelet theory. Grossman and Morlet  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 . 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 , finance  and astronomy . 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 . 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.

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 . The Daubechies wavelet Galerkin finite element was then used to analyse the bending of plates and beams  giving rise to the formulation of a wavelet based beam finite element  and two dimensional Daubechies wavelet plate finite element  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  and different multiresolution scales . The Daubechies plate finite element was developed by Diaz et al. for the static analysis of plates based on Mindlin-Reissner plate theory , where shear deformation is taken into consideration through the thickness of the plate, and compared it with Kirchhoff plate theory formulations . 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 , the wavelet based spectral finite element to study elastic wave propagation in 1-D connected waveguides  and also to investigate the thermal stress distribution along the vertical direction of the tank wall . 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 . Chui and Quak  constructed the semi-orthogonal B-spline 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 C0 type plane elastomechanics element and Mindlin plate element  as well as truncated conical shell wavelet finite elements . Xiang et al.  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.  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  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  and the WFEM was compared with an analytical wavelet approach using coiflets for the analysis of vehicle-bridge interaction for fast moving loads . Furthermore, the Daubechies and BSWI wavelets were used to construct a functionally graded beam wavelet finite element under various moving load conditions [29, 30].

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  and thin plate WFEs  for static and free vibration analysis. The Hermite Cubic Spline Wavelet on the Interval (HCSWI), polynomial wavelets  and the second generation wavelets  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 .

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.

## 2. Wavelet and multiresolution analysis

Wavelets are a class of basic functions that represent functions locally, both in space and time, and allow for the analysis of functions to be carried out at different resolutions (scales) . The wavelet basis emanates from a set of wavelet coefficients associated with a particular location in time and different multiresolution scales. The scaling and wavelet functions stem from multiresolution analysis (MRA), which is a key and desirable property of wavelets, and refers to the simultaneous appearance of multiple scales in function decompositions in the Hilbert space L2Rusing a sequence of closed subspaces Vj, which is represented mathematically as :

V2V1V0V1V2E1

Therefore in principle, in order for multiresolution to occur, the closed subspaces Vjsatisfy the following properties:

jZVj¯=L2RE2
jZVj=0E3
f2x=f2xxfVjf2Vj+1jZE4
fnx=fxnfV0fnV0nZE5

The orthogonal complement subspace Wjof Vjcontains the additional ‘detail’ for subspace Vj+1i.e., Vj+1=V0W0W1W2Wj. The union of the subspaces Vjleads to the space L2Rfrom the condition in Eq. (2) . The scaling ϕxL2Rand wavelet ψxL2Rfunctions correspond to the subspaces Vjand Wjrespectively. The difference between current subspace Vj and subsequent subspace Vj+1is represented by the wavelet space Wjwhich becomes automatically orthogonal to all other Wjfork<jdue to the inclusion in and orthogonality to Vj. For the fundamental space V0, the scaling function ϕxand its translates ϕxkproduce an orthonormal basis for V0. The orthonormal basis for the next space V1is the rescaled function 2ϕ2xk. Thus, the orthonormal basis of Vjis defined as:

ϕkjx=2j2ϕ2jxkkZE6

Provided Eq. (6) and the above mentioned properties are satisfied, the wavelet orthonormal basis for subspace Wjat scale j is

ψkjx=2j2ψ2jxkkZE7

The orthogonal subspaces Wjresult from the decomposition of L2Rand subsequently the functions within these subspaces inherit the scale and shift invariance properties from the scaling function subspaces Vjand are orthonormal . The projections of a function fL2Rat scale j in the subspaces Vjand Wj, defined as Pjfand Qjfrespectively, are expressed as:

Pjf=kakjϕkjxQjf=kbkjψkjxE8

where akjand bkjare coefficients in the subspaces Vjand Wjrespectively. Thus, if all the conditions described above are met, then the scaling and wavelet functions satisfy 

ϕxdx0ψxdx=0E9

### 2.1. Daubechies wavelet

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’ relation :

ϕLx=k=0L1pLkϕL2xkE10
ψLx=k=0L1qLkϕL2xkE11

The scaling and wavelet functions have the supports 0L1and 1L2L2respectively. The normalised wavelet function filter coefficients qLkand scaling function filter coefficients pLkhave the relation qLk=1kpL1k. The multiresolution scaling and wavelet basis functions corresponding to the subspaces Vj and Wj are defined as:

ϕL,kjx=2j2ϕL2jxkE12
ψL,kjx=2j2ψL2jxkE13

The scaling and wavelet functions defined in Eqs. (10)(13) satisfy the following properties :

ϕLxdx=1E14
ϕL,kjxϕL,ljxdx=δk,lE15
ψL,kjxψL,kjxdx=δk,lE16
ϕL,kjxψL,ljxdx=0E17
xmψLxdx=0m=0,1,,L21E18

Certain wavelet families have no explicit formulation, as is the case with the Daubechies wavelets. Therefore, Eq. (10) gives rise to a system of equations that require a normalising equation obtained from Eq. (14) to evaluate the scaling functions. The Daubechies wavelet of order L has L21vanishing moments from property (18) and consequently the scaling functions at scale j can represent a polynomial of order xmwhere 0mL21, i.e., 

xm=kMkj,mϕL,kjxE19

The coefficients Mkj,mdenote the moments of the scaling function and it translates at Vj. The derivatives of the Daubechies wavelet scaling functions are evaluated by differentiating the refinement Eq. (10) m times, and are obtained as :

ϕLmx=2mk=0L1pkϕLm2xkE20

A normalising condition is required to evaluate Eq. (20) which is obtained from the moments of the scaling functions.

k=k=kmϕmxk=m!E21

### 2.2. Daubechies connection coefficients

As earlier mentioned, the Daubechies functions cannot be computed analytically and their derivatives are highly oscillatory, particularly at low wavelet orders and/or high order derivatives. Therefore, the integral of the products of the scaling functions and/or derivatives are computed as what is commonly known as connection coefficients . There are two forms of connection coefficients that are of relevance to this study; the multiscale two-term connection coefficient Γa,bk,lj,d1,d2and multiscale connection coefficient Υkj,m. We define the two-term connection coefficient 

where a and b are the orders of the scaling function at multiresolution j, while the values d1and d2denote the order of the derivative of the scaling functions. X0,1x=10x10otherwiseis the characteristic function. The formulation presented is a modified algorithm of that described in  and allows for the evaluation of the connection coefficients for different values of a and b at different multiresolution scales j. From the ‘two-scale’ relation presented in Eq. (10),

ϕL2jξk=rprϕL2j+1ξ2krE23

Differentiating Eq. (23) m times

2jmϕLm2jξk=2j+1mrprϕLm2j+1ξ2krE24

Substituting Eq. (24) into Eq. (22) and applying the ‘two-scale’ relation of the characteristic function, the two-term connection coefficient can be expressed as:

Γa,bk,lj,d1,d2=2d1+d21r,spar2kpbs2l+par2k+2jpbs2l+2jΓr,sj,d1,d2E25

where 2ak,r2j1and 2bl,s2j1. Eq. (25) can be expressed in matrix form as:

Γja,ba+2j2b+2j2×1=2d1+d21Pa,ba+2j2b+2j2×a+2j2b+2j2Γja,ba+2j2b+2j2×1E26

where the square matrix Pa,bcontains the filter coefficients as expressed in Eq. (25) and Γja,bcontains the connection coefficients. To uniquely determine the connection coefficients, normalising conditions are required to generate a sufficient number of inhomogeneous equations via the multiscale moment condition from Eq. (19)

ξm=2j2kMLkj,mϕL2jξkE27

Defining the second form of the connection coefficient

Υkj,m=2j201xmϕL2jξk=2j2X01ξξmϕL2jξkE28

Substituting Eq. (27) into (28)

Υkj,m=2jlMlj,mX01xϕL2jxlϕL2jxkdxE29

However,

ΓL,Lk,lj,0,0=2jX01xϕL2jxlϕL2jxkdxE30

Thus

Υkj,m=lMlj,mΓL,Lk,lj,0,0E31

where ΓL,Lk,lj,0,0are the two-term connection coefficients with a=b=Land d1=d1=0and Mlj,mare the moments earlier described.

### 2.3. B-spline wavelets on the interval [0,1] (BSWI)

The BSWI are a family of wavelets that emanate from Basis splines functions (B-Splines) and the basic functions in subspace Vjof order m and scale j>0are expressed as 

Bm,kjx=tk+mjtkjtkjtk+mjftx+m1E32

with the knot sequence

tkjk=m+12j+m1tkjtk+1jE33

tkjtk+1jtk+mjt, is the mth divided difference of the truncated power function tx+m1with respect to variable t. The general B-splines take the form

Bm,kjx=xtkjtk+m1jtkjBm1,kjx+tk+mjxtk+mjtk+1jBm1,k+1jxB1,kjx=1kxk+10otherwiseE34

and have support Bm,kjxsupp=tkjtk+mj. The B-spline basis function has simple knots inside the unit interval and m-tuple knots at 0 and 1, as expressed in Eq. (33). The knots at 0 ad 1 coalesce and form multiple knots for BSWI while the internal knots are simple hence smoothness is unaffected. For the knot sequence on [0,1], tkjis given as :

tkj=0m+1k<12jk1k<2j12jk2j+m1E35

The number of inner scaling functions present in the formulation of BSWI is determined by the scale j. There must be at least one inner scaling function on the interval [0,1] and this gives rise to the minimum value of j necessary to ensure this condition is met and is defined as j0:

2j02m1E36

The basis Bm,kjxfrom the inner knots corresponds to the mth cardinal B-splines, Nmx, at multiresolution j :

Nmx=m01mtx+m1E37
ϕm,kjx=Bm,kjx=Nm2jxk0k<2jm+1E38

where ϕm,kjxis the BSWI scaling function which can be differentiated m times. The corresponding B-wavelet with support suppψm,kjx=k2jk+2m12jis expressed as:

ψm,kjx=12m1l=02m21lN2ml+1B2m,2i+lj+1,mxE39

B2m,kj+1,mxis the mth derivative for the B-spline of order 2m and scale j+1and can be evaluated explicitly from Eq. (34). Given that the requirement j>j0ensures at least one inner B-wavelet is present, the scaling and wavelet function of the BSWI are obtained as :

ϕm,kjx=Bm,kj02jj0xBm,0j02jj0x2j0kBm,2jkmj012jj0x    m+1k10i2jm2ji2j+m1E40
ψm,kjx=ψm,kj02jj0xψm,0j02jj0x2j0kψm,2jk2m+1j012jj0x   m+1k10i2jm2ji2j+m1E41

and the scaling function derivatives can be evaluated directly by differentiating Eq. (40).

## 3. The wavelet finite element method

### 3.1. Axial rod wavelet finite element

Assume each WFE is divided into equal segments, ns, connected by r=ns+1elemental nodes, as shown in Figure 1, with axial deformation ui. The total number of degrees of freedom (DOFs) within each WFE is denoted by n=rfor n,rN. Vector ue=u1u2ur1urTcontains all the axial DOFs in physical space, as illustrated in Figure 2(a), where ui=uxirepresents the elemental node axial deformation DOF at node i corresponding to coordinate position xi. The nodal natural coordinates is ξi=xix1Le(0 ≤ ξi ≤ 1, 1 ≤ i ≤ r). The Daubechies and BSWI scaling functions ϕz,kjxare used as the interpolating functions and for a family of order z at multiresolution scale j, the axial deformation

uξ=k=h2j1az,kjϕz,kjξE42

contains the unknown wavelet coefficients az,kj. This gives rise to the vector uecontaining the axial deformations at all elemental nodes in physical space.

uen×1=Rrwn×naen×1E43

The matrix Rrw=Φzjξ1Φzjξ2Φzjξr1ΦzjξrTcontains the scaling function vectors Φzjξiapproximating the axial deformation at the corresponding elemental nodes and ae=az,hjaz,h+1jaz,2j2jaz,2j1jT. The axial deformation at any point along the rod element can be generalised as:

uξ=Φzjξ1×nTrwn×nuen×1E44

The matrix Trw=Rrw1is the axial rod wavelet transformation matrix with the scripts r and w denoting rod and wavelet respectively. The wavelet based axial rod shape functions can be evaluated as Nr,eξ=ΦzjξTrwwithin each element.

Suppose the axial rod is subjected to nodal point loads fxiand distributed loadingfdx, then the potential energy within the axial rod Πacan be generally expressed as :

Πa=0lEA2duxdx2dxiuxifxi0lfdxuxdxE45

where E is the Young’s modulus, A is the cross-sectional area and l is the length of the rod. Therefore, given the relation highlighted in Eq. (44), the axial stain energy Ueawithin each WFE of length Leis expressed in natural coordinates as:

Uea=12EALeueT01TrwTdΦzjξTdΦzjξTrwueE46

The stiffness matrix of the rod element in wavelet space, kr,ewis computed using the first derivative of the scaling functions and is symmetric.

kr,ewn×n=01ΦzjξTΦzjξE47

In order for one to obtain the stiffness matrix in physical space, the element properties and transformation matrix Trware applied to the wavelet space stiffness matrix in Eq. (47).

kr,epn×n=EALeTrwn×nTkr,ewn×nTrwn×nE48

The load vector containing the axial point loads of the WFE in physical space is obtained as:

fr,en,pn×1=iTrwTΦzjξiTfxiE49

and the equivalent nodal load vector for the distributed load fdxin physical space is

fr,ed,pn×1=Le01fdξTrwTΦzjξTE50

When applying the Daubechies wavelet family, the WFE has a total of n=2j+L2DOFs. The wavelet space stiffness matrix is evaluated from the multiscale two-term connection coefficients Γa,bk,lj,d1,d2a=b=Land d1=d1=1and is given as:

kr,ew2j+L2x2j+L2                                     D=22jΓj,1,1E51

where 22jis the normalising factor and the matrix Γj,1,1has the entries ΓL,Lk,lj,1,1for the limits 2Lk,l2j1. Similarly, the distributed forces acting on the element require the form Υkj,mfor limits 2Lk,l2j1of connection coefficients and the value of m depends on the order of the function fdxof the forces. In the case of the BSWI formulations, the total DOFs is n=2j+m1and the condition jj0must be satisfied. Therefore, the wavelet space stiffness matrices of the BSWI axial rod are computed as:

kr,ew2j+m1×2j+m1                                       BS=01ΦmjξTΦmjξE52

### 3.2. Euler Bernoulli beam wavelet finite element

According to Euler Bernoulli beam theory, it is assumed that the shear deformation effects are neglected because before and after bending occurs, the plane cross-sections remain plane and perpendicular to the axial centroidal axis of the beam. The beam WFE of length Le, is divided into ns equally spaced elemental segments connected by r elemental nodes at coordinate values xix1xrand iNas illustrated in Figure 1. The WFE has the transverse displacement v and rotation θ taken into account, with corresponding transverse forces fyand moments m´respectively. The transverse displacement and rotation DOFs must be present at each elemental end node to ensure inter-element compatibility [4, 5, 6]. However, the DOFs at the internal elemental nodes can be tailored according to the desired requirements and this in turn will affect the total number of elemental segments and nodes present in each element. In this case the internal WFE nodes only have the transverse displacement present and the total number of DOFs within each beam element is n as illustrated in Figure 2(b). Therefore, there are n2displacement DOFs and 2 rotation DOFs in total for each WFE and consequently r=n2elemental nodes and ns=n3elemental segments. Let the vector {ve}=v1θ1v2v3vr2 vr1 vrθrTdenote all the physical DOFs within the beam element. The displacement and rotation DOFs corresponding to coordinate position xix1xriNand1irin local coordinates are denoted as vi=vxiand θi=θxi. The nodal natural coordinate ξi=xix1Le(0 ≤ ξi ≤ 1, 1 ≤ i ≤ r). The deflection and rotation at any point of the wavelet based beam finite element can be approximated by applying the wavelet scaling functions ϕz,kjxof order z at multiresolution scale j as interpolating functions.

vξ=k=h2j1bz,kjϕz,kjξθξ=vξx=1Lek=h2j1bz,kjϕz,kjξξE53

Therefore, the DOFs present within the entire beam element can be represented as

ven×1=Rbwn×nben×1E54

Rbw=Φzjξ11LeΦzjξ1Φzjξ2Φzjξr1Φzjξr1LeΦzjξrTand vector becontains the unknown wavelet coefficients bz,kjrepresenting the beam wavelet space DOFs.

From Eq. (54), the transverse displacement and rotation at any point of the beam element can be expressed as:

vξ=Φzjξ1×nTbwn×nven×1θξ=1LeΦzjξ1×nTbwn×nven×1E55

where Tbw=Rbw1is the beam wavelet transformation matrix which is used to obtain the wavelet based shape functions for the beam Nb,eξ=ΦzjξTbw. The potential energy Πbwithin a Euler Bernoulli beam subjected to concentrated forces fyi, distributed force fdxand bending moments m´ican be generally expressed as :

Πb=0lEI2d2vdx22dxifyivxi0lfdxvdxkm´kdvxkdxE56

where E is the Young’s modulus, I is the moment of inertia and l is the length of the beam. The strain energy Uebwithin each beam element of length Lecan expressed in terms of the approximation of the transverse displacement via scaling functions as highlighted in Eq. (55).

Ueb=12EILe3veT01TbwTd2Φzjξdξ2Td2Φzjξdξ2TbwveE57

This gives rise to the beam WFE stiffness matrix in wavelet space

kb,ewn×n=01ΦzjξTΦzjξE58

The vector Φzjξ=ϕz,hjξϕz,h+1jξϕz,2j2jξϕz,2j1jξcontains the second derivative of the scaling functions. Taking into account the material properties of the beam, the wavelet space stiffness matrix is transformed into physical space via the transformation matrix Tbw.

kb,epn×n=EILe3Tbwn×nTkb,ewn×nTbwn×nE59

The transverse kinetic energy of the beam element is expressed as

Λeb=12ρALe01v̇ξTv̇ξE60

where v̇ξ=vξt, ρis the density and A is the cross-sectional area of the beam. Applying the scaling functions to approximate the displacements within the beam, the kinetic energy becomes

Λeb=v̇eT12ρALe01TbwTΦzjξTΦzjξTbwv̇eE61

The mass matrix in physical space of the Euler Bernoulli beam element, mb,ep, can be evaluated as:

mb,ep=ρALeTbwT01ΦzjξTΦzjξTbwE62

The vectors containing the element concentrated point loads, bending moments and equivalent distributed loads in physical space respectively are subsequently evaluated as:

{fb,en,p}(n×1)=i=1r[Tbw](n×n)T{Φzj(ξi)}(n×1)Tfyi{fb,em,p}(n×1)=k[Tbw](n×n)T{Φzj(ξk)}(n×1)Tm´k{fb,ed,p}(n×1)=Le01fd(ξ)[Tbw](n×n)T{Φzj(ξ)}TdξE63

In various engineering problems, the loading conditions analysed vary in location and/or magnitude with respect to time, e.g., a train travelling over a track, and this is generally referred to as moving load problems. Assume a moving load of magnitude P travels across a beam element, as illustrated in Figure 3, from the left at a constant speed of c m·s−1 and is represented by the function xt=xx0. δxis the Dirac Delta function and x0is the distance travelled by the moving load at time t. The potential work of the load at this instant at position ξ0=x0Lein natural coordinates is [1, 30]:

Ωebξ0=01ξξ0vξ=PveTTbwTΦzjξ0TE64

Therefore, the element load vector in physical space is evaluated as

fb,ep,pt=PTbwT tΦzjξ0TE65

Assuming the moving load transverses to a new position ξ0within the same WFE, the numerical values of the shape functions, and consequently load vector, will change accordingly. All other WFEs representing the system with no loading present have zero entries within the load vectors at that particular time t. When the moving load is acting on a new WFE, the scaling functions corresponding to the WFE subjected to the moving load are used to obtain the load vector for that particular element.

When applying the Daubechies wavelet family of order L at multiresolution j, the total DOFs within a single element is n=2j+L2and for this specific layout, the total number of elemental nodes is r=2j+L4and corresponding elemental segments ns=2j+L5. The Daubechies wavelet space stiffness and mass matrices of the Euler Bernoulli beam WFE are obtained from the connection coefficients and are expressed as:

kb,ew2j+L2×2j+L2                                      D=24jΓj,2,2E66
mb,ew2j+L2×2j+L2                                      D=Γj,0,0E67

Correspondingly, the connection coefficients of the form Υkj,mfor 2Lk2j1are used to evaluated the distributed loads and the value of m is based on the load function fdx. For the BSWI family of order m and at scale j, there are n=2j+m1total DOFs, r=2j+m3elemental nodes and ns=2j+m4elemental segments within the each WFE for this layout. The stiffness and mass matrices in wavelet space can be evaluated directly and are obtained as:

kb,ew2j+m1×2j+m1                                      BS=01ΦmjξTΦmjξE68
mb,ew2j+m1×2j+m1                                      BS=01ΦmjξTΦmjξE69

### 3.3. Transversely varying functionally graded Euler Bernoulli beam wavelet finite element

Functionally graded materials are a recent evolution of composite materials where the material constituents, hence properties, vary continuously in the desired spatial directions. The need for such revolutionary materials arose to overcome limitations of conventional composite materials, for instance, desirable properties would diminished when applied to highly intense thermal environments or material debonding due to increased stress concentration at material interfaces . In the formulation of the wavelet based functionally grade beam as presented in Figure 4(a), of height h, length l and width b, the material distribution is modelled based on the power law of transverse gradation 

Py=PloPratio1yh+12n+1E70 Figure 4.(a) Cross-section of transversely varying functionally graded beam. (b) Effective Young’s modulus variation of steel-alumina functionally graded beam for different n. (c) Functionally graded beam layout.

As illustrated in Figure 4(b), the transverse variation of the effective material properties P(y) (Young’s modulus) can be infinitely altered via the non-negative volume fraction power law exponent, n. Pratiois the ratio of the upper and lower surface material properties Puand Plorespectively.

The beam WFE has axial deformation uiand transverse deflection viDOFs at all elemental nodes and rotation θiDOFs only present at elemental end nodes with corresponding axial forces fxi, transverse forces fyiand bending moments θias illustrated in Figure 4(c). The wavelet scaling functions are implemented as interpolating functions and the axial deformation, deflection and rotation at any point of the beam element are described by Eqs. (42) and (53) respectively. However, in order to ensure that the defined DOFs are positioned correctly, the layout of the element determines the order of scaling functions selected. In this case, the order of the scaling functions selected to approximate the axial displacement is z2if the scaling function order approximating the bending DOFs is z. The vector containing the total number of DOFs, s, present in the functionally graded beam element is he=u1v1θ1u2v2u3v3ur1vr1urvrθrTand subsequently

u(ξ)={Φz2j(ξ)}(1×s)a{ce}(s×1)v(ξ)={Φzj(ξ)}(1×s)t{ce}(s×1)θ(ξ)=v(ξ)x=1Lev(ξ)ξ=1Le{Φzj(ξ)}(1×s)t{ce}(s×1)E71

where the vector cecontains the unknown wavelet space element DOFs and

Φz2jξ1×s    a=ϕz2,hjξ00ϕz2,h+1jξ00ϕz2,2j1jξ00Φzjξ1×s    t=0ϕz,ijξϕz,i+1jξ00ϕz,2j2jξϕz,2j1jξΦzjξ1×s     t=0ϕz,ijξϕz,i+1jξ00ϕz,2j2jξϕz,2j1jξE72

Therefore, the DOFs present within the entire beam element can be represented as

hes×1=Rpws×sces×1E73

and consequently

u(ξ)={Φzj(ξ)}(1×s)a[Tpw](s×s){he}(s×1)v(ξ)={Φzj(ξ)}(1×s)t[Tpw](s×s){he}(s×1)θ(ξ)=1Le{Φzj(ξ)}(1×s)t[Tpw](s×s){he}(s×1)E74

The wavelet transformation matrix Tpw=Rpw1. The strain energy of the functionally graded beam element, Ue, is defined as

Ue=b2h2h201E(y)[1Le(u(ξ)ξ)T(u(ξ)ξ)yLe2(2v(ξ)ξ2)T(u(ξ)ξ)yLe2(u(ξ)ξ)T(2v(ξ)ξ2)+y2Le3(2v(ξ)ξ2)T(2v(ξ)ξ2)]dξdyE75

where Leis the length of the element and Eythe effective Young’s modulus obtained from Eq. (70). Let

EeA=h2h2Eydy=h2h2EuElyh+12n+EldyEeB=h2h2yEydy=h2h2yEuElyh+12n+EldyEeC=h2h2y2Eydy=h2h2y2EuElyh+12n+EldyE76

EeA, EeBand EeCdenote axial, axial-bending coupling and bending stiffness of the WFE respectively. The wavelet based physical space elemental stiffness matrix of the beam, kew, is

kews×s      A=01bEeALeTpwTΦz2jξξTaΦz2jξξaTpwkews×s     B=01bEeBLe2TpwT2Φzjξξ2TtΦz2jξξaTpwkews×s     C=01bEeBLe2TpwTΦz2jξξTa2Φzjξξ2tTpwkews×s     D=01bEeCLe3TpwT2Φzjξξ2Tt2Φzjξξ2tTpwkeps×s=kepAkepBkepC+kepDE77

The kinetic energy of the functionally graded beam element, Λe, is defined as

Λe=12obdzh2h201ρ(y)(Le(u˙(ξ,t)u˙(ξ,t))y(u˙(ξ,t)v˙(ξ,t)x)y(v˙(ξ,t)ξu˙(ξ,t))+y2Le(v˙(ξ,t)xv˙(ξ,t)x)+Le(v˙(ξ,t)v˙(ξ,t)))dξdyE78

ρyis the effective density also obtained from Eq. (70). Let the inertial coefficients be denoted as:

ρeA=h2h2ρydy=h2h2ρuρlyh+12n+ρldyρeB=h2h2ydy=h2h2yρuρlyh+12n+ρldyρeC=h2h2y2ρydy=h2h2y2ρuρlyh+12n+ρldyE79

The wavelet based physical space elemental mass matrix of the beam, mep, is

mews×s    A=01bρeALeTpwTΦz2jξTaΦz2jξaTpwTmews×s      B=01bρeBTpwTΦz2jξTaΦzjξξtTpwTmews×s     C=01bρeBTpwTΦzjξξTtΦz2jξaTpwTmews×s     D=01bρeCTpwTΦzjξξTtΦzjξξtTpwTmews×s     E=01bρeALeTpwTΦ¯zjξTtΦ¯zjξtTpwTmeps×s=mepAmepBmepC+mepD+mepEE80

## 4. Numerical examples

Example 1: A uniform axial cantilever rod (free-fixed) subjected to linear varying load qx=q0xhas a uniform cross sectional area,A=A0, Young’s Modulus, E=E0and length l. The exact solution for displacement at a particular point x can be obtained by solving ux=1EAoxPxdx=1E0A0q0x22dx. One WFE is used to represent the rod using Daubechies and BSWI WFEM approaches and the results are compared with the exact, h-FEM and p-FEM formulations. The governing equation of the system for FEM and WFEM is

KrUr=FrE81

where Kris the system stiffness matrix, Uris the system vector containing the DOFs and Fris the loading vector of the system. The axial deformation of the rod is analysed at the arbitrary point 0.1l and the rate of convergence of the different approaches is compared in Figure 5. The plot shows the absolute relative error of the axial deformation and corresponding number of DOFs. The FEM (h-FEM) solution involves increasing the number of elements, p-FEM involves increasing the order of the polynomials (one element only) and both Daubechies and BSWI WFEMs have the order and/or multiresolution scale j increased. The results show that although the rates of convergence of all the methods are similar, the WFEM approaches have a slightly improved rate with only one element employed.

Example 2: A simply supported two-stepped beam of length 2l has non-uniform flexural stiffness represented by the unequal cross sections; the bending stiffness of the right and left half is given as E1I1=E0I0and E2I2=4E0I0respectively. The entire beam is subjected to a uniformly distributed load q(x) = 1. The flexural stiffness function is expressed as :

ExIx=E0I01γĤxx0E82

where γ=0.75is defined as the decrement of discontinuity intensity and satisfies the condition 0γ1to ensure positivity of the flexural stiffness. Ĥxx0is the Heaviside function for 0x02l. The general analytical governing equation is

E0I01γĤxx0vx=qxE83

The FEM and WFEM governing equation is summarised as:

KbVb=FbE84

The vector Vbcontains the system DOFs within the entire beam, Kbis the beam stiffness matrix and Fbis the equivalent system load vector. The h-FEM (FEM-8; 8 elements), p-FEM of order 9 (p-FEM-9; 2 elements), Daubechies WFEM of order L=10and scale j=1(D101; 2 elements) and the BSWI WFEM of order m=3and scale j=3(BSWI33; 2 elements) are selected for comparison with the exact solution governed by Eq. (83). Each approach has a total of 18 DOFs within the beam. The deflection and rotation across the beam is presented in Figure 6(a) and (b) respectively. The percentage errors of the deflections are compared for the different approaches and presented in Figure 6(c). All numerical approaches describe the deflection and rotation across the beams very accurately. However, given that both the Daubechies and BSWI WFEM deflection solutions have a maximum error of 1.28% in comparison to 3.82% from the h-FEM and p-FEM approaches, the WFEMs exhibit better convergence to the exact solution. Furthermore, improved accuracy is attained with fewer elements implemented than the h-FEM and p-FEM and this results in reduced computational time.

Example 3: A steel-alumina functionally graded beam of length l and uniform cross-sectional area A=0.36m2 (height h=0.9m and width b=0.4m) is fully alumina at the upper surface and fully steel at the lower surface with material properties; Eu=3.9×1011Pa, ρu=3.96×103kg·m−3 and El=2.1×1011Pa, ρl=7.8×103kg·m−3 respectively (Eratio=EuEl; ρratio=ρuρl). E and ρ denote the Young’s modulus and density respectively. The slenderness ratio for the beam is lh=100. The free vibration of the steel-alumina beam is analysed for the boundary conditions pinned-pinned (PP), pinned-clamped (PC), clamped-clamped (CC) and clamped-free (CF), for different values of n in Eq. (70). The free vibration of the functionally graded beam is governed by 

Kω2MU´=0E85

The matrices Kand Mare the stiffness and mass matrices for the functionally graded beam, ωis the natural frequency and U´is the vector containing the DOFs within the entire beam. The ith non-dimensional frequency λiof the FGM beam is evaluated from the relation λi2=ωil212ρlElh212. The functionally graded beam is modelled for the different approaches using 2 Daubechies WFEs (L=12; j=0; 37 DOFs); one BSWI (m=5; j=4; 38 DOFs) WFE and 12 h-FEM elements (39 DOFs). The results of the first 3 non-dimensional natural frequencies of the beam are presented in Table 1 for different boundary conditions and material distributions. It is observed that all approaches give highly accurate results with respect to the reference (BSWI55), particularly for the fundamental frequencies. Furthermore, the BSWI WFEM solution exhibits better levels of accuracy than the Daubechies WFEM and h-FEM solutions for the higher frequencies. Both WFEM solutions achieve high levels of accuracy with the described layout of having the rotation DOFs present at elemental and nodes and using fewer elements that the h-FEM approach. Figure 5.Comparison of the convergence of the axial deformation at point x = 0.1l. Figure 6.(a) Deflection and (b) rotation (c) comparison of the deflection percentage error across a simply supported stepped beam subjected to a uniformly distributed load q(x) = 1.
n = 0n = 0.1n = 0.5n = 1n = 5n = 10n = 104
λ1PPBSWI554.344624.19433.849033.657953.371393.295043.33251
FEM4.344634.194313.849123.658113.37153.29513.33258
D1204.344624.19433.849033.657953.371393.295043.33251
BSWI544.344624.19433.849033.657953.371393.295043.33251
PCBSWI555.430225.242334.810794.571974.213814.118383.92681
FEM5.430245.242384.811124.572534.214194.118563.92682
D1205.430235.242344.81084.571974.213824.118393.92681
BSWI545.430225.242334.810794.571974.213814.118393.92681
CCBSWI556.541316.314985.795145.507455.076014.961054.73028
FEM6.541376.315095.795855.508675.076854.961454.73028
D1206.541326.314985.795145.507455.076014.961064.73028
BSWI546.541316.314985.795145.507455.076014.961054.73028
CFBSWI552.593182.503452.297372.183332.012291.966711.87523
FEM2.593182.503462.29742.183372.012321.966731.87523
D1202.593182.503452.297372.183332.012291.966711.87523
BSWI542.593182.503452.297372.183332.012291.966711.87523
λ2PPBSWI558.688718.388067.697547.315416.742376.589696.66461
FEM8.688948.388347.698447.316846.743386.590246.66537
D1208.689688.3897.69847.316236.743136.590436.66535
BSWI548.688718.388067.697547.315416.742386.589696.66461
PCBSWI559.774739.43658.659668.229777.585127.413357.06849
FEM9.775139.437028.661458.232637.587137.414427.06879
D1209.77659.438218.661248.231277.58657.41477.06977
BSWI549.774739.43658.659678.229777.585127.413357.06849
CCBSWI5510.859710.48399.620839.143228.427028.236197.85305
FEM10.860410.48489.623879.148088.430448.2387.85355
D12010.863610.48779.624339.146558.430098.239187.8559
BSWI5410.859710.48399.620839.143228.427028.236197.85305
CFBSWI556.491336.266715.750835.465345.037224.923154.69413
FEM6.491386.26685.751315.466155.037784.923424.69417
D1206.491346.266735.750845.465355.037234.923164.69415
BSWI546.491336.266715.750835.465345.037224.923154.69413
λ3PPBSWI5513.031712.580811.54510.971910.11253.295043.33251
FEM13.033412.582611.548910.977410.11663.29513.33258
D12013.046113.594711.557810.98410.12373.295043.33251
BSWI5413.031712.580811.54510.971910.11253.295043.33251
PCBSWI5514.117613.62912.50711.886110.955110.707110.209
FEM14.120113.631812.513111.89510.961710.71110.2108
D12014.144413.65512.530811.908810.976110.727510.2284
BSWI5414.117613.62912.50711.886110.955110.707110.209
CCBSWI5515.203414.677313.468912.800311.797711.530610.9942
FEM15.207114.681313.477912.813511.807411.536410.9968
D12015.266214.737913.524712.853311.846611.578311.0396
BSWI5415.203414.677313.46912.800311.797811.530610.9942
PCBSWI5510.861110.48539.622059.144378.428148.23737.85409
FEM10.861810.48619.624379.147978.43078.238727.85458
D12010.866710.49079.627029.14918.432498.241557.85814
BSWI5410.861110.48539.622069.144388.428158.23737.8541

### Table 1.

The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and boundary conditions.

Assume the same beam, with simply supported boundary conditions and length l=20m, is subjected to a moving load of magnitude P=1×105N travelling across at c m·s−1. The behaviour of the beam is described using Euler Bernoulli beam theory and is assumed to be undamped. The governing equation describing the dynamic behaviour of the system is given by :

MU¨t+KUt=FtE86

where U¨tand Utrepresent the system acceleration and displacement vectors at time t. Ftis the moving load vector. The deflection of the beam vxt, as the moving load travels across, is normalised as a non-dimensional parameter vxt/v0where v0=Pl348ElIis the deflection at the centre of the simply supported functionally graded beam when subjected to a static load of magnitude P at the centre. The maximum normalised deflection mid-span of the beam is analysed over a moving load velocity range 0<c300m·s−1 at increments of 1 m·s−1 to identify the critical velocity for the different variations of the constituent materials as illustrated in Figure 7. The results present are obtained from the BSWI (2 element; m=4; j=3; 37 DOFs) WFEM solution. The h-FEM (12 elements; 39 DOFs) and Daubechies (2 elements; L=12; j=0; 37 DOFs) WFEM solution gives similar results. The values of the critical moving load velocity and corresponding maximum non-dimensional displacement are presented in Table 2 for the different values of n for all approaches. The results are compared with those presented in . Both the Daubechies and BSWI WFE M solutions very accurately yield the correct values. Figure 7.Variation of the maximum non-dimensional vertical displacement at the centre of a simply supported steel-alumina beam with respect to moving load velocities, for different n.
nCritical velocity c m∙s−1Maxvl2tv0
Ref. FEMD120BSWI43Ref. FEMD120BSWI43
02522522522520.93280.93220.93230.9322
0.12352352350.98630.98640.9863
0.22222222222221.03441.03401.03401.0340
0.51981981981981.14441.14351.14371.1436
11791781781781.25031.24911.24951.2493
21641641641641.33761.33631.33681.3365
31571581581.37471.37511.3748
51511511521.42171.4221.4218
71481481481.45671.45701.4568
101451451451.49741.49761.4974
1041321321321321.73081.73091.7308

### Table 2.

The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and boundary conditions.

## 5. Conclusions

A generalised formulation framework for the construction of an axial rod, Euler Bernoulli beam and functionally graded two-dimensional wavelet based finite elements is presented. The Daubechies and BSWI families are selected due to their desirable properties, particularly compact support, ‘two-scale’ relation and multiresolution. It is illustrated via a set of numerical examples that the WFEMs perform exceptionally well when compared to conventional h-FEM and p-FEM where high levels of accuracy are achieved with fewer elements required and the approaches converge more rapidly to the exact solution. Furthermore, the methods are able to accurately describe the behaviour of static and dynamic systems with singularities, variation in material properties and loading conditions present. This exhibits the vast potential of the method in the analysis of more complicated systems and the ability to alter the multiresolution scales without affecting the original mesh allows effective and efficient avenues solution accuracy improvement.

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Mutinda Musuva and Cristinel Mares (December 20th 2017). Multiscale Wavelet Finite Element Analysis in Structural Dynamics, Finite Element Method - Simulation, Numerical Analysis and Solution Techniques, Răzvan Păcurar, IntechOpen, DOI: 10.5772/intechopen.71882. Available from:

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