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Multiscale Wavelet Finite Element Analysis in Structural Dynamics

Written By

Mutinda Musuva and Cristinel Mares

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

DOI: 10.5772/intechopen.71882

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Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

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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.


  • multiresolution
  • wavelets
  • wavelet finite element (WFE)
  • eigenvalue analysis
  • moving load problem
  • functionally graded material (FGM)

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 [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, 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 [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.

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 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 [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].

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.


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) [36]. 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 L2R using a sequence of closed subspaces Vj, which is represented mathematically as [36]:


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


The orthogonal complement subspace Wj of Vj contains the additional ‘detail’ for subspace Vj+1 i.e., Vj+1=V0W0W1W2Wj. The union of the subspaces Vj leads to the space L2R from the condition in Eq. (2) [36]. The scaling ϕxL2R and wavelet ψxL2R functions correspond to the subspaces Vj and Wj respectively. The difference between current subspace Vj and subsequent subspace Vj+1 is represented by the wavelet space Wj which becomes automatically orthogonal to all other Wjfork<j due to the inclusion in and orthogonality to Vj. For the fundamental space V0, the scaling function ϕx and its translates ϕxk produce an orthonormal basis for V0. The orthonormal basis for the next space V1 is the rescaled function 2ϕ2xk. Thus, the orthonormal basis of Vj is defined as:


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


The orthogonal subspaces Wj result from the decomposition of L2R and subsequently the 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 fL2R at scale j in the subspaces Vj and Wj, defined as Pjf and Qjf respectively, are expressed as:


where akj and bkj are coefficients in the subspaces Vj and Wj respectively. Thus, if all the conditions described above are met, then the scaling and wavelet functions satisfy [8]


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 [8]:


The scaling and wavelet functions have the supports 0L1 and 1L2L2 respectively. The normalised wavelet function filter coefficients qLk and scaling function filter coefficients pLk have the relation qLk=1kpL1k. The multiresolution scaling and wavelet basis functions corresponding to the subspaces Vj and Wj are defined as:


The scaling and wavelet functions defined in Eqs. (10)(13) satisfy the following properties [8]:


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 L21 vanishing moments from property (18) and consequently the scaling functions at scale j can represent a polynomial of order xm where 0mL21, i.e., [37]


The coefficients Mkj,m denote 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 [12]:


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


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 [37]. There are two forms of connection coefficients that are of relevance to this study; the multiscale two-term connection coefficient Γa,bk,lj,d1,d2 and multiscale connection coefficient Υkj,m. We define the two-term connection coefficient [30]


where a and b are the orders of the scaling function at multiresolution j, while the values d1 and d2 denote the order of the derivative of the scaling functions. X0,1x=10x10otherwise is the characteristic function. The formulation presented is a modified algorithm of that described in [4] 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),


Differentiating Eq. (23) m times


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:


where 2ak,r2j1 and 2bl,s2j1. Eq. (25) can be expressed in matrix form as:


where the square matrix Pa,b contains the filter coefficients as expressed in Eq. (25) and Γja,b contains 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)


Defining the second form of the connection coefficient


Substituting Eq. (27) into (28)






where ΓL,Lk,lj,0,0 are the two-term connection coefficients with a=b=L and d1=d1=0 and Mlj,m are 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 Vj of order m and scale j>0 are expressed as [20]


with the knot sequence


tkjtk+1jtk+mjt, is the mth divided difference of the truncated power function tx+m1 with respect to variable t. The general B-splines take the form


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], tkj is given as [38]:


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:


The basis Bm,kjx from the inner knots corresponds to the mth cardinal B-splines, Nmx, at multiresolution j [38]:


where ϕm,kjx is the BSWI scaling function which can be differentiated m times. The corresponding B-wavelet with support suppψm,kjx=k2jk+2m12j is expressed as:


B2m,kj+1,mx is the mth derivative for the B-spline of order 2m and scale j+1 and can be evaluated explicitly from Eq. (34). Given that the requirement j>j0 ensures at least one inner B-wavelet is present, the scaling and wavelet function of the BSWI are obtained as [39]:

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+1 elemental 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=r for n,rN. Vector ue=u1u2ur1urT contains all the axial DOFs in physical space, as illustrated in Figure 2(a), where ui=uxi represents 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,kjx are used as the interpolating functions and for a family of order z at multiresolution scale j, the axial deformation


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


Figure 1.

Wavelet finite element layout.

Figure 2.

(a) Axial rod and (b) Euler Bernoulli beam wavelet finite element layout.

The matrix Rrw=Φzjξ1Φzjξ2Φzjξr1ΦzjξrT contains the scaling function vectors Φzjξi approximating 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:


The matrix Trw=Rrw1 is 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=ΦzjξTrw within each element.

Suppose the axial rod is subjected to nodal point loads fxi and distributed loadingfdx, then the potential energy within the axial rod Πa can be generally expressed as [40]:


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 Uea within each WFE of length Le is expressed in natural coordinates as:


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


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


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


and the equivalent nodal load vector for the distributed load fdx in physical space is


When applying the Daubechies wavelet family, the WFE has a total of n=2j+L2 DOFs. The wavelet space stiffness matrix is evaluated from the multiscale two-term connection coefficients Γa,bk,lj,d1,d2 a=b=L and d1=d1=1 and is given as:

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

where 22j is the normalising factor and the matrix Γj,1,1 has the entries ΓL,Lk,lj,1,1 for the limits 2Lk,l2j1. Similarly, the distributed forces acting on the element require the form Υkj,m for limits 2Lk,l2j1 of connection coefficients and the value of m depends on the order of the function fdx of the forces. In the case of the BSWI formulations, the total DOFs is n=2j+m1 and the condition jj0 must 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 xix1xr and iN as illustrated in Figure 1. The WFE has the transverse displacement v and rotation θ taken into account, with corresponding transverse forces fy and 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 n2 displacement DOFs and 2 rotation DOFs in total for each WFE and consequently r=n2 elemental nodes and ns=n3 elemental segments. Let the vector {ve}=v1θ1v2v3vr2 vr1 vrθrT denote all the physical DOFs within the beam element. The displacement and rotation DOFs corresponding to coordinate position xix1xriNand1ir in local coordinates are denoted as vi=vxi and 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,kjx of order z at multiresolution scale j as interpolating functions.


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


Rbw=Φzjξ11LeΦzjξ1Φzjξ2Φzjξr1Φzjξr1LeΦzjξrT and vector be contains the unknown wavelet coefficients bz,kj representing the beam wavelet space DOFs.

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


where Tbw=Rbw1 is the beam wavelet transformation matrix which is used to obtain the wavelet based shape functions for the beam Nb,eξ=ΦzjξTbw. The potential energy Πb within a Euler Bernoulli beam subjected to concentrated forces fyi, distributed force fdx and bending moments m´i can be generally expressed as [40]:


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


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


The vector Φzjξ=ϕz,hjξϕz,h+1jξϕz,2j2jξϕz,2j1jξ contains 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.


The transverse kinetic energy of the beam element is expressed as


where v̇ξ=vξt, ρ is 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


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


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 [41]. δx is the Dirac Delta function and x0 is the distance travelled by the moving load at time t. The potential work of the load at this instant at position ξ0=x0Le in natural coordinates is [1, 30]:


Figure 3.

Layout of a beam WFE subjected to a moving point load.

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 ξ0 within 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+L2 and for this specific layout, the total number of elemental nodes is r=2j+L4 and 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,m for 2Lk2j1 are 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+m1 total DOFs, r=2j+m3 elemental nodes and ns=2j+m4 elemental 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 [42]. 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 [43]


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. Pratio is the ratio of the upper and lower surface material properties Pu and Plo respectively.

The beam WFE has axial deformation ui and transverse deflection vi DOFs at all elemental nodes and rotation θi DOFs only present at elemental end nodes with corresponding axial forces fxi, transverse forces fyi and bending moments θi as 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 z2 if 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=u1v1θ1u2v2u3v3ur1vr1urvrθrT and 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 ce contains 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


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 Le is the length of the element and Ey the effective Young’s modulus obtained from Eq. (70). Let


EeA, EeB and EeC denote 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

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


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=q0x has a uniform cross sectional area,A=A0, Young’s Modulus, E=E0 and length l. The exact solution for displacement at a particular point x can be obtained by solving ux=1EAoxPxdx=1E0A0q0x22dx [40]. 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


where Kr is the system stiffness matrix, Ur is the system vector containing the DOFs and Fr is 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=E0I0 and E2I2=4E0I0 respectively. The entire beam is subjected to a uniformly distributed load q(x) = 1. The flexural stiffness function is expressed as [44]:


where γ=0.75 is defined as the decrement of discontinuity intensity and satisfies the condition 0γ1to ensure positivity of the flexural stiffness. Ĥxx0 is the Heaviside function for 0x02l. The general analytical governing equation is


The FEM and WFEM governing equation is summarised as:


The vector Vb contains the system DOFs within the entire beam, Kb is the beam stiffness matrix and Fb is 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=10 and scale j=1 (D101; 2 elements) and the BSWI WFEM of order m=3 and 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=0.36 m2 (height h=0.9 m and width b=0.4 m) is fully alumina at the upper surface and fully steel at the lower surface with material properties; Eu=3.9×1011 Pa, ρu=3.96×103 kg·m−3 and El=2.1×1011 Pa, ρl=7.8×103 kg·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 [45]


The matrices K and M are the stiffness and mass matrices for the functionally graded beam, ω is the natural frequency and U´ is the vector containing the DOFs within the entire beam. The ith non-dimensional frequency λi of 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

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=20 m, is subjected to a moving load of magnitude P=1×105 N 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 [45]:


where U¨t and Ut represent the system acceleration and displacement vectors at time t. Ft is the moving load vector. The deflection of the beam vxt, as the moving load travels across, is normalised as a non-dimensional parameter vxt/v0 where v0=Pl348ElI is 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<c300 m·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 [46]. 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. [46]FEMD120BSWI43Ref. [46]FEMD120BSWI43

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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Written By

Mutinda Musuva and Cristinel Mares

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