the construction of finite element multiwavelets for adaptive structural analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. 2011; 27:562–584Published online 24 September 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1320COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

The construction of finite element multiwaveletsfor adaptive structural analysis

Youming Wang, Xuefeng Chen∗,†, Yumin He and Zhengjia He

State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University,Xi’an 710049, People’s Republic of China

SUMMARY

A design method of finite element multiwavelets is proposed for adaptive analysis of structural problems.A multiresolution analysis for Lagrange and Hermite finite element space is discussed. New classes offinite element multiwavelets are constructed by the lifting scheme according to the operators of structuralproblems. Compared with classical wavelet methods, the finite element multiwavelet method is moreflexible and robust for multiscale structural analysis. Based on the operator-orthogonality of the finiteelement multiwavelets, we propose a new adaptive scheme for the finite element multiwavelet methodby adding new multiwavelets into the domain where the error estimator is larger than a given thresholdvalue. Numerical examples demonstrate that the finite element multiwavelets are flexible and are efficientbases in solving structural problems. Copyright q 2009 John Wiley & Sons, Ltd.

Received 17 January 2009; Revised 23 June 2009; Accepted 21 July 2009

KEY WORDS: finite element multiwavelets; lifting scheme; structural analysis; adaptive scheme

1. INTRODUCTION

Finite element method (FEM) is a widely popular numerical technique in solving differentialequations, structural problems, thermodynamics problems, etc. When traditional FEM is used tosolve complicated differential equations or engineering problems, that is, the problems with largegradients or scales, it has several disadvantages, that is, re-meshing initial elements, low efficiency,ill-conditioned elements, etc. Thus, different types of techniques were proposed to improve thecomputational performance of traditional FEM method. The hierarchical FEM [1, 2] is a commonmethod to solve the problems by adding hierarchical basis functions into the initial finite elementmodels instead of remeshing the initial meshes. However, a remarkable disadvantage of hierarchicalFEM is that the entire solution has to be re-computed when the detail and coarser parts of multiscalestiffness matrix are coupled across the scales.

In the last several years, there has been a growing interest in developing wavelet-based numericalalgorithms, which overcome the disadvantages of traditional FEM [3]. The good feature of waveletsis that they are more flexible to realize multiscale and adaptive algorithms. As the wavelet-based FEM incorporates the advantages of discrete approximation of FEM and the multiresolutionanalysis (MRA) and high approximation order of wavelet-based method, wavelet-based FEM has

∗Correspondence to: Xuefeng Chen, State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University,Xi’an 710049, People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 50875195Contract/grant sponsor: National Excellent Doctoral Dissertation of China; contract/grant number: 2007B33

Copyright q 2009 John Wiley & Sons, Ltd.

THE CONSTRUCTION OF FINITE ELEMENT MULTIWAVELETS 563

become a new numerical tool to solve multiscale [4], large deformation [3], crack problems [5], etc.Traditional wavelets are constructed by scaled and shifted versions of a single mother wavelet ona regularly spaced grid over a theoretically infinite or periodic domain. Therefore, they cannot beconstructed on finite meshes commonly encountered in finite element analysis. Second-generationwavelets constructed by the lifting scheme [6, 7] were developed to eliminate the restriction anddeficiency of traditional wavelets. As a generalization of biorthogonal wavelets, second-generationwavelets form a Riesz basis for L2 space, which are local in both space and frequency and canhave many vanishing polynomial moments without the translation and dilation invariance of theircousins.

In the last decades, second-generation wavelets based on the lifting scheme have gradually beenapplied in solving various mathematical and engineering problems. Vasilyev et al. [8–10] estab-lished second-generation wavelet collocation method to solve partial differential equations (PDE)over general geometries. A superior feature of the lifting scheme is that it provides the freedom andflexibility to build the wavelet bases depending on the application. Wang and Yang [11] developedan adaptive second-generation wavelet method to solve wave equations accurately. Pinho et al. [12]discussed aMRA for the discretization of Maxwell equations via second-generation wavelets, whichreduced the dimensionality and simplified the representation of nonlinear operators. Vasilyev [13]proposed a dynamically adaptive numerical method based on a general class of multi-dimensionalsecond-generation wavelets to solve multi-dimensional evolution problems with localized struc-tures. As a generalization of lifting scheme on single wavelet, Davis and Strela [14] generalizedlifting scheme to the construction of multiwavelets and proved that all compactly supportedbiorthogonal multiwavelet bases can be achieved by applying a finite sequence of simple liftingsteps to a simple initial basis. Goh et al. [15] constructed compactly supported biorthogonal multi-wavelets with optimum time–frequency resolution. Castrillon-Candas and Amaratunga [16, 17]developed spatially adaptive multiwavelets based on the lifting scheme to represent integral opera-tors sparsely on general geometries, which demonstrated high convergence rate in solving integralequations. D’Heedene et al. [18] built a lifting wavelet framework using stable completion forLagrange finite element basis function of any given order on unstructured meshes. Sudarshan andAmaratunga [19] proposed a lifting method called approximate Gram-Schmidt orthogonalizationmethod to construct operator- customized wavelets, which demonstrated their advantages in solvingsecond and fourth-order linear elliptic partial differential equations. He et al. [20] presented aconstruction method of lifting wavelets to solve field problems with changes in gradients andsingularities by designing suitable prediction operators and update operators.

The main drawback of the present wavelet methods is that the wavelets are seldom constructedespecially for structural analysis. Therefore, the main idea of this paper is to extend finite elementmultiwavelets to be generalized bases of different structural problems. The adaptive scheme forthe finite element multiwavelet method is also presented.

An outline of the paper is as follows. Section 2 introduces the multiresolution finite elementspace and the formulation of multilevel computation. Section 3 introduces the lifting scheme toconstruct finite element multiwavelets and the scale-decoupling condition for multilevel stiffnessmatrix. Section 4 discusses the construction of finite element multiwavelets for the structuralanalysis according to the operators of the structural problems. Section 5 presents adaptive schemefor finite element multiwavelet method according to the operator-orthogonality of finite elementmultiwavelets. Section 6 demonstrates the numerical performance of the finite element multiwaveletmethod and conclusions are drawn in Section 7.

2. MULTIRESOLUTION FINITE ELEMENT SPACE

2.1. Multiresolution analysis

The wavelets that are constructed by the lifting scheme are referred to as second-generationwavelets. The second-generation version of MRA is introduced in the following [6].

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2011; 27:562–584DOI: 10.1002/cnm

564 Y. WANG ET AL.

An MRA R of L2 is a sequence of closed subspaces R={Vj ⊂ L2| j ∈ J ⊂ Z}, such that

1. Vj ⊂Vj+1,2.⋃

j∈J Vj is dense in L2,3. for each j ∈ J , Vj has a Riesz basis given by scaling functions {� j,k |k∈K ( j)}, where j

is the level of resolution, J is an integer index set associated with resolution levels, K ( j)is some index set associated with scaling functions of level j , Vj denotes approximationspaces of level j . For each Vj , there exists a complement of Vj in Vj+1, namely as Wj . Letthe spaces Wj be spanned by wavelets, � j,m(x) for every m∈M( j), M( j)=K ( j+1)\K ( j)where M( j) is the difference set of K ( j+1) and K ( j). Furthermore, let l∈K ( j+1) be theindex at level j+1.

2.2. Lagrange and Hermite MRA

In Lagrange and Hermite multiresolution finite element space [16], the scaling functions are chosento be the Lagrange and Hermite interpolating polynomials. The wavelets are chosen to be detailinterpolating polynomials in the wavelet space. A multiresolution decomposition of a Lagrangefinite element space Vj at different levels of resolution is spatial hierarchy:

Vj =Wj−1⊕Vj−1=Wj−1⊕Wj−2⊕·· ·⊕W0⊕V0

This hierarchical framework can be extended to two-dimensional multiresolution basis for any orderLagrange basis on a nested mesh with irregularly spaced vertices. Similarly, Hermite hierarchicalbases can be constructed in Hermite multiresolution finite element space.

As the finite element spaces are nested, the relation between scaling function � j,k and wavelet� j,m at level j and j+1 satisfies refinement equations of the form

� j,k =∑lh j,k,l� j+1,l (1)

� j,m =∑lg j,m,l� j+1,l (2)

where h j,k,l and g j,m,l are referred to as low-pass and high-pass filters, respectively. A MRAallows that a sufficiently smooth function f j ∈Vj can be decomposed into its projection on acoarse approximation space V0 along with the projections at multiple levels of wavelet spaces

f j = ∑k∈K ( j)

b j,k� j,k = ∑k∈K (0)

b0,k�0,k+j−1∑i=0

∑m∈M(i)

�i,m�i,m (3)

where b j,k and � j,m are the projection coefficients of f (x) in the space Vj and Wj respectively.Equation (3) means that the function f (x) can be approximated with the projection f j (x) in Vjand the projection eventually captures all the details of the initial function f (x) as scale j getslarger (i.e. j →∞), such as

limj→∞‖ f (x)− f j (x)‖=0 (4)

2.3. Refinement equations of Lagrange and Hermite finite element wavelets

The refinement equations of Lagrange and Hermite finite element wavelets are the foundation ofconstructing new kinds of Lagrange and Hermite finite element multiwavelets. The Lagrange andHermite wavelet coefficients are solved by the simultaneous equations of the scaling functionsand wavelets at random vertices on the two adjacent levels. Let C j+1 be the coefficient matrix atthe level j+1, the nodal values of the wavelets and scaling functions at the chosen vertices aredenoted by � j ={� j,k :k∈K ( j)} and � j ={� j,m :m∈M( j)} respectively, the coefficients C j+1



-1 0 1 2 30

0.5

1

k1 k2 k3m1 m2

Figure 1. The refinement relation for a linear Lagrange scaling function.

can be computed from the following simultaneous equations

C j+1

[� j+1

� j+1

]=[� j ] (5)

The refinement relation of different scaling functions may be obtained by solving Equation (5).In Lagrange multiresolution finite element space, there is one degree of freedom on every nodeof the scaling functions and wavelets. Therefore, the coefficient of each Lagrange wavelet on onenode is a constant. Figure 1 shows the refinement relation of a piecewise-linear scaling functionin the form

� j,k2 =� j+1,k2 + 12 (� j+1,m1

+� j+1,m2) (6)

where the coefficient matrix C j+1 has the form

C j+1=[1 12

12 ]

and the scaling function at level j on k2 has a compact support [k1,k3], which is denoted by thereal line; the scaling function at level j+1 on k2 is supported on a half interval of the scalingfunction at level j , i.e. [m1,m2]; the wavelets are compactly supported on the interval betweentwo adjacent k-nodes, i.e. [k1,k2] and [k2,k3], which satisfy the following relationship [6]:

� j,m =� j+1,m (7)

Similarly, many new wavelets can be derived using the higher-order scaling functions in themultiresolution finite element space. The refinement relation for quadratic scaling function shownin Figure 2 is[

� j,k2

� j,k3

]=[

� j+1,k2

� j+1,k3

]+[ 3

434

− 18

34

][� j+1,m1

� j+1,m2

]+[

0 0

− 18

34

][� j+1,m4

� j+1,m3

](8)

which denotes the scaling functions in the centre and on the boundary of an element. The coefficientmatrix C j+1 for the quadratic scaling function has the form

C j+1=[1 0 3

434 0 0

0 1 − 18

34 − 1

834

]

As Hermite scaling functions always have continuous derivative at the nodes, each node of thescaling functions and wavelets generally has more than one degree of freedom, which means that


566 Y. WANG ET AL.

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

k1 k2 k3m1m2

-1 -0.5 0 0.5 1 1.5 2 2.5 3-0.2

0

0.2

0.4

0.6

0.8

1

k1k3

k5m1 m2m3

m4k2k4

(a) (b)

Figure 2. The refinement relation for quadratic Lagrange scaling functions.

0 0.5 1 1.5 2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k1k2

k3m1 m2

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

0.15

k1 k2 k3m1 m2

(a) (b)

Figure 3. The refinement relation for cubic Hermite scaling functions.

the coefficient of a Hermite wavelet on a certain node is a square matrix. The refinement relationof cubic Hermite scaling functions shown in Figure 3(a)–(b) is

� j,k2 =� j+1,k2 +

⎡⎢⎢⎣

1

2

h

8

− 3

2h−1

4

⎤⎥⎥⎦� j+1,m1

+

⎡⎢⎢⎣

1

2−h

83

2h−1

4

⎤⎥⎥⎦� j+1,m2

(9)

where h is an element length and the coefficient matrix is given

C j+1=

⎡⎢⎢⎣1 0

1

2

h

8

1

2−h

8

0 1 − 3

2h−1

4

3

2h−1

4

⎤⎥⎥⎦

Similarly, the refinement relation of quintic Hermite scaling functions shown in Figure 4 hasthe form

� j,k2 =� j+1,k2 +

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1

2

5h

32

h2

64

−15

8h− 7

16− h

32

0 − 3

2h−1

4

⎤⎥⎥⎥⎥⎥⎥⎥⎦

� j+1,m1+

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1

2−5h

32

h2

6415

8h− 7

16

h

32

03

2h−1

4

⎤⎥⎥⎥⎥⎥⎥⎥⎦

� j+1,m2(10)



0 0.5 1 1.5 2-0.4

0

0.4

0.8

1.2

k1 k

2k

3m

1m

2

0 0.5 1 1.5 2-0.2

-0.1

0

0.1

0.2

k1 k

2k

3m1

m2

0 0.5 1 1.5 2-0.01

-0.005

0

0.005

0.01

0.015

0.02

k1

k2

k3m

1m

2

(a)

(c)

(b)

Figure 4. The refinement relation for quintic Hermite scaling functions.

where each node of scaling functions and wavelets has three degrees of freedom and the coefficientmatrix is given and the coefficient matrix is given

C j+1=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 0 01

2

5h

32

h2

64

0 1 0 −15

8h− 7

16− h

32

0 0 1 0 − 3

2h−1

4

1

2−5h

32

h2

6415

8h− 7

16

h

32

03

2h−1

4

⎤⎥⎥⎥⎥⎥⎥⎥⎦

2.4. Multilevel formulation

Consider a common structural problem. A general system of equations by finite element discretiza-tion and the principle of minimum potential energy can be denoted as [21]

K0u0= f0 (11)

where K0 is the finite element stiffness matrix at initial scale j =0, f0 the force vector, u0 thefinite element solution. If we approximate the displacement field by using of the multiresolutionrepresentation in Equation (3), we can derive the following relation:

u j+1=∑ku j,k� j,k+

∑md j,m� j,m (12)

where u j,k and d j,m are the coefficients of the scaling functions and wavelets in the multiresolutionfinite element space respectively. Substituting Equation (11) with Equation (12) , we can obtainthe following system of equations:

K j+1u j+1= f j+1 (13)


568 Y. WANG ET AL.

where K j+1=a(� j+1,l ,� j+1,l ′) is the nodal finite element stiffness matrix with respect to nodal land l ′, l ′ ∈K ( j+1); a(•,•) are the operators derived from the finite element models. The multilevelstiffness matrix K j+1 can be written in the two-level form

K j+1=[

K j Ka, j

Kb, j Kc, j

](14)

where the individual entries in K j+1 are given as

K j [k,k′] = a(� j,k,� j,k′) (nodal finite element matrix at level j) (15)

Ka, j [k,m] = a(� j,k,� j,m) (interaction matrix between level jand j+1) (16)

Kb, j [k,m] = a(� j,m,� j,k)= KTa, j [k,m] (17)

Kc, j [m,m′] = a(� j,m,� j,m′) (detail matrix at level j+1) (18)

where the node set k′ ∈K ( j), m′ ∈M( j).Similarly, the force vector f can be denoted in the multiresolution finite element space as

f j+1=[f j

fc

](19)

where f j is the force vector at scale j , fc is the force vector corresponding to the wavelets.When computing engineering problems in the multilevel finite element space, it is much desirable

that the multilevel stiffness matrix are scale-decoupling, which means that the details not haveany influence on the coarser solution at all. The scale-decoupling property of multilevel stiffnessmatrix may permit us to efficiently compute the contribution of different scales independent ofeach other, which can be satisfied by the operator-orthogonality between the scaling functions andwavelets

Ka, j [k,m]=a(� j,k,� j,m)=0 (20)

Therefore, the sufficient condition for the scale-decoupling of multilevel stiffness matrix is toconstruct new wavelets orthogonal with the operators in the engineering problems.

3. LIFTING SCHEME

3.1. Lifting scheme

Lifting scheme [6, 7] was presented by Sweldens to custom design new kinds of wavelets withspecific properties, that is, increasing the vanishing moments, ensuring symmetry and compactsupport, etc. In a random MRA, a new compact wavelet � j,m can be constructed by adding

neighboring scaling functions to a primitive wavelet,�oldj,m , which is chosen to be a simple scaling

function � j+1,m as

� j,m =�oldj,m−∑

ks j,k,m� j,k =� j+1,m−∑

ks j,k,m� j,k (21)

where s j,k,m are the lifting coefficients. The lifting scheme provides a simple and flexible tool tocustom design different types of second-generation wavelets by choosing appropriate the liftingcoefficients s j,k,m with user-defined properties. The introduction of lifting scheme into multiwavelettheory has led to the construction of a new family of multiwavelets by adding the wavelets intothe lifting procedure [14]

� j,m =∑m′

r j,m,m′� j+1,m−∑ks j,k,m′� j,k (22)



where r j,m,m′ and s j,k,m′ are lifting coefficients of the wavelets and scaling functions, respectively,m′ is the set of vertices contributing to the construction of finite element multiwavelets, which isthe subset of m.

3.2. Finite element multiwavelets using lifting scheme

According to the operator-orthogonality between the scaling functions and wavelets, the liftingcoefficients of finite element multiwavelets are computed by

R j S j =[a(� j,k∗,� j,k)a(� j,k∗,� j+1,m)][−s j,k,m′

r j,m,m′

]=0 (23)

where � j,k∗ are all the scaling functions on a given domain � j , k∗ the nodes on the domain� j , � j,k the subset of scaling functions that are all interior in the domain � j , R j the interactionor coupling matrix between the scaling functions and wavelets, S j the lifting coefficient matrixof the scaling functions and wavelets. The lifting coefficients can be computed by computing abasis for the null space of the interaction matrix R j . The finite element multiwavelets can beconstructed based on the three basic principles of multiresolution finite element space as (1) themultiwavelets based on the lifting scheme are compactly supported, (2) the multiwavelet space iscompletely complementary with the scaling space and (3) the lifting coefficient vectors are linearlyindependent.

3.3. Decoupling condition

To make the multilevel stiffness matrix scale-decoupling, the operator-orthogonality between thescaling functions and wavelets has to be ensured based on the vanishing moments of the wavelets.In Lagrange multiresolution finite element space, if a wavelet is zero-valued at the integrationboundaries, the derivative of the wavelet will have one vanishing moment. In addition, eachvanishing moment in the wavelet will result in an additional vanishing moment in the derivativeof the wavelet, ∫

�

��

�xdx = �

∣∣∣∣∣x2

x1=0 (24)

∫�x��

�xdx = x�

∣∣∣∣∣x2

x1−∫

��dx=0 ⇔

∫�

�dx=0 (25)

∫�xn

��

�xdx = xn�|x2x1 −

∫�nxn−1�dx=0 ⇔

∫�xn−1�dx=0 (26)

which is referred to as the inheritance of vanishing moment property [18]. In Hermite multires-olution finite element space, the inheritance of vanishing moments of Hermite multiwavelets canalso be derived as∫

�

�2��x2

dx = ��x

�

∣∣∣∣∣x2

x1=0 ⇔ ��

�x

∣∣∣∣∣x2

x1=0 (27)

∫�x�2��x2

dx = x��

�x

∣∣∣∣∣x2

x1−�

∣∣∣∣∣x2

x1=0 ⇔ �

∣∣∣∣∣x2

x1=0 (28)

∫�xn

�2��x2

dx = xn��

�x

∣∣∣∣∣x2

x1−nxn−1�

∣∣∣∣∣x2

x1+n(n−1)

∫�xn−2�dx=0 ⇔

∫�xn−2�dx=0 (29)

It can be easily verified that Equations (27) and (28) are naturally satisfied for Hermite multi-wavelets. Equation (29) implies that an order-n Hermite wavelet has to be lifted with n−1 vanishingmoments.


570 Y. WANG ET AL.

4. CONSTRUCTION OF FINITE ELEMENT MULTIWAVELETS

In this section, various types of finite element multiwavelets are constructed by the lifting schemeaccording to the operators of structural problems in the multiresolution finite element space.

4.1. Multiwavelets for bar and cable problems

The total potential energy for the finite element models of bars and cables can be expressed in theform [22]

�e(u)=∫ xb

xa

E A

2

(du

dx

)2

dx−∫ xb

xau f dx−∑

iuei Q

ei (30)

where E denotes the material modulus, A the cross-sectional area, u the displacement, f thedistributed force, Qi lump force, uei is the acting position of lump force and the element length isle= xb−xa . Applying the principle of minimum of total potential energy, ��e=0, we can obtainthe following system of equations:

Keue= f e+Qe (31)

The stiffness matrix of bars in Lagrange MRA on the scale j ( j�0, j ∈ Z) can be denoted as

K j+1=

⎡⎢⎢⎢⎢⎣

∫ xb

xaEA

d� j,k1

dx

d� j,k2

dxdx

∫ xb

xaEA

d� j,k1

dx

d� j,m2

dxdx

∫ xb

xaEA

d� j,m1

dx

d� j,k2

dxdx

∫ xb

xaEA

d� j,m1

dx

d� j,m2

dxdx

⎤⎥⎥⎥⎥⎦ (32)

The distributed and lump forces on the scale j are

f ei =∫ xb

xaf [� j,k � j,m]Tdx (33)

Qe =∑iQe

i [� j,k � j,m]T (34)

The operator derived from the bar and cable problems is

a(� j,m,� j,k)=∫ xb

xaEA

d� j,m

dx

d� j,k

dxdx (35)

For the uniform tension problems of a cable, there is a similar operator by substituting theconstant E A. According to the operator of bar or cable problems, we can construct finite elementmultiwavelets based on the lifting scheme to ensure that the scaling functions and wavelets areoperator-orthogonal. For linear scaling functions, it can be verified that the piecewise-linear waveletsare naturally operator-orthogonal with the scaling functions with respect to the operator of rodproblems. To satisfy the operator-orthogonality between quadratic scaling functions and wavelets,two quadratic multiwavelets are constructed on the support of one element. Figure 5 shows twomultiwavelets with only one vanishing moment and Figure 6 illustrates two multiwavelets with twovanishing moments. Based on the basic principle of selecting appropriate lifting coefficients forcubic scaling functions, three multiwavelets are required on the support of one element. Figure 7shows three multiwavelets with two, three and four vanishing moments, respectively.

4.2. Multiwavelets for plane truss problems

As each member of a plane truss in the local or member coordinate system is oriented differentlywith respect to structural or global coordinate system, it is necessary to transform the forcedisplacement relations that are derived in local coordinate system to the global coordinate system.



-1 -0.5 0 0.5 1-0.5

0

0.5

1

-1 -0.5 0 0.5 1-0.5

0

0.5

1

(a) (b)

Figure 5. Quadratic finite element multiwavelets with one vanishing moments for rod problems.

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

(a) (b)

Figure 6. Quadratic finite element multiwavelets with two vanishing moments for rod problems.

Consider a uniform bar element oriented at angle � measured counterclockwise from the positivex-axis shown in Figure 8. The relationship between the displacements and forces in the global andlocal coordinate systems is in the form of

�e=T e�e (36)

where �e={ue1ve1ue2ve2} and �e={ue1ve1ue2ve2} are the nodal displacement vectors in the local and

global coordinate systems, respectively, T e is the transformation matrix in the form

T e=

⎡⎢⎢⎢⎢⎣

cos� sin� 0 0

−sin� cos� 0 0

0 0 cos� sin�

0 0 −sin� cos�

⎤⎥⎥⎥⎥⎦ (37)

The element stiffness matrix in global coordinate system and local coordinate system has therelation form

Ke=(T e)TKeT e (38)

From Equation (38), we can conclude that the operator for the finite element models of planetrusses has the same formulation as that of the bar or cable model. Hence the construction ofmultiwavelets for the plane truss problems is the same as those of bar or cable problems.


572 Y. WANG ET AL.

-1 -0.5 0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1-0.5

0

0.5

(a)

(c)

(b)

Figure 7. Cubic finite element multiwavelets for rod problems with: (a) two vanishing moments; (b) threevanishing moments; and (c) three vanishing moments.

x

1

2

ex

eu2

u e1

ev2

y

eu2

eu1

ev2

v e1

ve1

ye

Figure 8. A plane truss oriented at an angle with respect to the global coordinate system.

4.3. Multiwavelets for Timoshenko beam problems

Considering the influence of shear deformation, the total potential energy functional of Timoshenkobeam element �e=(xa, xb) is

�e(w,�) =∫ xb

xa

[EI

2

(d�

dx

)2

+GAKs

2

(dw

dx+�

)2

+ 1

2c f w

2−wq

]dx

−w(xa)Qe1−�(xa)Q

e2−w(xb)Q

e3−�(xb)Q

e4 (39)



where EI and GAKs are the constant of the beam element, w and � are the transverse deflectionand rotation, respectively, Qe

i is the generalized forces of the respective generalized displace-ments (w,�) at the ends of the elements, q is the distributed force, c f is the elastic foundationmodulus. Consider Timoshenko beam problems in Lagrange multiresolution finite element space.Let ��e(w,v)=0, the operators of Timoshenko beam problems can be derived as

a11(�1j,m,�1

j,k) =∫ xb

xa

(GAKs

d�1j,m

dx

d�1j,k

dx+c f �

1j,m�1

j,k

)dx (40)

a12(�1j,m,�2

j,k) =∫ xb

xaGAKs

d�1j,m

dx�2

j,k dx (41)

a21(�2j,m,�1

j,k) =∫ xb

xaGAKs

d�2j,m

dx�1

j,k dx (42)

a22(�2j,m,�2

j,k) =∫ xb

xa

(E I

d�2j,m

dx

d�2j,k

dx+GAKs�

2j,m�2

j,k

)dx (43)

where �1j,k and �2

j,k are the scaling functions corresponding to the deflection and rotation offinite element model, respectively. To ensure the scaling functions are operator-orthogonal withthe wavelets, we can use the lifting scheme to construct two types of Lagrange multiwaveletsoperator-orthogonal with the two types of scaling functions.

4.4. Multiwavelets for the Euler–Bernoulli beam problems

According to the theoretical assumption that plane cross-sections perpendicular to the axis of thebeam remain plane and perpendicular to the axis after deformation, the total potential energy ofthe Euler–Bernoulli beam element �e=(xa, xb) is given by

�e(w) =∫ xb

xa

[E I

2

(d2w

dx2

)2

+ 1

2c f w

2−wq

]dx−w(xa)Q

e1−w(xb)Q

e3

−(

−dw

dx

)∣∣∣∣xa

Qe2−

(−dw

dx

)∣∣∣∣xb

Qe4 (44)

where EI is the constant of the beam element, w is the transverse deflection, the cross-sectionrotation �=dw/dx , q the distributed force, Qe

i (i=1, . . . ,4) the generalized forces of the respectivegeneralized displacements (w,�) at the ends of the elements. The operator of the Euler–Bernoullibeam problems can be derived as

a(� j,k,� j,m)=∫ xb

xa

(EI

d2� j,k

dx2d2� j,m

dx2+c f � j,k� j,m

)dx (45)

Two independent multiwavelets can be constructed for Euler–Bernoulli beam problems to beoperator-orthogonal with the scaling functions. For simplicity, let c f =0, it is certified that the cubicwavelets are naturally operator-orthogonal with the scaling functions. For quintic Hermite scalingfunctions, we can construct three multiwavelets on the support of two elements with elementallength h=1, which are shown in Figure 9(a)–(c).

4.5. Multiwavelets for plane frame problems

Based on the plane frame theory, a plane frame can be equivalent with a bar element and a beamelement shown in Figure 10. The local or frame system coordinate and the structural or globalcoordinate system are related according to

�e=T e�e (46)


574 Y. WANG ET AL.

0 0.5 1 1.5 2-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2-0.4

-0.2

0

0.2

0.4

0.6

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(a)

(c)

(b)

Figure 9. Quintic Hermite multiwavelets for the Euler–Bernoulli beam problems.

x

z

1

2

ex

eu2

eu1

ew1

ew2

ez

eu1

u2

ew2

ew1

e1

e1

e2

e2

Figure 10. A plane frame oriented at an angle with respect to the global coordinate system.

where �e={ue1we

1�1ue2w

e2�2} and �e={ue1we

1�1ue2w

e2�2}are the nodal displacement vectors in the

local and global coordinate systems, respectively, T e is the transformation matrix of the form

T e=

⎡⎢⎢⎢⎢⎢⎣

cos� sin� 0−sin� cos� 0 0

0 0 1cos� sin� 0

0 −sin� cos� 00 0 1

⎤⎥⎥⎥⎥⎥⎦ (47)

where the angle � is measured clockwise from the global x-axis.



The element stiffness matrix in global coordinate system and local coordinate system satisfythe relation

Ke=(T e)TKeT e (48)

From Equation (48) we can see that the plane frame models have two kinds of operators includingthe operators of bar and the Euler–Bernoulli beam element model. The multiwavelets for the planeframe problems can be constructed as those of the Euler–Bernoulli beam problems, respectively.

5. ADAPTIVE SCHEME FOR FINITE ELEMENT MULTIWAVELET METHOD

5.1. Error estimation

The error estimator of the finite element multiwavelet solution is key parameter to test the accuracyof the finite element multiwavelet method. The error estimator � j is chosen to be the uniformnorm of the difference e j between the finite element multiwavelet solution u j+1 and u j at twolevels j+1 and j , respectively, in the form

� j =‖e j‖∞ =max |u j+1− u j | (49)

As the scale gets larger, the approximate error estimator � j will become smaller as well as thewavelet coefficients. To establish a unified standard of error controlling values, the relative errorestimate on the level j are defined in the nondimensional form

ε j = ‖e j‖∞max |u j+1| = max |u j+1− u j |

max |u j+1| (50)

The relative error estimation also can indicate the convergence rate of finite element multiwaveletsolution on each level. To indicate the relative error in an interval or local domain � j,k betweenthe neighboring k-nodes, that is, node k and k+1,

� j,k =N j,k N j,k+1 (51)

where N denotes the node, a new local relative error estimator is defined as

j,k = |u j+1.l − u j.k |max |u j+1.l | (52)

where j,k denotes the convergence rate of the finite element multiwavelet solution in the localdomain � j,k . According to the refinement relation in Equation (12), the finite element multiwaveletsolution can be obtained by using of all the multiwavelets on the interval � j,k . As the scalebecomes larger, it can be ensured that the error estimator becomes small to satisfy a randomthreshold value.

5.2. Adaptive scheme

Given a threshold value and order-n scaling functions, structural problems can be solved by theadaptive scheme of finite element multiwavelet method in the following procedures:

(1) Solve the structural problems on initial coarse meshes, K0u0= f0.(2) Derive the operators of the structural problems, construct the finite element multiwavelets

by the lifting scheme according to the operators of the problems.(3) Form the detail matrix Kc, j , and solve the finite element multiwavelet coefficients d j,m

from the equation Kc, j d j,m = fc. Compute the local relative error estimators j,k in the localdomains � j,k .

(4) If j,k�, mark the local domains for finite element multiwavelet refinement.(5) Compute the detail matrices and force vectors of the finite element multiwavelets and add

them into multilevel stiffness matrix and force vectors respectively, GOTO 3.


576 Y. WANG ET AL.

6. NUMERICAL EXAMPLES

In this section we present numerical experiments to demonstrate the efficiency and flexibilityof the finite element multiwavelet method. As common structural problems, rod, Timoshenkobeam and Euler–Bernoulli beam problems are solved by uniform [20] and adaptive finite elementmultiwavelet method, respectively.

Example 1Figure 11 shows an axial rod subjected to distributed loading q(x)=q0x cos(5x)/L , q0=1, rodlength L=1 and the physical parameter E A=1.

Given a threshold value =0.1%, the problem is solved by uniform and adaptive refinementscheme using quadratic and cubic finite element multiwavelets, respectively. Tables I and II illustratethe error estimator and relative error estimator by quadratic and cubic multiwavelet methods usinguniform and adaptive refinement strategy, respectively. Figure 12 shows the relative error of axialdeformation using uniform and adaptive multiwavelet method with increasing number of levelsand degrees of freedoms, respectively. It can be seen that numerical solution of the problems usinguniform and adaptive refinement method has the same convergence rate, but adaptive refinementmethod approximates the analytic solution with fewer degrees of freedom. Obviously, the analyticsolution of the problem can be approximated by the cubic multiwavelets with faster convergencerate than quadratic multiwavelets.

Example 2Figure 13 shows a rectangle cross-section cantilever beam subjected to distributed loading. Thephysical parameters are: elastic modulus E=8×1010N/m2, shear modulus G=2×109N/m2,width B=0.625m, height H =1m, shear correction coefficient Ks =5/6, length L=10m,

O

q(x)

L

x

Figure 11. A torsional rod subjected to distributed loading.

Table I. Quadratic finite element multiwavelet solution of displacement for rod.

DOF for uniform DOF for adaptive Error estimator Relative errorSpace refinement refinement (10−2) (%)

V 0 ( j =0) 3 3 — —W0 ( j =0) 2 2 0.55422 22.920W1 ( j =1) 4 4 0.07609 2.786W2 ( j =2) 8 6 0.03788 1.387W3 ( j =3) 16 2 0.00666 0.244W3 ( j =4) 32 — 0.00094 0.034

Table II. Cubic finite element multiwavelet solution of displacement for rod.


V 0 ( j =0) 4 4 — —W0 ( j =0) 3 3 0.80106 3.237W1 ( j =1) 6 6 0.55665 2.062W2 ( j =2) 12 3 0.04382 0.160W3 ( j =3) 24 — 0.00274 0.010



0 1 2 310-2

10-1

100

101

102

j

Quadratic Uniform RefinementQuadratic Adaptive RefinementCubic Uniform RefinementCubic Adaptive Refinement

5 10 15 20 25 30 3510-1

100

101

102

DOF


Rel

ativ

e E

rror

(%

)R

elat

ive

Err

or (

%)

(a)

(b)

Figure 12. Convergence of axial deformation for rod using uniform and adaptive multiwavelet methodwith (a) number of levels and (b) degrees of freedom.

q0

q(x) F0

x

M0

w

L

Figure 13. A cantilever beam subjected to distributed loading.

distributed loading q(x)=q0(1−x/L),q0=106N, lump force F0=105N and moment M0=103Nm, respectively.

To overcome shear locking problems caused by the increasing value of length–height ratio L/H ,the consistent interpolation is used to ensure that the approximation of transverse deflection w

and rotation � is interpolated by two scaling functions in the same polynomial order. To solvethe Timoshenko problems, two types of consistent interpolation functions are chosen to derive the


578 Y. WANG ET AL.

multiwavelets: (1) w and � are interpolated by quadratic and linear scaling functions respectively;(2) w and � are interpolated by cubic and quadratic scaling functions respectively. The liftingcoefficients can be solved by the following two simultaneous equations

a11(�1j,m,�1

j,k) = 0 (53a)

a12(�1j,m,�2

j,k) = 0 (53b)

a21(�2j,m,�1

j,k) = 0 (54a)

a22(�2j,m,�2

j,k) = 0 (54b)

If the deflection and rotation are selected to be quadratic and linear scaling functions respec-tively, the construction method of finite element multiwavelets is the same as that of rod element.However, it is impossible to impose more vanishing moments on initial linear wavelets for thelinear scaling functions that are naturally operator-orthogonal with the linear wavelets. Considerthe construction of high-order scaling functions. Equation (53b) is naturally satisfied for the multi-wavelets constructed from Equation (53a) always have sufficient number of vanishing momentsto be operator-orthogonal with the scaling functions in Equation (53b). The finite element multi-wavelets constructed from Equation (54a) has the same property as Equation (53a). The sufficientcondition of Equation (54a) is that the multiwavelets must have one more vanishing moments tomake the new finite element multiwavelets be orthogonal with the scaling functions of deflec-tion. Since quadratic multiwavelets with two or more vanishing moments are operator-orthogonalwith cubic scaling functions, we use the quadratic multiwavelets with two vanishing momentsto solve Timoshenko beam problems, which ensure that the multiscale stiffness matrices are scale-orthogonal.

Given a threshold value =0.15% for adaptive solution of deflection, the Timoshenko beamproblem is solved by uniform and adaptive refinement strategy. Tables III and IV illustrate the errorestimator and relative error estimator of deflection by quadratic and cubic multiwavelet methodusing uniform and adaptive refinement strategy, respectively. Figure 14 shows the convergencerate of deflection by the uniform and adaptive multiwavelet with increasing number of levels and

Table III. Quadratic finite element multiwavelet solution of deflection for Timoshenko beam.


V 0 ( j =0) 5 5 — —W0 ( j =0) 4 4 0.64297 6.257W1 ( j =1) 8 8 0.16440 1.575W2 ( j =2) 16 14 0.04133 0.394W3 ( j =3) 32 — 0.01035 0.099

Table IV. Cubic finite element multiwavelet solution of deflection for Timoshenko beam.


V 0 ( j =0) 7 7 — —W0 ( j =0) 6 6 0.28000 2.692W1 ( j =1) 12 9 0.07000 0.668W2 ( j =2) 24 3 0.01750 0.167W3 ( j =3) 48 — 0.00438 0.042



0 1 2 310-2

10-1

100

101

102

j


Rel

ativ

e E

rror

(%

)

(a)

(b)

10 20 30 40 5010

-1

100

101

DOF


Rel

ativ

e E

rror

(%

)

Figure 14. Convergence rate of deflection for Timoshenko beam using uniform and adaptive multiwaveletmethod with (a) number of levels and (b) degrees of freedom.

Table V. Quadratic finite element multiwavelet solution of rotation for Timoshenko beam.


V 0 ( j =0) 3 3 — —W0 ( j =0) 2 2 0.22234 19.856W1 ( j =1) 4 4 0.07390 6.599W2 ( j =2) 8 6 0.02108 1.883W3 ( j =3) 16 6 0.00562 0.502W4 ( j =4) 32 — 0.00145 0.129

degrees of freedoms. It can be concluded that adaptive refinement scheme needs fewer degrees offreedom to approximate the analytic solution compared with that of uniform refinement.

Given a threshold value =0.2% for adaptive solution of rotation, the Timoshenko beam problemis solved by adaptive refinement strategy. Tables V and VI illustrate the error estimator and relativeerror estimator of rotation by quadratic and cubic multiwavelet methods using uniform and adaptiverefinement strategy, respectively. Figure 15 shows the convergence rate of rotation by the uniformand adaptive multiwavelet with increasing number of levels and degrees of freedoms. The adaptiverefinement scheme shows its advantage over uniform refinement in solving structural problemswith less computational cost.


580 Y. WANG ET AL.

Table VI. Cubic finite element multiwavelet solution of rotation for Timoshenko beam.


V 0 ( j =0) 5 5 — —W0 ( j =0) 4 4 0.38281 3.457W1 ( j =1) 8 8 0.09424 0.844W2 ( j =2) 16 4 0.02347 0.210W3 ( j =3) 32 — 0.00587 0.052

0 1 2 310-2

10-1

100

101

102

j


Rel

ativ

e E

rror

(%

)

(a)

(b)5 10 15 20 25 30 35

10-1

100

101

102

DOF


Rel

ativ

e E

rror

(%

)

Figure 15. Convergence of rotation for Timoshenko beam using uniform and adaptive multiwavelet methodwith (a) number of levels and (b) degrees of freedom.

Example 3Figure 16 shows a simply supported beam with uniform section subjected to a distributed loadingq(x)=q0xex , q0=106N, beam length L=10m, E I =5×109Nm2.

We adopt cubic and quintic Euler–Bernoulli beam scaling functions to solve the problem.For adaptive solution of deflection, we select a threshold value =0.01% and =0.001% forthe multiscale computation with cubic and quintic Hermite scaling functions. It can be verifiedthat cubic Hermite wavelets are naturally operator-orthogonal with Hermite scaling functions.



O

q(x)

x

L

Figure 16. A simply supported beam subjected to a distributed loading.

Table VII. Cubic finite element multiwavelet solution of deflection for the Euler–Bernoulli beam.

DOF for uniform DOF for adaptive Relative errorSpace refinement refinement Error estimator (%)

V 0 ( j =0) 9 9 — —W0 ( j =0) 4 4 7.2710 35.112W1 ( j =1) 8 6 1.2745 6.145W2 ( j =2) 16 6 0.1450 0.687W3 ( j =3) 32 6 0.0125 0.059W3 ( j =4) 64 — 0.0009 0.004

Table VIII. Quintic finite element multiwavelet solution of deflection for the Euler–Bernoulli beam.


V 0 ( j =0) 9 9 — —W0 ( j =0) 7 7 0.25963 1.2538W1 ( j =1) 13 6 0.01276 0.0615W2 ( j =2) 25 5 0.00036 0.0017W3 ( j =3) 49 — 0.00000 0.0000

Tables VII and VIII illustrate the error estimator and relative error estimator of deflection by cubicand quintic multiwavelet method using uniform and adaptive refinement strategy, respectively.Figure 17 shows the convergence rate of deflection by the uniform and adaptive multiwaveletmethod with increasing number of levels and degrees of freedoms. It can be concluded that uniformand adaptive refinement for multiwavelets have the same convergence rate, but adaptive refinementscheme needs fewer degrees of freedom to approximate the analytic solution.

For adaptive solution of rotation, we select the threshold value =0.01% and =0.001% for themultiscale computation with cubic and quintic Hermite scaling functions. Table IX and X illustratethe error estimator and relative error estimator of rotation by cubic and quintic multiwavelet methodsusing uniform and adaptive refinement strategy, respectively. Figure 18 shows the convergencerate of deflection by the uniform and adaptive multiwavelet methods with increasing number oflevels and degrees of freedoms, respectively. Therefore, the adaptive refinement scheme shows itsadvantages over uniform refinement with less computational cost.

It can be concluded that by designing proper multiwavelets according to the operators ofstructural problems, the coarser solution can be adaptively refined by adding details over severallevels till the relative errors decrease to the threshold value while the solution converges to theanalytic solution fast.


582 Y. WANG ET AL.

0 1 2 310-6

10-4

10-2

100

102

j

Cubic Uniform RefinementCubic Adaptive RefinementQuintic Uniform RefinementQuintic Adaptive Refinement

Rel

ativ

e E

rror

(%

)

(a)

(b)10 20 30 40 50 60 70

10-4

10-3

10-2

10-1

100

101

DOF


Rel

ativ

e E

rror

(%

)

Figure 17. Convergence of deflection for the Euler–Bernoulli beam using uniform and adaptive multiwaveletmethods with (a) number of levels and (b) degrees of freedom.

Table IX. Cubic finite element multiwavelet solution of rotation for the Euler–Bernoulli beam.


V 0 ( j =0) 9 9 — —W0 ( j =0) 4 4 1.4307 16.025W1 ( j =1) 8 6 0.2737 3.066W2 ( j =2) 16 6 0.0318 0.356W3 ( j =3) 32 4 0.0028 0.031W4 ( j =4) 64 — 0.0002 0.002

7. CONCLUSIONS

The finite element multiwavelets are constructed flexibly and efficiently based on the lifting schemeaccording to the operators of structural problems. The most remarkable property of finite elementmultiwavelets is the operator-orthogonality, which allows the scale-decoupling of multilevel stiff-ness matrix and adaptive analysis of structural problems. Compared with uniform finite element



Table X. Quintic finite element multiwavelet solution of rotation for the Euler–Bernoulli beam.


V 0 ( j =0) 9 9 — —W0 ( j =0) 7 7 0.21383 2.3951W1 ( j =1) 13 13 0.02128 0.2383W2 ( j =2) 25 6 0.00099 0.0111W3 ( j =3) 49 — 0.00003 0.0003

0 1 2 3

10-4

10-2

100

102

j


10 20 30 40 50 60 7010-3

10-2

10-1

100

101

DOF


Rel

ativ

e E

rror

(%

)R

elat

ive

Err

or (

%)

(a)

(b)

Figure 18. Convergence of rotation for the Euler–Bernoulli beam using uniform and adaptive multiwaveletmethod with (a) number of levels and (b) degrees of freedom.

multiwavelet method, the adaptive finite element multiwavelet method can lead to faster conver-gent rate in solving structural problems. The numerical examples indicate that the finite elementmultiwavelets can be efficiently and flexibly applied to the adaptive structural analysis. Furtherresearch is to construct specific finite element multiwavelets for solving engineering problems withhigher dimension, general domains, etc.


584 Y. WANG ET AL.

ACKNOWLEDGEMENTS

We would like to thank two anonymous referees for several important comments and suggestions thathelped in improving the clarity and presentation of our original manuscript. The work in this article issupported by National Natural Science Foundation of China (No. 50875195) and a Foundation for theAuthor of National Excellent Doctoral Dissertation of China (No. 2007B33).

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the construction of finite element multiwavelets for adaptive structural analysis

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