Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 591–600

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

An adaptive inverse iteration algorithm using interpolatingmultiwavelets for structural eigenvalue problems

Youming Wang a,b, Xuefeng Chen a,n, Zhengjia He a

a State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of Chinab Zhuzhou Times New Material Technology Co., Ltd., Zhuzhou 412007, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 24 May 2009

Received in revised form

25 October 2009

Accepted 24 August 2010Available online 15 September 2010

Keywords:

Structural eigenvalues

Stable completion

Interpolating multiwavelets

Adaptive inverse iteration algorithm

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2010.08.007

esponding author.

ail address: xautroland@163.com (X. Chen).

a b s t r a c t

Structural eigenvalues have been broadly applied in modal analysis, damage detection,

vibration control, etc. In this paper, the interpolating multiwavelets are custom

designed based on stable completion method to solve structural eigenvalue problems.

The operator-orthogonality of interpolating multiwavelets gives rise to highly sparse

multilevel stiffness and mass matrices of structural eigenvalue problems and permits

the incremental computation of the eigenvalue solution in an efficient manner.

An adaptive inverse iteration algorithm using the interpolating multiwavelets is

presented to solve structural eigenvalue problems. Numerical examples validate the

accuracy and efficiency of the proposed algorithm.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The structural eigenvalue problems are widely researched in many engineering areas such as mechanical, civil andaerospace engineering. As a very common approach, the finite element method (FEM) has drawn much interest because ofits effective and convenient computing capability especially in structural eigenvalue analysis [1]. However, traditionalfinite element model has to be remeshed until sufficient accuracy is reached, which leads to the low computation efficiencyor convergent rate. The deficiency of the FEM becomes much explicit in the accurate computation of cracked structures,such as cracked beams [2–4] and rotors [5–7], complex structural eigenvalue problems [8,9], etc.

The drawback of FEM can be overcome by the introduction of wavelet theory. The wavelet method can be viewed as anumerical method in which the approximating function is defined by use of a multiresolution technique based on scalingor wavelet functions. The advantage of the wavelet method is that the analysis solution of engineering problems can beapproximated by increasing the scale of wavelet analysis instead of re-meshing the initial model. In the last few decades,the wavelet method becomes a new numerical method in the structural eigenvalue solution [10–13]. Unfortunately,traditional wavelets are only suited for the problems defined on regularly spaced grids or meshes because the wavelets areinvariant under translation and dilation.

The restriction and deficiency of traditional wavelets were eliminated by the second generation wavelets. As ageneralization of biorthogonal wavelets, second generation wavelets are local in both space and frequency and can haveseveral vanishing moments without the translation and dilation invariance of their cousins and can be custom designed forcomplex domains and irregular samples. Recently, the lifting scheme [14,15] and the stable completion [16] are the twomain methods to design the interpolating multiwavelets for engineering problems. Davis et al. [17] generalized lifting

ll rights reserved.

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600592

scheme to construct multiwavelets by applying a finite sequence of simple lifting steps to a simple initial basis.The generalization of the multiwavelet concept to second generation wavelets has led to the construction of a new familyof Lagrangian and Hermite interpolating multiwavelet bases, which are piecewise polynomial of any given order andflexible to build on irregularly spaced, unstructured meshes over bounded domains [19,21]. Sudarshan et al. [18]constructed Hermite multiwavelets based on the lifting scheme and presented a multiscale subspace iteration approach forsolving eigenvalue problems, which showed the computational efficiency over a single level subspace iteration methodand FEM. D’Heedene et al. [19] built a lifting wavelet framework using stable completion for Lagrange finite element basisfunction of any given order on unstructured meshes.

In this paper, we propose a new adaptive inverse iteration algorithm using interpolating multiwavelets to solvestructural eigenvalue problems. An outline of the paper is as follows. Section 2 describes the construction of interpolatingmultiwavelets based on the stable completion method and the formulation of multilevel eigenvalue equations. Section 3presents interpolating multiwavelet method (IMM) and adaptive inverse iteration algorithm (AIIA) using interpolatingmultiwavelets. Section 4 demonstrates the numerical performance of the proposed method and conclusions are drawn inSection 5.

2. Multilevel eigenvalue computation

2.1. Multiresolution finite element space

We first briefly introduce the multiresolution analysis and multiresolution finite element space. The multiresolutionanalysis [14] of wavelets is an important property in the multilevel approximation of engineering problems.The hierarchical scaling functions are constructed by first defining a nested sequence of close subspace {Vj}jAZ with theproperties: ?CV�1CV0CV1C?CL2(R), [jAZVj=L2(R) and \jAZVj=0. For a given subspace Vj, a complementary subspace

Wj can be defined so that Vj=Vj�1�Wj�1. Each subspace Vj and Wj is spanned by a set of scaling functions fj,k and wavelets

jj,m, respectively: Vj=span{fj,k}kAK(j) and Wj=span{jj,m}mAM(j), where K(j)is some index set associated with scalingfunctions of level j and M(j)=K(j+1)\K(j). The relation between the scaling function and wavelet at level j and j+1 satisfiesrefinement equations of the form

fj,k ¼X

l2Kðjþ1Þ

hj,k,lfjþ1,l ð1Þ

jj,m ¼X

l2Kðjþ1Þ

rj,m,lfjþ1,l ð2Þ

where hj,k,l and rj,m,l are referred to low-pass and high-pass filters, respectively. The refinement relation for the scalingfunctions in different levels has the form

fj,k ¼fjþ1,kþX

m2MðjÞ

hj,k,mjj,m ð3Þ

The multiresolution finite element space can be seen as a generalized hierarchical framework constructed from awavelet perspective of finite element space. As stated in Ref. [20], it is possible to construct a valid multiresolution analysisof V provided the interpolating functions in the finite element space are complete and compatible. Hence, the scalingfunctions fj,k can be simply chosen to be the finite element interpolating functions. The multiresolution analysis forLagrangian and Hermite finite element space was introduced in Refs. [19,21].

2.2. Stable completion

The stable completion [16,19] is a new lifting method to construct interpolating multiwavelets with specific definedproperties, such as vanishing moments, compact support, etc. It assembles a new compact wavelet as a linear combinationof several primitive wavelets and interior scaling functions

jj,mu¼X

m

gj,m,mufjþ1,m�X

k

sj,k,mufj,k ð4Þ

where m0AM(j), gj,m,m’is the stable completion coefficients and sj,k,m0 the lifting coefficients. Compared to the lifting schemein Ref. [18], the stable completion provides us more flexibility and freedom to design the multiwavelets by choosing stablecompletion or lifting coefficients. The pair of coefficients can be determined by computing a basis for the null space of thelocal interaction matrix Rj in the operator-orthogonal relation equations

Rj

�sj,k,mu

gj,m,mu

" #¼ aðfj,k�,fj,kÞ aðfj,k�,fjþ1,mÞh i �sj,k,mu

gj,m,mu

" #¼ 0 ð5Þ

where fj,kn are all the scaling functions on a given domain Oj, fj,k are the subset of scaling functions that are all interior inthe domain Oj and a(�,�) is the bilinear operator with respect to the scaling functions and scaling functions or wavelets,

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600 593

which is derived from the multiresolution finite element models. To construct general interpolating multiwavelets withhigh-order vanishing moments, a new vector a(x,�) including a series of ascending order monomials may be added into R0jin the operator-orthogonal relation equations

Ruj�sj,k,mu

gj,m,mu

" #¼

aðfj,k�,fj,kÞ aðfj,k�,fjþ1,mÞ

aðx,fj,kÞ aðx,fjþ1,mÞ

" #�sj,k,mu

gj,m,mu

" #¼ 0 ð6Þ

where x¼

x0

^

xx�1

264

375 is the homogeneous matrix with respect to x in one-dimensional problems, which ensures x vanishing

moments on interpolating multiwavelets.

2.3. The formulation of multilevel eigenvalue equation

The eigenvalue equation

Kl�o2Ml¼ 0 ð7Þ

can be discretized by interpolating multiwavelets at level j and the generalized eigenvalue problems can be obtained as

Kjþ1ljþ1 ¼o2jþ1Mjþ1ljþ1 ð8Þ

where Kj +1 and Mj + 1 are multilevel stiffness and mass matrices at level j+1, lj + 1 and oj +1 are the eigenvectors andeigenvalues at level j+1. The multilevel stiffness matrices Kj + 1 of eigenvalue equation can be denoted in the two-level form

Kjþ1 ¼Kj Kb,j

Kd,j Kc,j

" #ð9Þ

with the individual entries being given as

Kj½p,q� ¼ aðfj,p,fj,qÞ ð10Þ

Kb,j½p,q� ¼ aðfj,p,fj,qÞ ð11Þ

Kd,j½p,q� ¼ aðjj,p,fj,qÞ ¼ KTb,j½q,p� ð12Þ

Kc,j½p,q� ¼ aðjj,p,jj,qÞ ð13Þ

where p and q are the index sets of scaling functions and wavelets in the multilevel stiffness matrix. In the solution ofstructural eigenvalue problems using interpolating multiwavelets, it is desirable that the operator matrices Kb,j be zero-valued, which permits the only required entries Kc,j to directly add into the multilevel stiffness matrices Kj + 1. Generally, theinterpolating multiwavelets have to be constructed with more vanishing moments to satisfy the operator-orthogonality inthe multilevel mass matrices Mj +1. Therefore, the interpolating multiwavelets can be custom designed based on the stablecompletion method to obtain highly sparse multilevel stiffness and mass matrices, which are favorable for the adaptivesolution of eigenvalue problems.

3. Adaptive inverse iteration algorithm

Based on the operator-orthogonality of interpolating multiwavelets, an IMM and AIIA are presented to solve eigenvalueproblems in the multiresolution finite element space.

3.1. Error estimation

The error estimator is the important parameter to control the approximate process in the adaptive inverse iterationalgorithm. The global error estimator on the global domain Oj indicates the convergence rate of the adaptive algorithm,which can be defined as

ej ¼oref�oj

oref

�������� ð14Þ

where oref is the reference or exact solution, oj is the adaptive solution at the scale j. Because the reference or exactsolution may not be found in general engineering eigenvalue problems, the eigenvalue solution at level j+1 is used toreplace the exact eigenvalue solution and the global error estimator has another form

ej ¼ojþ1�oj

ojþ1

�������� ð15Þ

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600594

and the local error estimator on the local domain Xnj (n=1, y, N) is given

tj,n ¼ojþ1,n�oj,n

ojþ1,n

�������� ð16Þ

Similarly, the iteration error estimator for the IMM can be defined as

Zj,i ¼~o2

j ðiþ1Þ� ~o2j ðiÞ

~o2j ðiþ1Þ

���������� ð17Þ

where ~o jðiÞ is the eigenvalue solution at the iterative step i and the scale j.

3.2. The procedures of IMM and AIIA

Based on the operator-orthogonality of interpolating multiwavelets, an adaptive inverse iteration algorithm ispresented to solve structural eigenvalue problems. The initial condition is given as: a steering parameter 0ryr1,an initial domain Oj, an initial scale j=0, a global tolerance e and an iteration tolerance g. The IMM for eigenvalue problemsis programmed as follows:

IMM1: Select initial vector X0j at level j, let Yj ¼MjX

0j . The initial eigenvector can be chosen randomly, but not orthogonal

to lj, i.e. satisfy the equation ðX0j Þ

T Mjlja0.IMM2: Solve the equation KjX

1j ¼ Yj. The scale-decoupling property of multilevel stiffness matrices ensures that it is

unnecessary to factorize the stiffness matrices at each level.IMM3: Construct new stiffness and mass matrices

~K j ¼ ðX1j Þ

T Yj ð18Þ

and let Y uj ¼MjX1j , then

~Mj ¼ X1j

� �TY uj: ð19Þ

IMM4: Compute the approximate value of o2j

~o2j ¼

~K j= ~Mj: ð20Þ

IMM5: Test whether iteration error estimator Zj,i satisfies Zj,iog. If it does, GOTO IMM7.IMM6: Normalize the eigenvectorX1

j , let Yj ¼ Y uj= ~Mj1=2

, which makes the new X0j satisfy the equation ðX0

j ÞT MjX

0j ¼ 1. The

details at each level are added into the eigenvectors and the stiffness and mass matrices at each level, GOTO IMM2.IMM7: Output the eigenvalue o2

j ¼~o2

j and normalize the eigenvector lj

lj ¼ X1j =

~Mj1=2: ð21Þ

The use of IMM in the multilevel eigenvalue solution becomes very convenient because the interpolating multiwaveletsat each level can be directly added into the multilevel stiffness and mass matrices. However, a certain number ofmultiwavelets contributing insignificantly to the eigenvalue solution may be added to the initial eigenvalue equations. Toimprove the solution efficiency of IMM, the adaptive inverse iteration algorithm is proposed in the following:

AIIA1: Solve the eigenvalue problem using the IMM on the domain Oj, j= j+1.AIIA2: Calculate local error estimators tj,n and let tmax

j ¼maxðtj,nÞ.AIIA3: If tj,nZytmax

j , add the wavelet bases into the local domains and form new stiffness and mass matrices.AIIA4: Test if the global error estimation ej satisfies ejoe. If it does, GOTO AIIA5; else GOTO AIIA1.AIIA5: Output the eigenvalue oj and the normalized eigenvector lj.

The operator-orthogonality of multilevel stiffness and mass matrices ensures the incremental computation ofeigenvalue solution by the IMM and AIIA. Since the steering parameter y of AIIA is chosen randomly, an increasing numberof interpolating multiwavelets can be added into the local domains and the convergence rate of the eigenvalue solution canbe adjusted to one’s needs. Hence, the AIIA is more flexible than the IMM in solving eigenvalue problems.

4. Numerical examples

In this section, we present two numerical experiments to demonstrate the accuracy and efficiency of the AIIA. For thecomparison of eigenvalue solution, the multiscale subspace iteration algorithm (MSIA) [18] using the interpolatingmultiwavelets constructed by the lifting scheme is introduced. The two algorithms are used to solve beam and plane frameeigenvalue problems and executed by MATLAB software on AMD Athlon 64 processor and 512 MB of main memory.

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600 595

The steering parameter for the AIIA is chosen to be y=1 and the iteration tolerance is g=1�10�7. The convergence rate andruntime of the two algorithms are illustrated in the following numerical examples.

Example 1. Fig. 1 shows a four-step cantilever beam with rectangle cross-section. The physical parameters are: elasticmodulus E=5�1010 N/m2, shear modulus G=8�109 N/m2, width B=0.625 m, height H=1 m, shear correction coefficientKs=5/6, length L=1 m and the density is r=7.9�103 kg/m3, respectively.

The Timoshenko beam model is constructed for this problem and the consistent interpolation is used in the eigenvaluesolution. The operators in multilevel stiffness matrix of Timoshenko beam are

a11 j1j,m,f1

j,k

� �¼

Z xb

xa

GAKs

dj1j,m

dx

df1j,k

dxdx ð22Þ

a12ðj1j,m,f2

j,kÞ ¼

Z xb

xa

GAKs

dj1j,m

dxf2

j,k dx ð23Þ

a21ðj2j,m,f1

j,kÞ ¼

Z xb

xa

GAKs

dj2j,m

dxf1

j,k dx ð24Þ

a22ðj2j,m,f2

j,kÞ ¼

Z xb

xa

EIdj2

j,m

dx

df2j,k

dxþGAKsj2

j,mf2j,k

!dx ð25Þ

and the operators in multilevel mass matrix of Timoshenko beam are

b1ðjj,m,fj,kÞ ¼

Z xb

xa

rAj1j,mf

1j,k dx ð26Þ

b2ðjj,m,fj,kÞ ¼

Z xb

xa

rIj2j,mf

2j,k dx ð27Þ

where I is the second moment of area.According to the operators of Timoshenko beam eigenvalue problem, we can construct two types of Lagrangian

multiwavelets, j1j,m and j2

j,m, based on the stable completion to satisfy the operator-orthogonality in Eqs. (22)–(27). The

global error estimator of AIIA and MSIA is chosen to be Eq. (14) and the initial Timoshenko beam model is discretized byfour individual elements on the four steps. Fig. 2(a–c) illustrates a linear multiwavelet with no vanishing moments, aquadratic multiwavelet with two vanishing moments and a cubic multiwavelet with three vanishing moments,respectively. Fig. 3 shows the convergence rate of first-order eigenvalue solutions for the case of MSIA and AIIA usingquadratic Lagrangian multiwavelets with increasing number of degrees of freedom (DOFs), respectively. Fig. 4 comparesthe convergence rate of fourth-order eigenvalue solutions by the two algorithms using cubic Lagrangian multiwaveletswith increasing number of DOFs, respectively. It can be seen that the convergence rate of AIIA is faster than that of theMSIA in the eigenvalue solution of Timoshenko beam problem. Tables 1 and 2 show the comparison of global errorestimation and runtime (t) for the typical first-order and fouth-order eigenvalue solutions in Figs. 3 and 4, respectively.

16EI 9EI 4EI EI

LLLL

Fig. 1. A four-step cantilever beam.

-1 -0.5 0 0.5 10

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 1-1

-0.5

0

0.5

1

Fig. 2. Multiwavelets for Timoshenko beam problems (a) linear multiwavelet, (b) quadratic multiwavelet and (c) cubic multiwavelet.

0 20 40 60 80 100-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

DOFs

ln ||

e||

Multiscale subspace iteration algorithmAdaptive inverse iteration algorithm

Fig. 3. Convergence of first-order eigenvalue solutions for the Timoshenko beam.

0 20 40 60 80 100 120 140 160 180-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

DOFs

ln ||

e||

Multiscale subspace iteration algorithmAdaptive inverse iteration algorithm

Fig. 4. Convergence of fourth-order eigenvalue solutions for the Timoshenko beam.

Table 1Comparison of first-order eigenvalue solutions of Timoshenko beam.

Multiscale subspace iteration algorithm Adaptive inverse iteration algorithm

j f1 (Hz) DOFs ej (%) t j f1 (Hz) DOFs ej (%) t

1 52.385 14 0.69 0.042 3 52.128 23 0.19 0.068

2 52.142 26 0.22 0.091 6 52.045 47 0.03 0.154

3 52.058 50 0.06 0.172 9 52.035 74 0.01 0.230

4 52.036 98 0.01 0.336 10 52.033 80 0.01 0.278

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600596

We can observe that the computational cost of AIIA is less than the MSIA given the same convergence rate in solvingTimoshenko beam problem.

Example 2. Fig. 5 shows a plane frame with rectangle cross-section. The physical parameters are: elastic modulusE=1�1011 N/m2, width B=0.625 m, height H=1 m, length L=1 m and the density is r=7.9�103 kg/m3, respectively.

A plane frame can be equivalent with a bar and Euler–Bernoulli beam, which means there are two kinds of operatorswith respect to the bar and Euler–Bernoulli beam problems. The operators in multilevel stiffness and mass matrix of

Table 2Comparison of fourth-order eigenvalue solutions of Timoshenko beam.

Multiscale subspace iteration algorithm Adaptive inverse iteration algorithm

j f4 (Hz) DOFs ej (%) t j f4 (Hz) DOFs ej (%) t

1 326.862 22 0.51 0.074 4 325.314 37 0.04 0.115

2 325.323 42 0.04 0.137 8 325.202 82 0.00 0.413

3 325.207 82 0.00 0.425 11 325.200 127 0.00 0.854

4 325.200 162 0.00 1.283 12 325.200 137 0.00 0.981

L

L L

Fig. 5. A plane frame.

0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2-0.4

-0.2

0

0.2

0.4

Fig. 6. Multiwavelets for Euler–Bernoulli beam problems: (a) cubic Hermite multiwavelet and (b) quintic Hermite multiwavelet.

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600 597

Euler–Bernoulli beam are

a1ðjj,m,fj,kÞ ¼

Z xb

xa

EIdj2

j,m

dx

df2j,k

dxdx ð28Þ

b1ðjj,m,fj,kÞ ¼

Z xb

xa

rAjj,mfj,kþrIdjj,m

dx

dfj,k

dx

� �dx ð29Þ

and the operators in multilevel stiffness and mass matrix of bar are

a2ðjj,m,fj,kÞ ¼

Z xb

xa

EAdjj,m

dx

dfj,k

dxdx ð30Þ

b2ðjj,m,fj,kÞ ¼

Z xb

xa

rAjj,mfj,k dx ð31Þ

The Hermite and Lagrangian multiwavelets are constructed to satisfy the operator-orthogonality of Euler–Bernoullibeam and bar problems independently. It can be verified that the cubic Hermite wavelets for Euler–Bernoulli beamproblems are naturally operator-orthogonal to the scaling functions. To satisfy the operator-orthogonality in Eq. (28), thequintic Hermite multiwavelets can be constructed to be with two vanishing moments. Fig. 6(a–b) shows a typical cubicHermite multiwavelet with zero vanishing moment and a quintic Hermite multiwavelet with two vanishing moments,

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600598

respectively. The initial plane frame model is discretized by three individual elements and the cubic and quinticinterpolating multiwavelets are used to solve the eigenvalue problems. Fig. 7 compares the convergence rate of first-ordereigenvalue solutions for the case of MSIA and AIIA using cubic multiwavelets with increasing number of DOFs, respectively.Fig. 8 illustrates the convergence rate of fifth-order eigenvalue solutions using quintic multiwavelets with increasingnumber of DOFs, respectively. It can be observed that the convergence rate of AIIA is higher than that of the MSIA. Tables 3and 4 give the global error estimation and runtime for the first-order and fifth-order eigenvalue solutions in Figs.7 and 8,

0 10 20 30 40 50 60 70 80 90-10

-9

-8

-7

-6

-5

-4

DOFs

ln ||

e||

Multiscale subspace iteration allgorithmAdaptive inverse iteration algorithm

Fig. 7. Convergence of first-order eigenvalue solutions of the plane frame.

0 20 40 60 80 100 120-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

DOFs

ln ||

e||

Multiscale subspace iteration algorithmAdaptive inverse iteration algorithm

Fig. 8. Convergence of fifth-order eigenvalue solutions of the plane frame.

Table 3Comparison of first-order eigenvalue solutions of plane frame.

Multiscale subspace iteration algorithm Adaptive inverse iteration algorithm

j f1 (Hz) DOFs ej (%) t j f1 (Hz) DOFs ej (%) t

1 409.929 12 1.30 0.057 3 405.539 21 0.21 0.092

2 405.985 21 0.32 0.103 4 405.117 27 0.11 0.124

3 405.006 39 0.08 0.249 5 404.800 45 0.03 0.267

4 404.767 75 0.02 0.596 6 404.723 81 0.00 0.634

Table 4Comparison of fifth-order eigenvalue solutions of plane frame.

Multiscale subspace iteration algorithm Adaptive inverse iteration algorithm

j f5 (Hz) DOFs ej (%) t j f5 (Hz) DOFs ej (%) t

1 1686.827 16 6.54 0.089 3 1600.260 32 1.08 0.138

2 1614.661 28 1.99 0.150 4 1593.190 40 0.63 0.296

3 1591.707 52 0.54 0.445 5 1587.377 60 0.26 0.490

4 1585.277 100 0.13 0.933 7 1584.415 104 0.08 0.962

Y. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 591–600 599

respectively. It can be seen that the computational cost of AIIA is less than the MSIA given the same convergence rate insolving the plane frame eigenvalue problem. For general structural eigenvalue problems, such as two- or three-dimensional problems, we can construct different kinds of high-dimensional interpolating multiwavelets based on thestable completion method to satisfy the operator-orthogonality and use the AIIA to solve the problems accurately andefficiently.

5. Conclusions

According to the operators of structural eigenvalue problems, a new class of interpolating multiwavelets is flexiblyconstructed by the stable completion method. The superior feature of interpolating multiwavelets is the operator-orthogonality in the multilevel eigenvalue computation, which permits the incremental computation of eigenvaluesolution and reduces the computational complexity. An IMM and AIIA using interpolating multiwavelets are proposed tosolve the structural eigenvalue problems. The advantage of AIIA over IMM is that an increasing number of interpolatingmultiwavelets can be added into the local domains based on the local error estimation and the convergence rate of theeigenvalue solution can be adjusted to one’s needs. Numerical examples show that the AIIA is an accurate and efficient toolin solving structural eigenvalue problems. The further research is the construction of high-dimensional interpolatingmultiwavelets using the stable completion method and the proposed method will be applied to the eigenvalue solution ofa wide range of structures, such as rotors, plates, shells.

Acknowledgement

This work is supported by National Natural Science Foundation of China (No. 50875195, 51035007) and a Foundationfor the Author of National Excellent Doctoral Dissertation of China (No. 2007B33).

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