THE BOUNDARY COLLOCATION METHOD WITH MESHLESS CONCEPT FOR ACOUSTIC EIGENANALYSIS OF TWO-DIMENSIONAL CAVITIES USING RADIAL BASIS FUNCTION

Journal of Sound and Vibration (2002) 257(4), 667–711 doi:10.1006/jsvi.5038, available online at http://www.idealibrary.com on

THE BOUNDARY COLLOCATION METHOD WITH MESHLESS CONCEPT FOR ACOUSTIC EIGENANALYSIS OF TWO-DIMENSIONAL CAVITIES USING RADIAL BASIS FUNCTION J. T. Chen, M. H. Chang, K. H. Chen and S. R. Lin Department of Harbor and River Engineering, National Taiwan Ocean University, 202 Keelung, Taiwan, Republic of China. E-mail:[email protected] (Received 30 July 2001, and in final form 8 January 2002) In this paper, a meshless method for the acoustic eigenfrequencies using radial basis function (RBF) is proposed. The coefficients of influence matrices are easily determined by the two-point functions. In determining the diagonal elements of the influence matrices, two techniques, limiting approach and invariant method, are employed. Based on the RBF in the imaginary-part kernel, the method results in spurious eigenvalues which can be separated by using the singular value decomposition (SVD) technique in conjunction with the Fredholm alternative theorem. To understand why the spurious eigenvalues occur, analytical study in the discrete system by discretizing the circular boundary is conducted by using circulants. By using the SVD updating terms and documents, the true and spurious eigensolutions can be extracted out respectively. Several examples are demonstrated to see the validity of the present method. # 2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The mesh generation of a complicated geometry is time consuming in the stage of model creation for engineers in dealing with the engineering problems by using the numerical methods, such as the finite difference method (FDM), finite element method (FEM) and boundary element method (BEM). In the recent years, researchers have paid attention to the meshless method which the element is free. The initial idea of meshless method dates back to the smooth particle hydrodynamics (SPH) method for modelling astrophysical phenomena [1]. Several meshless methods have also been reported in the literature, for example, the element-free Galerkin method [2] and the reproducing kernel method [3]. For acoustics, the integral equations have been utilized to solve the interior and exterior problems for a long time. Several approaches, e.g., complex-valued boundary element method [4], multiple reciprocity method (MRM) [5–7], and the real-part boundary element method [5, 8] have been developed for acoustic problems. To solve acoustic problems by using the complex-valued BEM, the influence coefficient matrix would be complex arithematics [9, 10]. Therefore, Tai and Shaw [11] employed only the real-part kernel to solve the eigenvalue problems and to avoid the complex-valued computation. The computation of the real-part kernel method or the MRM [11, 12] has some advantages, but it still faces both the singular and hypersingular integrals. To avoid the singular and hypersingular integrals, De Mey [13] used imaginary-part kernel to solve the eigenvalue 0022-460X/02/$35.00

# 2002 Elsevier Science Ltd. All rights reserved.

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problems. At the same time, De Mey also found the spurious eigensolutions but he did not discuss them analytically. Kang et al. [14–16] proposed the non-dimensional dynamic influence function (NDIF) method to solve the eigenproblem of an acoustic cavity. Later, Chen et al. [17] commented that the NDIF method is a special case of imaginary-part BEM. Nevertheless, spurious eigensolutions are inherent in the imaginary-part BEM, realpart BEM and MRM. Numerically speaking, the spurious eigensolutions result from the rank deficiency of the coefficient matrix which is less than 2N; where 2N is the number of boundary unknowns. This implies the fewer number of constraint equations making the solution space larger. Mathematically speaking, the spurious eigensolutions for interior acoustics and fictitious solutions for exterior acoustics arise from an ‘‘improper approximation of the null space of operator’’. In this paper, we will employ the imaginary-part kernel to solve the acoustic eigenproblems. In solving the problem numerically, elements are not required and only boundary nodes are necessary. Both the collocation and source points are distributed on the boundary only. Besides, the kernel function is composed of two-point function which is a kind of radial basis function. To examine the reason why spurious eigenvalues occur in the above methods, we will employ the SVD updating technique in conjunction with the Fredholm alternative theorem to overcome this difficulty by assembling the dual equations [18–20]. The SVD updating terms and updating documents will be employed to extract out the true and spurious eigenvalues for two-dimensional cavities respectively. For a special case of circular cavity, the spurious eigensolutions will be analytically predicted in the discrete system of circulants. Finally, the true eigenvalues for a circular cavity will be derived analytically by approaching the discrete system to the continuous system using the analytical properties of circulants [21].

2. MESHLESS FORMULATION USING RADIAL BASIS FUNCTION OF THE IMAGINARY-PART KERNEL

The governing equation for an interior acoustic problem is the Helmholtz equation as follows: ðr2 þ k2 Þuðx1 ; x2 Þ ¼ 0;

ðx1 ; x2 Þ 2 D;

ð1Þ

where r2 is the Laplacian operator, D is the domain of the cavity and k is the wave number which is angular frequency over the speed of sound. The boundary conditions can be either the Neumann or Dirichlet type. The radial basis function is expressed by Gðxi ; sj Þ ¼ jðjsj  xi jÞ;

ð2Þ

where xi and sj are the ith collocation and jth source points, respectively. The Euclidean norm jsj  xi j is referred to as the radial distance between the collocation and source points. The two-point function (jðjsj  xi jÞ) is called radial basis function since it depends on the radial distance between xi and sj : By considering the imaginary-part kernel of ð1Þ fundamental solution for the Helmholtz equation (Uðs; xÞ ¼ ImfiH0 ðkrÞg) with globally supported radial basis function, we can choose the four kernels in the dual formulation [5, 22], Uðs; xÞ ¼ J0 ðkjs  xjÞ ¼ J0 ðkrÞ; Tðs; xÞ ¼

@Uðs; xÞ J1 ðkrÞyi ni ; ¼ k @ns r

ð3Þ ð4Þ

THE BOUNDARY COLLOCATION METHOD

J1 ðkrÞyj n% j @Uðs; xÞ ; ¼k @nx r
 kJ2 ðkrÞyi yj ni n% j J1 ðkrÞni n% i @ 2 Uðs; xÞ Mðs; xÞ ¼ ¼k þ ; @ns @nx r2 r Lðs; xÞ ¼

669 ð5Þ ð6Þ

where r  js  xj is the distance between the source and collocation points; ni is the ith component of the outnormal vector at s; n% i is the ith component of the outnormal vector at x; Jn denotes the first-kind Bessel function of the nth order, and yi  si  xi ; i ¼ 1; 2: Based on the dual formulation [23] for the indirect method, we can represent the acoustic field solution by Single-layer potential approach: X uðxi Þ ¼ Uðsj ; xi Þfj ! fui g ¼ ½Uij ffj g; ð7Þ matrix form

j

tðxi Þ ¼

X @Uðsj ; xi Þ @nx

j

fj

Double-layer potential approach: X Tðsj ; xi Þcj uðxi Þ ¼ j

tðxi Þ ¼

X @Tðsj ; xi Þ j

@nx

cj

!

fti g ¼ ½Lij ffj g:

ð8Þ

!

fui g ¼ ½Tij fcj g;

ð9Þ

!

fti g ¼ ½Mij fcj g;

ð10Þ

matrix form

matrix form

matrix form

where ffj g and fcj g are the generalized unknowns by using the single- and double-layer potential approaches respectively. By adopting the two bases, Jm ðkxÞ and its derivative J0m ðkxÞ; we can decompose the imaginary-part kernel functions into the separate forms, ( P U I ðy; 0Þ ¼ 1 Jm ðkRÞJm ðkrÞcosðmyÞ; R > r; Uðs; xÞ ¼ Pm¼1 ð11Þ 1 E U ðy; 0Þ ¼ m¼1 Jm ðkrÞJm ðkRÞcosðmyÞ; R5r; ( P 0 T I ðy; 0Þ ¼ 1 R > r; m¼1 kJm ðkRÞJm ðkrÞcosðmyÞ; Tðs; xÞ ¼ P ð12Þ 1 0 E T ðy; 0Þ ¼ m¼1 kJm ðkrÞJm ðkRÞcosðmyÞ; R5r; ( P LI ðy; 0Þ ¼ 1 kJm ðkRÞJ0m ðkrÞcosðmyÞ; R > r; Pm¼1 Lðs; xÞ ¼ ð13Þ 1 E L ðy; 0Þ ¼ m¼1 kJm ðkrÞJ0m ðkRÞcosðmyÞ; R5r; ( P 0 2 0 M I ðy; 0Þ ¼ 1 R > r; m¼1 k Jm ðkRÞJm ðkrÞcosðmyÞ; P Mðs; xÞ ¼ ð14Þ 1 0 E 2 0 M ðy; 0Þ ¼ m¼1 k Jm ðkrÞJm ðkRÞcosðmyÞ; R5r; where the superscripts ‘‘I’’ and ‘‘E’’ denote the interior and exterior domains, x ¼ ðr; 0Þ and s ¼ ðR; yÞ in polar coordinate. By superimposing 2N lumped density along the boundary, we have the four influence matrices 3 2 a1;2 a1;3    a1;2N1 a1;2N a1;1 6 a a2;2 a2;3    a2;2N1 a2;2N 7 7 6 2;1 7 6 7 6 a3;1 a a    a a 3;2 3;3 3;2N1 3;2N ð15Þ ½Uij  ¼ 6 7; 6 . .. 7 .. .. .. .. 7 6 . 4 . . 5 . . . . a2N;1

a2N;2

a2N;3



a2N;2N1

a2N;2N

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J. T. CHEN ET AL.

2

b1;2

b1;3



b1;2N1

b2;2

b2;3



b2;2N1

b3;2 .. . b2N;2

b3;3 .. . b2N;3

 .. . 

b3;2N1 .. . b2N;2N1

c1;1 6 c 6 2;1 6 6 ½Lij  ¼ 6 c3;1 6 . 6 . 4 .

c1;2 c2;2

c1;3 c2;3

 

c1;2N1 c2;2N1

c3;2 .. .

c3;3 .. .

 .. .

c3;2N1 .. .

c2N;1

c2N;2

c2N;3

   c2N;2N1

d1;1

d1;2

d1;3



d2;2 d3;2 .. . d2N;2

d2;3 d3;3 .. . d2N;3

   d2;2N1    d3;2N1 .. .. . .    d2N;2N1

b1;1

6 b 6 2;1 6 6 ½Tij  ¼ 6 b3;1 6 . 6 . 4 . b2N;1 2

2

6 d 6 2;1 6 6 ½Mij  ¼ 6 d3;1 6 . 6 . 4 . d2N;1

d1;2N1

b1;2N

3

b2;2N 7 7 7 b3;2N 7 7; .. 7 7 . 5

ð16Þ

b2N;2N 3 c1;2N c2;2N 7 7 7 c3;2N 7 7; .. 7 7 . 5

ð17Þ

c2N;2N d1;2N

3

d2;2N 7 7 7 d3;2N 7 7; .. 7 7 . 5 d2N;2N

ð18Þ

where the elements can be obtained by ai;j ¼ Uðsj ; xi Þ;

bi;j ¼ Tðsj ; xi Þ;

ð19; 20Þ

ci;j ¼ Lðsj ; xi Þ;

di;j ¼ Mðsj ; xi Þ:

ð21; 22Þ

Then, we can determine ffj g and fcj g by satisfying the boundary conditions.

3. CALCULATION FOR THE DIAGONAL ELEMENTS IN THE FOUR INFLUENCE # PITAL’S RULE AND INVARIANT METHOD MATRICES USING THE L’HO

The diagonal elements in the influence matrices where the radial distance is zero (r ¼ 0 when i ¼ j) can be solved by using the L’Ho# pital’s rule. Considering the asymptotic behavior and the recurrence relations of Bessel functions, we can obtain the diagonal elements as follows: lim Uðs; xÞ ¼ lim J0 ðkrÞ ¼ 1; s!x

r!0

J1 ðkrÞyi ni r!0 r ðJ ðkrÞ  J2 ðkrÞÞyi ni 0 ¼ lim k2 ¼ 0; s1 !x1 2

ð23Þ

lim Tðs; xÞ ¼ lim  k s!x

ð24Þ

s2 !x2

J1 ðkrÞyj n% j r %j 2 ðJ0 ðkrÞ  J2 ðkrÞÞyj n ¼ lim k ¼ 0; s1 !x1 2

lim Lðs; xÞ ¼ lim k s!x

r!0

s2 !x2

ð25Þ

THE BOUNDARY COLLOCATION METHOD

671




 kJ2 ðkrÞyi yj ni n% j J1 ðkrÞni n% i þ s!x r!0 r2 r ðJ0 ðkrÞ  J2 ðkrÞÞni n% i k2 ¼ ; ð26Þ ¼ lim 0 þ k2 s1 !x1 2 2 s2 !x2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where r ¼ ðs1  x1 Þ þ ðs2  x2 Þ2 : For a circular case, the eigenvalues for the influence matrices in equations (15)–(18) can be obtained by Z p 1 X 1 l‘ ¼ cosð‘yÞ Jm ðkRÞJm ðkrÞcosðmyÞr dy r 4 y p m¼1 lim Mðs; xÞ ¼ lim k

¼ 2NJ‘ ðkrÞJ‘ ðkrÞ; m‘ ¼

1 r4y

Z

‘ ¼ 0; 1; . . . ; ðN  1Þ; N; 1 X

p

cosð‘yÞ p

1 r4y

Z

p

cosð‘yÞ p

Z

‘ ¼ 0; 1; . . . ; ðN  1Þ; N;

1 X

ð28Þ

Jm ðkRÞJ0m ðkrÞcosðmyÞr dy

m¼1

¼ 2NðkÞJ‘ ðkrÞJ0‘ ðkrÞ; 1 k‘ ¼ r4y

J0m ðkRÞJm ðkrÞcosðmyÞr dy

m¼1

¼ 2NðkÞJ0‘ ðkrÞJ‘ ðkrÞ; n‘ ¼

ð27Þ

p

cosð‘yÞ p

‘ ¼ 0; 1; . . . ; ðN  1Þ; N;

1 X

ð29Þ

J0m ðkRÞJ0m ðkrÞcosðmyÞr dy

m¼1

¼ 2Nðk2 ÞJ0‘ ðkrÞJ0‘ ðkrÞ;

‘ ¼ 0; 1; . . . ; ðN  1Þ; N;

ð30Þ

where 4y ¼ 2p=2N: According to the addition theorem for the Bessel function and putting the same position for the two points, we have 1 ¼ J20 ðkrÞ þ 2

1 X

J2m ðkrÞ:

ð31Þ

Jm ðkrÞJ0m ðkrÞ:

ð32Þ

m¼1

By taking derivative with respect to r; we have 0 ¼ J0 ðkrÞJ00 ðkrÞ þ 2

1 X m¼1

Using the invariant property for the influence matrices, the first invariant is the sum of all the eigenvalues 2Na0 ¼ lðN1Þ þ    þ l1 þ l0 þ l1 þ    þ lN 1 X ¼ 2N J‘ ðkrÞJ‘ ðkrÞ;

ð33Þ

‘¼1

2Nb0 ¼ mðN1Þ þ    þ m1 þ m0 þ m1 þ    þ mN 1 X ¼ 2Nk J0‘ ðkrÞJ‘ ðkrÞ; ‘¼1

ð34Þ

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J. T. CHEN ET AL.

2Nc0 ¼ nðN1Þ þ    þ n1 þ n0 þ n1 þ    þ nN 1 X ¼ 2Nk J‘ ðkrÞJ0‘ ðkrÞ;

ð35Þ

‘¼1

2Nd0 ¼ kðN1Þ þ    þ k1 þ k0 þ k1 þ    þ kN 1 X ¼ 2Nk2 J0‘ ðkrÞJ0‘ ðkrÞ:

ð36Þ

‘¼1

By substituting equations (31) and (32) into equations (33)–(36), we obtain a0 ¼ 1; b0 ¼ 0; c0 ¼ 0; and d0 ¼ k2 =2 if the imaginary-part kernel was considered. Hence, the indeterminate forms of diagonal elements are easily determined from the first invariant as well as the values obtained by using the L’Ho# pital’s rule.

4. THE RELATIONS OF KERNELS AND MATRICES BETWEEN THE DIRECT AND INDIRECT METHODS

In solving the boundary-value problem using BEM, two methods, direct method and indirect method are employed. The direct method is derived by using the Green identity in terms of the unknowns which are the actual physical quantities on the boundary. In the direct method, we have Z Z E 0¼ T ðs; xÞuðsÞ dBðsÞ  U E ðs; xÞtðsÞ dBðsÞ; ð37Þ B

0¼

Z

B

M E ðs; xÞuðsÞ dBðsÞ  B

Z

LE ðs; xÞtðsÞ dBðsÞ;

ð38Þ

B

where uðsÞ and tðsÞ are the potential and its normal derivative on the boundary and x is outside the domain. A key point of the indirect method is to represent a solution which satisfies the governing equation. The unknown densities are determined by matching the boundary conditions. Based on the superposition principle for the potentials, we have Single-layer potential approach: Z uðxÞ ¼ U I ðs; xÞfðsÞ dBðsÞ; ð39Þ B

tðxÞ ¼

Z

LI ðs; xÞfðsÞ dBðsÞ:

ð40Þ

T I ðs; xÞcðsÞ dBðsÞ;

ð41Þ

M I ðs; xÞcðsÞ dBðsÞ;

ð42Þ

B

Double-layer potential approach: uðxÞ ¼

Z B

tðxÞ ¼

Z B

where fðsÞ and cðsÞ are the single- and double-layer unknown densities, respectively. By discretizing equations (37) and (38), we have the linear algebraic equations for the direct method ½TijE fuj g ¼ ½UijE ftj g;

ð43Þ

673

THE BOUNDARY COLLOCATION METHOD

Table 1 The distributed and concentrated-type of imaginary-part method for the single- and double-layer potential approaches Distributed-type

Concentrated-typey

Single-layer potential approach

Dirichlet problem: R uðxÞ ¼ B U I ðs; xÞfðsÞ dBðsÞ Neumann problem: R tðxÞ ¼ B LI ðs; xÞfðsÞ dBðsÞ

Dirichlet problem: P I j U ðsj ; xi ÞAj ¼ ðSMÞij Aj Neumann problem: P tðxi Þ ¼ j LI ðsj ; xi ÞAj ¼ ðSMx Þij Aj

Double-layer potential approach

Dirichlet problem: R uðxÞ ¼ B T I ðs; xÞcðsÞ dBðsÞ Neumann problem: R tðxÞ ¼ B M I ðs; xÞcðsÞ dBðsÞ

Dirichlet problem: P I j T ðsj ; xi ÞBj ¼ ðSMs Þij Bj Neumann problem: P tðxi Þ ¼ j M I ðsj ; xi ÞBj ¼ ðSMsx Þij Bj

y

uðxi Þ ¼

uðxi Þ ¼

NDIF method by Kang et al. [14–16] is the special case.

½MijE fuj g ¼ ½LEij ftj g;

ð44Þ

where fuj g and ftj g are the potential and its normal derivative on the boundary B: By discretizing equations (39)–(42), we have the linear algebraic equations for the indirect method. Single-layer potential approach: uðxi Þ ¼ ½UijI ffj g;

ð45Þ

tðxi Þ ¼ ½LIij ffj g:

ð46Þ

uðxi Þ ¼ ½TijI fcj g;

ð47Þ

tðxi Þ ¼ ½MijI fcj g;

ð48Þ

Double-layer potential approach:

where ffj g and fcj g are the single- and double-layer unknown densities, respectively. The distributed-type and concentrated-type of imaginary-part method for the single- and double-layer potential approaches are shown in Table 1. By considering the degenerate kernels equations (11)–(14) and comparing with equations (43)–(48), we can find the following relations between the interior and exterior kernels from equations (11)–(14), i.e., UijE ¼ UijI or U E ðs; xÞ ¼ U I ðx; sÞ;

ð49Þ

TijE ¼ LIij or T E ðs; xÞ ¼ LI ðx; sÞ;

ð50Þ

LEij ¼ TijI or LE ðs; xÞ ¼ T I ðx; sÞ;

ð51Þ

MijE ¼ MijI or M E ðs; xÞ ¼ M I ðx; sÞ:

ð52Þ

674

Table 2 SVD updating technique for the true and spurious eigensolutions for a circular cavity using the direct and indirect methods Boundary-value problem

True eigensolution

Spurious eigensolution

Density function Single-layer potential approach

Double-layer potential approach

Jm ðkrÞ ¼ 0

Jm ðkrÞ ¼ 0

(Figure 1(a))

(Figure 1(b))

Jm ðkrÞ ¼ 0 (Figure 1(a))

Neumann problem

True eigensolution

Spurious eigensolution

True and spurious eigenvalues

J0m ðkrÞ

¼0

J0m ðkrÞ ¼ 0

(Figure 2(a))

(Figure 2(b))

(Figure 2(a)) Note: U I ¼ U E ; LI ¼ T E ; T I ¼ LE and M I ¼ M E :



SVD updating term

SVD updating document



Indirect method UE LE



E

E

L

M





SVD updating term 

SVD updating document

(Figure 1(b))

J0m ðkrÞ ¼ 0

Jm ðkrÞ ¼ 0

Direct method

J0m ðkrÞ

¼0

(Figure 2(b))

SVD updating term

SVD updating document



U

TE ME

E

T

E

SVD updating term 

SVD updating document



UI TI



(Figure 1(c))  I  T MI (Figure 1(d))  I  L MI (Figure 2(c))  I  U LI (Figure 2(d))

J. T. CHEN ET AL.

Dirichlet problem

Eigensolutions

675

THE BOUNDARY COLLOCATION METHOD

5. DERIVATION OF TRUE AND SPURIOUS EIGENSOLUTIONS FOR THE CICULAR CAVITY USING THE DEGENERATE KERNELS

In the circular cavity, we have the degenerate kernel functions in equations (11)–(14). We assume the single- and double-layer density functions on the boundary as 1 X fðsÞ ¼ ðAn cosðnyÞ þ Bn sinðnyÞÞ; ð53Þ n¼1

cðsÞ ¼

1 X

ðCn cosðnyÞ þ Dn sinðnyÞÞ;

ð54Þ

n¼1

where An ; Bn ; Cn and Dn are coefficients. For the Dirichlet problem, equation (39) reduced to Z 0¼ U I ðs; xÞfðsÞ dBðsÞ ¼

Z

B 2p

1 X

Jm ðkRÞJm ðkrÞ cosðmyÞ

0 m¼1

1 X

ðAn cosðnyÞ þ Bn sinðnyÞÞ r dy;

ð55Þ

n¼1

by using equation (53). By considering the orthogonality of trigonometric function, we have ( Z 2p p; m ¼ n; cosðmyÞ cosðnyÞ dy ¼ ð56Þ 0; m=n; 0 Z

2p

cosðmyÞ sinðnyÞdy ¼ 0:

ð57Þ

0

According to equation (56) and equation (57), equation (55) simplifies to Z 2p X 1 1 X Jm ðkRÞJm ðkrÞcosðmyÞ ðAn cosðnyÞ þ Bn sinðnyÞÞ r dy; 0¼ ¼p

0 m¼1 1 X

n¼1

Am Jm ðkRÞJm ðkrÞ:

ð58Þ

m¼1

Table 3 Zeros of the Bessel functions and its derivative, Jn ðkÞ and J0n ðkÞ Problem

Eigenvalues

1

2

3

4

5

Dirichlet problem

Jn ðkÞ J1 ðkÞ J2 ðkÞ Jn ðkÞ J4 ðkÞ J5 ðkÞ

24042 38317 51356 63802 75883 87715

55201 70156 84172 97610 110647 123386

86537 101735 116198 130152 143725 157002

117915 133237 147959 162234 176160 189801

149309 164706 179598 194094 208269 222178

Neumann problem

J00 ðkÞ J01 ðkÞ J02 ðkÞ J03 ðkÞ J04 ðkÞ J05 ðkÞ

0 18412 30542 42012 53175 64156

38317 53314 67071 80152 92824 105199

70156 85363 99695 113459 126819 139872

101735 117060 131703 145858 159641 173128

133237 148636 163475 177887 191960 205755

676

J. T. CHEN ET AL.

Thus, we obtain the eigenequation Jm ðkRÞJm ðkrÞ ¼ 0 by using the single-layer potential methods for the interior Dirichlet problem. Similarly, we also obtain Z T I ðs; xÞcðsÞ dBðsÞ 0¼ ¼

Z

B 2p

1 X

0 m¼1

kJ0m ðkRÞJm ðkrÞcosðmyÞ

1 X

ðCn cosðnyÞ þ Dn sinðnyÞÞ r dy

ð59Þ

n¼1

by using the double-layer potential approach. By considering equation (59) and the orthogonality of trigonometric function, we have

0¼

Z

2p

1 X

0 m¼1 1 X

kJ0m ðkRÞJm ðkrÞcosðmyÞ

1 X

ðCn cosðnyÞ þ Dn sinðnyÞÞ r dy

n¼1

Cm J0m ðkRÞJm ðkrÞ:

¼ pk

ð60Þ

m¼1

Thus, we obtain the eigenequation J0m ðkRÞJm ðkrÞ ¼ 0 by using the double-layer potential method for the interior Dirichlet problem. In the Neumann problem, we have Z 0¼ LI ðs; xÞfðsÞ dBðsÞ ¼

Z

B 2p

1 X

0 m¼1

kJm ðkRÞJ0m ðkrÞcosðmyÞ

1 X

ðAn cosðnyÞ þ Bn sinðnyÞÞ r dy

ð61Þ

n¼1

by substituting equations (13) and (53) into equation (40). By considering the orthogonality of trigonometric function, equation (61) simplifies to Z 2p X 1 1 X 0¼ kJm ðkRÞJ0m ðkrÞcosðmyÞ ðAn cosðnyÞ þ Bn sinðnyÞÞ r dy 0 m¼1 1 X

n¼1

Am Jm ðkRÞJ0m ðkrÞ:

¼ pk

ð62Þ

m¼1

Therefore, we obtain the eigenequation Jm ðkRÞJ0m ðkrÞ ¼ 0 by using the single-layer potential method for the interior Neumann problem. Similarly, we use the double-layer density functions to obtain Z 0¼ M I ðs; xÞcðsÞ dBðsÞ ¼

Z

B 2p

1 X

0 m¼1

k2 J0m ðkRÞJ0m ðkrÞcosðmyÞ

1 X

ðCn cosðnyÞ þ Dn sinðnyÞÞ r dy:

ð63Þ

n¼1

Figure 1. (a) The minimum singular value for different wave numbers by using the single-layer potential approach for the Dirichlet problem. (b) The minimum singular value for different wave numbers by using the double-layer potential approach for the Dirichlet  problem. (c) The minimum singular value for different wave numbers using the SVD updating term U T for the Dirichlet problem. (d) The minimum singular value for   different wave numbers using the SVD updating document T M for the Dirichlet problem. (e) The former three interior modes by using the single-layer potential approach for the Dirichlet problem. (f) The former three interior modes matrix by using the double-layer potential approach for the Dirichlet problem. (g) The former three analytical interior modes for the Dirichlet problem.

677

THE BOUNDARY COLLOCATION METHOD

0

T T

T

T

K5

6.215

T

J 3(1)

K3

(6.380 )

5.130 J 2(1)

-10

K4

(5.136) 5.52 0 J 0(2)

K2

(5.520)

3.83 0 J 1(1)

Log σ1

K1

(3.832)

2.405

-20

J 0(1)

(2.405)

-30

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 10 ∆k : 0.005

-40 0

2

4

6

k

(a) 10

0

T

S

S

S

S

T T S

T

T

Log σ1

K1

2.405 J 0(1)

-10

K

K3

K2

K

3.055 3.83 0 J' 2(1)

J

5.135

K4

5.375

K5

6.295

J 2(1) J 3(1) J (2) ) 0 (6.380) J' 3(1) (5.136 (5.520)

4.20 5

(1)

1 (2.405) (3.054) (3.832)

(4.201)

-20

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 10 ∆ k : 0.005

-30 0

(b)

2

4

k

6

678

J. T. CHEN ET AL. 10

0

T

T

T

T T K1

-10

J 0(1)

Log σ1

K4 J 3(1) J 2(1) 5.52 0 (6.380) J 0(2) (5.136)

J 1(1)

(3.832)

(2.405)

6.368

5.136

3.832

2.405

K5

K3

K2

(5.520)

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-40

0

2

4

(c)

6

k 10

S

S

0

S

S

K2

K1

-10

3.05 4

Log σ1

J' 2(1)

(3.054 )

K3

3.832

4.201

J' 0(1)

J' 3(1)

(3.832) (4.201)

K4

5.331 J' 1(2)

(5.331)

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-40

0

(d)

2

4

k Figure 1. Continued.

6

THE BOUNDARY COLLOCATION METHOD

Figure 1. Continued.

679

680

J. T. CHEN ET AL.

Figure 1. Continued.

THE BOUNDARY COLLOCATION METHOD

Figure 1. Continued.

681

682

J. T. CHEN ET AL.

By considering equation (63) and the orthogonality of trigonometric function, we have Z 2p X 1 1 X 0¼ k2 J0m ðkRÞJ0m ðkrÞcosðmyÞ ðCn cosðnyÞ þ Dn sinðnyÞÞ r dy 0 m¼1 1 X 2

n¼1

Cm J0m ðkRÞJ0m ðkrÞ:

¼ pk

ð64Þ

m¼1

Therefore, we obtain the eigenequation J0n ðkRÞJ0n ðkrÞ ¼ 0 by using the double-layer potential method for the interior Neumann problem. All the relations of true and spurious eigenvalues by using the single- and double-layer potential approaches are summarized in Table 2.

6. METHOD TO EXTRACT OUT THE TRUE EIGENSOLUTIONS

The SVD technique is an important tool in linear algebra. A matrix A with dimension M  N can be decomposed into a product of an orthogonal matrix C (N  N), a diagonal matrix S (M  N) with positive or zero elements, and an orthogonal matrix F (M  M), ½AMN ¼ ½FMM ½SMN ½CTNN ;

ð65Þ

where the superscript ‘‘T’’ is the transpose, F and C are both orthogonal in the sense that their column vectors are orthogonal,

T

fi fj ¼ dij ;

ð66Þ

ci cj ¼ dij ;

ð67Þ

T

where F F ¼ C C ¼ I and dij is the Kronecker delta symbol. Besides, we solve a homogeneous equation and obtain a non-trivial solution from a column vector fci g of C such that the singular value (si ) is zero. For the direct method in the discrete system, equations (43) and (44) are represented by Singular equation: ðUT methodÞ ½T E fug ¼ ½U E ftg ¼ 0;

ð68Þ

Hypersingular equation: ½M E fug ¼ ½LE ftg ¼ 0:

ðLM methodÞ

ð69Þ

Table 4 True and spurious eigensolutions for the Dirichlet and Neumann problems using the present formulation Indirect formulation Single-layer potential approach True mode Spurious mode Double-layer potential approach True mode Spurious mode Note: 05a5r; 05f52p: y Denotes a near-zero solution.

Dirichlet problem

Neumann problem

pJn ðkaÞJn ðkrÞcosðnfÞy pJ ðkaÞJ ðkrÞcosðnfÞy

pkJ0n ðkaÞJn ðkrÞcosðnfÞ pkJ0 ðkaÞJ ðkrÞcosðnfÞy

pJn ðkaÞJ0n ðkrÞcosðnfÞ pJ ðkaÞJ0 ðkrÞcosðnfÞy

pk2 J0n ðkaÞJ0n ðkrÞcosðnfÞy pk2 J0 ðkaÞJ0 ðkrÞcosðnfÞy

n

n

n

n

n

n

n

n

683

THE BOUNDARY COLLOCATION METHOD

For the Dirichlet problem, equations (68) and (69) are combined to " # ½U E  ftg ¼ f0g: ½LE 

ð70Þ

By using the SVD technique, the two matrices in equation (70) are decomposed into X U U T ½U E  ¼ ½FU ½SU ½CU T or ½UjE  ¼ sU j ffj gfcj g ; j

½LE  ¼ ½FL ½SL ½CL T

or ½LEj  ¼

X

sLj ffLj gfcLj gT :

ð71Þ

j

For the linear algebraic system, t is one column vector fci g in the ½C matrix corresponding to the zero singular value (si ¼ 0). By viewing t as a fci g vector in the right unitary matrix ½C; equation (70) reduces to  E U fci g ¼ f0g;  E L fci g ¼ f0g; ð72Þ since

X

U U T sU j ffj gfcj g fci g ¼ f0g

j

X

sLj ffLj gfcLj gT fci g ¼ f0g

j

!

cU j cj ¼dij

!

cLjcj ¼dij

U sU i ffi g ¼ f0g

sLi ffLi g ¼ f0g;

ði no sumÞ; ði no sumÞ;

ð73Þ

L E where ffi g and fci g are the orthonormal bases, sU j and sj are the singular values of ½U  E and ½L  matrices, respectively. We can easily extract out the true eigensolutions L (sU i ¼ si ¼ 0) since there exists the same eigensolution (t ¼ fci g) in the Dirichlet problem using equations (70)–(73) together. In a similar way, equations (68) and (69) are combined to " # ½T E  fug ¼ f0g ð74Þ ½M E 

for the Neumann problem. We can easily extract the true eigensolutions; X sTj ffj gfcj gT fci g ¼ f0g ! sTi ffi g ¼ f0g ði no sumÞ; ci cj ¼dij

j

X j

T sM j ffj gfcj g fci g ¼ f0g

!

ci cj ¼dij

sM i ffi g ¼ f0g

ði no sumÞ;

ð75Þ

since there exists the same solution (u ¼ fci g) corresponding to the zero singular values (sTi ¼ sM i ¼ 0) by using equations (74) and (75). According to the relations between the direct method and indirect method (equations (49)–(52)), we can extend to extract out the true solutions in the indirect method. For the Dirichlet problem, equation (70) changes to " # " # ½U E  ½U I  U E ¼U I ftg ¼ f0g LE ¼T I ftg ¼ f0g; ð76Þ ½LE  ½T I  after using equations (49)–(51). By using the SVD technique and equation (76), we have X T sU ! sU j ffj gfcj g fci g ¼ f0g i ffi g ¼ f0g ði no sumÞ; j

ci cj ¼dij

684

J. T. CHEN ET AL.

X

sTj ffj gfcj gT fci g ¼ f0g

j

!

ci cj ¼dij

sTi ffi g ¼ f0g

ði no sumÞ;

ð77Þ

T I I where fci g is an orthonormal basis; sU j and sj are the singular values of ½U  and ½T  matrices, respectively. We can easily extract the true solutions in the Dirichlet problem by T using equation (77) for the zero singular values of sU i ¼ si ¼ 0: In a similar way, equation (74) changes to " # " # ½T E  ½LI  T E ¼LI ftg ¼ f0gM E ¼M I ftg ¼ f0g; ð78Þ ½M E  ½M I 

after using equations (50)–(52). By using the SVD technique and equation (78), we have X sLj ffj gfcj gT fci g ¼ f0g ! sLi ffi g ¼ f0g ði no sumÞ; cicj ¼dij

j

X

T sM j ffj gfcj g fci g ¼ f0g

j

!

cicj ¼dij

sM i ffi g ¼ f0g

ði no sumÞ;

ð79Þ

I I where fci g is an orthonormal basis, sLj and sM j are the singular values of ½L  and ½M  matrices, respectively. We can easily extract out the true eigensolutions in the Neumann problem by using equation (79) for the zero singular values of sLi ¼ sM i ¼ 0:

6.1. EXTRACTING OUT THE TRUE EIGENSOLUTIONS BY USING THE SVD UPDATING TERM FOR A CIRCULAR CASE

For a circular cavity subject to the Dirichlet boundary condition, we obtain the two eigenequations Single-layer approach : Ji ðkrÞJi ðkrÞ ¼ 0 ði no sumÞ; Double-layer approach : Ji ðkrÞJ0i ðkrÞ ¼ 0

ði no sumÞ:

ð80Þ ð81Þ

The true eigensolution Ji ðkrÞ ¼ 0 satisfies both equations and the spurious eigensolutions J0i ðkrÞ ¼ 0 satisfies only one of the equations by using the double-layer potential approach for the Dirichlet problem. To obtain an overdetermined system for the Dirchlet problem, we can combine ½U I  and ½T I  matrices by using the updating terms, " # ½U I  ½C ¼ ð82Þ ½T I  4N2N:

Since the eigensolution is non-trivial, the rank of ½C must be smaller than 2N: Therefore, the 2N singular values for ½C matrix must have at least one zero value. Based on the equivalence between the SVD technique and the least-squares method in the mathematical

Figure 2. (a) The minimum singular value for different wave numbers by using the single-layer potential approach for the Neumann problem. (b) The minimum singular value for different wave numbers by using the double-layer potential approach for the Neumann  problem. (c) The minimum singular value for different wave numbers by using the SVD updating term L M for the Neumann (d) The minimum singular value  problem.  for different wave numbers by using the SVD updating document U L for the Neumann problem. (e) The former three interior modes by using the single-layer potential approach for the Neumann problem. (f) The former three interior modes by using the double-layer potential approach for the Neumann problem. (g) The former three analytical interior modes for the Neumann problem.

685

THE BOUNDARY COLLOCATION METHOD 10

0

T

T

T S

ST S S

T

S K

K

2.405

-10

J 0(1)

K3

K2

K1

4.205

3.055 3.830 J' 2(1)

5. 135 J 2(1)

(2.404) (3.054) (3.832) (4.201)

Log σ1

6.295

J' 4(1)

J' 3(1) (5 .1 36)(5.318)

J' 0(1)

K5

K4

5.340

J' 5(1)

(6.416)

-20

-30

-40 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 10 ∆ k : 0.005

-50 0

2

4

6

k

(a) 10

0

T

T

T

T

K4

4.180

-10 K1

Log σ1

J' 3(1) K 3 (4.201)

K2

-20

3.055

3.830

T

T

K5

5.330 J' 1(2)

(5.331)

K6

6.415

1.840

J' 2(1)

J' 0(1)

J' 5(1)

J' 1 (1)

(3.054)

(3.831 )

(6.415)

(1.841)

-30

-40 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 10 ∆ k : 0.005

-50 0

(b)

2

4

k

6

686

J. T. CHEN ET AL. 10

T

T

T

0

T

K1

-10

3.054

Log σ1

J' 2(1)

(3.054)

K2

K3

3.832

4.201

J' 0(1)

J' 3(1)

J' 1(2)

(3.832) (4.201)

(5.331)

K4

5.331

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-40 0

2

4

6

k

(c) 0

S

S

S

S

K5

S

6.367 J 3(1)

-10

(6.380) K3 K2

3.830 J1

Log σ1

K1

(3.832)

2.405

-20

(1)

5.13 5 J 2(1)

K4

(5.136) 5.520 J 0(2)

(5.520)

J 0(1)

(2.405)

-30

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-40 0

(d)

2

4

k Figure 2. Continued.

6

THE BOUNDARY COLLOCATION METHOD

Figure 2. Continued.

687

688

J. T. CHEN ET AL.

Figure 2. Continued.

THE BOUNDARY COLLOCATION METHOD

Figure 2. Continued.

689

690

J. T. CHEN ET AL.

essence, we have h

I T

I T

½U 

½T 

" # i ½U I  ½T I 

¼ ½U I 2 þ ½T I 2 :

ð83Þ

For the special case of a circular cavity in the Dirchlet problem, we can decompose ½U I  and ½T I  into 2

6 . 6 I ½U  ¼ ½F6 4

2

3

..

7 7 T 7½F ; 5

l‘ ..

. 3

..

6 . 6 I ½T  ¼ ½F6 4

ð84Þ

7 7 T 7½F 5

m‘ ..

ð85Þ

.

where ½F is a modal matrix which can be chosen as either one of the following two matrices; 2 61 6 6 61 6 6 6 1 1 6 U1 ¼ pffiffiffiffiffiffiffi6 6 2N 6 .. 6. 6 6 61 6 4 1

2p

2p

ðei 2N Þ0

ðei 2N Þ0

2p ðei 2N Þ1

2p ðei 2N Þ1

2p

ðei 2N Þ2 .. .

2p

ðei 2N Þ2 .. .

2p

ðei 2N Þ2N2

ðei 2N Þ2N1

2p

ðei 2N Þ2N1

1

0

ðei 2N Þ2N2

2p 2p

ðei

2ðN1Þp 0 2N Þ

  .. .. . . .. .. . . .. .. . . .. .. . .

ðei

2ðN1Þp 2N2 2N Þ

 

ðei

2ðN1Þp 2N1 2N Þ

2ðN1Þp ðei 2N Þ1

ðei

2ðN1Þp 2 2N Þ

.. .

ðei

2Np 0 2N Þ

3

7 7 7 7 7 7 2Np i 2N 2 7 ðe Þ 7 7 ð86Þ; 7 .. 7 . 7 7 7 2Np i 2N 2N27 ðe Þ 7 5 2Np 2N1 ðei 2N Þ 2N2N 2Np ðei 2N Þ1

U2 ¼ 2

1

6 6 2p 61 cosð2N Þ 6 6 6 4p Þ 1 cosð2N 1 6 pffiffiffiffiffiffiffi6 6 . . 2N 6 . .. 6. 6 6 6 1 cosð2pð2N2ÞÞ 4 2N Þ 1 cosð2pð2N1Þ 2N

2p sinð2N Þ 4p sinð2N Þ .. .

sinð2pð2N2Þ Þ 2N sinð2pð2N1Þ Þ 2N

3   0 1 7 .. .. 7 2pN 7 . sinð2pðN1Þ . Þ cosð Þ 2N 2N 7 7 .. .. 7 4pðN1Þ . sinð 2N Þ . cosð4pN 7 2N Þ 7 7 .. .. .. .. 7 . . . . 7 7 7 .. .. pð4N4ÞðNÞ 7 . sinðpð4N4ÞðN1Þ . Þ cosð Þ 5 2N 2N       sinðpð4N2ÞðN1Þ Þ cosðpð4N2ÞðNÞ Þ 2N 2N

2N2N

ð87Þ

691

THE BOUNDARY COLLOCATION METHOD

By substituting equations (84) and (85) into (83), we obtain 3 2 .. 7 6 . 7 T 6 T 2 2 7½F : 6 ½C ½C ¼ ½F6 ð88Þ ðl‘ þ m‘ Þ 7 5 4 .. . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Therefore, we have the singular values of l‘ þ m2‘ ; ‘ ¼ 0; 1; . . . ; ðN  1Þ; N: Only the true eigenvalues (l‘ ¼ m‘ ¼ 0) have dips in the figure of s1 versus k; i.e., the zeros of eigensolutions (J‘ ðkrÞ ¼ 0) are obtained for the Dirichlet problem. For a circular cavity subject to the Neumann problem, we obtain the two eigenequations Single-layer approach : Ji ðkrÞJ0i ðkrÞ ¼ 0 Double-layer approach : J0i ðkrÞJ0i ðkrÞ ¼ 0

ði no sumÞ; ði no sumÞ:

ð89Þ

J0i ðkrÞ

¼ 0 satisfies both equations and the spurious eigensolutions The true eigensolution Ji ðkrÞ ¼ 0 satisfies only one of equations in the single-layer potential approach for the Neumann problem. We can combine ½LI  and ½M I  matrices by using the updating terms to obtain an overdetermined system: " # LI ½C ¼ ð90Þ MI 4N2N

for the Neumann problem. Since the eigensolution is non-trivial and the rank of ½C must be smaller than 2N: Therefore, the 2N singular values for ½C matrix must have at least one zero value. Based on the equivalence between the SVD technique and the least-squares method in the mathematical essence, we have " # h i ½LI  I T I T ð91Þ ¼ ½LI 2 þ ½M I 2 : ½L  ½M  ½M I  For the special case of a circular cavity of the Neumann problem, we can decompose ½LI  and ½M I  into 2 3 .. . 6 7 6 7 T n‘ ½LI  ¼ ½F6 ð92Þ 7½F ; 4 5 .. .

Table 5 The former five exact eigenvalues for a two-dimensional square cavity subject to the Dirichlet and Neumann boundary conditions Problem Dirichlet problem Neumannproblem

Eigenvalues pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmn ¼ p m2 þ n2 ðm; n ¼ 1; 2; 3; . . .Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmn ¼ p m2 þ n2 ðm; n ¼ 0; 1; 2; 3; . . .Þ

1

2

3

4

5

44429

70248

88858

99346

113271

31416

44429

62831

70248

88858

692

J. T. CHEN ET AL.

2

3

..

6 . 6 I k‘ ½M  ¼ ½F6 4

..

7 7 T 7½F : 5

ð93Þ

.

By substituting equations (92) and (91) into equation (91), we obtain 2 3 .. 6 . 7 6 7 T ðn2‘ þ k2‘ Þ ½CT ½C ¼ ½F6 ð94Þ 7½F : 4 5 .. . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Therefore, we have the singular values of n‘ þ k‘ ; ‘ ¼ 0; 1; . . . ; ðN  1Þ; N: Only the true eigenvalues (n‘ ¼ k‘ ¼ 0) have dips in the figure of s1 versus k; i.e., the zeros of eigensolutions (J0‘ ðkrÞ ¼ 0) are obtained for the Neumann problem. 7. METHOD TO FILTER OUT THE SPURIOUS EIGENSOLUTIONS

Fredholm’s Alternative Theorem (1) Nonsingular system: The equation ½Hfgg ¼ fpg has a unique solution fgg ¼ ½H1 fpg when the determinant of H is not zero. Besides, the equation has a trivial solution fgg ¼ f0g if and only if the only continuous solution to the homogeneous equation ½Hfgg ¼ f0g:

ð95Þ

(2) Singular system: Alternatively, the equation has at least one solution if the homogeneous adjoint equation has at least one solution {fi } such that ð96Þ ½Hy ff g ¼ f0g; i

where ½Hy is the transpose conjugate matrix of [H] [24]. If the matrix H is real, the transpose conjugate is equal to transpose only, i.e., ½Hy ¼ ½HT : Moreover, a necessary and sufficient solvability condition [25] is that the constraint ðfpgy ffi g ¼ 0Þ must be satisfied. Then, the general solution can be written as Nr X ci ffi g; ð97Þ fgg ¼ fg% g þ i¼1

where {g% } is a particular solution and Nr is the rank of matrix [H]. The ci are arbitrary constants and {fi } are bases. Moreover, the particular solution is zero for the interior eigenproblems. The result implies that spurious eigenvalues are imbedded for both the Neumann and Dirichlet problems once the numerical method is chosen.

Figure 3. (a) The minimum singular value for different wave numbers by using the single-layer potential approach for the Dirichlet problem. (b) The minimum singular value for different wave numbers by using the double-layer potential approach for the Dirichlet   problem. (c) The minimum singular value for different wave numbers by using the SVD updating term U T for the Dirichlet problem. (d) The minimum singular value for   different wave numbers by using the SVD updating document T M for the Dirichlet problem. (e) The former three interior modes by using the single-layer potential approach for the Dirichlet problem. (f) The former three interior modes by using the double-layer potential approach for the Dirichlet problem. (g) The former three analytical interior modes for the Dirichlet problem.

693

THE BOUNDARY COLLOCATION METHOD 0 T

T

T

T

K4

9.600

K3

8.86 0

-10

K 22 K2

K1

(8.8858)

7.030

4.440

Log σ1

K 31

(9.9346)

K 21

K 11

(7.0248)

(4.4429)

-20

-30

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-40 0

2

4

6

8

10

k

(a) 5

S

0

T

T

S T

S S

T

-5

K3

Log σ1

8.610

S

K1

K2

4.440

7.060

-10

K 11

K 22

K4

9.870

(8.8858 ) K 31 (9.934)

K 21

(4.4429 )

(7.0248 )

-15

-20 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.001

-25 0

(b)

2

4

6

k

8

10

694

J. T. CHEN ET AL. 10

0

T

T

K3 8.610

T K2 6.990

K1 4.440 K11 (4.4429)

Log σ1

-10

K21 (7.0248)

T K4 9.960

K22 (8.8858) K 31 (9.9346)

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-40 0

2

4

6

8

10

k

(c) 10

0

S

S

S

K5 9.070

S K4 7.050

Log σ1

S

-10

K2 4.470 K21 (4.4429)

K1 3.140 K11 (3.1416)

K32 (8.8858)

K3 K31 6.280 (7.0248) K22 (6.2831)

-20

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-30 0

(d)

2

4

6

k Figure 3. Continued.

8

10

THE BOUNDARY COLLOCATION METHOD

Figure 3. Continued.

695

696

J. T. CHEN ET AL.

Figure 3. Continued.

THE BOUNDARY COLLOCATION METHOD

Figure 3. Continued.

697

698

J. T. CHEN ET AL.

By employing the LM formulation in the direct method, we have LM formulation: ½M E  fug ¼ ½LE  ftg ¼ fpg:

ð98Þ

E T

Hence the eigensolution for t is non-trivial, we have fi and ½L  fi ¼ f0g such that equation (98) reduces to fuT g½M E T ffi g ¼ 0:

ð99Þ

Since u is arbitrary for boundary excitation, we have ½M E T ffi g ¼ f0g:

ð100Þ

According to Fredholm’s alternative theorem in the real-matrix system, we have the homogeneous solutions in the linear algebraic equation if there exists a vector ffi g which satisfies " #   ½LE T ð101Þ ffi g ¼ f0g or ffi gT ½LE  ½M E  ¼ f0g: E T ½M  It indicates that the two matrices have the same spurious mode ffi g corresponding to the zero singular value. By using the SVD technique, the two matrices in equation (101) are decomposed into X ½LE T ¼ ½CL ½SL ½FL T or fLEj g ¼ sLj fcLj gffLj gT ; ð102Þ j

½M E T ¼ ½CM ½SM ½FM T

or

fMjE g ¼

X

M M T sM j fcj gffj g :

ð103Þ

j

By substituting equation (103) into equation (101), we have X sLj fcj gffj gT ffi g ¼ f0g ! sLi fci g ¼ f0g fi fj ¼dij

j

X j

T sM j fcj gffj g ffi g ¼ f0g

!

fi fj ¼dij

sM i fci g ¼ f0g

ði no sumÞ; ði no sumÞ;

ð104Þ

where ffi g and fci g are the orthonormal bases, sLi and sM i are the zero singular values of ½LE  and ½M E  matrices respectively. We can easily extract out the eigensolutions since there exists the same spurious boundary mode ffi g corresponding to the zero singular values (sLi ¼ sM i ¼ 0). Similarly, we can employ the UT formulation in the direct method, UT formulation: ½U E  ftg ¼ ½T E  fug ¼ fqg:

ð105Þ

E T

Since the eigensolution for u is non-trivial, we have ffi g and ½T  ffi g ¼ f0g such that equation (105) reduces to ftT g½U E T ffi g ¼ 0:

ð106Þ

Figure 4. (a) The minimum singular value for different wave numbers by using the single-layer potential approach for the Neumann problem. (b) The minimum singular value for different wave numbers by using the double-layer potential approach for the Neumann  problem. (c) The minimum singular value for different wave numbers using the SVD updating term L M for the Neumann problem. (d) The minimum singular value for   different wave numbers using the SVD updating document U L for the Neumann problem. (e) The former three interior modes by using the single-layer potential approach for the Neumann problem. (f) The former three interior modes by using the double-layer potential approach for the Neumann problem. (g) The former three analytical interior modes for the Neumann problem.

699

THE BOUNDARY COLLOCATION METHOD 10

T 0

S

T

T T

S

K5

T

8.610

T

K2

-10

4.44 0

Log σ1

K1

3.140

K 21

(4.4429 )

K3

K4

6.280

6.980

K 22

K 31

K6

K 32

(8.8858 )

9.960 K 32

(6.283 1) (7.0248 )

(9.9346)

K 11

(3.1416 ) -20

-30

T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-40

0

2

4

6

8

10

k

(a) 10

T

0

T

T T

K5

9.030

K4

Log σ1

K 32

7.04 0

T

-10

K2

K 31

4.470

(7.0248 )

(8.8858 )

K 21

(4.4429 )

K3

6.280

K1

K 22

3.140

(6.2831 )

K 11

(3.141 6)

-20

T: Trueeigen value S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-30 0

(b)

2

6

4

k

8

10

700

J. T. CHEN ET AL. 10

0

T

T K3

T

8.610 K2

6.990

K1

-10

4.440

K 21

Log σ1

K 11

K 22

T K4

9.96 0

(8.8858 ) K 31 (9.9346)

(7.0248 )

(4.4429 )

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-40 0

2

4

6

8

10

k

(c) 10

S

S

0

K3

K4

8.61 0

9.870

S K2 K1

-10

6.99 0

4.440

K 22

K 31

(8.8858) (9.9346)

K 21

K 11

Log σ1

S

(7.0248)

(4.4429)

-20

-30 T: True eigenvalue S: Spurious eigenvalue (): Analytical solution

Number of nodes : 12 ∆ k : 0.01

-40 0

(d)

2

4

6

k Figure 4. Continued.

8

10

THE BOUNDARY COLLOCATION METHOD

Figure 4. Continued.

701

702

J. T. CHEN ET AL.

Figure 4. Continued.

THE BOUNDARY COLLOCATION METHOD

Figure 4. Continued.

703

704

J. T. CHEN ET AL.

Since {t} is arbitrary for boundary excitation, we have ½U E T ffi g ¼ f0g:

ð107Þ

According to Fredholm’s alternative theorem, we have the homogeneous solutions in the linear algebraic equation if there exists a vector ffi g which satisfies " #   ½U E T ð108Þ ffi g ¼ f0g or ffi gT ½U E  ½T E  ¼ f0g: E T ½T  It indicates that the two matrices have the same spurious boundary mode ffi g corresponding to the zero singular value. By using the SVD technique, equation (108) is expressed as ½U E T ¼ ½C½SU ½FT ;

ð109Þ

½T E T ¼ ½C½ST ½FT :

ð110Þ

By substituting equation (110) into equation (108), we have X T sU ! sU j fcj gffj g ffi g ¼ f0g i fci g ¼ f0g fifj ¼dij

j

X

sTj fcj gffj gT ffi g ¼ f0g

j

!

fifj ¼dij

ði no sumÞ;

sTi fci g ¼ f0g ði no sumÞ;

ð111Þ

T where ffi g and fci g are the orthonormal bases, sU i and si are the zero singular values of E E ½U  and ½T  matrices, respectively. We can easily extract out the eigensolutions since there exists the same spurious boundary mode ffi g corresponding to the zero singular T values (sU i ¼ si ¼ 0). According to the relations of the matrices between the direct method and indirect method (equations (49)–(52)), we can filter out the spurious eigensolutions in the indirect method by using " # " # I T ½LE T ½T  E I ð112Þ ffi g ¼ f0g ML E ¼T ffi g ¼ f0g: ¼M I ½M E T ½M I T

By employing the SVD technique, we obtain X sTj fcj gffj gT ffi g ¼ f0g !

fi fj ¼dij

j

X j

T sM j fcj gffj g ffi g ¼ f0g

!

fi fj ¼dij

sTi fci g ¼ f0g

ði no sumÞ;

sM i fci g ¼ f0g

ði no sumÞ;

ð113Þ

where ffi g and fci g are the orthonormal bases, sTi and sM i are the zero singular values of ½T I  and ½M I  matrices, respectively. We can easily extract out the eigensolutions since there exists the same spurious boundary mode ffi g corresponding to the zero singular values (sTi ¼ sM i ¼ 0). In a similar way, we have the influence matrices by using the indirect method and equation (108) is transformed to " # " # ½U E T ½U I T U E ¼U I ffi g ¼ f0g T E ¼LI ffi g ¼ f0g: ð114Þ ½T E T ½LI T

705

THE BOUNDARY COLLOCATION METHOD

By using the SVD technique, we obtain X T sU j fcj gffj g ffi g ¼ f0g j

X

sLj fcj gffj gT ffi g ¼ f0g

j

!

sU i fci g ¼ f0g

ði no sumÞ;

!

sLi fci g ¼ f0g

ði no sumÞ;

fi fj ¼dij

fi fj ¼dij

ð115Þ

L I where ffi g and fci g are the orthonormal bases, sU i and si are the singular values of ½U  I and ½L  matrices, respectively. We can easily extract out the eigensolutions since there exists the same spurious boundary mode ffi g corresponding to the zero singular values L (sU i ¼ si ¼ 0).

7.1. FILTERING OUT THE SPURIOUS EIGENSOLUTIONS BY USING SVD UPDATING DOCUMENT FOR A CIRCULAR CASE

In solving a circular cavity problem using the double-layer potential approach, we obtain the eigenequations Dirichlet problem : Ji ðkrÞJ0i ðkrÞ ¼ 0

ði no sumÞ;

ð116Þ

Neumann problem : J0i ðkrÞJ0i ðkrÞ ¼ 0

ði no sumÞ:

ð117Þ

The spurious eigensolution J0i ðkrÞ are embedded in both the Dirichlet and the Neumann problems. Based on the dual formulation, the ½T I  and ½M I  matrices have the same spurious eigenvalues. In order to extract the spurious eigenvalues, we can combine the ½T I  and ½M I  matrices by using the updating documents,   ð118Þ ½D ¼ ½T I  ½M I  2N4N: Similarly, we have 

I

I

½T  ½M 

" #  ½T I T ½M I T

¼ ½T I 2 þ ½M I 2 :

ð119Þ

For the circular cavity, the spurious eigenvalues are both embedded in the transposes of ½T I  and ½M I  matrices according to equations (11)–(14). The singular values for ½D must have at least one zero singular value. To determine the singular values for ½D; we have 2 3 .. . 6 7 6 7 T ðm2‘ þ k2‘ Þ ½D½DT ¼ ½F6 ð120Þ 7½F : 4 5 .. . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By plotting the minimum singular values of m2‘ þ k2‘ ; ‘ ¼ 0; 1; . . . ; ðN  1Þ; N versus k; we can filter out the spurious eigenvalues (n‘ ¼ k‘ ¼ 0) where dips occur, i.e., the zeros of eigensolutions (J0‘ ðkrÞ ¼ 0) are obtained by using the double-layer potential approach. In solving a circular cavity by using the single-layer potential approach, we obtain Dirichlet problem: Ji ðkrÞJi ðkrÞ ¼ 0

ði no sumÞ;

ð121Þ

Neumann problem: Ji ðkrÞJ0i ðkrÞ ¼ 0

ði no sumÞ:

ð122Þ

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J. T. CHEN ET AL.

The spurious eigensolution Ji ðkrÞ are embedded in both the Dirichlet and the Neumann problems. In a similar way, we can combine the ½U I  and ½LI  matrices by using the updating documents,   ½D ¼ ½U I  ½LI  2N4N: ð123Þ Similarly, we have 

½U I  ½LI 



½U I  ½LI 

T

¼ ½U I 2 þ ½LI 2 :

ð124Þ

For the special case of a circular cavity, the spurious eigenvalues are embedded in the transpose of ½U I  and ½LI  matrices according to equations (11)–(14). The singular values for ½D must have at least one zero singular value. To determine the singular values for ½D; we have 2 3 .. 6 . 7 6 7 T T 2 2 7½F : ½D½D ¼ ½F6 ð125Þ ðl þ n Þ ‘ ‘ 6 7 4 5 .. . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By plotting the minimum singular values of l2‘ þ n2‘ ; ‘ ¼ 0; 1; . . . ; ðN  1Þ; N versus k; we can filter out the spurious eigenvalues (l‘ ¼ n‘ ¼ 0) where dips occur, i.e., the zeros of eigensolutions (J‘ ðkrÞ ¼ 0) are obtained. All the relations between the direct and indirect methods are shown in Table 2. We can easily extract out the true and spurious eigenvalues in the indirect method or the direct method by using the SVD updating terms and documents.

8. NUMERICAL RESULTS AND DISCUSSIONS

Case 1: Circular cavity (Dirichlet case). A circular cavity with a radius ðr ¼ 1 mÞ subjected to the Dirichlet boundary condition ðu ¼ 0Þ is considered. In this case, an analytical solution is available as follows: Eigenequation: Jm ðkn rÞ ¼ 0;

m; n ¼ 0; 1; 2; 3 . . . :

Eigenmode: uða; yÞ ¼ Jm ðkn aÞeimy ;

04a4r; 04y42p; m; n ¼ 0; 1; 2; 3; . . . :

By collocating 10 nodes on the circular boundary, two results by using the single- and double-layer potential approaches are obtained. The exact eigenvalues of a circular cavity subject to the Dirichlet boundary condition are shown in Table 3. Figure 1(a) shows the minimum singular value versus k using the single-layer potential approach. The former four true eigenvalues are obtained as shown in Figure 1(a) by considering the near zero minimum singular values if only the single-layer potential method is chosen. Figure 1(b) shows the minimum singular value versus k using the double-layer potential approach. The true eigenvalues occur at the positions of zeros for Jm ðkn rÞ while the spurious eigenvalues occur at the positions of zeros for J0m ðkn rÞ if the double-layer approach is chosen. In order to extract out the true eigenvalues, we can combine the ½U I  and ½T I  by using the SVD updating term. The minimum singular value versus k is shown in Figure 1(c), it is found that dips occur only at the positions of true eigenvalues. In a similar way, we can combine the ½T I  and ½M I  by using the SVD updating document in order to filter

THE BOUNDARY COLLOCATION METHOD

707

out the spurious eigenvalues. The minimum singular value versus k is shown in Figure 1(d), it is found that the dips occur only at the positions of spurious eigenvalues. The first three interior modes obtained by using the single- and double-layer potential approaches are shown in Figures 1(e) and 1(f), and are compared well with the analytical modes in Figure 1(g). It is observed that the nodal line can rotate in case of the eigenvalue of multiplicity more than one. There are two independent eigenmodes with respect to the same eigenvalue. The two modes can be linearly combined by any constants and can differ by the phase lag. This is the reason why the one mode can be rotated to another mode. A similar explanation can also be found in Reference [26]. It is also found that the result of case 1 in Figure 1(e) does not match the analytical solution very well. The reason is that the amplitudes of the modes are very small since they are multiplied by a near-zero value. However, the nodal line can be identified. To demonstrate this point, the exact solutions for the mode using the present method are shown to have the near-zero solutions in Table 4. The near-zero solutions are present if the single-layer approach was applied to solve the Dirichlet problem. Case 2: Circular cavity (Neumann case). A circular cavity with a radius ðr ¼ 1 mÞ subjected to the Neumann boundary condition ðt ¼ 0Þ is considered. In this case, an analytical solution is available as follows: Eigenequation: J0m ðkn rÞ ¼ 0;

m; n ¼ 0; 1; 2; 3; . . . :

Eigenmode: uða; yÞ ¼ Jm ðkn aÞeimy ;

04a4r; 04y42p; m; n ¼ 0; 1; 2; 3; . . . :

By collocating 10 nodes on the circular boundary, two results by using the single- and double-layer potential approaches are obtained. The exact eigenvalues of a circular cavity subject to the Neumann boundary condition are shown in Table 3. Figure 2(a) shows the minimum singular value versus k using the single-layer potential approach. The former four true eigenvalues are obtained as shown in Figrue 2(a) by considering the near-zero minimum singular values if only the single-layer potential method is chosen. Figure 2(b) shows the minimum singular value versus k using the double-layer potential approach. The true eigenvalues occur at the positions of zeros for J0m ðkn rÞ while the spurious eigenvalues occur at the positions of zeros for Jm ðkn rÞ if the double-layer approach is chosen. In order to extract out the true eigenvalues, we combine the ½LI  and ½M I  by using the SVD updating term. The minimum singular value versus k is shown in Figure 2(c), it is found that the dips occur only at the positions of true eigenvalues. In a similar way, we combine the ½U I  and ½LI  by using the SVD updating document in order to filter out the spurious eigenvalues. The minimum singular value versus k is shown in Figure 2(d), it is found that the dips occur only at the positions of spurious eigenvalues. The first three interior modes obtained by using the single- and double-layer potential approaches are shown in Figure 2(e) and Figure 2(f), and are compared well with the analytical modes in Figrue 2(g) for the nodal lines. It is found that the nodal line can rotate in case of the eigenvalue of multiplicity more than one. There are two independent eigenmodes with respect to the same eigenvalue. The two modes can be linearly combined by any constants and can differ by the phase lag. This is the reason why the one mode can be rotated to another mode. A similar explanation can also be found in Reference [26]. It is also found that the result of case 2 in Figure 2(f) does not match the analytical solution very well. The reason is that the amplitudes of the modes are very small since they are multiplied by a near-zero value. However, the nodal line can be identified. To demonstrate this point, the exact solutions for the mode using the present method are shown to have the near-zero

708

J. T. CHEN ET AL.

solutions in Table 4. The near-zero solutions are present if the double-layer approach was applied to solve the Neumann problem. Case 3: Square cavity (Dirichlet case). A square cavity with each side of a unit length ðL ¼ 1Þ subjected to the Dirichlet boundary condition ðu ¼ 0Þ is considered. In this case, an analytical solution is available as follows: Eigenvalues: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 kmn ¼ þ p; m; n ¼ 1; 2; 3; . . . : L L Eigenmode: sin

mpx npx sin ; m; n ¼ 1; 2; 3; . . . : L L

By collocating 12 nodes on the boundary, two results by using the single- and doublelayer potential approaches are obtained. The exact eigenvalues of a square cavity subject to the Dirichlet boundary condition are shown in Table 5. Figure 3(a) shows the minimum singular value versus k using the single-layer potential approach. The former four true eigenvalues are obtained as shown in Figure 3(a) by considering the near-zero minimum singular values if only the single-layer potential method is chosen. Figure 3(b) shows the minimum singular value versus k using the double-layer potential approach. The true qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eigenvalues occur at the positions of kmn ¼ ðm=LÞ2 þ ðn=LÞ2 while the spurious eigenvalues occur if the double-layer approach is chosen. In order to extract out the true eigenvalues, we combine the ½U I  and ½T I  by using the SVD updating term. The minimum singular value versus k is shown in Figure 3(c), it is found that the dips occur only at the positions of true eigenvalues. In a similar way, we combine the ½T I  and ½M I  by using the SVD updating document in order to filter out the spurious eigenvalues. The minimum singular value versus k is shown in Figure 3(d), it is found that the dips occur only at the positions of spurious eigenvalues. The first three interior modes using the single- and double-layer approaches are shown in Figure 3(e) and Figure 3(f) and are compared with analytical modes in Figure 3(g). For the eigenvalue of multiplicity more than one, some discrepancy for the nodal line is found since the eigensolution can be superimposed by two independent eigensolutions of the same eigenvalue. Case 4: Square cavity (Neumann case). A square cavity with each side of a unit length ðL ¼ 1Þ subjected to the Neumann boundary condition ðt ¼ 0Þ is considered. In this case, an analytical solution is available as follows: Eigenvalues: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r m2  n 2 kmn ¼ þ p; m; n ¼ 0; 1; 2; 3; . . . : L L Eigenmode: cos

mpx npx cos ; L L

m; n ¼ 0; 1; 2; 3; . . . :

By collocating 12 nodes on the boundary, two results by using the single- and doublelayer approaches are obtained. The exact eigenvalues of a square cavity subject to the Neumann boundary condition are shown in Table 5. Figure 4(a) shows the minimum singular value versus k using the single-layer potential approach. The former four true eigenvalues are obtained as shown in Figure 4(a) by considering the near-zero minimum singular values if only the single-layer potential method is chosen. Figure 4(b) shows the minimum singular value versus k using the double-layer potential approach. The true

Table 6 The true and spurious eigenvalues for circular and square cavities using the single- and double-layer potential approaches Boundary-value problem

True eigensolution

Spurious eigensolution

Neumann problem

True eigensolution

Spurious eigensolution

Circular cavity Single-layer potential approach

Double-layer potential approach

Jm ðkrÞ ¼ 0

Jm ðkrÞ ¼ 0

(Figure 1(a))

(Figure 1(b))

Jm ðkrÞ ¼ 0

J0m ðkrÞ ¼ 0

(Figure 1(a))

(Figure 1(b))

J0m ðkrÞ ¼ 0

J0m ðkrÞ ¼ 0

(Figure 2(a))

(Figure 2(b))

Jm ðkrÞ ¼ 0

J0m ðkrÞ ¼ 0

(Figure 2(a))

(Figure 2(b))

Square cavity Single-layer potential approach

kmn

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p ¼ þ L L

ðm; n ¼ 1; 2; 3; . . .Þ (Figure 3(a)) kmn ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p þ L L

ðm; n ¼ 1; 2; 3; . . .Þ (Figure 3(a)) kmn ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 þ p L L

ðm; n ¼ 0; 1; 2; 3; . . .Þ (Figure 4(a)) kmn

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p ¼ þ L L

kmn

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p ¼ þ L L

ðm; n ¼ 1; 2; 3; . . .Þ (Figure 3(b)) kmn ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p þ L L

m; n ¼ 0; 1; 2; 3; . . .Þ (Figure 3(b)) kmn ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 þ p L L

m; n ¼ 0; 1; 2; 3; . . .Þ (Figure 4(b)) kmn

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2  n 2 p ¼ þ L L

ðm; n ¼ 0; 1; 2; 3; . . .Þ (Figure 4(b))

709

ðm; n ¼ 1; 2; 3; . . .Þ (Figure 4(a))

Double-layer potential approach THE BOUNDARY COLLOCATION METHOD

Dirichlet problem

Eigensolution

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J. T. CHEN ET AL.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eigenvalues occur at the positions of kmn ¼ ðm=LÞ2 þ ðn=LÞ2 while the spurious eigenvalues occur if the double-layer approach is chosen. In order to extract out the true eigenvalues, we combine the ½LI  and ½M I  by using the SVD updating term. The minimum singular value versus k is shown in Figure 4(c), it is found that the dips occur only at the positions of true eigenvalues. In a similar way, we combine the ½U I  and ½LI  by using the SVD updating document in order to filter out the spurious eigenvalues. The minimum singular value versus k is shown in Figure 4(d), it is found that the dips occur only at the positions of spurious eigenvalues. The former three interior modes for the single- and double-layer potential approaches are shown in Figure 4(e) and Figure 4(f), and are compared with the analytical modes in Figure 4(g). For the eigenvalue of multiplicity more than one, some discrepancy for the nodal line is found since the eigensolution can be superimposed by two independent eigensolutions of the same eigenvalue.

9. CONCLUSIONS

We have developed a new meshless method by using the imaginary-part kernel. The imaginary part in the complex-valued fundamental solution was chosen as a radial basis function to approximate the solution. Although this method is very simple by using only two-point function for the influence matrices, it results in spurious eigensolutions as shown in Table 6. Two approaches, the SVD updating terms and updating documents in conjunction with dual formulation, were proposed to extract out the true eigensolutions and to filter out the spurious eigensolutions, respectively, as shown in Table 2. Both cases, circular and square cavities, subject to the Dirichlet and the Neumann boundary conditions, were demonstrated to check the validity of the meshless formulation. ACKNOWLEDGMENTS

Financial support from the National Science Council under Grant No. NSC-90-2211-E019-006 for National Taiwan Ocean University is gratefully acknowledged.

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