National Taiwan Ocean University Department of Harbor and River Engineering,
Keywords: Spurious eigenvalues; SVD; CHEEF; MRM.
In this paper, the eigensolutions are derived by using seven determinants of the direct searching approach. Seven determinants include the complex-valued, real and imaginary parts of determinant using the complex-valued kernel as well as the determinant by using the real-part, and imaginary-part kernels as well as the multiple reciprocity method (MRM). It is found that spurious eigensolutions of the real -part BEM match well with those of the MRM for the one-dimensional case.
The idea of the Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) in conjunction with the singular value decomposition (SVD) technique can be applied to suppress the occurrence of spurious eigenvalues.
Possible failure CHEEF points are also examined. Updatin g terms and updating documents are employed to extract out the true and spurious eigenvalues. Two and three dimensional cases are straight forward to be extended. Their spurious eigenvalues are also explored.
The application of eigenanalysis is gradually increasing for vibration and acoustics. The demand for eigenanalysis calls for an efficient and reliable method of computation for eigenvalues and eigenmodes. Over the last three decades since 1974, several boundary element formulations have been employed to solve the eigenproblems [1], e.g., determinant searching method, internal cell method, dual reciprocity method, particular integral method and multiple reciprocity method. In this paper, we will focus on the determinant searching method with emphasis on spurious eigenvalues due to the real-part BEM and the MRM.
Spurious and fictitious solutions stem from non-uniqueness problems. They appear in different aspects in computational mechanics. First of all, hourglass modes in the finite element method (FEM) using the reduced integration occur due to rank deficiency [2]. Also, loss of divergence-free constraint for the incompressible elasticity also results in spurious modes. In the other side of numerical solution for the differential equation using the finite difference method
(FDM), the spurious eigenvalue also appears due to discretization [3-5]. In the real-part BEM [6] or the
MRM formulation [7-12], spurious eigensolutions occur in solving eigenproblems. Even though the complex-valued kernel is adopted, the spurious eigensolution also occurs for the multiply-connected problem [13] as well as the appearance of fictitious frequency for the exterior acoustics [14]. In this paper, a
1 simple case of one-dimensional rod will be demonstrated to see how spurious eigensolutions occur and how they can be suppressed. Although the one-dimensional case is simple [15-19], it provides the insight to understand how the spurious eigenvalue behaves and how they can be suppressed from the education point of view.
In the literature review, we can find seven alternatives to solve eigenproblems using the direct-searching scheme. Tai and Shaw [20] employed the determinant of complex-valued BEM. De Mey [21] revisited this problem in 1976. Later, De Mey [22] proposed a simplified approach by using only the real-part or imaginary-part kernel where he found that spurious solutions were imbedded as well as the ill-posed matrix appeared. In a similar way of using the real-part kernel, Hutchinson [23-24] solved the free vibration of plate. Also, Yasko [25] as well as Duran et al. [26] employed the real-part kernel approach. It is interesting to find that Kang et al. [27] proposed an imaginary-part kernel approach using the collocation approach as commented by Chen et al. [28]. Yeih et al. [9] found that the MRM is nothing but the real-part BEM. This is the reason why spurious eigenvalues are inherent in the two methods. The chronology list of the literature survey is shown in Table 1. All the indicators in the determinant searching method will be employed to solve a simple one-dimensional problem in this paper.
In the recent years, the SVD technique has been applied to solve problems of continuum mechanics [29] and fictitious-frequency problems [14]. Two ideas,

Indicator 1 det[ C ]
Indicator 2
Abs{det[ C ]}
Indicator 3
Im{det[ C ]}
Indicator 4
Re{det[ C ]}
Indicator 5 det[ I ]
Indicator 6 det[ R ]
Indicator 7
MRM updating term and updating document [14], were successfully applied to extract the true and spurious solution, respectively. Also, the CHIEF [30] and CHEEF
[31-32] methods were employed to suppress the occurrence of fictitious frequency and spurious eigenvalue, respectively. Based on these successful experiences, the CHEEF concept will be employed to study the spurious eigenvalue of the one-dimensional where
2
( )
 dx
2
( )

0 , x

D , (1)
( ) is the axial displacement of the rod, k is the wave number, D is the domain of 0 < x < L .
The case of the mixed-type is considered in Fig. 1 as shown below. eigenproblem.
In this paper, eigenproblems of one-dimensional case will be explored by using seven indicators for the direct-searching scheme in details. Spurious modes in the u (0)=0 t ( L )=0
Figure 1 A rod subject to the mixed-type boundary condition. real-part BEM formulation will be examined through the
SVD technique and will be suppressed by using the
CHEEF idea. Although a one-dimensional case is studied analytically, two and three dimensional cases can be where

. By introducing the auxiliary dx system of the fundamental solution, we have straightforward extended by only changing the degenerate kernel.
The rest of this paper is organized as follows. In
¶ 2 ( , )
+
¶ x
2
2
( , ) = d ( x s ) ,
(2)
Section 2, we propose a dual formulation for the eigenproblem of a rod. In Section 3, Seven indicators for the direct-searching scheme are employed to solve eigenproblems. The occurrence of spurious eigenvalues are suppressed by using the SVD technique in where x , d is the Dirac-delta function, x is the field point, and s is the source point, ( , ) is a conjunction with the CHEEF idea in Section 4. Updating terms and updating documents is in Section 5. Section 6 extends the one-dimensional case to two and three dimensional cases. Finally, a conclusion is made.
with the following governing equation
Consider the eigenproblems of free vibration for a rod subject to boundary conditions as shown in Fig. 1 complex-valued fundamental solution as


1
2 ik e
 
, (3) and can be expressed in terms of degenerate kernel
 





2

1 ik
1
 iks e
 e
,
, x
 s x
 s
. (4)
2 ik
By employing the Green’s third identity and integration by parts, we can derive the boundary integral equations as
Table 1 Literature review for eigenproblems using the direct searching scheme in BEM.
1974 1976 1977 1985 1997 1999 2000 2001 Present
Tai and
De Mey
Shaw
Duran et al.
Tai and
Shaw
Tai and
Shaw
Tai and
Shaw
Tai and
Shaw
De Mey
De Mey
De Mey
Hutchinson and Wong
Chen and
Wong
Kang et al.
Yeih et al.
Yasko
Duran et al.
2





 x
( )

( , ) ( )

 x

0
,
T (0,

) (0) 1 ( ,

) ( )
.

U (0,

) (0)

( , ) ( )

0
(5)
0 < s < L .
Then we exchange x with s to each other of Eq. (5), and obtain
( )


( , ) ( )

( , ) ( )
 s

0
¶
, 0 < x < L . (6) where = .
¶ s
Conventionally, by collocating the field point x close to the boundary 0

and L

for Eq. (6), we have
1

T

(0, 0 ) u (0)

T L

( , 0 ) ( )
, (7)

U (0, 0 ) (0)

U L
 t L

0
(8)
By assembling two Eqs. (7) and (8) into a matrix form, we have



1

T (0, 0 )
T (0, L

) 1



T L L

)



 u (0) 
 

 
 
U (0, 0 ) ( , 0 )
 
U (0, L ) ( , )



 t (0)
  
  
.
(9)
Substituting the kernel functions into Eq. (9), we have










1
1
2
2 ik
2
1
2 e
1 e
 ikL ik
 ikL

1
2 e
 ikL
1
2

1
2 ik e

1
2 ik



  u (0)

  


 ikL

 
  t (0)
  
0
.
(10)
For the case of mixed-type with B.C. u (0)=0 and t ( L )=0,
Eq. (10) simplifies to




1
2
1 ik e
 ikL
2 ik

1
2 e
1
2
 ikL

  t (0)
 

0
  
. (11)
Owing to the introduction of degenerate kernels of Eq. (4),
BIE for the domain point can be expressed as
( )
 
( , ) ( )

( , ) ( )
 s

0
, x

, (12) and the null-field BIE is
0
 
( , ) ( )

( , ) ( )
 s

0 x
 
, 0 ]

[ L , ) ,
,
(13) where U and T kernels must be represented in a correct form by using the expression of degenerate kernels of Eq.
(4). The collocation point can locate on the real boundary for two Eqs. (12)-(13). Mathematically speaking, the domain of Eqs. (12)-(13) is a closed set instead of an open set in the conventional BEM formulation of Eq. (6) using the closed-form fundamental solution.
3
According to Eq. (11), we use seven determinants to obtain the eigenvalue. Seven indicators to find the eigenvalue by using determinants in the direct searching scheme are shown below:
Indicator 1. Complex determinant using the complex-valued kernel BEM
Indicator 2. Absolute value of determinant using the complex-valued kernel BEM
Indicator 3. Imaginary-part of determinant using the complex-valued kernel BEM
Indicator 4. Real-part of determinant using the complex-valued kernel BEM
Indicator 5. Determinant using the imaginary-part kernel
BEM
Indicator 6. Determinant using the real-part kernel BEM
Indicator 7. Determinant using the MRM[7]
Based on the seven indicators, the true and spurious eigensolutions and the corresponding boundary eigenvectors are shown in Table 2. The first column in the
Table 2 denotes the Indicator for the direct-searching determinant. The second column shows the rank of the influence matrix for the true and spurious cases. True and spurious eigenvalues are found in the third and sixth columns and their corresponding boundary eigenvectors are listed in the fourth and seventh columns, respectively.
Then their analytical solutions are listed in the fifth (true) and eighth (spurious) columns. The last column shows the treatment for the spurious eigenvalue.
According to the Table 2, we find that the Indicators 4 to 7 also results in spurious eigenvalues. Since the MRM is nothing but the real-part BEM [7-12], Indicators 6 and
7 result in the same spurious eigenvalue as shown in
Table 2. We employ the CHEEF concept to suppress the occurrence of spurious eigenvalues as elaborated on later in the next section.
Because the real-part BEM lacks for the constraining condition of imaginary-part information, spurious eigenvalues appear in the case of mixed-type by using the
Indicator 6 in Table 2. In order to filter out spurious eigenvalues, we employ the CHEEF concept. A point, c , in the complementary domain, which is called a CHEEF point as shown in Fig. 2 and 3 is selected to provide a constraint. Therefore we have the null-field equation by using the Indicator 6. The constraints of UT equation for the CHEEF point are
0

U
R c t

T L c u L c
 
, 0)
 
,
,
(14) where U
R
(0, c ) and T
R
( L , c ) are the real-parts of U (0, c ) and
T ( L , c ) respectively. By assembling Eqs. (11) and (14) into the matrix form, and by selecting a right ( c > L ) CHEEF point, we have the UT equation:

1
2
3
4
＊
[
Rank of
(1
，
2)
Table 2 Eigensolutions for the case of mixed-type: u (0)=0 and t ( L )=0 using seven approaches.
( ), ( )
(T
，
S)
(1
，
NA)
(1
，
NA)
(1
，
NA)
]
True
Eigenvalue
( )
(2 n

1)

2 L
(2 n

1)

2 L
(2 n

1)

2 L
(2 n

1)

2 L
True boundary
Eigenvector п о t u
(0)
(1) п ю


(2 n

1)

2 L n

1


(2 n

1)

2 L n

1


(2 n

1)

2 L n

1


(2 n

1)

2 L n

1 sin sin sin sin
True analytical
Solution
(2 n

1)
(2 n

1)
(2 t
( ) n
2
2

2
2
L
L
1)
L
(2 n

1)
L



 x x x x
Spurious
Eigenvalue
( k s
)
NA
NA
NA n

L
Spurious boundary
Eigenvector п о t u
(0)
(1) п ю
NA
NA
NA
×
Rank=2
Spurious analytical
Solution s
( )
NA
NA
NA
×
Rank=2
Treatment
NA
NA
NA
Taking the absolute value
5
＊
(1
，
1)
(2 n

1)

2 L


(2 n

1)

2 L n

1
×
Null equation n

L
×
Null CHEEF equation
6
＊
7
＊
(1
，
1)
(1
，
1)
(2 n

1)
 x









0 sin(
2
2 k kL k sin( ck )
)
＊
Indicators 4,5,6 and 7 result in spurious eigensolutions.
 cos(
1
2 kL )

2 cos[( c

) ]


2





 t (0)
( L c ) .

0
 
,
(15) we have where
K
   
T , (17)
[ ] is the influence matrix in Eqs. (15) and (16),
[ ] is a left unitary matrix constructed by the left singular vectors ( diagonal line as
  
1 2 3
), which has singular values
[ ] is a diagonal matrix

1
and

2 allocated in a
If the CHEEF point c is selected in the left side ( c <0). We have the UT equation:








 0 sin( kL )
2 k
2 k
 cos(
1
2
2
2 kL ) sin( ck ) cos[( c

) ]





 t (0)

,
(16)
-
∞
u(0)=0
0
t(L)=0
L c
∞
Figure 2 A CHEEF point , c , in the right side.
(
(2 n

1)

2 L
(2 n

1)

2 L c 0) .


(2 n

1)

2 L n

1


(2 n

1)

2 L n

1 sin sin
(2 n

1)
It is interesting to find that the left right
( c 0) and
( L c ) CHEEF points yield the same row but negative to each other due to the CHEEF point as shown in Eqs. (15) and (16). Based on the SVD decomposition,
2
2
L
L
 x
-
∞ n n c

L

L
u(0)=0
0
1
  sin( sin( n

L n

L x x )
) CHEEF
CHEEF
t(L)=0
L
∞
Figure 3 A CHEEF point , c , in the left side.
4

in which

2
 
1
[ ]
 


0
0
2

0
0
1
,

2
and




, (18)

are the functions of c , k
1 and L , and
 T
is the matrix transpose of a right unitary matrix constructed by the right singular vectors
(
 
). As we can see in Eq. (18), there exist at most 2
1
,
2 nonzero singular values. The rank of influence matrices are equal to 2 at most. Since the boundary eigenvector is nontrivial, the rank of influence matrices must be one.
Therefore the minimum singular value

1
must be zero.
We show zero contour figures of mixed-type for

1 versus ( k , c ) by using the UT equation as shown in Figure
4, where the length L of a rod is 1. The k axis is the wave number and the c axis is the location of CHEEF point in
Figure 3. It is found that no matter what value c is,

is
1 always zero for the true eigenvalue of n

N . For the spurious eigenvalue of failure CHEEF points appear when k
S k
T

(2 n

1)

,
2 L
 n

L
, n

N , k
S c = n

, n

N , since the CHEEF constraint is trivial which can not filter out spurious eigenvalues. True and spurious modes by using the UT equation are shown in Figs. 5 and 6. It indicates that a true eigenvalue results in a true mode in the domain and null field in the complementary domain, while the spurious case has a nontrivial field in the complementary domain. The location of zero response in the complementary domain happens to be that of the failure CHEEF point.
Now we differentiate Eq. (6) with respect to the field uT x
1
3
2.8
2.6
2.4
c failure CHEEF failure CHEEF point
Spurious failure CHEEF point
Spurious failure CHEEF
2.2
True True True point point
(2,0) k
2
3
2
2 5
2
Figure 4 Zero contour plot of the minimum singular value versus (k,c) by using the UT equation. (zero nodal line only). point x , the hypersingular integral equation ( LM equation) is
( )
 
( , ) ( )

( , ) ( )
 s

0
, 0 < x < L . (19)
¶ ¶ where = and = .
¶ x
By collocating of the field point x close to the boundary
0

and L

, we have
 
M

1 L (0, 0 ) t (0)

L L
 t L

0
, (20)
M (0,

) (0)

( ,

) ( )

L (0,

) (0) 1 L L L

) t L

0
, (21)
By assembling two Eqs. (20) and (21) into a matrix form and substituting the kernel functions, we have the LM equation



 ik ik e
2
 ikL
2
 




1
2
1
2 e
 ikL
 ik

2
 ik
1
2 e
2
 ikL



  u (0)

 e
 ikL
1
2



 
 t (0)   
0
, (22) true mode
-3 null field
0.5
null field
2
3
4
3
2
8
3
3
-0.5
-1 true node
Figure 5 True mode of case 3 by using the UT equation ( k
T uT x
=1.5π, L =3). nontrivial field Spurious mode nontrivial field
1
0.5
6 x x
-3
12
5
9
5
6
5
3
5
-0.5
3
18
5
24
5
27
5
6
-1 failure CHEEF point
Figure 6 Spurious mode of case 3 by using the UT equation ( k
S
=5π/3, L =3).
5

Now we arrange Eqs. (10) and (22) by using the indicator
6, we have

 t (0) u L



 u (0)

, (23)

 t (0) u L
 u (0)
, (24) where [ ] , [ ] , [ ] and D are influence matrices after rearrangement. In order to extract out true and spurious eigenvalues, we utilize updating terms and updating documents, respectively. Therefore we combine
[ ] with [ ] and [ ] with B . which can be written as





0 sin( kL )
2 k

A B

  






1
2 cos( kL )
2






0 sin( kL )
2 k
 cos( kL )
2
1
2

1
2 cos( kL )
2

 cos(
1
2
0
2 kL ) k sin( kL )
2











, (25)
 sin( kL )
2
0 k 



.
(26)
By employing SVD for
A
 
and

A B
 respectively, the minimum singular values versus k can be obtained.
The figures of minimum singular values versus k are shown in Figs. 6 and 7 for the unit length rod. The k value at drops in Figs 7 and 8 shows the true and spurious eigenvalues, respectively.
Following the simple example of one-dimensional rod, it is straight forward to extend two and three dimensional cases as shown in Figs. 9 and 10. The eigenequations by using the Indicators 5 and 6 for the two and three dimensional cases are shown in Tables 3 and 4.
By changing the degenerate kernels in Eq. (4) to the two-dimensional circular case, we have п пп i =
p
2 i е
Ґ m = 0 e m
J m
( k r ) H (1) m
( kR )
= нп пп U e ( , ) =
q
f )), R і r
, (27)
2 p i е
Ґ m = 0 e m
H (1) m
( k r ) J m
( kR ) where x по
 and s

( , ) q f )), > R
, J m and H
(1) m are the m th order cylindrical Bessel function and the cylindrical Hankel function of the first kind, respectively,
0.7
min
0.6
0.5
0.4
0.3
0.2
0.1
k
2
3
2
2
5
2
3
7
2
4 9
2
5
Figure 7 Extraction of true eigenvalues using SVD updating terms. min
0.6
0.5
0.4
0.3
0.2
0.1
k
3 2 5 3 7 4 9 5
2 2 2 2 2
Figure 8 Extraction of spurious eigenvalues using SVD updating terms. a
Figure 9 Two-dimensional case (circle membrane). a
Figure 10 Three-dimensional case (spherical cavity). the superscripts “ i ” and “ e ” denote the interior and exterior cases for the kernel expressions, respectively, and e m
is the Neumann factor e m
= м по
1,
2, m = m
=
0
1, 2, L
Ґ
, (28)
6

Table 3 Occurring mechanism of 2-D true and spurious eigenequations by using the real-part and imaginary-part BEM.
Dirichlet
UT
LM
(imaginary-part)
(imaginary-part)
UT
LM
(real-part)
(real-part)
{
{
J
J
J
J
} [
} [
J
J
Y
Y
]
]
=
=
0
0
0
0
Neumann
{ J ў
} [ J ] = 0
{ J ў
} [ J 0
{ J ў
} [ Y ] = 0
{ J ў
} [ Y 0
where H = J + iY
.
Table 4 Occurring mechanism of 3-D true and spurious eigenequations by using the real-part and
imaginary-part BEM.
UT (imaginary-part)
LM (imaginary-part)
Dirichlet j j = 0
{ j } [ j ] 0
Neumann
{ } [ j ] = 0
{ j ў
} [ j 0
UT (real-part)
LM (real-part) j y = 0
{ j } [ y ] 0
{ } [ y ] = 0
{ j ў
} [ y 0 where h = j + iy , and the equations inside { } and [ ] denote the true and spurious eigenequations, respectively.
For the three-dimensional spherical case, we have









 U i  ik
 



U e  ik





 n



0

 n

0 m

2 n

1
 n 

0
 m
P n m

 n
 m n
 m


!
!
cos
 cos
 
P n m
 m
 
 cos

 j n
  h n
   
,
 
,

2 n

1
 m n 

0
 m
P n m

 n
 m n
 m


!
!
cos
 cos
 
P n m m
 
 cos

 j n
  h n
 
,
 
,
,
(29) where x
 and s

( , , ) , j and n h n
  are the n th order spherical Bessel function of the first kind and the n th order spherical Hankel function of the second kind, respectively, and P m is the associated n
Lengendre polynomial. It is interesting to find that J ,
Y , J ў
, Y ў
and j , y , j ў
, y ў
appear for the two and three dimensional cases of Tables 3(a) and (b), once the real and imaginary-part BEM are employed. [33-34]
A simple example of one dimensional eigenproblem was demonstrated to show that seven indicators of determinant by using the direct searching scheme can obtain the possible solution. Spurious eigensolutions in the real-part BEM were studied analytically and were compared with those of MRM. The CHEEF idea was applied to filter out spurious eigenvalues. Possible failure points were also examined. SVD updating term and document were employed to extract out true and spurious eigenvalues, respectively. Extensions to two and three dimensional cases were also made. The sequence of one-dimensional example gives an insight to understand the mechanism of spurious eigensolutions.
The financial support (July to October, 2007) from the
National Taiwan University to the undergraduate student
(JWL) is gratefully acknowledged.
[1] A. Ali, C. Rajakumar and SM. Yunus, “Advances in acoustic eigenvalue analysis using boundary element method,” Comp .
Stru ., 56(5), pp. 837-847, (1995).
[2] J.R. Winkler and J.B. Davies, “Elimination of spurious modes in finite element analysis,” J .
Comp.
Phy ., vol. 56, pp. 1-14, (1984).
[3] M.D. Greenberg, Advanced engineering mathematics , Second edition. Prentice-Hall: (1998).
[4] H. Fujiwara, “High-accurate numerical computation with multiple-precision arithmetic and spectral method,” Taiwan-Japan Joint Workshop on Inverse
Problem ., Institute of Mathematics, Academia
Sinica, Nankang, Taipei (2007).
[5] S. Zhao, “On the spurious solutions in the high-order finite difference methods for eigenvalue problems,”
Comput. Methods Appl. Mech. Engrg ., (2007) in press.
[6] S.R. Kuo, J.T. Chen and C.X Huang, “Analytical study and numerical experiments for true and
7

spurious eigensolutions of a circular cavity using the real-part dual BEM,” Int. J. Numer. Meth.
Engng ., ,48, 1401-1422, (2000).
[7] J.T. Chen and F.C. Wong, “Analytical derivations for one-dimensional eigenproblems using dual boundary element method and multiple reciprocity method,”
Engng .
Anal. Bound. Elem ., 20, 25-33, (1997).
[8] J.T. Chen and F.C. Wong, “Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition,” J. Sound. Vibr .,
217(1), 75-95, (1998).
[9] W. Yeih, J.T. Chen, K.H. Chen and F.C.Wong, “A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation,” Adv. Engng. Soft ., Vol. 29, No. 1, pp. 1-6,
(1998).
[10] W. Yeih, J.T. Chen and C.M. Chang, “Applications of dual MRM for determining the natural frequencies and natural modes of an Euler-Bernoulli beam using the singular value decomposition method,”
Engng.
Anal. Bound. Elem ., 23, 339-360, (1999).
[11] W. Yeih, J.R. Chang, C.M. Chang and J.T. Chen,
“Applications of dual MRM for determining the natural frequencies and natural modes of a rod using the singular value decomposition method,” Adv.
Engng. Soft ., 30, 459-468, (1999).
[12] J.T. Chen, S.R. Kuo, I.L. Chung and C.X. Huang,
“Study on the true and spurious eigensolutions of two-dimentional cavities using the multiple reciprocity method,” Engng. Anal. Bound. Elem ., 27,
655-670, (2003).
[13] J.T. Chen, L.W. Liu and H.K. Hong, “Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply-connected problem,” Proc. Roy. Soc .,
London Series A, 459, 1891-1925, (2003).
[14] J.T. Chen, I.L. Chen and K.H. Chen, “Treatment of rank deficiency in acoustics using SVD,” J. Comp.
Acous ., Vol. 14, No. 2, 157-183, (2006).
[15] F. Hartmann, Introduction to boundary elements-theory and applications . Berlin: Springer:
(1989).
[16] P.K. Banerjee, and R. Butterfield, Boundary element methods in engineering science . London:
McGraw-Hill: (1981).
[17] J.T. Chen, K.S. Chou and C.C. Hsieh, “Derivation of stiffness and flexibility for rods and beams by using dual integral equations,” Engng. Anal. Bound. Elem .,
(2007) in press.
[18] C.P. Providakis and D.E. Beskos, “Dynamic analysis of beams by the boundary element method,” Comp.
Stru ., 22(6), 957-964, (1986).
[19] E.J. Sapountzakis and G.C. Tsiatas,
“Flexural-torsional vibrations of beams by BEM,”
Comput. Mech ., 39, 409-417, (2007).
[20] G.R.C Tai and R.P. Shaw, “Helmholtz-equation eigenvalues and eigenmodes for arbitrary domains,”
J. Acoust. Soc. Am ., September Vol. 56, No.3,(1974).
[21] G. De Mey, “Calculation of eigenvalues of the
Helmholtz equation by an integral equation,” Int. J.
Numer. Meth. Engng ., Vol.10, 59-66, (1976).
[22] G. De Mey, “A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation,” Int. J. Numer. Meth. Engng ., Vol.11,
1340-1342, (1977).
[23] J.R. Hutchinson, “Boundary method for time dependent problem,” Proc. 5th Engineering
Mechanics Division Conf., ASCE ., WY , pp.136-139,
(1984).
[24] J.R Hutchinson and G.K.K Wong, “The boundary element method for plate vibrations,” Proc. ASCE
7th Int. Conf. Elec. Comp ., pp.297-311, (1979).
[25] M. Yasko, “BEM with the real-valued fundamental solutions for the Helmholtz equation,” Proc. 7th Int.
Congress of Sound. Vibr ., Germany, 2037-2044,
(2000).
[26] M. Duran, M. Miguez and J.C. Nedelec, “Numerical stability in the calculation of eigenfrequencies using integral equations,” J. Comp. Appl. Math ., 130,
323-336, (2001).
[27] S.W. Kang, J.M. Lee and Y.J Kang, “Vibration analysis of arbitrarily shaped membranes using non-dimensional dynamic influence function,” J.
Sound. Vibr ., 234(1), 455-470, (1999).
[28] J.T. Chen, M.H. Chang, I.L. Chung and Y.C. Cheng,
“Comment on “Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point matching”,” J. Acoust. Soc. Am ., 111 (1), Pt.1,
Jan, (2002).
[29] J.T. Chen, C.F. Lee and S.Y. Lin, “A new point of view for the polar decomposition using singular value decomposition,” Int. J. Comp. Numer. Anal .,
2(3), 257-264, Appl 2002.
[30]I.L. Chen, J.T. Chen and M.T. Liang, “Analytical study and numerical experiments for radiation and scattering problems using the CHIEF method,” J.
Sound. Vibr ., Vol. 248, No.5, pp.809-828, (2001).
[31] J.T. Chen, I.L. Chen, S.R. Kuo and M.T. Liang, “A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method,” J.
Acoust. Soc. Am ., 109 (3), March (2001).
[32]W.M. Lee and J.T. Chen, “Analytical study and numerical experiments of true and spurious eigensolutions of free vibration of circular plates using real-part of BIEM,” Engng. Anal. Bound. Elem .,
(2007)(Accepted).
[33]J.T. Chen and H.K. Hong, Boundary Element Method .
Taipei: New World Press, (1992). (in Chinese)
[34]J.T. Chen, M.H. Chang, K.H. Chen and I.L. Chen,
“Boundary collocation method for acoustic eigenanalysis of three dimensional cavities using radial basis function,” Comp. Mech ., Vol.29, No.4-5, pp.392-408, (2002).
URL: http://msvlab.hre.ntou.edu.tw/
8