On the spurious eigensolutions for the real -part boundary
element method
Adviser:Jeng-tzong Chen professor
Jia-Wei Lee
National Taiwan Ocean University Department of Harbor and River Engineering,
Keywords: Spurious eigenvalues; SVD; CHEEF; MRM.
ABSTRACT
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. Updating 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.
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,
INTRODUCTION
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
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
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
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
straightforward extended by only changing the
degenerate kernel.
The rest of this paper is organized as follows. In
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
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.
d 2 u ( x)
(1)
 k 2 u ( x)  0 , x  D ,
dx2
where u ( x) 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.
t(L)=0
u(0)=0
Figure 1 A rod subject to the mixed-type boundary
condition.
du ( x)
where t ( x) 
. By introducing the auxiliary
dx
system of the fundamental solution, we have
¶ 2U ( x, s)
+ k 2U ( x, s) = d( x - s) ,
(2)
¶ x2
- ¥ < x< ¥ ,
where d is the Dirac-delta function, x is the field
point, and
s
is the source point, U ( x, s)
is a
complex-valued fundamental solution as
1 ik x  s
e
,
(3)
2ik
and can be expressed in terms of degenerate kernel
 1 ikx iks
 2ik e e , x  s
DUAL FORMULATION FOR
U ( x, s )  
.
(4)
ONE-DIMENSIONAL EIGENPROBLEM
 1 e ikx eiks , x  s
 2ik
Consider the eigenproblems of free vibration for a
rod subject to boundary conditions as shown in Fig. 1
By employing the Green’s third identity and integration
with the following governing equation
by parts, we can derive the boundary integral equations
as
Table 1 Literature review for eigenproblems using the direct searching scheme in BEM.
1974
Indicator 1
det[ C ]
1976
1977
U ( x, s ) 
1985
1997
1999
2000
Tai and
2001
Duran
De Mey
Shaw
et al.
Indicator 2
De Mey
Abs{det[C]}
Indicator 3
Tai and
Im{det[C]}
Shaw
Indicator 4
Tai and
Re{det[C]}
Shaw
Indicator 5
Tai and
Kang
De Mey
det[ I ]
Shaw
Indicator 6
Tai and
et al.
Hutchinson
Duran
De Mey
det[ R ]
Shaw
Yasko
and Wong
et al.
Indicator 7
Chen and
Yeih
MRM
Wong
et al.
2
Present
xL
DIRECT SEARCHING SCHEME FOR THE
EIGENVALUES USING SEVEN
INDICATORS
 U ( x, s )

u (s)  
u ( x)  U ( x, s)t ( x) 
,
(5)
 x
 x 0
0< s< L.
Then we exchange x with s to each other of Eq. (5), and
obtain
u ( x)  T (s, x)u( s)  U ( s, x)t ( s)  s 0 , 0 < x < L .
sL
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.
(6)
¶ U ( s, x )
.
¶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 )u ( L)
,
(7)
U (0, 0 )t (0)  U ( L, 0 )t ( L)  0
where T ( s, x) =
T (0, L )u (0)  1  T ( L, L )  u ( L)
.
(8)
U (0, L )t (0)  U ( L, L )t ( L)  0
By assembling two Eqs. (7) and (8) into a matrix form,
we have
1  T (0, 0 ) T ( L, 0 )   u (0) 



 
 T (0, L ) 1  T ( L, L )  u ( L) 
(9)
 U (0, 0 ) U ( L, 0 )   t (0)  0

   .

 
 U (0, L ) U ( L, L )  t ( L)  0
Substituting the kernel functions into Eq. (9), we have
1
 1

 e ikL 
 2
 u (0) 
2



1  u ( L) 
  1 eikL
 2
2 
(10)
1 ikL 
 1

e
 2ik
  t (0)  0 
2ik


   .
1  t ( L)  0 
 1 eikL

 2ik
2ik 
For the case of mixed-type with B.C. u(0)=0 and t(L)=0,
Eq. (10) simplifies to
1
 1

 e  ikL 
 2ik
 t (0)  0 
2


(11)
   .
1  u ( L)  0 
 1 e ikL
 2ik
2 
Owing to the introduction of degenerate kernels of Eq. (4),
BIE for the domain point can be expressed as
u ( x)  T (s, x)u( s)  U ( s, x)t ( s)  s 0 , x  0, L ,
sL
FILTERING OUT SPURIOUS
EIGENVALUES IN THE REAL-PART BEM
BY USING THE CHEEF CONCEPT
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 (0, c)t (0)  TR ( L, c)u( L) ,
(14)
c  (,0)  ( L, ) ,
where UR(0,c) and TR(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:
(12)
and the null-field BIE is
0  T (s, x)u (s)  U (s, x)t (s)  s 0 ,
sL
(13)
x  (,0]  [ L, ) ,
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
1
2
3
4
5
＊
＊
Table 2 Eigensolutions for the case of mixed-type:u(0)=0 and t(L)=0 using seven approaches.
True
Spurious
Rank
True
Spurious
boundary
boundary
True
Spurious
of
analytical
analytical
Eigenvector
Eigenvector
Eigenvalue
Eigenvalue
Solution
Solution
[U (k ), T (k )]
ïìï t (0)ïüï
ïìï t (0)ïüï
(kt )
(ks )
ut ( x)
us ( x )
í
ý
í
ý
(T，S)
ïîï u (1)ïþï
ïîï u (1)ïþï
 (2n  1) 


 (2n  1) x 
(2n  1)
(1，NA)
NA
NA
NA

 2L
 sin 

2L
2L


 (1) n 1 


Treatment
NA
(1，NA)
(2n  1)

2L
 (2n  1) 


 2L

 (1) n 1 


 (2n  1) x 
sin 

2L


NA
NA
NA
NA
(1，NA)
(2n  1)

2L
 (2n  1) 


 2L

 (1) n 1 


 (2n  1) x 
sin 

2L


NA
NA
NA
NA
(1，2)
(2n  1)

2L
 (2n  1) 


 2L

 (1) n 1 


 (2n  1) x 
sin 

2L


n
L
×
×
Rank=2
Rank=2
(2n  1)

2L
 (2n  1) 


 2L

 (1) n 1 


×
Null equation
n
L
0 
 
1 
 (2n  1) 


 2L

 (1) n 1 


 (2n  1) x 
sin 

2L


n
L
1 
 
0 
sin(
n
x)
L
CHEEF
 (2n  1) 


 2L

 (1) n 1 


 (2n  1) x 
sin 

2L


n
L
1 
 
0 
sin(
n
x)
L
CHEEF
(1，1)
6
＊
(1，1)
(2n  1)

2L
7
＊
(1，1)
(2n  1)

2L
Taking
the absolute
value
×
Null
CHEEF
equation
＊Indicators 4,5,6 and 7 result in spurious eigensolutions.
we have
cos(kL) 

0



(17)
[ K ]  [][][]T ,
2
0 


t
(0)


sin(
kL
)
1
where
is
the
influence
matrix
in
Eqs.
(15)
and
(16),
[
K
]





  0  ,


u ( L)   
2k
2
(15)
[] is a left unitary matrix constructed by the left


0 
sin(
ck
)
cos[(
c

L
)
k
]
singular vectors ( 1 , 2 , 3 ), [] is a diagonal matrix



 2k

2
which has singular values  1 and  2 allocated in a
( L  c  ) .
diagonal line as
If the CHEEF point c is selected in the left side (c<0). We
have the UT equation:
cos(kL) 

t(L)=0
u(0)=0
0



2
0 


c ∞
-∞ 0
L
1
  sin( kL)
  t (0)   0  ,

  


u ( L)   
2k
2
(16)
Figure 2 A CHEEF point , c, in the right side.


0 
sin(
ck
)
cos[(
c

L
)
k
]


2k
2


(  c  0) .
It is interesting to find that the left (  c  0) and
right ( 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,
u(0)=0
-∞ c
0
t(L)=0
L ∞
Figure 3 A CHEEF point , c, in the left side.
4
 2 0 
[]   0  1  ,
(18)
 0 0 
in which  2  1 ,  2 and  1 are the functions of c, k
c
3
2.6
and L, and []T is the matrix transpose of a right
unitary matrix constructed by the right singular vectors
(  1 , 2 ). As we can see in Eq. (18), there exist at most 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.
failure CHEEF
point
point
Spurious
Spurious
2.4
failure CHEEF
2.2
True
(2,0)
failure CHEEF
True
point
True
point
3
2
2
5
2
2
k
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
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,  1 is
always zero for the true eigenvalue of kT 
failure CHEEF
2.8
t ( x)   M (s, x)u(s)  L( s, x)t ( s)  s 0 , 0 < x < L .
sL
(19)
¶ U ( s, x )
¶ U ( s, x )
and M ( s, x) =
.
¶x
¶ x¶ s
By collocating of the field point x close to the boundary
0 and L , we have
M (0, 0 )u (0)  M ( L, 0  )u ( L)
,
(20)
 1  L(0, 0 )  t (0)  L( L, 0  )t ( L)  0
(2n  1)
,
2L
where
n
, n N ,
L
failure CHEEF points appear when 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.
n  N . For the spurious eigenvalue of kS 
L ( s, x ) =
M (0, L )u (0)  M ( L, L )u ( L)
 L(0, L )t (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 ikL 
 2
 u (0) 
2



ik
ik
UPDATING TERMS AND UPDATING
 e ikL
 u ( L) 

 2
2 
DOCUMENTS TO EXTRACT OUT TRUE
,
(22)
1 ikL 
 1
AND SPURIOUS EIGENVALUES
 e 

 t (0)  0 
Now we differentiate Eq. (6) with respect to the field
2
 2

 
1  t ( L)  0 
  1 e ikL
 2
2 
true mode
uT x
1
0.5
null field
null field
x
2
3
-3
4
3
2
8
3
3
6
-0.5
-1
true node
Figure 5 True mode of case 3 by using the UT equation (kT=1.5π, L=3).
uT x
nontrivial field
1
Spurious mode
nontrivial field
0.5
x
-3
12
5
9
5
6
5
3
5
3
18
5
21
5
24
5
27
5
-0.5
-1
failure CHEEF point
Figure 6 Spurious mode of case 3 by using the UT equation (kS=5π/3, L=3).
5
6
Now we arrange Eqs. (10) and (22) by using the indicator
6, we have
 t (0) 
u (0) 
 A     B    ,
(23)
u
(
L
)


 t ( L) 
min
0.7
0.6
0.5
 t (0) 
u (0) 
(24)
   D 
,
u ( L) 
 t ( L) 
where [ A] , [ B] , [C ] 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
[ A] with [C ] and [ A] with [ B] . which can be
written as
cos(kL) 

0



2


1
  sin( kL)


 A 
2k
2
(25)
,
C   
1
k sin(kL) 
  



2
2
 cos(kL)



0
2


 A B 
0.4
C  
0.3
0.2
0.1
2
3
2
2
5
2
7
2
3
4
9
2
5
k
Figure 7 Extraction of true eigenvalues using SVD
updating terms.
min
0.6
0.5
0.4
0.3
0.2
0.1
1
sin( kL) 

0



(26)
2
2k 

.
cos( kL)
  sin(kL)

0


2k
2
 A
By employing SVD for   and  A B respectively,
C 
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.
cos(kL)

2
1
2
2
3
2
2
5
2
7
2
3
4
9
2
5
k
Figure 8 Extraction of spurious eigenvalues using SVD
updating terms.
a
EXTENSIONS TO 2-D AND 3-D
EIGENPROBLEMS
Figure 9 Two-dimensional case (circle membrane).
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
¥
ìï i
ïï U ( s, x) = - p i å em J m (k r ) H m(1) (kR )
ïï
2 m= 0
ïï
cos( m(q - f )), R ³ r
ïï
, (27)
U ( s, x ) = í
¥
ïï e
- pi
(1)
ïï U ( s, x) =
å em H m (k r ) J m (kR)
2 m= 0
ïï
ïï
cos(m(q - f )), r > R
ïî
where x  (  ,  ) and s  ( R, ) , J m and H m(1) are
the mth order cylindrical Bessel function and the
cylindrical Hankel function of the first kind, respectively,
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
em is the Neumann factor
ìï 1, m = 0
em = ïí
,
(28)
ïïî 2, m = 1, 2, L ¥
6
Table 3 Occurring mechanism of 2-D true and spurious eigenequations by using the real-part and
imaginary-part BEM.
Dirichlet
Neumann
UT(imaginary-part)
{ J }[ J ] = 0
{ J ¢}[ J ] = 0
LM(imaginary-part)
{ J }[ J ¢] = 0
{ J ¢}[ J ¢] = 0
UT(real-part)
{ J }[Y ] = 0
{ J ¢}[Y ] = 0
LM(real-part)
{ J }[Y ¢] = 0
{ J ¢}[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.
Dirichlet
Neumann
UT(imaginary-part)
{ j}[ j] = 0
{ j ¢}[ j ] = 0
LM(imaginary-part)
{ j }[ j ¢] = 0
{ j ¢}[ j ¢] = 0
UT(real-part)
{ j}[ y] = 0
{ j ¢}[ y ] = 0
LM(real-part)
{ j }[ y ¢] = 0
{ j ¢}[ 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
the real-part BEM were studied analytically and were

compared with those of MRM. The CHEEF idea was
 i
U  ik  2n  1
applied to filter out spurious eigenvalues. Possible failure

n 0
points were also examined. SVD updating term and

n
document were employed to extract out true and spurious
n

m
!



m
cos  m    
eigenvalues, respectively. Extensions to two and three



 n  m !
m 0

dimensional cases were also made. The sequence of

one-dimensional example gives an insight to understand
Pnm  cos   Pnm cos  jn  k  

the mechanism of spurious eigensolutions.

2
hn   k   ,    ,

,
U ( s, x )  
(29)

ACKNOWLEDGEMENTS
 e
The financial support (July to October, 2007) from the
U  ik  2n  1
n 0

National Taiwan University to the undergraduate student
n

n  m !
(JWL) is gratefully acknowledged.


m
cos  m    



 n  m !
m 0

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
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URL: http://msvlab.hre.ntou.edu.tw/
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