Study on the degenerate
problems in BEM
邊界元素法中退化問題之探討
指導教授：陳正宗 教授
研究生：林書睿
國立臺灣海洋大學河海工程研究所結構組
碩士班畢業論文口試
時間: 5/21, 2002
地點: 河工二館307室
1
Motivation
Four pitfalls in BEM


Why numerical instability occurs in BEM ?
(1) degenerate scale
(2) degenerate boundary
(3) fictitious frequency
Why spurious eigenvalues appear ?
(4) true and spurious eigenvalues
Mathematical essence—rank deficiency
(How to deal with ?)
2
Outlines





Degenerate scale for torsion bar
problems
Degenerate boundary problems
True and spurious eigensolution
for interior eigenproblem
Fictitious frequency for exterior
acoustics
Conclusions and further research
3
Four pitfalls in BEM
1.Degenerate scale for torsion bar
problems
2.Degenerate boundary problems
3.True and spurious eigensolution for
interior eigenproblem
4.Fictitious frequency for exterior
acoustics
4
The degenerate scale for torsion bar using BEM
T
ad

Error (%)
of 125
torsional
rigidity
5
0
ad
a
Previous approach : Try and error on a
Present approach : Only one trial
5
Determination of the degenerate scale by trial and error
1.20
0.12
0.15
0.80
0.08
0.10
a
1
1
1
0.04
0.40
2
0.05
2.35
1.31
0.00
0.00
0.50

1.00
Fig.2-13 The minimum singular value 1 of [U] versus semiaxes 
for the interior potential problem with an elliptic domain.
0.60
0.80
1.00
1.20
1.40
a
Fig.2-19 The minimum singular value  1 of [U] versus radius a
for the interior potential problem with a semicircular domain.
0.00
2.10
2.20
2.30
2.40
2.50
a
Fig.2-20 The minimum singular value  1 of [U] versus parameter a
for the interior potential problem with a sector domain.
Direct searching for the degenerate scale
Trial and error---detecting zero singular value by using SVD
[Lin (2000) and Lee (2001)]
6
Degenerate scale for torsion bar problems with
arbitrary cross sections


An analytical way to determine the
degenerate scale
The existence of degenerate scale for the
two-dimensional Laplace problem
xd
x
f ( x1 , x2 )  0
U ( s, x)  ln( r )
s
sd
x1 x2
, )0
d d
U ( s, x)  ln( r )
f(
B
Bd
d : expansion ratio
7
Degenerate scale for torsion bar problems with
arbitrary cross sections
 U (s, x) (s)dB(s)  u( x)
B
1
where  1 ( s) is boundary density function
0   U ( sd , xd ) 1 ( sd )dB ( sd )
Bd
( x1, x2 )  ( x1d , x2d )  ( x1, x2 )d
Mapping
properties
dB( s )  dB( sd )  dB( s)d
U ( s, x)  U ( sd , xd )  U ( s, x)  ln d
1 (s) 1 (sd ) 1 (s)d
Expansion ratio d  e

1
,
where   B 1 (s)dB(s)
8

Adding a rigid body term c in the fundamental solution
xd
x x
g ( 1c , 2c )  0
e e
U (s, x)  ln( r )  c
g ( x1 , x2 )  0
xd *
U ( s, x)  ln( r )  c
sd
Dd *
sd *
*
Bd
Dd
u ( x)   [U ( s, x)  c] 1 ( s )dB( s ),
Bd
Bd
0   * [U ( sd * , xd * )  c] 1 ( sd * )dB( sd * )
Bd
 d * ln( d * )   1 ( s )dB( s )  cd *   1 ( s )dB( s )
Bd
d
*

Bd
0
Bd
U ( s, x) 1 ( s )dB( s )
For arbitrary cross section, expansion ratio is
d *  ec
9
Degenerate scale for torsion bar problems with
arbitrary cross sections
0.5
Ellipcal cross section
Conventional BEM (UT formulation)
Adding a rigid body term (c=1.0)
Normal scale
  3.0,   1.0 (  3 )
0.4
U ( s, x)  ln( r )
d e

1

0.3

Regularized
c=1.0
U * (s, x)  ln( r )  c
0.2
Original degenerate scale
  1.5,   0.5


U (s, x)  ln( r )  c
*
Unregularized
U ( s, x)  ln( r )
0.1
New degenerate scale
Shifting
0.184
  0.552,   0.184
0.50
0
0
0.4
d *  ec
0.8
1.2

1.6
2
10
Degenerate scale for torsion bar problems with
arbitrary cross sections
0.5
Normal scale
Square cross section
Conventional BEM (UT formulation)
Adding a rigid body term (c=1.0)
0.4
a  1.0
( ): exact solution of the degenerate scale
U ( s, x)  ln( r )
d e

1

0.3

Regularized
U * (s, x)  ln( r )  c
Original degenerate scale
a  0.85
2a
0.2
U * (s, x)  ln( r )  c
2a
Unregularized
U ( s, x)  ln( r )
0.1
Shifting
0.31
d *  ec
New degenerate scale
a  0.31
0.85
0
0
0.4
0.8
1.2
a
1.6
2
11
Determination of the degenerate scale for the two-dimensional Laplace problems
Cross
Section
Normal scale
R  2.0
Torsional
rigidity

G
2
R
b

R
G
Reference
equation
3 3
2  2
2a
2h

  3.0,   1.0
4
a
h
3
2a
a  2.0 , b 
2
a
3
2Ga4 k2
h  3.0
a  1.0
3 4
h
45
Ga4 k1
G
u( x)   U (s, x) 1 (s)dB(s) , where , x on B, . [U ]{ }  {1}
B
1
    1 (s)dB(s) 1.4480 ( ln( 2) )
B
1.4509
(
1
)
ln( 2)
1.5539 (N.A.)
2.6972 (N.A.)
6.1530 (6.1538)
Expansion ratio
d e

1

Degenerate
scale
0.5020 (0.5)
0.5019 (0.5)
0.5254 (N.A.)
0.6902 (N.A.)
0.8499 (0.85)
R=1.0040
(1.0)
   = 2.0058 (2.0)
a=1.0508 (N.A.)
h=2.0700 (N.A.)
a=0.8499 (0.85)
Note: Data in parentheses are exact solutions.
Data marked in the shadow area are derived by using the polar coordinate.
12
Three regularization techniques
to deal with degenerate scale
problems in BEM



Hypersingular formulation (LM equation)
Adding a rigid body term (U*(s,x)=U(s,x)+c)
CHEEF concept

D
13
Numerical results
cross
section

Torsion
rigidity

Normal scale
( =3.0,  =1.0)
method
Analytical
solution
c=1.0
c=2.0
Degenerate scale
( =1.5,  =0.5)
 3  3
2 2
2a
Square
Normal scale
Degenerate scale
(a=1.0)
(a=0.85)
16k1Ga 4
16k1Ga 4
G
8.4823
0.5301
2.249
1.174
8.7623 (3.30%)
-0.8911 (268.10%)
2.266 (0.76%)
2.0570 (75.21%)
LM
formulation
Add a rigid
body term
Ellipse
 3  3
2 2
G
U T
Conventional BEM
2a
0.4812 (9.22%)
Regularization
techniques are
not necessary.
CHEEF concept
Note: data in parentheses denote error.
0.5181 (2.26%)
0.5176 (2.36%)
0.5647 (6.53%)
CHEEF POINT
(2.0, 2.0)
1.1472 (2.31%)
Regularization
techniques are
not necessary.
1.1721(0.19%)
1.1723 (0.17%)
1.1722 (0.18%)
CHEEF POINT
(5.0, 5.0)
14
Numerical results
cross
section
Torsion
rigidity
2h
a
Keyway
Triangle
method
Analytical
solution
Normal scale
h=3.0
Degenerate scale
h=2.07
Normal scale
(a=2.0)
Degenerate scale
(a=1.05)
3.1177
0.7067
12.6488
0.9609
2Ga4 k2
2Ga4 k2
G
U T
Conventional BEM
3 4
h
45
3.1829 (2.09%)
LM
formulation
Add a rigid
body term
b
3
h
c=1.0
c=2.0
G
3 4
h
45
1.1101 (57.08%)
12.5440 (0.83%)
0.6837 (3.25%)
Regularization
techniques are
not necessary.
CHEEF concept
Note: data in parentheses denote error.
0.7035 (0.45%)
0.7024 (0.61%)
0.7453 (5.46%)
CHEEF POINT
(15.0, 15.0)
1.8712 (94.73%)
0.9530 (0.82%)
Regularization
techniques are
not necessary.
0.9876 (2.78%)
0.9879 (2.84%)
0.9272 (3.51%)
CHEEF POINT
(20.0, 20.0)
15
Four pitfalls in BEM
1.Degenerate scale for torsion bar
problems
2.Degenerate boundary problems
3.True and spurious eigensolution for
interior eigenproblem
4.Fictitious frequency for exterior
acoustics
16
Degenerate boundary problems

Multi-domain BEM
u
( 2  k 2 )u( x)  0
C
uc1
1
c
Subdomain 1

u
C
r=1
u=0
t 1f
C
u 2f
t 2f
C
C
C
C
C
C
u c2
Dual BEM
[T ]{u}  [U ]{t}
( 2  k 2 )u( x)  0
C
r=1
t 2f
u 2f
u c2

C
uc1
Subdomain 2
u=0
r=1
interface
Subdomain 2
2

u1f
C
uc1
Subdomain 1
(  k )u( x)  0
2
t 1f
1
f
u=0
[ M ]{u}  {L}{t}
17

Conventional BEM in conjunction with SVD
Singular Value Decomposition
[U ]M P  []M M []M P []
H
P P
Rank deficiency originates from two sources:
(1). Degenerate boundary
(2). Nontrivial eigensolution
Nd=5
Nd=4
Nd=5
18
1
UT BEM + SVD
(Present method)

d
N + 1
0.1
0.01
N
d
1
versus k
0.001
0
2
4
6
8
6
8
6
8
k
1e-014

Multi-domain BEM
1e-015
det[U(k)]
1e-016
1e-017
1e-018
Determinant versus k
1e-019
1e-020
0
2
4
k
1e-034
Dual BEM
1e-035
d e t [ K UL]

Determinant versus k
1e-036
1e-037
1e-038
0
2
4
k
19
Two sources of rank deficiency (k=3.09)
0.8
0.4
-0.4
0
{}
{}

0.4
0
0
-0.4
-0.4
0.4
-0.8
0
20
40
60
80
0
20
Element label
80
0
-0.4
0
40
60
80
{ 4}, ( 4  0)
0
20
40
60
Element label
{ 5 }, ( 5  0)
Degenerate boundary
80

0
None trivial sol.
-0.4
-0.4
Element label
60
 sin( )
2
0.4
{}
{}
0
20
40
{ 3}, ( 3  0)
0.4
0
20
Element label
{ 2}, ( 2  0)
0.4
{}
60
Element label
{1}, ( 1  0)
Nd=5
40
80
0
20
40
60
80
Element label
{ 6 }, ( 6  0)
Eigensolution
20
UT BEM+SVD
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
-0.6
-0.6
-0.6
-0.8
-0.8
-0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8
-0.6
-0.4
-0.2
k=3.09
0
0.2
0.4
0.6
0.8
-0.8
-0.6
-0.4
k=3.84
-0.2
0
0.2
0.4
0.6
0.8
k=4.50
FEM (ABAQUS)
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
-0.6
-0.6
-0.6
-0.8
-0.8
-0.8
-0.6
-0.4
-0.2
0
k=3.14
0.2
0.4
0.6
0.8
-0.8
-0.8
-0.6
-0.4
-0.2
0
k=3.82
0.2
0.4
0.6
0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
k=4.48
0.4
0.6
0.8
21
Four pitfalls in BEM
1.Degenerate scale for torsion bar
problems
2.Degenerate boundary problems
3.True and spurious eigensolution for
interior eigenproblem
4.Fictitious frequency for exterior
acoustics
( 2  k 2 )u( x)  0
D
22
True and spurious eigensolution
for Interior eigenproblem
Complex-valued BEM --- true eigenvalues
Spurious eigenvalues?



MRM
Real-part BEM
Imaginary-part BEM
23
SVD structure for the influence matrices (true)
k=kt
Dirichlet problem
0   0 
 


2
(U )
 [{ (UL) }]T
[{1 }]N  N 
1
N N

  


0



N  N N

Neumann problem
0   0 
 


2
(T )
 [{ (TM ) }]T
[{1 }]N  N 
1
N N

  


0



N  N N

[U e ] [Te ]
[ Le ] [ M e ]
0   0 
0   0 
 

 



2
2
(M )
 [{ (UL) }]T

 [{ (TM ) }]T
[{1( L ) }]N  N 
1
N  N [{1 }] N  N
1
N N


  
  




0



0



N  N N
N  N N


24
SVD structure for the influence matrices (spurious)
k=ks
Singular formulation
0   0 
0   0 
 

 



2
2
(UT )
(
U
)
T
(
UT
)
 [{ }] [{ }] 
 [{ (T ) }]T
[{1 }]N  N 
1
NN
1
N N
1
N N


  
  




0



0



N  N N
N  N N


[U e ] [Te ]
[ Le ] [ M e ]
0   0 
0   0 
 

 



2
2
( LM )
(
L
)
T
(
LM
)
 [{ }] [{ }] 
 [{ ( M ) }]T
[{1 }]N  N 
1
NN
1
N N
1
N N


  
  




0



0



N  N N
N  N N


Hypersingular formulation
25
True eigensolution for interior eigenproblems
Real-part BEM
SVD updating technique
Dirichlet problem
Neumann problem
u 0
t 0
10
10
1


1
1
1
0.1
0.01
0.1
20 constant elements
U 
L 
 
0
0
1
2
3
4
5
k
k
J n (k )  0
T 
M 
 
1
2
3
4
5
k
k
J n' (k )  0
26
True eigensolution for interior eigenproblems
Imaginary-part BEM
SVD updating technique
Dirichlet problem
Neumann problem
u 0
t 0
10
10
8 constant elements
1
0.1
0.1
0.01
0.01
0.001
0.001
1
0.0001


1
1
1e-005
1e-006
0.0001
1e-005
1e-006
1e-007
1e-007
1e-008
1e-008
1e-009
1e-009
U 
L 
 
0
1
2
3
4
5
k
k
J n (k )  0
T 
M 
 
0
1
2
3
4
5
k
k
J n' (k )  0
27
Spurious eigensolution for interior eigenproblems
Real-part BEM
The Fredholm alternative theorem and SVD updating technique
Singular formulation
Hypersingular formulation
10
10
1
1


1
0.1
1
0.01
0.001
0.1
0
1
2
3
4
5
0
1
k
20 constant elements
U T 
k
Yn (k )  0
2
3
4
5
k
L
M
k
Yn' ( k )  0
28
Spurious eigensolution for interior eigenproblem
Imaginary-part BEM
The Fredholm alternative theorem and SVD updating technique
Singular formulation
Hypersingular formulation
10
10
1
0.1
0.1
0.01
0.01
0.001
0.001
1
0.0001
0.0001


1
1
1e-005
1e-006
1e-005
1e-006
1e-007
1e-007
1e-008
1e-008
1e-009
1e-009
0
1
2
3
4
0
5
1
8 constant elements
U T 
k
J n (k )  0
2
3
4
5
k
k
L
M
k
J n' ( k )  0
29
Four pitfalls in BEM
1.Degenerate scale for torsion bar
problems
2.Degenerate boundary problems
3.True and spurious eigensolution for
interior eigenproblem
4.Fictitious frequency for exterior
acoustics
2
UT method
LM method
Burton & Miller method
1
t(a,0)
0
-1
-2
0
2
4
ka
6
8
30
Fictitious frequency for exterior acoustics




Mathematical structure for updating matrix
Source of numerical instability---zero
division by zero
A criterion to check the validity of CHIEF
points
Numerical example
31
Mathematical structure for updating matrix
[U i (k f )]H 

{ j }  {0},
H
[Ti (k f )] 
j  1, 2,, P
[U i ]{ (jU ) }  0{ j }  {0}
[Ti ]{
(T )
j
}  0{ j }  {0}
Proof
{ (jU ) }  { (jT ) }  { Dj }
The subscript i denotes the use of interior degenerate kernel
for exterior problem
32
Mathematical structure for updating matrix
For the Dirichlet eigenproblem
[U e (kt )] D
[ L (k )] { j }  {0}
 e t 
D
u 0
U e (s, x)  U i ( x, s) or [U e ]  [U i ] symmetry
[Ue (kt )]{ Dj }  [Ui (k f )]{ Dj }  0
[ Le (kt )]{ Dj }  [Ti (k f )]{ Dj }  0
{ (jU ) }  { (jT ) }  { Dj }
Le (s, x)  Ti ( x, s) or [ Le ]  [Ti ] transpose symmetry
[U i ]{ Dj }  0{ j }
[U i ]H { j }  0{ Dj }
[Ti ]{ Dj }  0{ j }
[Ti ]H { j }  0{ Dj }
[U i (k f )]H 
 Dj 
{ j }N 1  0 D 


H
[Ti (k f )]  2 N  N
 j 2 N 1
33
Mathematical structure for updating matrix
Multiplicity: P
1   2     P  0
[U i ]H 
 H 
[Ti ]  2 N  N
[0]P P
 

D
D
{ 1 }{ P } { P 1}{ 2 N }
 0
 D


D
{ 1 }{ P } { P 1}{ 2 N } 2 N 2 N  0
 

 0
True boundary mode for
the Dirichlet eigenproblem






0
 
N
H


{

}

{

}
{

}

{

}

1
P
P 1
N N N
0
 

0  2 N  N
Fictitious boundary mode
34
Source of numerical instability---zero division by zero
N
{u}    i { i(T ) }  [ (T ) ]{ }
i 1
N
{t}    i { i(U ) }  [ (U ) ]{ }
For the Dirichlet problem
i 1
[ (U ) ][(U ) ]{ }  {b*}
By pre-multiplying the regular modes{i
(U )
Solvable
i 
1
}H ,
(U ) H
*
{

}
{
b
},
i
(U )
i
 i(U )  0,
P 1  i  N
P 1  i  N
P 1  i  N
35
Source of numerical instability---zero division by zero
By pre-multiplying the fictitious modes {i(U ) }H ,
Unsolvable
i 
{i(U ) }H {b*}

(U )
i
,
0
0
1 i  P
1 i  P
{i(U ) }H {b*}  {i(U ) }H [T ]{u }  0
([T ]H {i(U ) }  0)
36
A criterion to check the validity of CHIEF points
[U ] 
[T ] 
 w {t}   v {u}




Adding CHIEF points
New constraints
 w  {t}  v  {u}
(U PP )ij  wi  { (jU ) },
[U PP
 1 
 1 
 
  (U PK )ij  wi  { (jU ) },
U PK ]    [TPP TPK ]  
(T )
(
T
)

v

{

i
j },
 
  PP ij
 N
 N  (T )  v  { (T ) },
PK ij
i
j
1  i, j  P
P  1  i, j  N
1  i, j  P
P  1  i, j  N
 1 
 1 
 P 1 
 
 


[U PP ]    [TPP TPK ]    [U PK ]    { f }
 
 
 
 P
 P
 N 
37
A criterion to check the validity of CHIEF points
 w1  {
det
(U )
1
}   w1  {

 wP  {

(U )
1
(U )
P
}
0

}   wP  {
(U )
P
}
The selected P CHIEF points are valid
{ (jU ) }  { (jT ) }  { Dj } (No change)
For the Neumann problem
 wi    vi 
38
Numerical example
: valid
: invalid
Real-part
Imaginary-part
0.8
0.6
J 0( 2 )  5.53
0.4
0.2
0
-1
-0.5
0
0.5
1
-0.2
1.2
1
0.8
-0.4
-0.6
0.5
-0.8
0.4
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0
0
-0.4
0.8
-0.5
0.6
-0.8
0.4
-1
0.2
-1.2
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
0
-0.2
J1(1)  3.84
-0.4
-0.6
-0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
39
Conclusions and further research
Conclusions




A more efficient technique was proposed to directly
determine the degenerate scale since only one normal scale
needs to be computed.
The conventional BEM in conjunction with SVD was applied
to deal with rank-deficiency (degenerate boundary and
nontrivial eigensolution) for the degenerate boundary
eigenproblem.
By using the Fredholm alternative theorem and SVD
techniques in conjunction with the dual formulations, the true
and spurious eigenvalues in the complex-valued formulation,
the real-part, the imaginary-part BEMs and MRM were sorted
out successfully.
In order to overcome the rank-deficiency problem due to
fictitious frequency, the CHIEF method was reformulated in a
unified manner by using the Fredholm alternative theorem
40
and SVD technique.
Further research



Although the degenerate scale occurs in the Dirichlet
problem of simply two-dimensional Laplace problems
by using the BEM, there is no proof of the occurrence
of degenerate scale for the problem with the mixedtype boundary condition.
The main drawback of the imaginary-part BEM seems
to produce ill-conditioned matrices. While this is
sometimes the case, it is hoped that further research
can alleviate the drawback.
Whether the spurious (fictitious) modes in the UT and
LM formulations are the same or not needs further
investigation.
41