dual bem since 1986 j t chen ( 陳正宗特聘教授 )
DESCRIPTION
Dual BEM since 1986 J T Chen ( 陳正宗特聘教授 ). National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering. Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan 8:30-9:50, Nov. 19, 2006. Outlines. Overview of BEM and dual BEM - PowerPoint PPT PresentationTRANSCRIPT
1
Dual BEM since 1986
J T Chen ( 陳正宗特聘教授 )
Dual BEM since 1986
J T Chen ( 陳正宗特聘教授 )
Department of Harbor and River Engineering
National Taiwan Ocean University, Keelung, Taiwan
8:30-9:50, Nov. 19, 2006
National Taiwan Ocean University
MSVLABDepartment of Harbor and River
Engineering
2
Overview of BEM and dual BEM
Mathematical formulation Hypersingular BIE Nonuniqueness and its treatments Degenerate scale True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics)
Conclusions
Outlines
3
BEM USA (3423) , China (1288) , UK, Japan,
Germany, France, Taiwan (551), Canada, South Korea, Italy (No.7)
Dual BEM (Made in Taiwan) UK (119), USA (90), Taiwan (69), China
(46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3)
(ISI information Nov.06, 2006)
Top ten countries of BEM and dual BEM
台灣加油 FEM Taiwan (No.9/1311)
4
FEM USA, China, Japan, France, Germany, England,
South Korea, Canada, Taiwan, Italy Meshless methods USA, China, Singapore, Japan, Spain, Germany,
Slovakia, England, France, Taiwan FDM USA, Japan, China, England, France, Germany,
Canada, Taiwan, South Korea, Italy (ISI information Nov.06, 2006)
Top ten countries of FEM, FDM and Meshless methods
5
BEM Aliabadi M H (UK, Queen Mary College) Mukherjee S (USA, Cornell Univ.) Chen J T (Taiwan, Ocean Univ.) 56 篇 Tanaka M (Japan, Shinshu Univ.) Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) 43 篇 Power H (UK, Univ Nottingham) (ISI information Nov.06, 2006)
Top three scholars on BEM and dual BEM
NTOU/MSV 加油
6
Overview of numerical methods
Finite Difference M ethod Finite Element M ethod Boundary Element M ethod
M esh M ethods M eshless M ethods
Numerical M ethods
6
PDE- variational
IEDE
Domain
Boundary
MFSTrefftz method
MLS, EFG
7
Number of Papers of FEM, BEM and FDM
(Data form Prof. Cheng A. H. D.)
126
8
Growth of BEM/BIEM papers
(data from Prof. Cheng A.H.D.)
9
Advantages of BEM
Discretization dimension reduction Infinite domain (half plane) Interaction problem Local concentration
Disadvantages of BEM Integral equations with singularity Full matrix (nonsymmetric)
北京清華
10
BEM and FEM
(1)边界元法和各种无网格法等在许多工 程领域都属于有限元法的一个补充。
(2) 它的发展既受到有限元法的启发，又受到
有限元法的制约。只有在其优势领域，作
为补充才有实际意义。如果只是有限元能
计算的它也能算，而且结果吻合，那就这
种补充没有必要 (Quoted from Prof. Yao)Degenerate boundary(Crack growth) ! 第一場
Fast multipole ! 第二場
NTUCE
11
• Finite element method Boundary element method
What Is Boundary Element Method ?
NTUCE
1 2
3
45
6
1 2
geometry node the Nth constantor linear element
N
12
Dual BEM
Why hypersingular BIE is required
(potential theory)
NTUCE
1 2
3
4
56
7
8
1 2
3
4
56
7
8
910
Artifical boundary introduced !
BEM
Dual integral equations needed !
Dual BEM
Degenerate boundary
13
Some researchers on Dual BEMChen(1986)
Hong and Chen (1988 ） 71 citings ASCE
Portela and Aliabadi (1992) 188 citings IJNME
Mi and Aliabadi (1994)
Wen and Aliabadi (1995)
Chen and Chen (1995) 新竹清華黎在良等 --- 斷裂力學邊界數值方法 (1996)
周慎杰 (1999)
Chen and Hong (1999) 76 citings ASME
Niu and Wang (2001)
Yu D H, Zhu J L, Chen Y Z, Tan R J …
NTUCE
14
Dual Integral Equations by Hong and Chen(1984-1986)
NTUCE
Singular integral equation Hypersingular integral equation
Cauchy principal value Hadamard principal value
Boundary element method Dual boundary element method
normal
boundarydegenerate
boundary
1969 1986 2006
15
Degenerate boundary
geometry node
the Nth constantor linear element
un0
un0
un0
u 1 u 1(0,0)
(-1,0.5)
(-1,-0.5)
(1,0.5)
(1,-0.5)
1 2
3
4
56
7
8 [ ]{ } [ ]{ }U t T u
[ ]{ } [ ]{ }L t M u
N
1.693-0.335-019.0019.0445.0703.0445.0335.0
0.334-1.693-281.0281.0045.0471.0347.0039.0
0.0630.638-193.1193.1638.0063.0081.0081.0
0.0630.638-193.1193.1638.0063.0081.0081.0
0.4710.045-281.0281.0693.1335.0039.0347.0
0.7030.445019.0019.0335.0693.1335.0445.0
0.4710.347054.0054.0039.0335.0693.1045.0
0.335-0.039054.0054.0347.0471.0045.0693.1
][
U
-1.107464.0464.0219.0490.0219.0107.1
1.107-785.07854.0000.0588.0519.0927.0
0.8881.326326.1888.0927.0927.0
0.8881.326326.1888.0927.0927.0
0.5880.000785.0785.0107.1927.0519.0
0.4900.219464.0464.0107.1107.1219.0
0.5880.519321.0321.0927.0107.1000.0
1.1070.927321.0321.0519.0588.0000.0
][
T
5(+) 6(+) 5(+) 6(-)
5(+)6(+)
5(+)6(+)
n s( )
0.805464.0464.0612.0490.0612.0805.0
0.805347.0347.0000.0184.0519.0927.0
0.8881.417417.1888.0511.0511.0
0.888-1.417-417.1888.0511.0511.0
0.1840.000347.0347.0805.0927.0519.0
0.4900.612464.0464.0805.0805.0612.0
0.1840.519458.0458.0927.0805.0000.0
0.8050.927458.0458.0519.0184.0000.0
][
L
000.41.600-400.0400.0282.0235.0282.0600.1
1.600-000.4000.1000.1333.1205.0062.0800.0
0.715-3.765-000.8000.8765.3715.0853.0853.0
0.7153.765000.8000.8765.3715.0853.0853.0
0.205-1.333-000.1000.1000.4600.1800.0061.0
0.236-0.282-400.0400.0600.1000.4600.1282.0
0.205-0.061-600.0600.0800.0600.1000.4333.1
1.600-0.800-600.0600.0061.0205.0333.1000.4
][
M
5(+) 6(+)5(+) 6(-)
5(+)6(-)
5(+) 6(-)
n x( ) n x( )
n s( )
16
The constraint equation is not enough to determine coefficients of p and q,
Another constraint equation is required
axwhenqpaaf
qpxxQaxxf
,)(
)()()( 2
axwhenpaf
pxQaxxQaxxf
,)(
)()()()(2)( 2
How to get additional constraints
17
Original data from Prof. Liu Y J
(1984)
crack
BEMCauchy kernel
singular
DBEMHadamard
kernelhypersingular
FMM
Large scaleDegenerate kernel
Desktop computer fauilure
(2000)Integral equation
1888
18
Fundamental solution
Field response due to source (space)
Green’s function Casual effect (time)
K(x,s;t,τ)
19
Green’s function, influence line and moment diagram
P
Force
Moment diagrams:fixed
x:observer
P
Force
Influence line s:moving
x:observer(instrument)
sG(x,s)
x
G(x,s)x sx=1/4
s
s=1/2
20
Two systems u and U
u(x)source
U(x,s)
s
Infinite domain
Domain(D)
Boundary (B)
21
Dual integral equations for a domain point(Green’s third identity for two systems, u and
U)
Primary field
Secondary field
where U(s,x)=ln(r) is the fundamental solution.
2 ( ) ( , ) ( ) ( )B
u x T s x u s dB s ( , ) ( ) ( ), BU s x t s dB s x D
2 ( ) ( , ) ( ) ( )B
t x M s x u s dB s ( , ) ( ) ( ), B
L s x t s dB s x D
sn
UxsT
),(xn
UxsL
),(xs nn
UxsM
2
),(
ut
n
22
Dual integral equations for a boundary point(x push to boundary)
Singular integral equation
Hypersingular integral equation
where U(s,x) is the fundamental solution.
B
sdBsuxsTVPCxu )()(),( . . .)( B
BxsdBstxsUVPR ),()(),( . . .
B
sdBsuxsMVPHxt )()(),( . ..)( B
BxsdBstxsLVPC ),()(),( . . .
sn
UxsT
),(xn
UxsL
),(xs nn
UxsM
2
),(
23
Potential theory
Single layer potential (U) Double layer potential (T) Normal derivative of single layer
potential (L) Normal derivative of double layer
potential (M)
24
Physical examples for potentialsP
M
U:moment diagram
M:shear diagram
T:moment diagram
L:shear diagram
Force Moment
25
Order of pseudo-differential operators Single layer potential
(U) --- (-1) Double layer
potential (T) --- (0)
Normal derivative of single layer potential (L) --- (0)
Normal derivative of double layer potential (M) --- (1)
2 2
0 0( , ) ( ) - ( , ) ( )L t d t CPV L t d
2
0( , ) ( ) ( )
( ( )) ''
( ( )) ''
M u d u
u u
D D u u
2 2
0 0( , ) ( ) ( , ) ( )T u d u CPV T u d
Real differential operator Pseudo differential operator
26
Calderon projector
2
2
1[ ] [ ] [ ][ ] [0]
4 [ ][ ] [ ][ ]
[ ][ ] [ ][ ]
1[ ] [ ] [ ][ ] [0]
4
I T U M
U L T M
M T L M
I L M U
27
How engineers avoid singularityHow engineers avoid singularity
BEM / BIEMBEM / BIEM
Improper integralImproper integral
Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity
Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary
Collocation Collocation pointpoint
Fictitious BEMFictitious BEM
Null-field approachNull-field approach
CPV and HPVCPV and HPVIll-posedIll-posed
Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)
Waterman (1965)Waterman (1965)
Achenbach Achenbach et al.et al. (1988) (1988)
28
• R.P.V. (Riemann principal value)
• C.P.V.(Cauchy principal value)
• H.P.V.(Hadamard principal value)
Definitions of R.P.V., C.P.V. and H.P.V.using bump approach
NTUCE
R P V x dx x x x. . . ln ( ln )z
1
1
- = -2 x=-1x=1
C P Vx
dxx
dx. . . lim
z zz 1
1 1
1
1 1 = 0
H P Vx
dxx
dx. . . lim
z zz 1
1
2
1
12
1 1 2 = -2
0
0
x
ln x
x
x
1 / x
1 / x 2
29
Principal value in who’s sense
Common sense Riemann sense Lebesgue sense Cauchy sense Hadamard sense (elasticity) Mangler sense (aerodynamics) Liggett and Liu’s sense The singularity that occur when the base
point and field point coincide are not integrable. (1983)
30
Two approaches to understand HPV
1
2 2-10
1lim =-2 y
dxx y
H P Vx
dxx
dx. . . lim
z zz 1
1
2
1
12
1 1 2 = -2
(Limit and integral operator can not be commuted)
(Leibnitz rule should be considered)
1
01
1{ } 2
t
dCPV dx
dt x t
Differential first and then trace operator
Trace first and then differential operator
31
Bump contribution (2-D)
( )u x
( )u xx
sT
x
sU
x
sL
x
sM
0
0
1( )
2t x
1 2( ) ( )
2t x u x
1( )
2t x
1 2( ) ( )
2t x u x
32
Bump contribution (3-D)
2 ( )u x
2 ( )u x
2( )
3t x
4 2( ) ( )
3t x u x
2( )
3t x
4 2( ) ( )
3t x u x
x
xx
x
s
s s
s0`
0
33
Successful experiences since 1986
34
Solid rocket motor ( 工蜂火箭 )
35
X-ray detection ( 三溫暖測試 )
36
FEM simulation
37
Stress analysis
38
BEM simulation
39
Shong-Fon II missile
40
IDF
41
Flow field
42
V-band structure (Tien-Gen missile)
43
FEM simulation
44
45
Seepage flow
46
Meshes of FEM and BEM
47
Screen in acoustics
U T L Mm ode 1.
m ode 2.
m ode 3.
0.00 0 .05 0 .10 0 .15 0 .200 .00
0 .05
0 .10
0 .00 0 .05 0 .10 0 .15 0 .200 .00
0 .05
0 .10
0.00 0.05 0.10 0.15 0.200.00
0.05
0.10
0.00 0.05 0.10 0.15 0.200.00
0.05
0.10
0.00 0.05 0.10 0.15 0.200.00
0.05
0.10
0.05 0.10 0.15 0.20
0.05
587.6 H z 587.2 H z
1443.7 H z 1444.3 H z
1517.3 H z 1516.2 H z
b
a
e
c
2 2 0u k u t0
t=0
t=0
t=0
t=0
t=0
48
Water wave problem with breakwaterWater wave problem with breakwater
Free water surface S
x
Top view
O
y
zO
xz
S
breakwater
breakwater
oblique incident water wave 0)~()~( 22 xuxu
49
Reflection and Transmission
0.00 0.40 0.80 1.20 1.60 2.00
kd
0.00
0.40
0.80
1.20
lRl a
nd lT
l
k h = 5D e ep w a te r th eo ry (U rse ll, 1 9 4 6 )M u lti-d o m a in B E M (L iu e t a l ., 1 9 8 2 )U T m e th o d o f D B E M (C o m b in e d L M , 3 2 0 e le m en ts)L M m e th o d o f D B E M (C o m b in e d U T , 3 2 0 e le m en ts)
R
T
50
Cracked torsion bar
T
da
51
IEEE J MEMS
52
53
54
Is it possible !
No hypersingularity !
No subdomain !
55
Dual BEM
Degenerate boundary problems
u=0r=1
0)()( 22 xukC
C
u=0r=1
0)()( 22 xukC C
CC
u=0r=1
0)()( 22 xukC
C
interface
Subdomain 1
Subdomain 2
Subdomain 1
Subdomain 2
1cu
1cu
1fu
1fu
2fu
2fu
2ft
1ft
2ft
1ft
2cu
2cu
1cu
1cu
C
C
C
C
Multi-domain BEM
}}{{}]{[
}]{[}]{[
tLuM
tUuT
56
Conventional BEM in conjunction with SVD
Singular Value DecompositionH
PPPMMMPMU ][][][][
Rank deficiency originates from two sources:
(1). Degenerate boundary
(2). Nontrivial eigensolution
Nd=5 Nd=5Nd=4
57
0 2 4 6 8
k
0.001
0.01
0.1
1
N d + 1
0 2 4 6 8
k
1e-020
1e-019
1e-018
1e-017
1e-016
1e-015
1e-014
d e t [ U ( k ) ]
0 2 4 6 8
k
1e-038
1e-037
1e-036
1e-035
1e-034
d e t [ K U
L ]
Dual BEM
UT BEM + SVD
(Present method)
versus k1dN
Determinant versus k
Determinant versus k
58k=3.14 k=3.82
-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
0
0.2
0.4
0.6
0.8
k=4.48
UT BEM+SVD
-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
0
0.2
0.4
0.6
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
0
0.2
0.4
0.6
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
0
0.2
0.4
0.6
0.8
k=3.09
-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
0
0.2
0.4
0.6
0.8
k=3.84
-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
0
0.2
0.4
0.6
0.8
k=4.50
FEM (ABAQUS)
59
BEM trap ?Why engineers should learn
mathematics ? Well-posed ? Existence ? Unique ?
Mathematics versus Computation
Some examples
60
Numerical phenomena(Degenerate scale)
Error (%)of
torsionalrigidity
a
0
5
125
da
Previous approach : Try and error on aPresent approach : Only one trial
T
da
Commercial ode output ?
61
Numerical and physical resonance
a
m
k
e i t
incident wave
e i t e i t
radiation
Physical resonance Numerical resonance
, if ufinite
( )
2 2
, if u finite lim00
k
m
62
Numerical phenomena(Fictitious frequency)
0 2 4 6 8
-2
-1
0
1
2UT method
LM method
Burton & Miller method
t(a,0)
1),( au0),( au
Drruk ),( ,0),()( 22
9
1),( au0),( au
Drruk ),( ,0),()( 22
9
A story of NTU Ph.D. students
63
Numerical phenomena(Spurious eigensolution)
0 1 2 3 4 5 6 7 8 9 10 11 12fre q u e n cy p a ra m e te r
1E-080
1E-060
1E-040
1E-020
de
t|SM
|
C -C annular p la teu, com plex-vauled form ulation
T<9.447>
T: T rue e igenvalues
T<10.370>
T<10.940>
T<9.499>
T<9.660>
T<9.945>
S<9.222>
S<6.392>
S<11.810>
S : Spurious e igenvalues
ma 1
mb 5.0
1B
2B
64
Some findings
`Laplace Helmholtz
Ling1947
Analytical solution
Bird & Steele1991
房營光 1995
Analytical solution
Lee &
Manoogian1992
Caulk1983 Naghdi1991
Analytical solution
Tsaur et al.2004
Analytical solution
Present method
Present method Tsaur et al.
?
65
Torsional rigidity
?
66
Stress concentration at point B
0.4 0.5 0.6 0.7
b
1.5
2
2.5
3
3.5
Sc
Theta=3*pi/8
Theta=pi/8
Theta=pi/4
Present
Present
method
method
Naghdi’s result
Naghdi’s result
ss
0.4 0.5 0.6 0.7
a
1.5
2
2.5
3
3.5
Sc
0.4 0.5 0.6 0.7
a
1.5
2
2.5
3
3.5
Sc
Steele &
Steele &
Bird
Bird
The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified.
--ASME Applied Mechanics Review
67
A half-plane problem with two alluvial valleys subject to the incident SH-wave
Canyon
Matrix
3aSH-Wave
房 [93] 將正弦和餘弦函數的正交特性使用錯誤，以至於推導出錯誤的聯立方程，求得錯誤的結果。
-- 亞太學報
68
Limiting case of two canyons
Present methodPresent method
Tsaur et al.’s results [103]Tsaur et al.’s results [103]
8/ 102, , / 2 /3IM MI
0 30 60 90
- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a
0
2
4
6
8
Am
plit
ud
e
- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a
0
2
4
6
8
Am
plit
ud
e
- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a
0
2
4
6
8
Am
plit
ud
e
- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x / a
0
2
4
6
8
Am
plit
ud
e
69
Inclusion
Matrixh
SH-Wave
a
xy
A half-plane problem with a circular inclusion subject to the incident SH-wave
70
Surface displacements of a inclusion problem under the ground surface
- 3 - 2 - 1 0 1 2 3
x/a
0
1
2
3
4
5
6
Am
plit
ud
e
- 3 - 2 - 1 0 1 2 3
0
1
2
3
4
5
6
- 3 - 2 - 1 0 1 2 3
x/a
0
1
2
3
4
5
6
Am
plit
ud
e
- 3 - 2 - 1 0 1 2 3
0
1
2
3
4
5
6
- 3 - 2 - 1 0 1 2 3
x/a
0
1
2
3
4
5
6
Am
plit
ud
e
- 3 - 2 - 1 0 1 2 3
0
1
2
3
4
5
6
- 3 - 2 - 1 0 1 2 3
x/a
0
1
2
3
4
5
6
Am
plit
ud
e
0
1
2
3
4
5
6-3 -2 -1 0 1 2 3
Present methodPresent method 2, / 1/ 6, / 2 /3I M I M
Tsaur et al.’s results [102]Tsaur et al.’s results [102]
Manoogian and Lee’s results [62]Manoogian and Lee’s results [62]
0 30 60 90
When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time.
--Private communication
71
Journals related to mechanics in Taiwan
SCI
SCIEI
72
National Taiwan Ocean University
MSVLABDepartment of Harbor and River
Engineering
73
Conclusions
• Introduction of dual BEM
• The role of hypersingular BIE was examined.
• Successful experiences in the engineering applications us
ing BEM were demonstrated.
• The trap of BIEM and BEM were shown
• Previous errors by other investigators were identified
74
The End
Thanks for your kind attention
76