bem introduction
DESCRIPTION
BEM introduction. J. T. Chen. Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan Sep. 21, 2006 First-class introduction for NTPU/MSV BEM course (lecture1-2006.ppt). Outlines. Overview of BIE and BEM Mathematical tools - PowerPoint PPT PresentationTRANSCRIPT
1
BEM introductionBEM introduction
J. T. Chen
Department of Harbor and River Engineering
National Taiwan Ocean University, Keelung, Taiwan
Sep. 21, 2006
First-class introduction for NTPU/MSV BEM course
(lecture1-2006.ppt)
2
Overview of BIE and BEM Mathematical tools Hypersingular BIE Degenerate kernel Circulants SVD updating term SVD updating document Fredholm alternative theorem Nonuniqueness and its treatments Degenerate scale Degenerate boundary True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics) Corner Conclusions and further research
Outlines
3
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
3
PDE- variational
IEDE
Domain
Boundary
MFSTrefftz method
MLS
4
Numeber of Papers of FEM, BEM and FDM
(Data form Prof. Cheng A. H. D.)
126
5
Growth of BEM/BIEM papers
(data from Prof. Cheng A.H.D.)
6
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
7
Original data from Prof. Liu Y J
(1984)
crack
BEMCauchy kernel
singular
DBEMHadamard
kernelhypersingular
FMM
Large scaleDegenerate kernel
Desktop computer fauilure
(2000)
8
Why engineers should learn mathematics ?
Well-posed ? Existence ? Unique ?
Mathematics versus Computation
Some examples
9
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 ?
10
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
11
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
12
Numerical phenomena(Degenerate boundary)
Singular integral equation
Hypersingular integral equation
Cauchy principal value Hadamard principal value
Boundary element method Dual boundary element method
degenerate
boundary
normal
boundary
1967-1984-2006
13
Numerical phenomena(Corner)
B B
B B
BxsdBstxsLVPCsdBsuxsMVPHxtxt
BxsdBstxsUVPRsdBsuxsTVPCxu
,)()(),(...)()(),(...)()sin()(
,)()(),(...)()(),(...)(
tt
u\ u \
D Boundary
14
Motivation
Numerical instability occurs in BEM ? (1) degenerate scale (2) degenerate boundary (3) fictitious frequency (4) corner Spurious eigenvalues appear ? (5) true and spurious eigenvalues
Five pitfalls in BEM
Mathematical essence—rank deficiency ?(How to deal with ?) nonuniqueness ?
15
Mathematical tools
Hypersingular BIE
Degenerate kernel
Circulants
SVD updating term
SVD updating document
Fredholm alternative theorem
16
Mathematical tools
Hypersingular BIE
(potential theory)
Degenerate kernel
Circulants
SVD updating term
SVD updating document
Fredholm alternative theorem
17
Two systems u and U
u(x)source
U(x,s)
s
Infinite domain
Domain(D)
Boundary (B)
18
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
19
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
),(
20
Potential theory
Single layer potential (U) Double layer potential (T) Normal derivative of single layer
potential (L) Normal derivative of double layer
potential (M)
21
Physical examples for potentialsP
M
U:moment diagram
M:moment diagram
T:moment diagram
L:shear diagram
Force Moment
22
Order of pseudo-differential operator 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
23
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
24
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)
25
• 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
26
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)
27
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
28
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
29
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
30
Hypersingular BIE
Degenerate kernel
31
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
32
Fundamental solution
Field response due to source (space)
Green’s function Casual effect (time)
K(x,s;t,τ)
33
Separable form of fundamental solution (1D)
-10 10 20
2
4
6
8
10
Us,x
2
1
2
1
(x) (s), s x
(s, x)
(s) (x), x s
i ii
i ii
a b
U
a b
=
=
ìïï ³ïïïï=íïï >ïïïïî
å
å
1(s x), s x
1 2(s, x)12
(x s), x s2
U r
ìïï - ³ïïï= =íïï - >ïïïî
-10 10 20
-0.4
-0.2
0.2
0.4
Ts,x
s
Separable Separable propertyproperty
continuocontinuousus
discontidiscontinuousnuous
1, s x
2(s, x)1
, x s2
T
ìïï >ïïï=íï -ï >ïïïî
34
Degenerate kernel (step1)
Step 1xsRxsU ln)ln(),(
S
x
Rx: variable
s: fixed
35
RmR
mRU
m
me
1)),(cos()(
1)ln(),,,(
RmRm
RRUm
mi
1)),(cos()(
1)ln(),,,(
Degenerate kernel (step2, step3)
x
s
A
B
Step 2
RA
iUeU
AStep 3
x
B
A
s
RB
iU
eUB
36
Mathematical tools
Hypersingular BIE
Degenerate kernel
Circulants
SVD updating term
SVD updating document
Fredholm alternative theorem
37
Engineering problems
Thin-airfoill
Aerodynamics
Seepage with
sheetpiles
1 2
3
4
56
7
8
oblique incident water wave
Free water surface
I II III
Pseudo boundary
O
S
Pseudo boundary
x
S
y
dC C
z
x
Top view
O~
38
Degeneracy of the Degenerate Boundary
geometry node
the Nth constantor linear element
un
0
un
0 un
0
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( )
39
The constraint equation is not enough to determine the coefficient p and q,
Another constraint equation is required
axwhenqpaaf
qpxxQaxxf
,)(
)()()( 2
axwhenpaf
pxQaxxQaxxf
,)(
)()()()(2)( 2
Theory of dual integral equations
40
Successful experiences
41
X-ray detection
42
FEM simulation
43
Stress analysis
44
BEM simulation
45
V-band structure (Tien-Gen missile)
46
FEM simulation
47
48
Shong-Fon II missile
49
IDF
50
Flow field
51
Seepage flow
52
Meshes of FEM and BEM
53
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
54
Water wave problemWater wave problem
Free water surface S
x
Top view
O
y
zO
xz
S
breakwater
breakwater
oblique incident water wave 0)~()~( 22 xuxu
55
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
56
Cracked torsion bar
T
da
57
58
59
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
60
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
61
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
Multi-domain BEM
UT BEM + SVD
(Present method)
versus k1dN
Determinant versus k
Determinant versus k
62
Two sources of rank deficiency (k=3.09)
Nd=5
0 20 40 60 80
E lem ent labe l
-0.4
0
0.4
0 20 40 60 80
E lem ent labe l
0 .4
0
-0.4
{}
0 20 40 60 80
E lem ent label
-0.8
-0.4
0
0.4
0.8
{}
0 20 40 60 80
E lem ent labe l
-0.4
0
0.4
{}
0 20 40 60 80
E lem ent labe l
-0.4
0
0.4
{}
0)( },{ 11
0 20 40 60 80
E lem ent labe l
-0.4
0
0.4
{}
)2
sin(
None trivial sol.
EigensolutionDegenerate boundary
0)( },{ 22 0)( },{ 33
0)( },{ 44 0)( },{ 55 0)( },{ 66
63k=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)
64
On the Mechanism of Fictitious Eigenvaluesin Direct and Indirect BEM
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
65
CHIEF and Burton & Miller method
ka0.00 2.00 4.00 6.00 8.00
-0.40
0.00
0.40
0.80
1.20
1.60
UT method Burton and Miller Analytical solution CHIEF method
t
66
Conclusions
• Introduction of BEM and BIEM
• The nonuniqueness in BIEM and BEM were reviewed an
d its treatment was addressed.
• The role of hypersingular BIE was examined.
• The numerical problems in the engineering applications
using BEM were demonstrated.
• Several mathematical tools, SVD, degenerate kernel, …,
were employed to deal with the problems.
67
The End
Thanks for your kind attention
