1

INTRODUCTION OF DUAL BEM/BIEM AND ITS ENGINEERING APPLICATIONS

J T Chen, Distinguished Prof.

Taiwan Ocean University

June 07, 11:00-12:00, 2007

(city-London2007.ppt)

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering

2

Overview of BEM and dual BEM ( 中醫式的工程分析法 )

Mathematical formulation Hypersingular BIE Successful experiences

Nonuniqueness and its treatments Degenerate scale True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics)

Conclusions

Outlines

3

BEM USA (3569) , China (1372) , UK, Japan,

Germany, France, Taiwan (580), Canada, South Korea, Italy (No.7)

Dual BEM (Made in Taiwan) UK (124), USA (98), Taiwan (70), China

(46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3)

(ISI information Apr.20, 2007)

Top ten countries of BEM and dual BEM

台灣加油 FEM Taiwan (No.9/1383)

4

FEM

USA(9946), China(4107), Japan(3519), France, Germany, England, South Korea, Canada, Taiwan(1383), Italy

Meshless methods

USA(297), China(179), Singapore(105), Japan(49), Spain, Germany, Slovakia, England, Portugal, Taiwan(28)

FDM

USA(4041), Japan(1478), China(1411), England, France, Canada, Germany, Taiwan (559), South Korea, India

(ISI information Apr.20, 2007)

Top ten countries of FEM, FDM and Meshless methods

5

BEM Aliabadi M H (UK, Queen Mary College) Chen J T (Taiwan, Ocean Univ.) 106 SCI papers Mukherjee S (USA, Cornell Univ.) Leisnic D(UK, Univ. of Leeds) Tanaka M (Japan, Shinshu Univ.) Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) Power H (UK, Univ. Nottingham) (ISI information Apr.20, 2007)

Top three scholars on BEM and dual BEM

NTOU/MSV 加油

6

Top 25 scholars on BEM/BIEM since 2001

北京清華姚振漢教授提供 (Nov., 2006)

NTOU/MSVTaiwan

7

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

7

PDE- variational IEDE

Domain

BoundaryMFS,Trefftz method MLS, EFG

開刀 把脈

針灸

8

Number of Papers of FEM, BEM and FDM

(Data form Prof. Cheng A. H. D.)

126

9

Growth of BEM/BIEM papers

(data from Prof. Cheng A.H.D.)

10

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

11

BEM and FEM

(1) BEM and meshless methods can be seen as a supplement of FEM.

(2) BEM utilizes the discretization concept of FEM as well as the limitation. Whether the supplement is needed or not depends on its absolutely superio

r area than FEM.

Ｃ rack & large scale problems

NTUCE

12

What Is Boundary Element Method ?

NTUCE

1 2

3

45

6

1 2

geometry nodethe Nth constantor linear element

N

西醫 郎中

13

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

14

Some researchers on Dual BEMChen(1986) 436 citings in total

Hong and Chen (1988 ） 74 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 AMR

Niu and Wang (2001)

Yu D H, Zhu J L, Chen Y Z, Tan R J …

NTUCE

cite

15

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

16

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( )

dependency

17

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

18

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

19

Fundamental solution

Field response due to source (space)

Green’s function Casual effect (time)

K(x,s;t,τ)

20

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

21

Two systems u and U

u(x)source

U(x,s)

s

Infinite domain

Domain(D)

Boundary (B)

22

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

23

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

),(

24

Potential theory

Single layer potential (U) Double layer potential (T) Normal derivative of single layer

potential (L) Normal derivative of double layer

potential (M)

25

Physical examples for potentialsP

M

U:moment diagram

M:shear diagram

T:moment diagram

L:shear diagram

Force Moment

26

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

27

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

28

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)

29

• 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

30

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)

31

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

32

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

33

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

34

Successful experiences since 1986

35

Solid rocket motor (Army 工蜂火箭 )

36

X-ray detection ( 三溫暖測試 )

Crack initiation crack growth

Stress reliever

37

FEM simulation

38

Stress analysis

39

BEM simulation

40

Shong-Fon II missile (Navy)

41

IDF (Air Force)

42

Flow field

43

V-band structure (Tien-Gen missile)

44

FEM simulation

45

46

Seepage flow

47

FEM (iteration No.49) BEM(iteration No.13)

Initial guessInitial guess

After iteration After iteration

Remesh areaRemesh line

Free surface seepage flow using hypersingular formulation

48

Meshes of FEM and BEM

49

Incomplete partition in room 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

50

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

51

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

52

Cracked torsion bar

T

da

53

IEEE J MEMS

54

55

56

Radiation and scattering problems

Nonuniform radiaton scattering

1),( au0),( au

Drruk ),( ,0),()( 22

32

5

Drruk ),( ,0),()( 22

2

57

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

58

   

Errorestimator

Error estimator

 

SolutionSolution

Strategy of adaptive BEM

Miller &Burton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][

HypersingularEquation

tu LMLM~~

][][

u,t u,t

21

59

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

Numerical solution: BEM Numerical solution: FEM

64 ELEMENTS 2791 ELEMENTS

Nonuniform radiation ： Dirichlet problem 2ka

9

60

Is it possible !

No hypersingularity !

No subdomain !

61

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

62

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

63

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

Sub domain

64k=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)

65

Disclaimer (commercial code)

The concepts, methods, and examples using

our software are for illustrative and educational purposes only. Our cooperation assumes no liability or responsibility to any person or company for direct or indirect damages resulting from the use of any information contained here.

inherent weakness ?

misinterpretation ? User 當自強

66

BEM trap ?Why engineers should learn

mathematics ? Well-posed ? Existence ? Unique ?

Mathematics versus Computation

equivalent ? Some examples

67

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

68

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 ?

69

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

70

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

71

TreatmentsSVD updating term

Burton & Miller method

CHIEF method

NN

cc

cc

SM

SMC

8162

1

cccc SMiSM21

NNN cCCUCUC

CCUCUC

UU

UU

UU

UU

C

8)4(2

*

2121

2121

22212221

12111211

22212221

12111211

][

Mathematical analysis and numerical study for free vibration of plate using BEM-71

a

b

1B

2B

72

Conclusions

• Introduction of dual BEM

• The role of hypersingular BIE was examined

• Successful experiences in the engineering applications us

ing BEM were demonstrated

• The traps of BIEM and BEM were shown

73

The End

Thanks for your kind attention

74

歡迎參觀海洋大學力學聲響振動實驗室烘焙雞及捎來伊妹兒

http://ind.ntou.edu.tw/~msvlab/http://140.121.146.149/

E-mail: jtchen@mail.ntou.edu.tw

75

76

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