1 Nonuniqueness problems in numerical methods J T Chen( 陳正宗 ), Life-time Distinguished Prof. Taiwan Ocean University August 8, 11:00-12:00 2008 高海大旗津校區,

1

Nonuniqueness problems in numerical methods J T Chen( 陳正宗 ) Life-time Distinguished Prof

Taiwan Ocean University

August 8 1100-1200 2008

高海大旗津校區高雄

(CFD15-2008-chenppt)

National Taiwan Ocean UniversityMSVLAB ( 海大河工系 )

Department of Harbor and River Engineering

2

Overview of numerical methods ( 中醫式的工程分析法 ) Nonuniqueness problems - review

BEM failure (mathematical degeneracy) Degenerate boundary (No subdomain and no hypersingularity) Degenerate scale True and spurious eigensolution (interior prob) Fictitious frequency (exterior acoustics)

Conclusions

Outline

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

PDE- variational IEDE

Domain

BoundaryMFSTrefftz method MLS EFG

開刀把脈

針灸

( 中醫式的工程分析法 )

4

BEM

USA China UK Germany France India Italy Iran Japan South Korea (Taiwan No11)

Dual BEM (Made in Taiwan)

UK USA Taiwan China Germany France Japan Brazil Australia Singapore (No3)

(ISI information updated March 21 2008)

Top ten countries of BEM and dual BEM

5

FEM USA China Germany France UK Japan India Tai

wan Turkey Italy (No8)

Meshless methods China USA Singapore Germany UK Taiwan Japan

Portugal Slovakia Australia (No6) FDM China USA Japan India France Taiwan Canada UK Italy South Korea (No6) (ISI information updated March 21 2008)

Top ten countries of FEM FDM and Meshless methods

6

BEM Zhang C (Germany) Sapountzakis E J (Greece) Sladek J and Sladek V (Slovakia twin) Chen J T (Taiwan Ocean Univ) 119 SCI papers gt 545 citing Mukherjee S (USA) Tanaka M (Japan) Dual BEM (Made in Taiwan) Aliabadi M H (UK Imperial College London) Chen J T (Taiwan Ocean Univ) Chen K H(Taiwan Ilan Univ) Power H (UK Univ Nottingham) (ISI information updated March 21 2008)

Active scholars on BEM and dual BEM

7

USA 劉毅軍教授

NTOUMSV Taiwan 海洋大學陳正宗終身特聘教授

北京清華大學工程力學系 -姚振漢教授

高海大造船系 -陳義麟博士

台大土木系 -楊德良終身特聘教授

宜蘭大學土木系陳桂鴻博士

北京清華姚振漢教授提供

Top 25 scholars on BEMBIEM since 2001

8

Number of Papers of FEM BEM and FDM

(Data form Prof Cheng A H D)

126

9

有限元素成長史

0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M

邊界元素成長史

0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel

Hadamard kernelBEM (no crack)

Dual BEM (crack)Small scale

Large scale

Early

Late

351

FMM(degenerate kernel)

NTOUMSV

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 amp large scale problems

NTUCE

12

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

13

BEM trap Why engineers should learn

mathematics Well-posed Existence Unique

Mathematics versus Computation

equivalent 馮康定理

Some examples

14

Nonuniqueness in numerical methods Nonlinear equation (spurious root) Finite difference method

spurious eigenvalue Finite element method amp meshless methods

spurious mode Boundary element method

spurious eigenvalues

fictitious frequency Boundary element method

degenerate scale

15

Nonuniqueness in solving nonlinear Eq Nonlinear equation (spurious root)

16

Why spurious solution occurs

2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗

兩邊平方後整理

再ㄧ次兩邊平方後整理

17

Nonuniqueness in FDM for ODE Finite difference method

solve first-order ODE

using Euler scheme (Greenberg 1998)

18

假根浮根溢根(Spurious Eigenvalue) 用中間差分的方法來逼近處理

2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -

igrave =iumliumliacuteiuml cent= + = -iumlicirc

0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19

假根浮根溢根(Spurious Eigenvalue)

h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -

igrave =iumliumliacuteiuml cent= + = -iumlicirc=

2xexact solution e -

1 2 3 4

-75

-5

-25

25

5

75

20

Nonuniqueness in FDM for eigenproblems Finite difference method

solve eigenproblem (S Zhao 2007)

spectral type

nonspectral type

rod beam and membrane

21

Nonuniqueness in FEM and meshless method

Hour glass mode (solid mechanics)

shear locking

incompressible (solid propellant grain) Solid mechanics

incompressible flow Fluid mechanics

reduced integration

Edge element-divergence free (electromagnetics)

22

Solid mechanics (spurious mode)

UCLA J S Chen 2008

Physics Mathematics

23

Nonuniqueness in BEM for degenerate boundary BEM with degenerate boundary

1 2

3

4

56

7

8

Cutoff wall crack Thin airfoil

24

What Is Boundary Element Method

NTUCE

1 2

3

45

6

1 2

geometry nodethe Nth constantor linear element

N

西醫郎中

25

Dual BEM

Why hypersingular BIE is required

(Two ways since 1986)

NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910

Artifical boundary introduced

BEM

Multi-domain

Dual integral equations needed

Dual BEM

Single-domain

Degenerate boundary

26

Some researchers on Dual BEM (1012)Chen (1986) 544 citings in total

Hong and Chen (1988 ） 78 citings ASCE EM

Portela and Aliabadi (1992) 212 citings IJNME

Mi and Aliabadi (1994)

Wen and Aliabadi (1995)

Chen and Chen (1995) 新竹清華 Yao (2005) 北京清華黎在良等 --- 斷裂力學邊界數值方法 (1996) 周慎杰 (1999)

Chen and Hong (1999) 88 citings ASME AMR

Niu and Wang (2001)

Kuhn G Wrobel L C Mukherjee S Tuhkuri J Gray L J

Yu D H Zhu J L Chen Y Z Tan R J hellip

NTUCE

cite

27

Dual Integral Equations by Hong and Chen(1984-1986)

NTUCE

Singular integral equation Hypersingular integral equation

Cauchy principal value Hadamard principal value

(Mangler principal value)Boundary element method Dual boundary element method

normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node

the Nth constantor linear element

un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29

The number of constraint equation is not enough to

determine coefficients of p and q

Another constraint equation is obtained by differential operator

axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2

How to get additional constraints

30

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

31

Successful experiences since 1986 (degenerate boundary)

32

Solid rocket motor (Army 工蜂火箭 )

33

X-ray detection ( 三溫暖測試 )

Crack initiation crack growth

Stress reliever

34

FEM simulation

35

Stress analysis

36

BEM simulation (Army)

37

Shong-Fon II missile (Navy)

38

V-band structure (Tien-Gen missile)

39

FEM simulation

40

41

Seepage flow (Laplace equation)

Sheet pileCutoff wall

42

Meshes of FEM and BEM

43

FEM (iteration No49) BEM(iteration No13)

Initial guessInitial guess

After iteration After iteration

Remesh areaRemesh line

Free surface seepage flow using hypersingular formulation

44

Incomplete partition in room acoustics(Helmholtz equation)

U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45

Water wave problem with breakwaterWater wave problem with breakwater(modified Helmholtz equation)(modified Helmholtz equation)

Free water surface S

x

Top view

O

y

zO

xz

S

breakwater

breakwater

oblique incident water wave 0)~()~( 22 xuxu

46

Reflection and Transmission

000 040 080 120 160 200

kd

000

040

080

120

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

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

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

53

Rank deficiency due to degenerate boundary and rigid body mode (SVD)

PhysicsMathematics 2d

SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12

Left unitary matrix Right unitary matrix

U

Spurious True

L

T

M

Rigid body mode

left unitary vector UK-1（）

-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列

left unitary vector LK-1（）

-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列

right unitary vector MK-1（）

000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列

right unitary vecto TK-1（）

-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列

1 spurious mode(fictitious mode)

(mathematics)

1 true mode rigid body mode

(physics)

54

SVD Technique (Google searching)

nnnmmmnm VUC

][][][][

[C] SVD decomposition

[U] and [V left and right unitary vectors

nm

nm

n

00

00

0

0

][ 1

11 nn

55

Physical meaning of SVD

1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後

假根真根Chen et al 2002 Int J Comp Numer Anal Appl

先拉再轉先轉再拉

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

0001

001

01

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


Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59

Nonuniqueness in BEM for exterior acoustics

BEM for exterior acoustics

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

m

k

60

Radiation and scattering problems

Nonuniform radiaton scattering

1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution

Strategy of adaptive BEM

Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][

HypersingularEquation

tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

Numerical solution BEM Numerical solution FEM

64 ELEMENTS 2791 ELEMENTS

Nonuniform radiation Dirichlet problem

2ka

9

64

Numerical phenomena(Fictitious frequency)

0 2 4 6 8

-2

-1

0

1

2UT method

LM method

Burton amp Miller method

t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9

A story of PhD students

65

Nonuniqueness in BEM for degenerate scale BEM with degenerate scale Gamma contour Critical value Logarithmic capacity Solvability of integral equations Invertibility of integral operator

66

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

Stokes Flowbiharmonic

TorsionLaplace

67

Nonuniqueness in BEM for multiply connected domain problem

Spurious eigensolution

68

Numerical phenomena (2-D)(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

Tlt9447gt

T T rue e igenvalues

Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt

S Spurious e igenvalues

ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a

BIEM Experiment Inner (spurious) ABAQUS

1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）

9961 990 helliphelliphellip


呂學育博士林羿州Fillipi JSV

Spuriouseigenvalue

70

TreatmentsSVD updating term


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-70

a

b

1B

2B

71

SVD structure for four influence matrices

U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM

1 spurious mode fictitious mode (mathematics)

1 true mode rigid body mode (physics)

The same

The same

The same The same

72

SVD updating technique ( 去蕪 [ ] 存精 ( ) 術 )

U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same[ ]()

73





Some examples

74

馮康定理同一個物理或工程問題可以有各種不同

的數學模式來模擬這些數學模式在物理上是等效的 (Equivale

nt) 但在離散後數值的實踐 (Implementation) 就不見得等效

唯有在不同的數值方法中儘量保有問題的基本特徵方是做計算數學與計算力學的最高指導原則 10524251052425105242510524251052425105242510524251052425105242510524251052425105242510524251052425105242510524251052425

104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions

bull Review of numerical methods ( 中醫式的工程分析法 )

bull Review of nonuniqueness problems in numerical method

s

bull Successful experiences in the engineering applications wi

th degenerate boundaries were demonstrated

bull Nonuniquness due to degenerate boundary degenerate s

cale spurious eigenvalue fictitious frequency is shown

bull SVD structures for the nonuniqueness are examined

76

Acknowledgementbull 台大土木系洪宏基終身特聘教授bull 中山科學院王政盛博士bull 高海大造船系陳義麟博士

bull 宜大土木系陳桂鴻博士 bull 中華技術學院機械系李為民博士bull 中華技術學院機械系呂學育博士bull 中山科學院全湘偉博士bull NTOUMSV group members

77

The End

Thanks for your kind attention

78

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httpindntouedutw~msvlabhttp140121146149

E-mail jtchenmailntouedutw

79

2

Overview of numerical methods ( 中醫式的工程分析法 ) Nonuniqueness problems - review

BEM failure (mathematical degeneracy) Degenerate boundary (No subdomain and no hypersingularity) Degenerate scale True and spurious eigensolution (interior prob) Fictitious frequency (exterior acoustics)

Conclusions

Outline

3




Numerical M ethods


Domain


開刀把脈

針灸


4

BEM






5






6



7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

3




Numerical M ethods


Domain


開刀把脈

針灸


4

BEM






5






6



7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

4

BEM






5






6



7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

5






6



7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

6



7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

7

USA 劉毅軍教授








8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

8



126

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

9


0

1000

2000

3000

4000

5000

6000

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

FE

M


0

200

400

600

800

1000

1200

1400

1600

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Year

Num

ber

of p

aper

s on

BE

M

March 21 2008

Cauchy kernel



Large scale

Early

Late

351


NTOUMSV

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

10

Advantages of BEM



北京清華

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

11

BEM and FEM



r area than FEM


NTUCE

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

12




inherent weakness


13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

13





Some examples

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

14






degenerate scale

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

15


16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

16


2 2

2 2

2

6 2 1

( 6 2 ) (1 )

( 3 5) (2 )

9 34 25 0

( 1)(9 25) 0

251

9

x x

x x

x x

x x

x x

x or x

1 1 6 2 1 1 1 ( )

25 25 252 6 2 1 1 ( )

9 9 9

Ox

x

K

spu

tru

riou

e

s

國中數學經驗



17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

17




18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

18


2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

n nn

n n n

y x y xy x

hhy x y x y x

y

y y hy h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

THORN - = -


0y hcent

X

y(x)

x0 x5 x1 x2 x3 x4

0 h 2h 3h 4h 5h

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

19


h=005 2 0 1y x y x y( ) ( ) ( )cent =- =

1 1

1 1

0

1 0 0

24

1

1 2

0 05

n nn

n n n

solution of FDM

y x y xy x

hhy x y x y x

y

y y hy h

h

( ) ( )( )

( ) ( ) ( )

+ -

+ -

-cent =

- = -



1 2 3 4

-75

-5

-25

25

5

75

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

20



spectral type

nonspectral type


21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

21



shear locking



reduced integration


22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

22


UCLA J S Chen 2008

Physics Mathematics

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

23


1 2

3

4

56

7

8


24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

24


NTUCE

1 2

3

45

6

1 2


N

西醫郎中

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

25

Dual BEM



NTUCE

1 2

3

4

56

7

8

1 2

3

4

56

7

8

910


BEM

Multi-domain


Dual BEM

Single-domain

Degenerate boundary

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

26








Niu and Wang (2001)



NTUCE

cite

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

27


NTUCE




normal

boundarydegenerate

boundary

1969 1986 2008

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

28

Degenerate boundary

geometry node


un0

un0

un0

u 1 u 1(00)

(-105)

(-1-05)

(105)

(1-05)

1 2

3

4

56

7

8 [ ] [ ] U t T u

[ ] [ ] L t M u

N

1693-0335-019001904450703044503350

0334-1693-281028100450471034700390

00630638-193119316380063008100810

00630638-193119316380063008100810

04710045-281028106931335003903470

07030445019001903350693133504450

04710347054005400390335069310450

0335-0039054005403470471004506931

][

U

-1107464046402190490021901071

1107-7850785400000588051909270

088813263261888092709270

088813263261888092709270

0588000078507850107192705190

0490021946404640107110712190

0588051932103210927010710000

1107092732103210519058800000

][

T

5(+) 6(+) 5(+) 6(-)

5(+)6(+)

5(+)6(+)

n s( )

0805464046406120490061208050

0805347034700000184051909270

088814174171888051105110

0888-1417-4171888051105110

0184000034703470805092705190

0490061246404640805080506120

0184051945804580927080500000

0805092745804580519018400000

][

L

00041600-400040002820235028206001

1600-0004000100013331205006208000

0715-3765-000800087653715085308530

07153765000800087653715085308530

0205-1333-000100010004600180000610

0236-0282-400040006001000460012820

0205-0061-600060008000600100043331

1600-0800-600060000610205033310004

][

M

5(+) 6(+)5(+) 6(-)

5(+)6(-)

5(+) 6(-)

n x( ) n x( )

n s( )

dependency

Nonuniqueness

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

29




axwhenqpaaf

qpxxQaxxf

)(

)()()( 2

axwhenpaf

pxQaxxQaxxf

)(

)()()()(2)( 2


30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

30


(1984)

crack

BEMCauchy kernel

singular

DBEMHadamard

kernelhypersingular

FMM




1888

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

31


32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

32


33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

33



Stress reliever

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

34

FEM simulation

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

35

Stress analysis

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

36


37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

37


38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

38


39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

39

FEM simulation

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

40

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

41



42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

42


43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

43






44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

44


U T L Mm ode 1

m ode 2

m ode 3

000 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

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

000 005 010 015 020000

005

010

005 010 015 020

005

5876 H z 5872 H z

14437 H z 14443 H z

15173 H z 15162 H z

b

a

e

c

2 2 0u k u t0

t=0

t=0

t=0

t=0

t=0

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

45



x

Top view

O

y

zO

xz

S

breakwater

breakwater


46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

46


000 040 080 120 160 200

kd

000

040

080

120

lRl a

nd lT

l


R

T

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

47

Cracked torsion bar

T

da

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

48

IEEE J MEMS

Comb drive

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

49

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

50

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

51

Is it possible

No hypersingularity

No subdomain

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

52

Dual BEM


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

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

53



SC

C

-12 -8 -4 0 4 8 12

-12

-8

-4

0

4

8

12


U

Spurious True

L

T

M

Rigid body mode


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-07

-06

-05

-04

-03

-02

-01

0

1 2 3 4 5 6 7 8 9 10

1數列


000501

01502

02503

03504

045

1 2 3 4 5 6 7 8 9 10

1數列


-04

-035

-03

-025

-02

-015

-01

-005

0

1 2 3 4 5 6 7 8 9 10

1數列


-08

-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


-06

-04

-02

0

02

04

06

08

1 2 3 4 5 6 7 8 9 10

1數列


(mathematics)


(physics)

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

54


nnnmmmnm VUC

][][][][



nm

nm

n

00

00

0

0

][ 1

11 nn

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

55


1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後



56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

56



PPPMMMPMU ][][][][




Nd=5 Nd=5Nd=4

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

57

0 2 4 6 8

k

0001

001

01

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



Sub domain

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

58k=314 k=382

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=448

UT BEM+SVD

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

-0 8 -0 6 -0 4 -0 2 0 02 04 06 08

-0 8

-0 6

-0 4

-0 2

0

02

04

06

08

k=309

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=384

-08 -06 -04 -02 0 02 04 06 08

-08

-06

-04

-02

0

02

04

06

08

k=450

FEM (ABAQUS)

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

59




a

m

k

e i t

incident wave

e i t e i t

radiation


if ufinite

( )

2 2

if u finite lim00

m

k

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

60



1)( au0)( au

Drruk )( 0)()( 22

32

5

Drruk )( 0)()( 22

2

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

61

Errorestimator

Error estimator

SolutionSolution


Miller ampBurton

SingularEquation

ut Mk

iTL

k

iU

~~

][][

tu UTUT~~

][][


tu LMLM~~

][][

ut ut

21

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

62

BEM FEM

Adaptive Mesh

- 1 - 1 0 1 1

- 1

- 1

0

1

1

5

DtN interface

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

63

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150

-150 -100 -050 000 050 100 150

-150

-100

-050

000

050

100

150




2ka

9

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

64


0 2 4 6 8

-2

-1

0

1

2UT method

LM method


t(a0)

1)( au0)( au

Drruk )( 0)()( 22

9

1)( au0)( au

Drruk )( 0)()( 22

9


65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

65


66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

66


Error ()of

torsionalrigidity

a

0

5

125

da


T

da



TorsionLaplace

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

67



68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

68



1E-080

1E-060

1E-040

1E-020

de

t|SM

|


Tlt9447gt


Tlt10370gt

Tlt10940gt

Tlt9499gt

Tlt9660gt

Tlt9945gt

Slt9222gt

Slt6392gt

Slt11810gt


ma 1

mb 50

1B

2B

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

69


x

y

z

a

05a

04a


1110 113 85357

2012 204 18649 （ 2 ）

2771 279 20985

3649 364 24775 （ 2 ）

4385 441 4263 2745

6421 640 6100 30505 （ 2 ）

7826 784 7820 3333

8492 854 8525 35002

9126 907 37471 （ 2 ）

9313 933 3993 （ 2 ）




Spuriouseigenvalue

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

70



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

][


a

b

1B

2B

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

71


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same

The same The same

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

72


U TL M

11

0 ULUT

11

0 TMUT

11

0 ULLM

11

0 TMLM



The same

The same


73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

73





Some examples

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

74





104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965 104996510499651049965104996510499651049965104996510499651049965104996510499651049965104996510499651049965

75

Conclusions



s






76



77

The End


78




79

75

Conclusions



s






76



77

The End


78




79

76



77

The End


78




79

77

The End


78




79

78




79

79

Documents

1 Nonuniqueness problems in numerical methods J T Chen( 陳正宗 ), Life-time Distinguished Prof. Taiwan Ocean University August 8, 11:00-12:00 2008 高海大旗津校區,