M M S S V V 0 MSVLAB, HRE, NTOU 博士班資格考口試報告報告人 : 李應德先生指導教授 : 陳正宗教授時間 : 2007 年 07 月 27 日地點 : 河工二館

MMSS VV

1MSVLAB, HRE, NTOU 博士班資格考口試報告

報告人 : 李應德先生指導教授 : 陳正宗教授時間 : 2007 年 07 月 27 日地點 : 河工二館 307 室

Null-field Integral Equation Approach Using Multipole Expansion

and Their Applications

博士班資格考口試報告

MMSS VV


Outlines

1. Introduction

2. Problem statements and formulation

3. Special treatment for some problems

4. Preliminary results

5. Multipole expansion

6. Further works

MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Numerical methods

Numerical methods Numerical methods

Finite Difference Method

Finite Difference Method

Finite Element Method

Finite Element Method

Boundary Element Method

Boundary Element Method Meshless Method Meshless Method

operation pulse acupuncture

MMSS VV


Pitfalls of conventional BEM / BIEM

Pitfalls of conventional BEM / BIEM

Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed systemIll-posed systemConvergence Convergence raterate

Bump contour

Limit process

Guiggiani (1995)

Gray and Manne (1993)

Fictitious BEM

T-matrix

Achenbach et al. (1988)

Waterman (1965)

Relative quantity

σ BEM

Kisu (1988)

Sladek (1991)

Linear order

Quadratic order

Cubic order

MMSS VV


Literature review (circular boundaries)

Key point Main application Author

Conformal mapping Torsion problemIn-plane electrostaticsAnti-plane elasticity

Chen & Weng (2001)Emets & Onofrichuk (1996)Budiansky & Carrier (1984)Steif (1989)Wu & Funami (2002)Wang & Zhong (2003)

Bi-polar coordinate Electrostatic potentialElasticity

Lebedev et al. (1965)Howland & Knight (1939)

Möbius transformation Anti-plane piezoelectricity & elasticity

Honein et al. (1992)

Complex potential approach Anti-plane piezoelectricity Wang & Shen (2001)

Those analytical methods are only limited to doubly connected regions.

Analytical solutions for problems with circular boundaries

MMSS VV


Literature review (Fourier series )

Author Main application Key pointLing

(1943)

Torsion of a circular tube

Caulk et al.

(1983)

Steady heat conduction with circular holes

Special BIEM

Bird and Steele

(1992)

Harmonic and biharmonic problems with circular holes

Trefftz method

Mogilevskaya et al.

(2002)

Elasticity problems with circular holes or inclusions

Galerkin method

However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary.

Fourier series approximation

MMSS VV


Literature review (Fourier series )

Author Main applicationSloan et al.

(1975)

Prove that it is equivalent to iterated Petrov-Galerkin approximation

Kress

(1985)

Prove it combined with integral equation

have convergence of exponential order

Chen et al.

(2005)

Applied it to solve engineering problem with circular boundaries

Chen et al.

(2006)

Link Trefftz method and method of fundamental solutions

However, its application in practical problems seem to have taken a back seat to other methods.

Degenerate kernel approximation

MMSS VV


Literature review (FMM )

Author Main application

Rokhlin (1983) Potential theory (First introducing)

Amini and Profit (1999) Scattering theory

Chen and Chen (2004) 2-D exterior acoustics

Liu et al. (2006) Combine with MFS to solve potential problem

MMSS VV


Present approach

)()()( sdBsxB ),( xsK

),( xsK e

Fundamental solution

No principal value

Advantages of present approach1. No principal value2. Well-posed system3. Exponential convergence4. Free of mesh

Degenerate kernel

CPV and HPV

xsxsK

xsxsKe

i

),,(

),,(

),( xsK i

sx ln

MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Problem statements

)()( 22 xuk

)()( 44 xu

)())(()(~

2

~xuGxuG

B0

B1

B2

B3

Bi

B4

a0

a1

a2

a3

a4

ai

x

y

0)( xuGoverning equation:

)(2 xu

Helmholtz problem:

Laplace problem:

Biharmonic problem:

BiHelmholtz problem:

Elasticity problem:

)(4 xu

MMSS VV


Inclusion problems

A circular bar with circular holes Each circular inclusion problem

B0

B1

B2

B3

Bi

B4

II11 ,

II22 ,

Ii

Ii ,II

33 ,

II44 ,

MM44 ,

B0

B1

B2

B3

Bi

B4

MM11 ,

MM22 ,

Mi

Mi ,MM

33 ,

M0

Satisfy 0)(2 xu

MMSS VV


Interior case Exterior case

x

xx

x

x x

Degenerate (separable) formDegenerate (separable) form

xsdBstxsUsdBsuxsTxuBB

),()(),()()(),()(2

BxsdBstxsUVPRsdBsuxsTVPCxuBB

),()(),(...)()(),(...)(

BcBB

xsdBstxsUsdBsuxsT ),()(),()()(),(0

B

BIE and null-field integral equation

s

s

n

sust

n

xsUxsT

rxsxsU

)()(

),(),(

lnln),(

c c

MMSS VV


Fundamental solution and Four kernels

),( xsU

),( xsM),( xsL

)(4

),( )1(0 krH

ixsU

rrxsU ln8

1),( 2

))]()((2

)()([8

),( 00002riIrKriJrY

ixsU

rxsU ln2

1),(

Helmholtz problem:

Laplace problem: Biharmonic problem:

BiHelmholtz problem:

Elasticity problem:

2

ln)43()1(8

1),(

r

yyr

GxsU ki

ki

Relationship of four kernels:

),( xsT

Fundamental solutions:

sn

sn

xn

xn

MMSS VV


NumericalAnalytical

Flowchart of present approach

The problem with circular boundaries

Null-field integral equation

Degenerate kernel for fundamental solution

Fourier series expansion for boundary density

Adaptive observer system in the boundary integrations

Collocating the point to boundary and matching boundary conditions

Linear algebraic system

Obtain the unknown Fourier coefficients

Boundary integral equation for the domain point

Displacement field

MMSS VV


Degenerate kernel and Fourier series

,,,2,1,,)sincos()(1

0 NkBsnbnaas kkn

kn

kn

kk

,,,2,1,,)sincos()(1

0 NkBsnqnpps kkn

kn

kn

kk

s

Ox

R

kth circularboundary

cosnθ, sinnθboundary distributions

,),(cos1

ln),;,(

,),(cos1

ln),;,(

),(

1

1

RmR

mRU

RmRm

RRU

xsU

m

m

e

m

mi

eU

x

iU

Expand fundamental solution by using degenerate kernel

Expand boundary densities by using Fourier series

In the real computation

M number of terms

MMSS VV


Keypoint of deriving degenerate kernels

1

1

1

1

m

mxmx

xdxx

1ln

1

1Laplace problem:

Helmholtz problem:

Elasticity problem:

221

22 )cos(2

)sinsin()coscos(

)sincos()sincos(

11

r

iyy

RR

RiR

iRRizz sx

),)1(cos(1

)cos(2

)coscos(

0222

1

mmR

RR

R

r

ym

m

),)1sin((1

)cos(2

)sinsin(

0222

2

mm

R

RR

R

r

ym

m

(Addition theorem)

m

mm mkJkRHkrH ))(cos()()()( )1()1(0

First derived

MMSS VV


collocation pointcollocation point

0 , 01 , 1k , k2 , 2

Adaptive observer system

MMSS VV


Collocation method and linear algebraic system

iI

I

M

M

10

II

MM

b

0

0

0

ψ

ψ

μ0μ0

0I0I

UT00

00UT

MNN

MN

MN

MN

MM

MN

MM

M

TTT

TTT

TTT

T

10

11110

00100

][

k

2

1

k

μ00

0μ0

00μ

][μ

)coscos()(

)coscos()(

)coscos()(

)coscos()(

12120

220

220

110

iM

ix

iM

iyi

iM

ix

iM

iyi

iix

iiyi

iix

iiyi

i

ee

ee

ee

ee

b

MNN

MN

MN

MN

MM

MN

MM

M

UUU

UUU

UUU

U

10

11110

00100

][

B0

B1

B2

B3

Bi

B4

Index of collocation circle

Index of routing circle

MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Transformation of tensor components

x

1

234

1’

2’

3’

4’

1, 2: transformed normal and tangential components

3, 4: original normal and tangential componentss

)()(),()()(),()(

2 sdBstxsLsdBsuxsMn

xuBB

x

MMSS VV


Domain superposition

Incident SH-wave

or

hard: 0

soft 0:

t

u

Incident SH-wave

(a) Incident wave field

or I It u

or R I R It t u u

(b) Radiation field

+=

MMSS VV


Image method for half-plane problem

(a) Real problem

Incident-SH wave

t=0

t=0

t=0

Incident-SH wave

t=0

t=0

Image incident-SH wave

(b) Extended problem

Image

MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Example 1: A circular bar with an eccentric inclusion

R1

R0

ex

3.001 RR

6.00 Rex

Ratio:

Torsional rigidity:

N

kB kD k

dBn

dDyxG1

22 )(

IMT GGG

0

1

GT : total torsion rigidity

GM : torsion rigidity of matrix

GI : torsion rigidity of inclusion

MMSS VV


Results of Example 1

0 10 20 30N u m b er o f F o u rier series

0.9624

0.9628

0.9632

0.9636

Tor

sion

al r

igid

ity

(2G

/ R 04 )

0 200000 400000 600000 800000 1000000

S h ea r m o d u lu s ()

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Tor

sion

al r

igid

ity

(2G

/ R 04 )

M u sk h e lish v ili fo rm u la

P re se n t m e th o d

Torsional rigidity versus number of Fourier series terms

Torsional rigidity versus shear modulus of inclusion

MMSS VV



Torsional rigidity of a circular bar with an eccentric inclusion

MMSS VV


Example 2: (limiting case)A circular bar with one circular hole

R1=0.3

R0=1

ex=0.5

0.10

01

MMSS VV


Torsional rigidity of a circular bar with an eccentric hole


MMSS VV


Stress calculation

t

0

tm

External diameter of the tube

D:

Dt

t

ttp m

tm: The maxium wall thickness

(eccentricity)

MMSS VV


Stress calculationalong outer and inner boundary

at boundaries for λ=0.3 and p=0.4z

(0.0%)

(0.1%)

(0.0%)

(0.0%)

(0.4%)

(0.0%)

(0.3%)

(0.0%)

(1.5%)

(0.6%)

MMSS VV


Stress calculationfor point in the center line

z alnog lines and for λ=0.3 and p=0.40

(0.0%)(0.1%)

(0.1%)

(0.1%)

(0.1%)

(0.3%)

(0.0%)(0.2%)

(0.5%)

(0.5%)

(0.0%)

(0.6%)

0

MMSS VV


Example 3: Four cylinders

a 2b

2b x

y

Incident wave

0);,,(2 tzyx

})(),(Re{),,,( tiezfyxtzyx

kh

hzkigAzf

cosh

)(cosh)(

0),()( 22 yxk

MMSS VV


Results of the hydordynamic force

0 2 4 6 8 10

k a

0

0.2

0.4

0.6

0.8

1

1.2

1.4

f (j)

C y lin d er 1C y lin d er 2C y lin d er 3C y lin d er 4

Perrey-Debain et al. Present method

This work was done some years ago, but my recollection was that we simply picked one example for comparison for scattering by multiple objects. We did find, however, that although the series solution in the paper is correct, some of the images in the Linton & Evans paper were incorrect. If you are using the images for comparison with your own work, it might be a good idea to check against the series solutions instead.

MMSS VV


Example 4: Stress concentrated factor

21 , tt

SS21 , tt

SS

hh tt 21 ,

=

+

MMSS VV


Deformation

=+

undeformed

deformed

cos

2

)1(3cos1

4cos

)1(2

222

1 G

Saa

G

Sa

G

Su

sin

23sin1

4sin

2

)21(2

222

2 G

Saa

G

Sa

G

Su

0rr

2cos2SS 0 r

MMSS VV


Example 5: Lamé problem

b

a

Pe

Pi

B1

B2

MMSS VV


Deformation

u n d efo rm edd efo rm ed

0111

2

1 sincos)1(2

1cos

)1(2

)21(cos

)1(4

)21(cos

)1(4

)21(),( a

bb

ac

ba

a

G

P

G

Pu ie

0111

2

2 cossin)1(2

1sin

)1(2

)21(sin

)1(4

21sin

)1(4

21),( a

bb

ac

ba

a

G

P

G

Pu ie

,)(

)()()(

222

20

220

22

ab

PbPaPPba iirr

,

)(

)()()(

222

20

220

22

ab

PbPaPPba ii

0)( r

MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Our goals

x

1

234

1’

2’

3’

4’

Tensor transformation

),(

),( jjR

x1

y1

xj

yj

xi

yi

x2

y2

),( 11 R

),( iiR

),( 22 RB1

B2

Bi

Bj

i

12

i

j1

2

j

os

os

os

os

Adaptive observer system

MMSS VV


Multipole expansion

R

c

mmmm

E

mmmm

I

RmkYkiJkRJi

RU

RmkRYkRiJkJi

RUxsU

,)),(cos()]()()[(2

),;,(

,)),(cos()]()()[(2

),;,(),(

LtikR

M

Mm

timcm

mtikE eekDHieL

ixsU c

1

)cos()()cos( )()(2

),(

Expanding the kernel function

cD

Lt 2where

MMSS VV


Fast algorithm

NNNNNNNN

NN

NN

VDUVDUVDU

VDUVDUVDU

VDUVDUVDU

A

2211

2222221212

1121211111

][

NNNNN

N

N

N V

V

V

DDD

DDD

DDD

U

U

U

00

00

00

00

00

00

2

1

21

22221

11211

2

1

Generalized Minimum RESidual Method (GMRES) Solver:

Generalized Conjugate RESidual Method

MMSS VV


Example 1: Two semi-circular canyons

Incident SH wave Reflected SH wave

3a

t=0t=0 t=0

t=0t=0

hv

ax

y

MMSS VV


- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x /a

0

1

2

3

4

5

6

7

8

|u |

hvU T fo rm u la tio n

L M fo rm u la tio n

0.10hv

Tsaur et al. Present method

MMSS VV


- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x /a

0

1

2

3

4

5

6

7

8

|u |

h vU T fo rm u la tio n


0.130hv


MMSS VV


0.160hv

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x /a

0

1

2

3

4

5

6

7

8

|u |




MMSS VV


0.190hv

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7x /a

0

1

2

3

4

5

6

7

8

|u |




MMSS VV


Example 2: A circular hole

hv

x

y

h=1.5a

a

t=0

t=0

MMSS VV


0.20hv

- 3 - 2 - 1 0 1 2 3x /a

0

1

2

3

4

5

6

|u |



Lee and Manoogian Present method

MMSS VV


0.230hv

- 3 - 2 - 1 0 1 2 3

x /a

0

1

2

3

4

5

6

|u |




MMSS VV


0.260hv

- 3 - 2 - 1 0 1 2 3

x /a

0

1

2

3

4

5

6

|u |




MMSS VV


0.290hv

- 3 - 2 - 1 0 1 2 3x /a

0

1

2

3

4

5

6

|u |




MMSS VV


Outlines

1. Introduction





6. Further works

MMSS VV


Example 1: Perfect interface to imperfect interface

R1

R0

ex

0

1

MMSS VV


Example 2: Two circular holes under a unified tension

R1

(0,0)

(-d,0)

T

T

s

MMSS VV


Example 3: An array of circular inclusions in an infinite plane

0x

y

0

MMSS VV


Example 4: A half-plane problem with a semi-circular hill

Incident SH wave

a

y

x

MMSS VV


Example 5: Two circular holes under the screw dislocation

⊕

a2

x

y

a1

d

MMSS VV


Example 6: A circular inclusion under the screw dislocation

22 ,G

11,G

⊕a

x

y

a

MMSS VV


Example 7: A circular hole under the edge dislocation

a

x

y

d

MMSS VV


Example 8: A half-plane problem with two alluvial valleys

Incident SH wave

a

3a

a

y

x

Alluvial

Matrix

MMSS VV


Thanks for your kind attention

The End

Documents

M M S S V V 0 MSVLAB, HRE, NTOU 博士班資格考口試報告 報 告 人 : 李應德 先生 指導教授 : 陳正宗 教授 時 間 : 2007 年 07 月 27 日 地 點 : 河工二館

M M S S V V 0 MSVLAB, HRE, NTOU 博士班資格考口試報告報告人 : 李應德先生指導教授 : 陳正宗教授時間 : 2007 年 07 月 27 日地點 : 河工二館