1 Null integral equations and their applications J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan June 04-06, 2007 BEM 29 in WIT Bem29-2007talk.ppt

1

Null integral equations and their Null integral equations and their applicationsapplications

J. T. Chen Ph.D. Taiwan Ocean University

Keelung, Taiwan

June 04-06, 2007BEM 29 in WIT

Bem29-2007talk.ppt

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering

2

Research collaborators Research collaborators

Dr. I. L. Chen Dr. K. H. ChenDr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. LeeDr. S. Y. Leu Dr. W. M. Lee Mr. Y. T. LeeMr. Y. T. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. HsiaoMr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr.P. Y. ChenMr. A. C. Wu Mr.P. Y. Chen Mr. J. N. Ke Mr. H. Z. Liao Mr. J. N. Ke Mr. H. Z. Liao

3URL: http://ind.ntou.edu.tw/~msvlab E-mail: [email protected] 海洋大學工學院河工所力學聲響振動實驗室 nullsystem2007.ppt`

Elasticity & Crack Problem

Laplace Equation

Research topics of NTOU / MSV LAB on null-field BIE (2003-2007)

Navier Equation

Null-field BIEM

Biharmonic Equation

Previous research and project

Current work

(Plate with circulr holes)

BiHelmholtz EquationHelmholtz Equation

(Potential flow)(Torsion)

(Anti-plane shear)(Degenerate scale)

(Inclusion)(Piezoleectricity)

(Beam bending)

Torsion bar (Inclusion)Imperfect interface

Image method(Green function)

Green function of half plane (Hole and inclusion)

(Interior and exteriorAcoustics)

SH wave (exterior acoustics)(Inclusions)

(Free vibration of plate)Indirect BIEM

ASME JAM 2006MRC,CMESEABE

ASMEJoM

EABE

CMAME 2007

SDEE

JCA

NUMPDE revision

JSV

SH wave

Impinging canyonsDegenerate kernel for ellipse

ICOME 2006

Added mass

李應德Water wave impinging circul

ar cylinders

Screw dislocation

Green function foran annular plate

SH wave

Impinging hillGreen function of`circular inc

lusion (special case:staic)

Effective conductivity

CMC

(Stokes flow)

(Free vibration of plate) Direct BIEM

(Flexural wave of plate)

4

Prof. C B Ling (1909-1993)Prof. C B Ling (1909-1993)Fellow of Academia SinicaFellow of Academia Sinica

He devoted himself to solve BVPs

with holes.

PS: short visit (J T Chen) of Academia Sinica 2006 summer `

C B Ling (mathematician and expert in mechanics)

5

OutlinesOutlines

Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation

Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density

Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation

Numerical examplesNumerical examples ConclusionsConclusions

6

MotivationMotivation

Numerical methods for engineering problemsNumerical methods for engineering problems

FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method

BEM / BIEM (mesh required)BEM / BIEM (mesh required)

Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate

Mesh free for circular boundaries ?Mesh free for circular boundaries ?

7

Motivation and literature reviewMotivation and literature review

Fictitious Fictitious BEMBEM

BEM/BEM/BIEMBIEM

Null-field Null-field approachapproach

Bump Bump contourcontour

Limit Limit processprocess

Singular and Singular and hypersingularhypersingular

RegulRegularar

Improper Improper integralintegral

CPV and CPV and HPVHPV

Ill-Ill-posedposed

FictitiFictitious ous

bounboundarydary

CollocatCollocation ion

pointpoint

8

Present approachPresent approach

1.1.No principal No principal valuevalue 2. Well-posed2. Well-posed

3. No boundary-laye3. No boundary-layer effectr effect

4. Exponetial converg4. Exponetial convergenceence

5. Meshless 5. Meshless

(s, x)eK

(s, x)iK

Advantages of Advantages of degenerate kerneldegenerate kernel

(x) (s, x) (s) (s)B

K dBj f=ò

DegeneratDegenerate kernele kernel

Fundamental Fundamental solutionsolution

CPV and CPV and HPVHPV

No principal No principal valuevalue

(x) (s)(x) (s) (s)B

db Baj f=ò 2

1 1( ), ( )

x s x sO O

- -

(x) (s)a b

9

Engineering problem with arbitrary Engineering problem with arbitrary geometriesgeometries

Degenerate Degenerate boundaryboundary

Circular Circular boundaryboundary

Straight Straight boundaryboundary

Elliptic Elliptic boundaryboundary

a(Fourier (Fourier series)series)

(Legendre poly(Legendre polynomial)nomial)

(Chebyshev poly(Chebyshev polynomial)nomial)

(Mathieu (Mathieu function)function)

10

Motivation and literature reviewMotivation and literature review

Analytical methods for solving Laplace problems

with circular holesConformal Conformal mappingmapping

Bipolar Bipolar coordinatecoordinate

Special Special solutionsolution

Limited to doubly Limited to doubly connected domainconnected domain

Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications

Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics

Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics

11

Fourier series approximationFourier series approximation

Ling (1943) - Ling (1943) - torsiontorsion of a circular tube of a circular tube Caulk et al. (1983) - Caulk et al. (1983) - steady heat conducsteady heat conduc

tiontion with circular holes with circular holes Bird and Steele (1992) - Bird and Steele (1992) - harmonic and harmonic and

biharmonicbiharmonic problems with circular hol problems with circular holeses

Mogilevskaya et al. (2002) - Mogilevskaya et al. (2002) - elasticityelasticity pr problems with circular boundariesoblems with circular boundaries

12

Contribution and goalContribution and goal

However, they didn’t employ the However, they didn’t employ the null-field integral equationnull-field integral equation and and degenerate kernelsdegenerate kernels to fully to fully capture the circular boundary, capture the circular boundary, although they all employed although they all employed Fourier Fourier series expansionseries expansion..

To develop a To develop a systematic approachsystematic approach for solving Laplace problems with for solving Laplace problems with multiple holesmultiple holes is our goal. is our goal.

13

Outlines (Direct problem)Outlines (Direct problem)





14

Boundary integral equation Boundary integral equation and null-field integral equationand null-field integral equation

Interior case Exterior case

cD

D D

x

xx

xcD

s

s

(s, x) ln x s ln

(s, x)(s, x)

n

(s)(s)

n

U r

UT

jy

= - =

¶=

¶

¶=

¶

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U dB Dj y= - Îò ò

(x) . . . (s, x) (s) (s) . . . (s, x) (s) (s), xB B

C PV T dB R PV U dB Bpj j y= - Îò ò

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB Dpj j y= - Îò ò

x x

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB D Bpj j y= - Î Èò ò

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U d D BBj y= - Î Èò ò

Degenerate (separate) formDegenerate (separate) form

15

Outlines (Direct problem)Outlines (Direct problem)




Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions

16

Gain of introducing the degenerate Gain of introducing the degenerate kernelkernel

(x) (s, x) (s) (s)B

K dBj f=ò

Degenerate kernel Fundamental solution

CPV and HPV

No principal value?

0

(x) (s)(x) (s) (s)jB

jja dBbj f

¥

=

= åò

0

0

(s,x) (s) (x), x s

(s,x)

(s,x) (x) (s), x s

ij j

j

ej j

j

K a b

K

K a b

¥

=

¥

=

ìïï = <ïïïï=íïï = >ïïïïî

å

åinterior

exterior

17

How to separate the regionHow to separate the region

18

Expansions of fundamental solution Expansions of fundamental solution and boundary densityand boundary density

Degenerate kernel - fundamental Degenerate kernel - fundamental solutionsolution

Fourier series expansions - boundary Fourier series expansions - boundary densitydensity

1

1

1( , ; , ) ln ( ) cos ( ),

(s, x)1

( , ; , ) ln ( ) cos ( ),

i m

m

e m

m

U R R m Rm R

UR

U R m Rm

rq r f q f r

q r f r q f rr

¥

=

¥

=

ìïï = - - ³ïïïï=íïï = - - >ïïïïî

å

å

01

01

(s) ( cos sin ), s

(s) ( cos sin ), s

M

n nn

M

n nn

u a a n b n B

t p p n q n B

q q

q q

=

=

= + + Î

= + + Î

å

å

19

Separable form of fundamental Separable form of fundamental solution (1D)solution (1D)

-10 10 20

2

4

6

8

10

Us,x

2

1

2

1

(x) (s), s x

(s, x)

(s) (x), x s

i ii

i ii

a b

U

a b

=

=

ìïï ³ïïïï=íïï >ïïïïî

å

å

1(s x), s x

1 2(s, x)12

(x s), x s2

U r

ìïï - ³ïïï= =íïï - >ïïïî

-10 10 20

-0.4

-0.2

0.2

0.4

Ts,x

s

Separable Separable propertyproperty

continuocontinuousus

discontidiscontinuousnuous

1, s x

2(s, x)1

, x s2

T

ìïï >ïïï=íï -ï >ïïïî

20-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

Separable form of fundamental Separable form of fundamental solution (2D)solution (2D)

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

Ro

s ( , )R q=

x ( , )r f=

iU

eU

r

1

1

1( , ; , ) ln ( ) cos ( ),

(s, x)1

( , ; , ) ln ( ) cos ( ),

i m

m

e m

m

U R R m Rm R

UR

U R m Rm

rq r f q f r

q r f r q f rr

¥

=

¥

=

ìïï = - - ³ïïïï=íïï = - - >ïïïïî

å

å

x ( , )r f=

21

Boundary density discretizationBoundary density discretization

Fourier Fourier seriesseries

Ex . constant Ex . constant elementelement

Present Present methodmethod

Conventional Conventional BEMBEM

22

OutlinesOutlines





23

Adaptive observer systemAdaptive observer system

( , )r f

collocation collocation pointpoint

24

OutlinesOutlines





25

Vector decomposition technique for Vector decomposition technique for potential gradientpotential gradient

zx

z x-

(s, x) 1 (s, x)(s, x) cos( ) cos( )

2

U ULr

pz x z x

r r f¶ ¶

= - + - +¶ ¶

(s, x) 1 (s, x)(s, x) cos( ) cos( )

2

T TM r

pz x z x

r r f¶ ¶

= - + - +¶ ¶

Special case Special case (concentric case) :(concentric case) :

z x=

(s, x)(s, x)

ULr r

¶=

¶(s, x)

(s, x)T

M r r¶

=¶

Non-Non-concentric concentric

case:case:

(x)2 (s, x) (s) (s) (s, x) (s) (s), x

(x)2 (s, x) (s) (s) (s, x) (s) (s), x

B B

B B

uM u dB L t dB D

uM u dB L t dB D

r r

ff

p

p

¶= - Î

¶¶

= - Î¶

ò ò

ò ò

n

t

nt

t

n

True normal True normal directiondirection

26

OutlinesOutlines





27

{ }

0

1

2

N

ì üï ïï ïï ïï ïï ïï ïï ïï ï=í ýï ïï ïï ïï ïï ïï ïï ïï ïî þ

t

t

t t

t

M

Linear algebraic equationLinear algebraic equation

[ ]{ } [ ]{ }U t T u=

[ ]

00 01 0

10 11 1

0 1

N

N

N N NN

é ùê úê úê ú= ê úê úê úê úë û

U U U

U U UU

U U U

L

L

M M O M

L

whwhereere

Column vector of Column vector of Fourier coefficientsFourier coefficients(Nth routing circle)(Nth routing circle)

0B1B

Index of Index of collocation collocation

circlecircle

Index of Index of routing circle routing circle

28

Physical meaning of influence Physical meaning of influence coefficientcoefficient

kth circularboundary

xmmth collocation point

on the jth circular boundary

jth circular boundary

Physical meaning of the influence coefficient )( mncjkU

cosnθ, sinnθboundary distributions

29

Flowchart of present methodFlowchart of present method

0 [ (s, x) (s) (s, x) (s)] (s)B

T u U t dB= -ò

Potential Potential of domain of domain

pointpointAnalytiAnalyticalcal

NumeriNumericalcal

Adaptive Adaptive observer observer systemsystem

DegeneratDegenerate kernele kernel

Fourier Fourier seriesseries

Linear algebraic Linear algebraic equation equation

Collocation point and Collocation point and matching B.C.matching B.C.

Fourier Fourier coefficientscoefficients

Vector Vector decompodecompo

sitionsition

Potential Potential gradientgradient

30

Comparisons of conventional BEM and present Comparisons of conventional BEM and present

methodmethod

BoundaryBoundarydensitydensity

discretizatiodiscretizationn

AuxiliaryAuxiliarysystemsystem

FormulatiFormulationon

ObservObserverer

systemsystem

SingulariSingularityty

ConvergenConvergencece

BoundarBoundaryy

layerlayereffecteffect

ConventionConventionalal

BEMBEM

Constant,Constant,linear,linear,

quadratic…quadratic…elementselements

FundamenFundamentaltal

solutionsolution

BoundaryBoundaryintegralintegralequationequation

FixedFixedobservobserv

erersystemsystem

CPV, RPVCPV, RPVand HPVand HPV LinearLinear AppearAppear

PresentPresentmethodmethod

FourierFourierseriesseries

expansionexpansion

DegeneratDegeneratee

kernelkernel

Null-fieldNull-fieldintegralintegralequationequation

AdaptivAdaptivee

observobserverer

systemsystem

DisappeaDisappearr

ExponentiaExponentiall

EliminatEliminatee

31

OutlinesOutlines





32

Numerical examplesNumerical examples

Laplace equation Laplace equation (EABE 2005, EABE 2007) (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM200(CMES 2005, ASME 2007, JoM200

7)7) (MRC 2007, NUMPDE revision)(MRC 2007, NUMPDE revision) Membrane eigenproblem Membrane eigenproblem (JCA)(JCA) Exterior acoustics Exterior acoustics (CMAME, SDEE (CMAME, SDEE )) Biharmonic equation Biharmonic equation (JAM, ASME 2006(JAM, ASME 2006)) Plate eigenproblem Plate eigenproblem (JSV )(JSV )

33

Laplace equationLaplace equation

A circular bar under torqueA circular bar under torque

(free of mesh generation)(free of mesh generation)

34

Torsion bar with circular holes Torsion bar with circular holes removedremoved

The warping The warping functionfunction

Boundary conditionBoundary condition

wherewhere

2 ( ) 0,x x DjÑ = Î

j

sin cosk k k kx yn

jq q

¶= -

¶ kB

2 2cos , sini i

i ix b y b

N N

p p= =

2 k

N

p

a

a

ab q

R

oonn

TorqTorqueue

35

Axial displacement with two circular Axial displacement with two circular holesholes

Present Present method method (M=10)(M=10)

Caulk’s data (1983)Caulk’s data (1983)ASME Journal of Applied MechASME Journal of Applied Mechanicsanics

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2-1.5-1-0.500.511.52

Dashed line: exact Dashed line: exact solutionsolution

Solid line: first-order Solid line: first-order solutionsolution

36

Torsional rigidityTorsional rigidity

?

37

Extension to inclusionExtension to inclusion

Anti-plane elasticity problemsAnti-plane elasticity problems

(free of boundary layer (free of boundary layer effect)effect)

38

Two circular inclusions with Two circular inclusions with centers on the centers on the yy axis axis

0 1 2 3 4 5 6 ( in radians)

- 2

0

2

4

Str

esse

s ar

ound

incl

usio

n of

rad

ius

r 1

Mzr /

Izr /

Mz /

Iz /

Hon

ein

Hon

ein

et a

l.et

al. ’

sdat

a (1

992)

’sda

ta (

1992

)

Present method (L=20)Present method (L=20)

Equilibrium of tractionEquilibrium of traction

39

Convergence test and Convergence test and boundary-layer effectboundary-layer effect analysisanalysis

2.04

2.08

2.12

2.16

2.2

2.24

2.28

Str

ess

co

nce

ntra

tion

fact

or

P . S . S te if (1989)Present m ethod

BEM -BEPO 2D

0

11 21 31 41 51 61 71

N um ber of degrees of freedom (nodes)0

0 5 10 15 20 25 30 35

N um ber of degrees of freedom (term s of Fourier series, L)0.01 0.1 1

/r1

1

10

z

/

Paul S . S te if (1989)P resent m ethod (L=10)P resent m ethod (L=20)

BEM -BEP O 2D (node=41)

2d

e

boundary-layer effectboundary-layer effect

40

Numerical examplesNumerical examples

Biharmonic equationBiharmonic equation (exponential convergence)(exponential convergence)

41

Plate problemsPlate problems

1B

4B

3B

2B1O

4O

3O

2O

Geometric data:

1 20;R 2 5;R

( ) 0u s 1B( ) 0s

1 (0,0),O 2 ( 14,0),O

3 (5,3),O 4 (5,10),O 3 2;R 4 4.R

( ) sinu s

( ) 1u s

( ) 1u s

( ) 0s

( ) 0s

( ) 0s

2B

3B

4B

and

and

and

and

on

on

on

on

Essential boundary conditions:

(Bird & Steele, 1991)

42

Contour plot of displacementContour plot of displacement

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

Present method (N=101)

Bird and Steele (1991)

FEM (ABAQUS)FEM mesh

(No. of nodes=3,462, No. of elements=6,606)

43

Stokes flow problemStokes flow problem

1

2 1R

e

1 0.5R

1B

Governing equation:

4 ( ) 0,u x x

Boundary conditions:

1( )u s u and ( ) 0.5s on 1B

( ) 0u s and ( ) 0s on 2B

2 1( )

e

R R

Eccentricity:

Angular velocity:

1 1

2B

(Stationary)

44

0 80 160 240 320 400 480 560 640

0.0736

0.074

0.0744

0.0748

0 80 160 240 320

Comparison forComparison for 0.5

DOF of BIE (Kelmanson)

DOF of present method

BIE (Kelmanson) Present method Analytical solution

(160)

(320)(640)

u1

(28)

(36)

(44)(∞)

Algebraic convergence

Exponential convergence

45

Contour plot of Streamline forContour plot of Streamline for

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Present method (N=81)

Kelmanson (Q=0.0740, n=160)

Kamal (Q=0.0738)

e

Q/2

Q

Q/5

Q/20-Q/90

-Q/30

0.5

0

Q/2

Q

Q/5

Q/20-Q/90

-Q/30

0

46

OutlinesOutlines




Numerical examplesNumerical examples Some findingsSome findings ConclusionsConclusions

47

Some findingsSome findings

`Laplace Helmholtz

LingLing19471947

Analytical Analytical solutionsolution

Bird & Bird & SteeleSteele19911991

房營光房營光 19951995

Analytical solutAnalytical solutionion

Lee & Lee &

ManoogianManoogian19921992

CaulkCaulk19831983

NaghdiNaghdi199199

11


Tsaur Tsaur et al.et al.20042004


Present Present methodmethod

Present method (semi-Present method (semi-analytical)analytical) Tsaur Tsaur et al.et al.

?

48

Stress concentration at point BStress concentration at point B

0.4 0.5 0.6 0.7

b

1.5

2

2.5

3

3.5

Sc

Theta=3*pi/8

Theta=pi/8

Theta=pi/4

Present

Present

method

method

Naghdi’s result

Naghdi’s result

ss

0.4 0.5 0.6 0.7

a

1.5

2

2.5

3

3.5

Sc

0.4 0.5 0.6 0.7

a

1.5

2

2.5

3

3.5

Sc

Steele &

Steele &

Bird

Bird

The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified.

--ASME Applied Mechanics Review

49

A half-plane problem with two alluvial A half-plane problem with two alluvial valleys subject to the incident SH-valleys subject to the incident SH-wavewave

Canyon

Matrix

3aSH-Wave

Tsaur et al. pointed out that Fang made a mistake of misusing the orthogonal relation.

50

Limiting case of two Limiting case of two canyons canyons

Present methodPresent method

Tsaur et al.’s results [103]Tsaur et al.’s results [103]

8/ 102, , / 2 /3IM MI

0 30 60 90

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

0

2

4

6

8

Am

plit

ud

e

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

0

2

4

6

8

Am

plit

ud

e

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

0

2

4

6

8

Am

plit

ud

e

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

0

2

4

6

8

Am

plit

ud

e

51

Inclusion

Matrixh

SH-Wave

a

xy

A half-plane problem with a circular A half-plane problem with a circular inclusion subject to the incident SH-inclusion subject to the incident SH-wave wave

52

Surface displacements of a inclusion Surface displacements of a inclusion problem under the ground surface problem under the ground surface

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

- 3 - 2 - 1 0 1 2 3

0

1

2

3

4

5

6

- 3 - 2 - 1 0 1 2 3

x/a

0

1

2

3

4

5

6

Am

plit

ud

e

0

1

2

3

4

5

6-3 -2 -1 0 1 2 3

Present methodPresent method 2, / 1/ 6, / 2 /3I M I M

Tsaur et al.’s results [102]Tsaur et al.’s results [102]

Manoogian and Lee’s results [62]Manoogian and Lee’s results [62]

0 30 60 90

When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time.

--Private communication

53

OutlinesOutlines





54

ConclusionsConclusions

A systematic approach using A systematic approach using degenerate kdegenerate kernelsernels, , Fourier seriesFourier series and and null-field integranull-field integral equationl equation has been successfully proposed has been successfully proposed to solve Laplace Helmholtz and Biharminito solve Laplace Helmholtz and Biharminic problems with circular boundaries.c problems with circular boundaries.

Numerical results Numerical results agree wellagree well with available with available exact solutions, Caulk’s data, Onishi’s dexact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for ata and FEM (ABAQUS) for only few terms only few terms of Fourier seriesof Fourier series..

55


Four previous results were Four previous results were examined.examined.

..`Laplace Helmholtz`

LingLing19471947


Bird & Bird & SteeleSteele19911991

房營光房營光 19951995


Lee & Lee & ManoogianManoogian

19921992

??? ?

56


Free of boundary-layer effectFree of boundary-layer effect Free of singular integralsFree of singular integrals Well posedWell posed Exponetial convergenceExponetial convergence Mesh-free approachMesh-free approach

57

The EndThe End

Thanks for your kind attentions.Thanks for your kind attentions.Your comments will be highly apprYour comments will be highly appr

eciated.eciated.

URL: URL: http://http://msvlab.hre.ntou.edu.twmsvlab.hre.ntou.edu.tw//

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Documents

1 Null integral equations and their applications J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan June 04-06, 2007 BEM 29 in WIT Bem29-2007talk.ppt