1 A semi-analytical approach for engineering problems with circular boundaries J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan Dec.23, 14:00-14:30,

1

A semi-analytical approach for A semi-analytical approach for engineering problems with circular engineering problems with circular boundariesboundaries

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

Keelung, Taiwan

Dec.23, 14:00-14:30, 2006南台科技大學教學研究大樓 T0111 演講廳

Diff2006.ppt

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering

1u =

1u =-1u =

1u =-

1u = 1u =-

第十五屆微分方程研討會 15th Diff. Equations

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. H. Z. Liao Mr. G. N. KehrMr. H. Z. Liao Mr. G. N. Kehr 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. Y. T. Lee Mr. Y. T. Lee

3

Top 25 scholars on BEM/BIEM since 2001Top 25 scholars on BEM/BIEM since 2001

北京清華姚振漢教授提供 (Nov., 2006)

4URL: 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 ``

Elasticity & Crack Problem李應德

Laplace Equation

Research topics of NTOU / MSV LAB on null-field BIE (2003-2006)

Navier Equation

Null-field BIEM

Biharmonic Equation

Previous research and project

Current work

蕭嘉俊 05’(Plate with circulr holes)

(Stokes’ flow)

BiHelmholtz EquationHelmholtz Equation

沈文成 05’(Potential flow)

(Torsion)(Anti-plane shear)(Degenerate scale)

吳安傑 06’(Inclusion)

(Piezoleectricity)

陳柏源 06’(Beam bending)

李應德Torsion bar(Inclusion)

廖奐禎、李應德Image method

(Green function)

柯佳男Green function of half plane

(Hole and inclusion)

陳佳聰 05’(Interior and exterior

Acoustics)

陳柏源 05’SH wave (exterior acoustics)

(Inclusions)

李為民、李應德(Plate vibration)

ASMEMRC,CMESEABE

ASMEJoM

EABE

CMAME revision

SDEE revision

JCA revision

NUMPDE revision

JSV revision

柯佳男SH wave

Impinging hill

廖奐禎

Degenerate kernel for ellipse

ICOME 2006

5

哲人日已遠典型在宿昔哲人日已遠典型在宿昔C B Ling (1909-1993)C B Ling (1909-1993)

省立中興大學第一任校長

林致平校長( 民國五十年 ~ 民國五十二年 )

林致平所長 ( 中研院數學所 )

林致平院士 ( 中研院 )

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

林致平院士 ( 數學力學家 )C B Ling (mathematician and expert in mechanics)

6

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

7

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 ?

8

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

9

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

(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

10

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)

11

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

12

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

13

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.

14

Outlines (Direct problem)Outlines (Direct problem)





15

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

16

• R.P.V. (Riemann principal value)

• C.P.V.(Cauchy principal value)

• H.P.V.(Hadamard principal value)

Definitions of R.P.V., C.P.V. and H.P.V.using bump approach

NTUCE

R P V x dx x x x. . . ln ( ln )z

1

1

- = -2 x=-1x=1

C P Vx

dxx

dx. . . lim

z zz 1

1 1

1

1 1 = 0

H P Vx

dxx

dx. . . lim

z zz 1

1

2

1

12

1 1 2 = -2

0

0

x

ln x

x

x

1 / x

1 / x 2

17

Principal value in who’s sensePrincipal value in who’s sense

Riemann sense (Common sense)Riemann sense (Common sense) Lebesgue senseLebesgue sense Cauchy senseCauchy sense Hadamard sense (elasticity)Hadamard sense (elasticity) Mangler sense (aerodynamics)Mangler sense (aerodynamics) Liggett and Liu’s senseLiggett and Liu’s sense The singularity that occur when the The singularity that occur when the base point and field point coincide are not base point and field point coincide are not

integrable. (1983)integrable. (1983)

18

Two approaches to understand Two approaches to understand HPVHPV

1

2 2-10

1lim =-2 y

dxx y

H P Vx

dxx

dx. . . lim

z zz 1

1

2

1

12

1 1 2 = -2

(Limit and integral operator can not be commuted)

(Leibnitz rule should be considered)

1

01

1{ } 2

t

dCPV dx

dt x t

Differential first and then trace operator

Trace first and then differential operator

19

Bump contribution (2-D)Bump contribution (2-D)

( )u x

( )u xx

sT

x

sU

x

sL

x

sM

0

0

1( )

2t x

1 2( ) ( )

2t x u x

1( )

2t x

1 2( ) ( )

2t x u x

20

Bump contribution (3-D)Bump contribution (3-D)

2 ( )u x

2 ( )u x

2( )

3t x

4 2( ) ( )

3t x u x

2( )

3t x

4 2( ) ( )

3t x u x

x

xx

x

s

s s

s0`

0

21

Outlines (Direct problem)Outlines (Direct problem)




Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions

22

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

23

How to separate the regionHow to separate the region

24

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

25

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

26-20 -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=

27

Boundary density discretizationBoundary density discretization

Fourier Fourier seriesseries

Ex . constant Ex . constant elementelement

Present Present methodmethod

Conventional Conventional BEMBEM

28

OutlinesOutlines





29

Adaptive observer systemAdaptive observer system

( , )r f

collocation collocation pointpoint

30

OutlinesOutlines





31

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

32

OutlinesOutlines





33

{ }

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

34

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

35

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

36

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

37

OutlinesOutlines





38

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) Eigen problem Eigen problem (JCA revision)(JCA revision) Exterior acoustics Exterior acoustics (CMAME, SDEE revision(CMAME, SDEE revision)) Biharmonic equation Biharmonic equation (JAM, ASME 2006(JAM, ASME 2006)) Plate vibration Plate vibration (JSV revision)(JSV revision)

39

Laplace equationLaplace equation

Steady state heat conduction Steady state heat conduction problemsproblems

Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane An infinite medium under antiplane

shearshear Half-plane problemsHalf-plane problems

40

0 90 180 270 360

Degr ee ( )

0

1

2

3

Rel

ativ

e er

ror

of f

lux

on t

he

sm

all

circ

le (

%)

B E M -B E P O2 D (N = 2 1 )

P r es ent met hod (M = 1 0 )

Tr efft z met hod (N T= 2 1 )

M FS (N M = 2 1 ) (a1 '= 3 .0 , a2 '= 0 .7 )

Relative error of flux on the small Relative error of flux on the small circlecircle

41

Convergence test - Parseval’s sum for Convergence test - Parseval’s sum for Fourier coefficientsFourier coefficients

0 4 8 12 16 20

Ter ms of Four ier s er ies (M )

1 0

1 1

1 2

1 3

1 4

1 5

Par

sev

al's

sum

0 4 8 12 16 20

Ter ms of Four ier s er ies (M )

2

2.4

2.8

3.2

3.6

Par

sev

al's

sum

22 2 2 2

00

1

( ) 2 ( )M

n nn

f d a a bp

q q p p=

+ +åò B&Parseval’s Parseval’s

sumsum

42





43

Electrostatic potential of wiresElectrostatic potential of wires

Hexagonal Hexagonal electrostatic electrostatic

potentialpotential

Two parallel cylinders Two parallel cylinders held positive and held positive and

negative potentialsnegative potentials

1u =- 1u =

2l

aa1u =

1u =-1u =

1u =-

1u = 1u =-

44





45

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

46

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

47

Torsional rigidityTorsional rigidity

?

48





49


Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation

50

Problem statementProblem statement

Doubly-connected domain

Multiply-connected domain

Simply-connected domain

51

Example 1Example 1

2 2( ) ( ) 0,k u x x D

2 2.0r

1B0u

2B

0u

1 0.5r

52

kk11 kk22 kk33 kk44 kk55

FEMFEM(ABAQUS)(ABAQUS)

2.032.03 2.202.20 2.622.62 3.153.15 3.713.71

BEMBEM(Burton & Miller)(Burton & Miller)

2.062.06 2.232.23 2.672.67 3.223.22 3.813.81

BEMBEM(CHIEF)(CHIEF)

2.052.05 2.232.23 2.672.67 3.223.22 3.813.81

BEMBEM(null-field)(null-field)

2.042.04 2.202.20 2.652.65 3.213.21 3.803.80

BEMBEM(fictitious)(fictitious)

2.042.04 2.212.21 2.662.66 3.213.21 3.803.80

Present methodPresent method 2.052.05 2.222.22 2.662.66 3.213.21 3.803.80

Analytical Analytical solution[19]solution[19]

2.052.05 2.232.23 2.662.66 3.213.21 3.803.80

The former five true eigenvalues The former five true eigenvalues by usinby using different approachesg different approaches

53

The former five eigenmodes by using prThe former five eigenmodes by using present method, FEM and BEMesent method, FEM and BEM

54



55

u=0 u=0

u=0

u=0 u=0

. .

.

.

.

x

y

cos( )8

ikre

2 2( ) ( ) 0,k u x x D

Sketch of the scattering problem (DirichSketch of the scattering problem (Dirichlet condition) for five cylinderslet condition) for five cylinders

56

k

-3 -2 -1 0 1 2 3-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(a) Present method (M=20) (b) Multiple DtN method (N=50)

The contour plot of the real-part The contour plot of the real-part solutions of total field forsolutions of total field for

57

The contour plot of the real-part The contour plot of the real-part solutions of total field forsolutions of total field for 8k

-3 -2 -1 0 1 2 3-2 .5

-2

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

2

2 .5

(a) Present method (M=20) (b) Multiple DtN method (N=50)

58

Fictitious frequenciesFictitious frequencies

0 2 4 6 8

-8

-4

0

4

8Present m ethod (M =20)

BEM (N =60)

59

Present methodPresent method

Soft-basin effect Soft-basin effect

/ 2 /3, 2I M

/ 1/ 2I Mc c / 1/3I Mc c / 2I Mc c

/x a /x a /x a

14 18

3

60



61

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)

62

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)

63

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)

64

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

65

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

66


Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation Plate vibrationPlate vibration

67

Free vibration of plateFree vibration of plate

68

Comparisons with FEMComparisons with FEM

69

BEM trap ?BEM trap ?Why engineers should learn Why engineers should learn

mathematics ?mathematics ? Well-posed ?Well-posed ? Existence ?Existence ? Unique ?Unique ?

Mathematics versus Mathematics versus Computation Computation

Some examplesSome examples

70

Numerical phenomenaNumerical phenomena(Degenerate scale)(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 ?

71

Numerical and physical resonanceNumerical and physical resonance

a

m

k

e i t

incident wave

e i t e i t

radiation

Physical resonance Numerical resonance

, if ufinite

( )

2 2

, if u finite lim00

k

m

72

Numerical phenomenaNumerical phenomena(Fictitious frequency)(Fictitious frequency)

0 2 4 6 8

-2

-1

0

1

2UT method

LM method

Burton & Miller method

t(a,0)

1),( au0),( au

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

9

1),( au0),( au

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

9

A story of NTU Ph.D. students

73

Numerical phenomenaNumerical phenomena(Spurious eigensolution)(Spurious eigensolution)

0 1 2 3 4 5 6 7 8 9 10 11 12fre q u e n cy p a ra m e te r

1E-080

1E-060

1E-040

1E-020

de

t|SM

|

C -C annular p la teu, com plex-vauled form ulation

T<9.447>

T: T rue e igenvalues

T<10.370>

T<10.940>

T<9.499>

T<9.660>

T<9.945>

S<9.222>

S<6.392>

S<11.810>

S : Spurious e igenvalues

ma 1

mb 5.0

1B

2B

74

OutlinesOutlines





75

ConclusionsConclusions

A systematic approach using A systematic approach using degenerate degenerate kernelskernels, , Fourier seriesFourier series and and null-field integnull-field integral equationral equation has been successfully propo has been successfully proposed to solve BVPs with sed to solve BVPs with arbitraryarbitrary circular circular holes and/or inclusions.holes and/or inclusions.

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

76

ConclusionsConclusions

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

77

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//

78

Documents

1 A semi-analytical approach for engineering problems with circular boundaries J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan Dec.23, 14:00-14:30,