null field integral approach for engineering problems with circular boundaries

1

Null field integral approach for Null field integral approach for engineering problems with circular engineering problems with circular boundariesboundaries

J. T. Chen Ph.D.終身特聘教授

Taiwan Ocean UniversityKeelung, Taiwan

Dec.26, 15:30-17:00, 2007交通大學機械系邀請演講

交大機械 2007.ppt

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering

1u =

1u =-1u =

1u =-

1u = 1u =-

2

Overview of numerical Overview of numerical methodsmethods

Finite Difference M ethod Finite Element M ethod Boundary Element M ethod

M esh M ethods M eshless M ethods

Numerical M ethods

2

PDE- variational IEDE

Domain

BoundaryMFS,Trefftz method MLS, EFG

開刀把脈

針灸

3

Number of Papers of FEM, BEM and Number of Papers of FEM, BEM and FDMFDM

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

126

4

Growth of BEM/BIEM Growth of BEM/BIEM paperspapers

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

5

BEMBEM USA USA (3569)(3569) , China , China (1372) (1372) , UK, , UK,

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

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

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

(ISI information Apr.20, 2007)(ISI information Apr.20, 2007)

Top ten countries of BEM and dual BEMTop ten countries of BEM and dual BEM

6

FEMFEM USA(9946), USA(9946), China(4107)China(4107), , Japan(3519)Japan(3519), France, Germany,, France, Germany,

England, South Korea, Canada, England, South Korea, Canada, Taiwan(1383), Taiwan(1383), ItalyItaly Meshless methodsMeshless methods USA(297), USA(297), China(179)China(179), Singapore(105), , Singapore(105), Japan(49)Japan(49), Spain,, Spain,

Germany, Slovakia, England, Portugal, Germany, Slovakia, England, Portugal, Taiwan(28)Taiwan(28) FDMFDM USA(4041), USA(4041), Japan(1478)Japan(1478), , China(1411)China(1411), England, France, , England, France,

Canada, Germany, Canada, Germany, Taiwan (559)Taiwan (559), South Korea, India, South Korea, India (ISI information Apr.20, 2007)(ISI information Apr.20, 2007)

Top ten countries of FEM, FDM and Meshless methodsTop ten countries of FEM, FDM and Meshless methods

7

BEMBEM Aliabadi M H (UK, Queen Mary College)Aliabadi M H (UK, Queen Mary College) Chen J TChen J T (Taiwan, Ocean Univ.) (Taiwan, Ocean Univ.) 113 SCI papers 479citing113 SCI papers 479citing Mukherjee S (USA, Cornell Univ.)Mukherjee S (USA, Cornell Univ.) Leisnic D(UK, Univ. of Leeds)Leisnic D(UK, Univ. of Leeds) Tanaka M (Japan, Shinshu Univ.)Tanaka M (Japan, Shinshu Univ.) Dual BEM (Made in Taiwan)Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London)Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) Chen J T (Taiwan, Ocean Univ.) Power H (UK, Univ. Nottingham)Power H (UK, Univ. Nottingham) (ISI information Apr.20, 2007)(ISI information Apr.20, 2007)

Active scholars on BEM and dual BEMActive scholars on BEM and dual BEM

8

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)

NTOU/MSVTaiwan

9URL: 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 Frame2007-c.ppt

Elasticity(holes and/or inclusions)

Laplace Equation

Research topics of NTOU / MSV LAB using null-field BIEs (2005-2007) (2008-2011)

Navier Equation

Null-field BIEM

Biharmonic Equation

Previous work (done)

Interested topic

Present project

(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

ASME JoMEABE

CMAME 2007

SDEE

JCA

NUMPDE revision

JSV

SH wave

Impinging canyonsDegenerate kernel for ellipse

ICOME 2006

Added mass

Water wave impinging circul

ar cylinders

Screw dislocation

Green function foran annular plate

SH wave

Impinging hill

Green function of`circular inclusion (special case:static)

Effective conductivity

CMC

(Stokes flow)

(Free vibration of plate) Direct BIEM

(Flexural wave of plate)

10

Research collaborators Research collaborators Dr. I. L. Chen (Dr. I. L. Chen ( 高海大高海大 ) Dr. K. H. Chen () Dr. K. H. Chen ( 宜蘭大學宜蘭大學 )) Dr. S. Y. Leu Dr. W. M. Lee (Dr. S. Y. Leu Dr. W. M. Lee ( 中華技術學院中華技術學院 )) Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh Mr. H. Z. Liao Mr. G. N. Kehr (2007)Mr. H. Z. Liao Mr. G. N. Kehr (2007) Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006)Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006) Mr. A. C. Wu Mr. P. Y. Chen (2005)Mr. A. C. Wu Mr. P. Y. Chen (2005) Mr. Y. T. Lee Mr. S. K. Kao Mr. Y. T. Lee Mr. S. K. Kao Mr. S. Z. Shieh Mr. Y. Z. LinMr. S. Z. Shieh Mr. Y. Z. Lin

11

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)

12

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)

13

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

14

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 ?除四舊

15

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

16

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

17

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 (Legendre polynomial)polynomial)

(Chebyshev poly(Chebyshev polynomial)nomial)

(Mathieu (Mathieu function)function)

18

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

19

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

20

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.

21

Outlines (Direct problem)Outlines (Direct problem)





22

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

23

• 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

24

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)

25

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

26

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

27

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

28

Outlines (Direct problem)Outlines (Direct problem)




Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions

29

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

30

How to separate the regionHow to separate the region

31

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

32

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

33-20 -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=

34

Boundary density discretizationBoundary density discretization

Fourier Fourier seriesseries

Ex . constant Ex . constant elementelement

Present Present methodmethod

Conventional Conventional BEMBEM

35

OutlinesOutlines





36

Adaptive observer systemAdaptive observer system

( , )r f

collocation collocation pointpoint

37

OutlinesOutlines





38

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

39

OutlinesOutlines





40

{ }

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

41

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

42

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

43

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

44

OutlinesOutlines





45

Numerical examplesNumerical examples

Laplace equation Laplace equation (EABE 2006, EABE2006,EABE 2007) (EABE 2006, EABE2006,EABE 2007) (CMES 2006, ASME 2007, JoM2007, CMC)(CMES 2006, ASME 2007, JoM2007, CMC) (MRC 2007, NUMPDE 2007,ASME 2007)(MRC 2007, NUMPDE 2007,ASME 2007) Eigen problem Eigen problem (JCA)(JCA) Exterior acoustics Exterior acoustics (CMAME 2007, SDEE 2008(CMAME 2007, SDEE 2008)) Biharmonic equation Biharmonic equation (JAM, ASME 2006, IJNME(JAM, ASME 2006, IJNME)) Plate vibration Plate vibration (JSV 2007)(JSV 2007)

46

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

47


Case Case 11

Case Case 22

1u =

0u =

1 2.5a =2 1.0a =

1u =

1u =

0u =

0 2.0R =

a

a

48

Case 1: Isothermal lineCase 1: Isothermal line

Exact Exact solutionsolution

(Carrier and (Carrier and Pearson)Pearson)

BEM-BEPO2DBEM-BEPO2D(N=21)(N=21)

FEM-ABAQUSFEM-ABAQUS(1854 (1854

elements)elements)

Present Present methodmethod(M=10)(M=10)-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

49

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

50

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

51





52

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 =-

53

Contour plot of potentialContour plot of potential

Exact solution (LebeExact solution (Lebedev et al.)dev et al.)

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

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

54

Contour plot of potentialContour plot of potential

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Onishi’s data Onishi’s data (1991)(1991)


55





56

Flow of an ideal fluid pass two Flow of an ideal fluid pass two parallel cylindersparallel cylinders

is the velocity of flow far is the velocity of flow far from the cylindersfrom the cylinders

is the incident angleis the incident angle

v¥

g

v¥

g

2l

a a

57

Velocity field in different incident Velocity field in different incident angleangle

-14 -12 -10 -8 -6 -4 -2 0 2 4-10

-8

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4-10

-8

-6

-4

-2

0

2

4

6

8

10


180g= o


135g= o

58





59

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

60

Axial displacement with two circular Axial displacement with two circular holesholes


Caulk’s data Caulk’s data (1983)(1983)

ASME Journal of Applied ASME Journal of Applied MechanicsMechanics -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

61

Torsional rigidityTorsional rigidity

?

62





63

Infinite medium under antiplane Infinite medium under antiplane shearshear

The displacementThe displacement

Boundary conditionBoundary condition

Total displacementTotal displacement

t

m

sw2 ( ) 0,sw x x DÑ = Î

( )sin

sw x

n

tq

m¶

=¶

sw w w¥= +

oonn

kB

64

Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11 (x axis) (x axis)

0 1 2 3 4 5 6 (in r adians )

- 2

0

2

4

6

8

z

/

(aro

un

d h

ole

wit

h r

ad

ius

a1

)

d/a1 = 0 .0 1

d/a1 = 0 .1

d/a1 = 2 .0

s ingle hole


Honein’s data (1Honein’s data (1992)992)Quarterly of Applied MathQuarterly of Applied Mathematicsematics

65

Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11

0 90 180 270 360

-2

0

2

4

R 0= 7 .5

R 0= 1 5 .0

R 0= 3 0 .0

0 90 180 270 360

-2

0

2

4

R 0= 7 .5

R 0= 1 5 .0

R 0= 3 0 .0



Steele’s data Steele’s data (1992)(1992)

Stress Stress approachapproach

Displacement Displacement approachapproachHonein’s data Honein’s data

(1992)(1992)5.35.34848

5.35.34949

4.64.64747

5.35.34545

13.1313.13%%

0.020.02%%

AnalytiAnalyticalcal

0.060.06%%

66

Extension to inclusionExtension to inclusion

Anti-plane piezoelectricity problemsAnti-plane piezoelectricity problems In-plane electrostatics problemsIn-plane electrostatics problems Anti-plane elasticity problemsAnti-plane elasticity problems

67

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

68

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

69

Patterns of the electric field for Patterns of the electric field for 00=2=2, , 11=9=9 and and 22=5=5

-1 .5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xE E¥¥=

cos 45 , sin 45x yE E E E¥ ¥¥ ¥= =o o

yE E¥¥=

Em

ets

& O

nofr

ichu

kE

met

s &

Ono

fric

huk

(199

6)(1

996)

Pre

sen

t m

eth

od (

L=20

)P

rese

nt

met

hod

(L=

20)

70

Three identical inclusions Three identical inclusions forming an equilateral triangleforming an equilateral triangle

12d r=x

yt ¥

2r

1r

2m

3m3r

0m

30o1m

30o

71

Tangential stress distribution around Tangential stress distribution around the inclusion located at the originthe inclusion located at the origin

Present method (L=20),Present method (L=20),agrees well with Gong’s data (1995)agrees well with Gong’s data (1995)

0 0.2 0.4 0.6 0.8 1

/

- 2

- 1

0

1

2

M z

/

1/ 0= 2/ 0= 3/ 0=0.0

1/ 0= 2/ 0= 3/ 0=0.5

1/ 0= 2/ 0= 3/ 0=2.0

1/ 0= 2/ 0= 3/ 0=5.0

1/ 0= 2/ 0= 3/ 0=

72





73

Half-plane problemsHalf-plane problems

Dirichlet boundary cDirichlet boundary conditionondition(Lebedev et al.)(Lebedev et al.)

Mixed-type boundary coMixed-type boundary conditionndition(Lebedev et al.)(Lebedev et al.)

0u =

1u =

1B

2B

0u =

1u

n

¶=

¶

1B

2B

h ha a

74

Dirichlet problemDirichlet problem



IsothermIsothermal lineal line

- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0

- 8

- 6

- 4

- 2

0

- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0

- 8

- 6

- 4

- 2

0

75

Mixed-type problemMixed-type problem



IsothermIsothermal lineal line

- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0

- 8

- 6

- 4

- 2

0

- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0

- 8

- 6

- 4

- 2

0

76


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

77

Problem statementProblem statement

Doubly-connected domain

Multiply-connected domain

Simply-connected domain

78

Example 1Example 1

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

2 2.0r

1B0u

2B

0u

1 0.5r

79

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

80

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

81


Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Water waveWater wave Biharmonic equationBiharmonic equation

82

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

83

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

84

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)

85

Fictitious frequenciesFictitious frequencies

0 2 4 6 8

-8

-4

0

4

8Present m ethod (M =20)

BEM (N =60)

86

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

87

Water wave impinging four Water wave impinging four cylinderscylinders

1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks

88

Maximum free-surface elevationMaximum free-surface elevation

Perrey-Debain et al. (JSV 2003 ) Present method (2007)

89


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

90

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)

91

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)

92

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)

93

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

94

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

95


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

96

Free vibration of plateFree vibration of plate

97

Comparisons with FEMComparisons with FEM

98

Disclaimer (commercial Disclaimer (commercial code)code)

The concepts, methods, and examples usThe concepts, methods, and examples using our software are for illustrative and eing our software are for illustrative and educational purposes only. Our cooperatioducational purposes only. Our cooperation assumes no liability or responsibility to n assumes no liability or responsibility to any person or company for direct or indirany person or company for direct or indirect damages resulting from the use of anect damages resulting from the use of any information contained here.y information contained here.

inherent weakness ?inherent weakness ? mismisinterpretation ? User interpretation ? User 當自強當自強

99

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

100

Numerical phenomenaNumerical phenomena(Degenerate scale)(Degenerate scale)

Error (%)of

torsionalrigidity

a

0

5

125

da

T

da

Commercial ode output ?

101

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

102

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

103

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

104

OutlinesOutlines





105

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..

106

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

107

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//

null field integral approach for engineering problems with circular boundaries

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