null field integral equation approach for engineering problems with circular boundaries

1

Null field integral equation approach Null field integral equation approach for engineering problems with circular for engineering problems with circular boundaries boundaries

J. T. Chen Ph.D.陳正宗終身特聘教授

Taiwan Ocean UniversityKeelung, Taiwan

June 22-23, 2007中山大學高雄

cmc2007.ppt

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering 應用數學系

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 BIEs (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:static)

Effective conductivity

CMC

(Stokes flow)

(Free vibration of plate) Direct BIEM

(Flexural wave of plate)

4

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

4

PDE- variational IEDE

Domain

BoundaryMFS,Trefftz method MLS, EFG

開刀把脈

針灸

國科會專題報導 : 中醫式的工程分析法

5

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)

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 Study of spurious solutionStudy of spurious solution SVD techniqueSVD technique 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

5. Meshless 5. Meshless

(s, x)eK

(s, x)iK

Advantages of Advantages of degenerate kerneldegenerate kernel

(x) (s, x) (s) (s)BK 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 (Legendre polynomial)polynomial)

(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)




Numerical examplesNumerical examples ConclusionsConclusions

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

Outlines (Direct problem)Outlines (Direct problem)




Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions

17

Gain of introducing the degenerate Gain of introducing the degenerate kernelkernel

(x) (s, x) (s) (s)BK dBj f=ò

Degenerate kernel Fundamental solution

CPV and HPV

No principal value?

0

(x) (s)(x) (s) (s)jBj

ja 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

18

How to separate the regionHow to separate the region

19

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

20

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

21-20 -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=

22

Boundary density discretizationBoundary density discretization

Fourier Fourier seriesseries

Ex . constant Ex . constant elementelement

Present Present methodmethod

Conventional Conventional BEMBEM

23

OutlinesOutlines





24

Adaptive observer systemAdaptive observer system

( , )r f

collocation collocation pointpoint

25

OutlinesOutlines





26

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

27

OutlinesOutlines





28

{ }

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

29

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

30

Flowchart of present methodFlowchart of present method

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

31

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

32

OutlinesOutlines





33

Numerical examplesNumerical examples

Laplace equation Laplace equation (EABE 2005, EABE 2007) (EABE 2005, EABE 2007) (CMES 2006, JAM-ASME 2007, JoM2007)(CMES 2006, JAM-ASME 2007, JoM2007) (CMA2007,MRC 2007, NUMPDE revision)(CMA2007,MRC 2007, NUMPDE revision) Membrane eigenproblem Membrane eigenproblem (JCA 2007)(JCA 2007) Exterior acoustics Exterior acoustics (CMAME 2007, SDEE 2007(CMAME 2007, SDEE 2007)) Biharmonic equation Biharmonic equation (JAM-ASME 2006(JAM-ASME 2006)) Plate eigenproblem Plate eigenproblem (JSV 2007)(JSV 2007)

34

Laplace equationLaplace equation

A circular bar under torqueA circular bar under torque

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

35

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

36

Axial displacement with two circular Axial displacement with two circular holesholes

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

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

37

Torsional rigidityTorsional rigidity

?

38

Extension to inclusionExtension to inclusion

Anti-plane elasticity problemsAnti-plane elasticity problems

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

39

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

40

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

41

Numerical examplesNumerical examples

Biharmonic equationBiharmonic equation (exponential convergence)(exponential convergence)

42

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)

43

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)

44

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)

45

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

46

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

47

OutlinesOutlines




Numerical examplesNumerical examples Discussions of spurious eigenvaluesDiscussions of spurious eigenvalues SVDSVD ConclusionsConclusions

48

Disclaimer (commercial Disclaimer (commercial code)code)

The concepts, methods, and examples usiThe concepts, methods, and examples using our software are for illustrative and edng our software are for illustrative and educational purposes only. ucational purposes only.

Our cooperation assumes no liability or reOur cooperation assumes no liability or responsibility to any person or company for sponsibility to any person or company for direct or indirect damages resulting from direct or indirect damages resulting from the use of any information contained here.the use of any information contained here.

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

49

Eccentric membrane Eccentric membrane (true and spurious eignevalues)(true and spurious eignevalues)

U T formulationSingular integral equations

L M formulationHypersingular formulation

spuriousspurious

Chen et al., 2001, Proc. Royal Soc. London Ser. A

50

SVD Technique (Google SVD Technique (Google searching) searching)

nnnmmmnm VUC

][][][][

[C] SVD decomposition

[U] and [V} left and right unitary vectors

nm

nm

n

,

00

00

0

0

][ 1

11 nn

51

Physical meaning of SVD Physical meaning of SVD

1 2 3 1 2 3 =[ ] =[ ]TF 變形前變形後

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

先拉再轉先轉再拉

52

[ ] tt

d

UC

U

(SVD updating terms)

Find true eigenvalue

SVD updating terms SVD updating terms SVD updating documentSVD updating document

52

[ ] d R LC U U (SVD updating document)Find spurious eigenvalue

Chen et al., 2003, Proc. Royal Soc. London Ser. A

53

Eccentric membrane Eccentric membrane (SVD updating for (SVD updating for truetrue eigenvalues) eigenvalues)

Dirichelet case

Neumann case T M

UL

54

Eccentric membrane Eccentric membrane (SVD updating for (SVD updating for spuriousspurious eigenvalues) eigenvalues)

U T

L M

55

Eccentric plate Eccentric plate

R1

R2

eo1 o2

Case 1: Geometric data:R1=1m

R2=0.4m

e=0.0～ 0.5mthickness=0.002mBoundary condition:Inner circle : clampedOuter circle: clamped

Figure 1. A clamped-clamped annular-like plate with one circular hole of radius 0.4 m

56

Eigenvalue versus eccentricity Eigenvalue versus eccentricity

0 0.1 0.2 0.3 0.4 0.54

5

6

7

8

9

10

Eccentricity

Fre

qu

en

cy p

ara

me

ter

First mode

Second mode

Third mode

Fourth mode

Spurious mode

Figure 2. Effect of the eccentricity e on the natural frequency parameter for the clamped- clamped annular-like plate (R1=1.0, R2 = 0.4)

57

True boundary eigenmode True boundary eigenmode

0 17 34 51-0.5

0

0.5

1

Re

al

0 17 34 51-5

0

5

10x 10

-6

Ima

gin

e

Fourier coefficients

Outer boundary

Outer boundary

Inner boundary

Inner boundary

T

Figure 5. Real and imagine part of Fourier coefficients for first true boundary mode

( T =6.1716, e = 0.2, R2 = 0.4m)

R1

R2

eo1 o2

R1

R2

eo1 o2

58

Spurious boundary eigenmode Spurious boundary eigenmode

0 17 34 51-1

-0.5

0

0.5

Re

al

0 17 34 51-4

-2

0

2x 10

-4

Ima

gin

e

Fourier coefficients

Inner boundary

Inner boundary

Outer boundary

Outer boundary

Figure 6. Real and imagine part of Fourier coefficients for first spurious boundary mode ( =7.9906, e = 0.2, R2 = 0.4m) T

R1

R2

eo1 o2

R1

R2

eo1 o2

59

OutlinesOutlines





60

ConclusionsConclusions

A systematic approach using A systematic approach using degenerdegenerate kernelsate kernels, , Fourier seriesFourier series and and null-fielnull-field integral equationd integral equation has been successf has been successfully proposed to solve boundary valuully proposed to solve boundary value problems with circular boundaries.e problems with circular boundaries.

Numerical results Numerical results agree wellagree well with avai with available exact solutions and FEM (ABAQUlable exact solutions and FEM (ABAQUS) for S) for only few terms of Fourier seriesonly few terms of Fourier series..

SpuriousSpurious eigenvalues are examined. eigenvalues are examined.

61

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

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The EndThe End

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

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null field integral equation approach for engineering problems with circular boundaries

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