null-field integral equation approach for structure problems with circular boundaries

Null-field integral equation approach for structure problems

with circular boundaries

第九屆結構工程研討會

Jeng-Tzong Chen, Ying-Te Lee, Wei-Ming Lee and I-Lin Chen

時間 : 2008 年 08 月 22~24 日地點 : 高雄國賓大飯店

第九屆結構工程研討會August 22-24, 2008, 國賓大飯店 , Kaohsiung

2

Outline

Introduction1.

2.

3.

Problem statement

Method of solution

Numerical examples4.

5. Concluding remarks


3

Motivation

Numerical methods for engineering problems

FDM / FEM / BEM / BIEM / Meshless method

BEM / BIEM

Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate

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


4

Motivation

BEM / BIEMBEM / BIEM

Improper integralImproper integral

Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity

Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary

Collocation Collocation pointpoint

Fictitious BEMFictitious BEM

Null-field approachNull-field approach

CPV and HPVCPV and HPVIll-posedIll-posed

Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)

Waterman (1965)Waterman (1965)

Achenbach Achenbach et al.et al. (1988) (1988)



5

Present approach

)()()( sdBsxB ),( xsK

),( xsK e

Fundamental solutionFundamental solution

No principal valueNo principal value

Advantages of present approach1. No principal value2. Well-posed model3. Exponential convergence4. Free of mesh

Degenerate kernelDegenerate kernel

CPV and HPVCPV and HPV

xsxsK

xsxsKe

i

),,(

),,(

),( xsK i

sx ln



6

Problem statement


Circular cavities and/or inclusions bounded in the domain

H

kkBB

0

xxu ,0)(


7

Problem statement


operator,Elasticity:)())(()(

operator,zBiHelmholt:)()(

operator,Laplacian:)(

)(

~

2

~

44

2

xuGxuG

xu

xu

xu

problemelasticitythefor)ln()43()1(8

1

problemzbiHelmholtthefor))()((2

)()(8

problemLaplacetheforln2

1

),(

2

00002

r

yyr

G

riIrKriJrYi

r

xsU

kiik

Governing Equation

Fundamental solution


8

Interior case Exterior case

cD

D D

x

xx

xcD

x x

Degenerate (separate) formDegenerate (separate) form

DxsdBstxsUsdBsuxsTxuBB

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

BxsdBstxsUVPRsdBsuxsTVPCxuBB

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

Bc

BBDxsdBstxsUsdBsuxsT ),()(),()()(),(0

B

Boundary integral equation and null-field integral equation

s

s

n

ss

n

xsUxsT

rxsxsU

)()(

),(),(

lnln),(



9

Degenerate kernel and Fourier series

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

0 NkBsnbnaasu kkn

kn

kn

kk

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

0 NkBsnqnppst kkn

kn

kn

kk


s

Ox

R

kth circularboundary

cosnθ, sinnθboundary distributions

eU

x

iU

Expand fundamental solution by using degenerate kernel

Expand boundary densities by using Fourier series

,),()(),(

,),()(),(

),(

0

0

sxsBxAxsU

sxxBsAxsU

xsU

jjj

E

jjj

I


10

Degenerate kernels

,,))(cos()(1

ln2

1),;,(

,,))(cos()(1

ln2

1),;,(

),(

1

1

RmR

mRU

RmRm

RRU

xsU

m

me

m

mi

,)],(cos[)]}()1()()[(2

)]()()[({8

1),;,(

,)],(cos[)]}()1()()[(2

)]()()[({8

1),;,(

),(

02

02

RmIiKRI

iJYRJRU

RmRIiRKI

RiJRYJRU

xsU

mm

mm

mmmmm

e

mm

mm

mmmmm

i

,,))1()1cos(()())2cos(()(12

1

))(cos()(1

ln)43()1(8

1

,,))1()1cos(()())2(cos()(12

1

))(cos()(1

ln)43()1(8

1

),(

0

1

0

1

0

1

0

1

11

RmmR

mmR

mR

mG

RmmR

mmR

mR

mG

xU

m

m

m

m

m

m

m

m

m

m

m

m

Laplace

problem

Helmholtz

problem

Elasticity

problem



11

collocation pointcollocation point

0 , 01 , 1k , k2 , 2

Adaptive observer system



12

Comparisons of conventional BEM and present method

Boundarydensity

discretization

Auxiliarysystem

FormulationObserver

systemSingularity Convergence

ConventionalBEM

Constant,linear,

quadratic…elements

Fundamentalsolution

Boundaryintegralequation

Fixedobserversystem

CPV, RPVand HPV

Linear

Presentmethod

Fourierseries

expansion

Degeneratekernel

Null-fieldintegralequation

Adaptiveobserversystem

Disappear Exponential



13

Problem with circular boundaries

Null-field BIE

Expansion

Degenerate kernel for Fundamental

solution

Fourier series for fictitious boundary

densities

Collocating the collocation point and matching the boundary conditions

Boundary integration in observer system

Linear algebraic system

SVD

Obtain the unknown Fourier coefficients

Mode shape

Frequency parameter

Flowchat



14

Case 1: A circular bar with an eccentric holeCase 2: A circular plate with two holesCase 3: Stress concentration factor problem

Numerical examples


15

Case 1: A circular bar with an eccentric hole

t


0

tm

External diameter of the tube

D:

Dt

t

ttp m

tm: The maxium wall thickness

(eccentricity)


16

Stress calculationalong outer and inner boundary

at boundaries for λ=0.3 and p=0.4z


(0.0%)

(0.1%)

(0.0%)

(0.0%)

(0.4%)

(0.0%)

(0.3%)

(0.0%)

(1.5%)

(0.6%)


17

Stress calculationfor point in the center line

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


(0.0%)(0.1%)

(0.1%)

(0.1%)

(0.1%)

(0.3%)

(0.0%)(0.2%)

(0.5%)

(0.5%)

(0.0%)

(0.6%)

0


18

Case 2: A circular plate with two holes



19

0 1 2 3 4 5 6 7 8 9 10 113

3.5

4

4.5

5

5.5

6

6.5

Terms of Fourier series ( )

Nat

ura

l fr

equ

ency

pa

ram

ete

r

First Mode

Second Mode

Third Mode

Fourth Mode

Fifth Mode

M 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

10-6

10-5

10-4

10-3

Min

imu

m s

ingu

lar

valu

e

Frequency parameter (λ)

3.177

4.529

4.699

6.514

7.205

7.644

5.751 5.957

Natural frequency parameter versus terms of Fourier series

The minimum singular value versus the frequency parameter

Results



20

The first five eigenvalues and eigenmodes



21

Case 3: Stress concentration factor problem


01 t 02 tBoundary conditions:

and


22

Domain superposition


=+

cos1 St 02 t cos1 St h 02 ht


23

Null-field BIE

B

I sdBSxsU )(cos),(0 11

B

N

n

hn

N

n

hn

hI sdBnbnaaxsT )(sincos),(1

1,1

1,1,011

B

N

n

hn

N

n

hn


2,1

2,2,021

B

I sdBSxsU )(cos),(0 12

B

N

n

hn

N

n

hn


1,1

1,1,012

B

N

n

hn

N

n

hn


2,1

2,2,022



24

,)1(

1,1 G

Saah

,01,1,2,12,01,0 hn

hn

hhh aaaaa ,3,2n

,2

)21(2,1 G

Sabh

,02,1,1,1 hn

hn

h bbb ,3,2n

3cos1

4cos

)1()(

2

222

1

aa

G

Sa

G

Sxu h

.3sin14

sin2

)21()(

2

222

2

aa

G

Sa

G

Sxuh

sincos

2

)1()( 1,11 b

bG

Sxu

.cossin2

)( 1,12 b

bG

Sxu


Similarly,

Results of case 3


25

1G 3.0 and

=+

Results of case 3



26

,cos2

)1(3cos1

4cos

)1(2

222

1

G

Saa

G

Sa

G

Su

.sin2

3sin14

sin2

)21(2

222

2

G

Saa

G

Sa

G

Su

4

222224

11

]4cos)23(2cos32[

Saaa

4

2222

22 2

]4cos)23(2cos[

Saa

2sin2

]2cos)46([4

2222

12

Saa

4

22222

2

]2cos)3()[(

Saa

rr

4

44222

2

]2cos)3()([

Saa

2sin

2

)32(4

4224 Saar

Results of case 3


0 rrr

2cos2SS )( a


27

Concluding remarks

A systematic approach was proposed for engineering problems with circular boundaries by using null-field integral equation in conjunction with degenerate kernel and Fourier series.

1.

2.

Only a few number of Fouries series terms for our examples were needed on each boundary.


Four advantages of our approach, (1) free of calculating principal value, (2) exponential convergence, (3) free of mesh and (4) well-posed model

3.

A general-purpose program for multiple circular boundaries of various radii, numbers and arbitrary positions was developed.

4.


28

The End

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null-field integral equation approach for structure problems with circular boundaries

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