torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral...

Torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral

approach

ICOME2006

Ying-Te Lee, Jeng-Tzong Chen and An-Chien Wu

Date: November 14-16

Place: Hefei, China

Second Asia-Pacific International Conference on Computational Methods in EngineeringNov. 14-16, 2006, Hefei, China

2

Outlines

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

Literature review

Key point Main application Author

Conformal mapping Torsion problemIn-plane electrostaticsAnti-plane elasticity

Chen & Weng (2001)Emets & Onofrichuk (1996)Budiansky & Carrier (1984)Steif (1989)Wu & Funami (2002)Wang & Zhong (2003)

Bi-polar coordinate Electrostatic potentialElasticity

Lebedev et al. (1965)Howland & Knight (1939)

Möbius transformation Anti-plane piezoelectricity & elasticity

Honein et al. (1992)

Complex potential approach Anti-plane piezoelectricity Wang & Shen (2001)

Those analytical methods are only limited to doubly connected regions.

Analytical solutions for problems with circular boundaries



7

Literature review

Author Main application Key pointLing

(1943)

Torsion of a circular tube

Caulk et al.

(1983)

Steady heat conduction with circular holes

Special BIEM

Bird and Steele

(1992)

Harmonic and biharmonic problems with circular holes

Trefftz method

Mogilevskaya et al.

(2002)

Elasticity problems with circular holes or inclusions

Galerkin method

However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary.

Fourier series approximation



8

Problem statement

B0

B1

B2

B3

Bi

B4

a0

a1

a2

a3

a4

ai

x

y


A circular bar with circular inclusions


9

Domain superposition


A circular bar with circular holes Each circular inclusion problem

B0

B1

B2

B3

Bi

B4

II11 ,

II22 ,

Ii

Ii ,II

33 ,

II44 ,

MM44 ,

B0

B1

B2

B3

Bi

B4

MM11 ,

MM22 ,

Mi

Mi ,MM

33 ,

M0

Satisfy 0)(2 xu


10

Interior case Exterior case

cD

D D

x

xx

xcD

x x

Degenerate (separate) formDegenerate (separate) form

DxsdBstxsUsdBsuxsTxuBB

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

BxsdBstxsUVPRsdBsuxsTVPCxuBB

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

BcBB

DxsdBstxsUsdBsuxsT ),()(),()()(),(0

B

Boundary integral equation and null-field integral equation

s

s

n

ss

n

xsUxsT

rxsxsU

)()(

),(),(

lnln),(



11

Degenerate kernel and Fourier series

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

0 NkBsnbnaas kkn

kn

kn

kk

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

0 NkBsnqnpps kkn

kn

kn

kk


s

Ox

R

kth circularboundary

cosnθ, sinnθboundary distributions

,),(cos1

ln),;,(

,),(cos1

ln),;,(

),(

1

1

RmR

mRU

RmRm

RRU

xsU

m

m

e

m

mi

eU

x

iU

Expand fundamental solution by using degenerate kernel

Expand boundary densities by using Fourier series


12

collocation pointcollocation point

0 , 01 , 1k , k2 , 2

Adaptive observer system



13

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



14

Case 1: A circular bar with an eccentric inclusion


R1

R0

ex

3.001 RR

6.00 Rex

Ratio:

Torsional rigidity:

N

kB kD k

dBn

dDyxG1

22 )(

IMT GGG

0

1

GT : total torsion rigidityGM : torsion rigidity of matrixGI : torsion rigidity of inclusion


15

Results of case 1

0 10 20 30N u m b er o f F o u rier series

0.9624

0.9628

0.9632

0.9636

Tor

sion

al r

igid

ity

(2G

/ R 04 )

0 200000 400000 600000 800000 1000000

S h ea r m o d u lu s ()

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Tor

sion

al r

igid

ity

(2G

/ R 04 )

M u sk h e lish v ili fo rm u la

P re se n t m e th o d

Torsional rigidity versus number of Fourier series terms

Torsional rigidity versus shear modulus of inclusion



16

Results of case 1


Torsional rigidity of a circular bar with an eccentric inclusion


17

Case 2: limiting case A circular bar with one circular hole


R1=0.3

R0=1

ex=0.5

0.10

01


18


Torsional rigidity of a circular bar with an eccentric hole

Results of case 2


19

Stress calculation

t


0

tm

External diameter of the tube

D:

Dt

t

ttp m

tm: The maxium wall thickness

(eccentricity)


20

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%)


21

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


22

Concluding remarks

A systematic approach was proposed for torsion problems with circular inclusions 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, and for more accurate results of torsional rigidity with error less than 2 %.


Four gains 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 inclusions of various radii, numbers and arbitrary positions was developed.

4.


23

The End

Thanks for your kind attention

Welcome to visit the web site of MSVLAB

http://ind.ntou.edu.tw/~msvlab


24

Torsion problem

0

xyxzyz


zyu zxv ),( yxw Displacement fields:

Strain components:

0 xyzyx )( yx

wxz

)( xy

wyz

Stress components:

0 xyzyx )( xy

wyz

02

2

2

2

yx

Equilibrium equation:

: the shear modulus : angle of twist per unit length

Following the theory of Saint-Venant torsion, we assume

)( yx

wxz

torsional rigidity of a circular bar with multiple circular inclusions using a null-field integral...

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