analysis of circular torsion bar with circular holes using null-field approach

九十四年電子計算機於土木水利工程應用研討會

1

Analysis of circular torsion bar with Analysis of circular torsion bar with circular holes using null-field circular holes using null-field approachapproach

研究生：沈文成指導教授：陳正宗教授時間： 15:10 ~ 15:25地點： Room 47450-3

2

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 Linear algebraic equationLinear algebraic equation

Numerical examplesNumerical examples ConclusionsConclusions

3

OutlinesOutlines





4

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 hypersiSingular and hypersingularngular

RegulRegularar

Improper Improper integralintegral

CPV and CPV and HPVHPV

Ill-Ill-posedposed

FictitiFictitious ous

bounboundarydary

CollocatCollocation ion

pointpoint

5

Present approachPresent approach

1. No principal 1. No principal valuevalue2. Well-2. Well-posedposed

(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 s

O O- -

(x) (s)a b

6

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)

7

Motivation and literature reviewMotivation and literature review

Analytical methods for solving Laplace problems with circ

ular 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

8

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

9

Contribution and goalContribution and goal

However, they didn’t employ the However, they didn’t employ the nnull-field integral equationull-field integral equation and and degedegenerate kernelsnerate kernels to fully capture the ci to fully capture the circular boundary, although they all ercular boundary, although they all employed mployed Fourier series expansionFourier series expansion..

To develop a To develop a systematic approachsystematic approach f for solving Laplace problems with or solving Laplace problems with mmultiple holesultiple holes is our goal. is our goal.

10

OutlinesOutlines





11

Boundary integral equation and null-Boundary integral equation and null-field integral equationfield integral equation

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB B

u T u dB U t dB Dp = - Îò ò0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT u dB U t dB D= - Îò ò

s

s

(s, x) ln x s ln

(s, x)(s, x)

(s)(s)

U r

UT

ut

= - =

¶=

¶

¶=

¶

n

n

x

D

xcD

x

D xcD

Interior Interior casecase

Exterior Exterior casecase

Null-field integral Null-field integral equationequation

12

OutlinesOutlines





13

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

14

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 x2(s, x)1, x s

2

T

ìïï >ïïï=íï -ï >ïïïî

15-20 -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

¥

=

¥

=


å

å

x ( , )r f=

16

Boundary density discretizationBoundary density discretization

Fourier Fourier seriesseries

Ex . constant Ex . constant elementelement

Present Present methodmethod

Conventional Conventional BEMBEM

17

OutlinesOutlines





18

Adaptive observer systemAdaptive observer system

( , )r f

collocation collocation pointpoint

19

OutlinesOutlines





20

OutlinesOutlines





21

{ }

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

22

Explicit form of each submatrix [Explicit form of each submatrix [UUpkpk] an] and vector {d vector {ttkk}}

0 1 11 1 1 1 1

0 1 12 2 2 2 2

0 1 13 3 3 3 3

0 1 12 2 2 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

c c s Mc Mspk pk pk pk pkc c s Mc Mspk pk pk pk pkc c s Mc Mspk pk pk pk pk

pk

c c s Mc Mspk M pk M pk M pk M pk

U U U U U

U U U U U

U U U U U

U U U U U

ff ff f

ff ff f

ff ff f

ff ff

é ù=ê úë ûU

L

L

L

M M M O M M

L 20 1 1

2 1 2 1 2 1 2 1 2 1

( )

( ) ( ) ( ) ( ) ( )M

c c s Mc Mspk M pk M pk M pk M pk MU U U U U

f

ff ff f+ + + + +

é ùê úê úê úê úê úê úê úê úê úê úê úê úë ûL

{ } { }0 1 1

Tk k k k kk M Mp p q p q=t L

1f

2f

3f

2Mf

2 1Mf +

Fourier Fourier coefficientscoefficients

Truncated Truncated terms of terms of

Fourier seriesFourier series

Number of Number of collocation pointscollocation points

23

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

24

Comparisons of conventional BEM Comparisons of conventional BEM and present methodand present method

BoundaryBoundarydensitydensity

discretizatdiscretizationion

AuxiliaryAuxiliarysystemsystem

FormulatFormulationion

ObserObserverver

systesystemm

SingularSingularityity

ConventiConventionalonal

BEMBEM

Constant,Constant,Linear,Linear,

QurdraturQurdrature…e…

FundameFundamentalntal

solutionsolution

BoundarBoundaryy

integralintegral

equationequation

FixedFixed

obserobserverver

systesystemm

CPV, RPCPV, RPVV

and HPVand HPV

PresentPresentmethodmethod

FourierFourier

seriesseries

expansioexpansionn

DegeneraDegeneratete

kernelkernel

Null-Null-fieldfield

integralintegral

equationequation

AdaptiAdaptiveve

obserobserverver

systesystemm

NoNo

principprincipalal

valuevalue

25

OutlinesOutlines





26

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

27

Torsion problemsTorsion problems

TorqTorqueue

TorqTorqueue

ab

qR

Eccentric Eccentric casecase

Two holes Two holes (N=2)(N=2)

a

b

q

R

a

28

Torsion problemsTorsion problems

a

a

ab q

R

TorqTorqueue

TorqTorqueuea

a

ab q

R

a

Three Three holes holes (N=3)(N=3)

Four holes Four holes (N=4)(N=4)

29

Torsional rigidity (Eccentric case)Torsional rigidity (Eccentric case)


(M=10)(M=10)

Exact Exact solutionsolution

BIEBIE

formulatioformulationn

0.200.20 0.978720.97872 0.978720.97872 0.978720.97872

0.400.40 0.951370.95137 0.951370.95137 0.951370.95137

0.600.60 0.903120.90312 0.903120.90312 0.903160.90316

0.800.80 0.824730.82473 0.824730.82473 0.824970.82497

0.900.90 0.761680.76168 0.761680.76168 0.762520.76252

0.980.98 0.667050.66705 0.665550.66555 0.667320.66732

42 /G Rmpb

R a-

30

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

31

Axial displacement with three Axial displacement with three circular holescircular holes


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2




32

Axial displacement with four circular Axial displacement with four circular holesholes


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2




33

Torsional rigidity (N=2,3,4)Torsional rigidity (N=2,3,4)

Number of Number of holesholes


(M=10)(M=10)

BIEBIE

formulatioformulationn

First-orderFirst-order

solutionsolution

22 0.86570.8657 0.86570.8657 0.86610.8661

33 0.82140.8214 0.82140.8214 0.82240.8224

44 0.78930.7893 0.78930.7893 0.79340.7934

42 /G Rmp

34

OutlinesOutlines





35

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 torsion probleully proposed to solve torsion problems with circular boundaries.ms with circular boundaries.

Numerical results Numerical results agree wellagree well with avai with available exact solutions and Caulk’s datlable exact solutions and Caulk’s data for a for only few terms of Fourier seriesonly few terms of Fourier series..

36

ConclusionsConclusions

Engineering problemsEngineering problems with with circular bocircular boundariesundaries which satisfy the which satisfy the Laplace eqLaplace equationuation can be solved by using the pro can be solved by using the proposed approach in a posed approach in a more efficient anmore efficient and accurate mannerd accurate manner..

37

The endThe end

Thanks for your kind attentions.Thanks for your kind attentions.

Your comments will be highly apprYour comments will be highly appreciated.eciated.

38

Derivation of degenerate kernelDerivation of degenerate kernel

Graf’s addition theoremGraf’s addition theorem Complex variableComplex variable

s xs ( , ) , x ( , )R z zq r f= = = =

x sln x s ln z z- = - Real Real partpart

x x xs x s s s

1s s s

1ln( ) ln[( )(1 )] ln( ) ln(1 ) ln( ) ( )m

m

z z zz z z z z

z z m z

¥

=

- = - = + - = - å

( )x

1 1 1 1s

1 1 1 1( ) ( ) ( ) [ ] ( ) cos ( )

im m m i m m

im m m m

z ee m

m z m Re m R m R

ff q

q

r r rq f

¥ ¥ ¥ ¥-

= = = =

= = = -å å å å

Real Real partpart

IfIf s xz z-

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

¥

=

¥

=


å

å

ln R

2

2 3

1

1ln(1 ) (1 )

11 1

( )2 31 m

m

x dx x x dxx

x x x

xm

¥

=

- =- =- + + +-

=- + + +

=-

ò ò

å

L

L

0k®

Bessel’s Bessel’s functionfunction

analysis of circular torsion bar with circular holes using null-field approach

Documents