The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 1

A study on half-plane Laplace problems with

a circular hole日期： 2004/08/22-24

地點：桃園大溪報告者：沈文成

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 2

Outlines

Motivations Boundary integral equations Degenerate kernels Numerical examples Conclusions

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Motivations

Heat conduction Pipe design Tunnels

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 4

Drawbacks of available methodsMethods drawbac

k

FEM mesh

BEM(truncated boundary)

mesh

BEM (image method)

mesh

MFS (Single-layer) node

MFS (Double-layer) node

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 5

Boundary integral equations

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

u s T x s u x dB x U x s t x dB x s D

0 ( , ) ( ) ( ) ( , ) ( ) ( ), e

B BT x s u x dB x U x s t x dB x s D

null-field integral equation

sx ： source points ： field point

x

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Degenerate kernels

x ： source point ； s ： field point

s

x

EU

rO

RIU

s

r

x

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 7

Image concept

( , )

( , )

( , )

x

s R

s R

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

u s T x s u x dB x U x s t x dB x s D interior integral equation

0 ( , ) ( ) ( ) ( , ) ( ) ( ),B B

eT x s u x dB x U x s t x dB x s D null-field integral equation

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Degenerate kernels

1

1

1ln ( ) cos ( ),

( , ) ln1

ln ( ) cos ( ),

m

m

m

m

R m Rm R

U s x rR

m Rm

1

1

1ln ( ) cos ( ),

( , ) ln1

ln ( ) cos ( ),

m

m

m

m

R m Rm R

U s x rR

m Rm

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 9

Modified degenerate kernels

1 1

( ; , ) ln ln

1 1ln ( ) cos ( ) ln ( ) cos ( )

IG

m m

m m

U x s s r r

Rm R m

m m R

1 1

( ; , ) ln ln

1 1ln ( ) cos ( ) ln ( ) cos ( )

EG

m m

m m

U x s s r r

R m R mm R m R

homogeneous Neumann boundary condition

homogeneous Dirichlet boundary condition0u =

0t =

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 10

Present method

0 [ ( ; , ) ( ) ( ; , ) ( )] ( )G GBT x s s u x U x s s t x dB x

Degenerate kernel

Fourier series

Null-field equation

Algebraic equation Fourier Coefficients

Potential

Analytical

Numerical

The 7th National Conference on Structure Engineering MSVLAB, HRE, NTOU 11

Collocation points

By choosing M terms of Fourier series, we have 2M+1 collocation points on the circle.

01

( ) ( cos sin )M

n nn

u x a a n b n

2M+1 terms

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Problem statements

Dirichlet Mixed-type2 2 2 2 2 2

2 2

( ) 41( , ) ln[ ]

ln[( ) / ] ( )

x c y c yu x y

h c a x c y

2 2c h a

0

10 0

( 1)( , ) [ sinh cos ]

sinh 2 cosh

nn

n

Q eu n n

k n n

0cosh /h a

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Boundary flux on the circle

0 90 180 270 360

degree ( )

0

0.2

0.4

0.6

0.8

1

t()

Analytica l so lu tion

P resent m ethod (M =5)

P resent m ethod (M =10)

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Illustrations (Dirichlet)

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-8.00

-6.00

-4.00

-2.00

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-8.00

-6.00

-4.00

-2.00

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-8.00

-6.00

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0.00

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0.00

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

- 8

- 6

- 4

- 2

0

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-8.00

-6.00

-4.00

-2.00

MFS (Double-layer potential method)

BEM with truncated boundary

Present method

MFS (Single-layer potential method)

BEM using image method

Analytical solution (Lebedev)

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Illustrations (Mixed-type)

MFS (Double-layer potential method)

BEM with truncated boundary

Present method

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-8.00

-6.00

-4.00

-2.00

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-8.00

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-4.00

-2.00

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0.00

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-8.00

-6.00

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-2.00

0.00

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

- 8

- 6

- 4

- 2

0

-8 .00 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00-10.00

-8.00

-6.00

-4.00

-2.00

MFS (Single-layer potential method)

BEM using image method

Analytical solution (Lebedev)

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Conclusions

A novel method using degenerate kernels, Fourier series expansion and null-field equation has been developed.

Boundary flux were obtained to converge to exact solution for only ten terms of Fourier series.

The results of present method agree with analytical solution of Lebedev.

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The End

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