DUAL BOUNDARY INTEGRAL EQUATIONS

FOR HELMHOLTZ EQUATION AT A CORNER

USING CONTOUR APPROACH AROUND

SINGULARITY

Report: C.E. Lin

Number: M98520025

Adviser: J.T. Chen

Date: Jan.07.2010NTOU HRE

Outlines

1. Dual Integral formulation of BEM for Helmholtz equation with a corner

2. Discussions on the Laplace and He-lmholtz equations at a corner

3. Conclusions

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Dual Integral formulation of BEM for Helmholtz equation with a corner

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'0 ( , ) ( ) ( , ) ( ) ( )

B BV x x u x U x x x dB x

'0 ( , ) ( ) ( , ) ( ) ( )

B BM x x u x L x x x dB x

'0 ( , ) ( ) ( , ) ( ) ( )t t

B BM x x u x L x x x dB x

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5

(1)0( , ) ( ) ( ) ( )( )

2B

iU x x x dB x H k v v

Single layer potential:

Double layer potential:

( , ) ( ) ( ) ( ) ( )BV x x u x dB x u x v v

Normal derivative of single layer potential:

'( , ) ( ) ( ) ( ) ( )BL x x x dB x c x du x

Normal derivative of double layer potential:

'( , ) ( ) ( ) ( ) ( )BM x x x dB x c x du x

Tangent derivative of single layer potential:

'( , ) ( ) ( ) ' ( ) ( )t

BL x x x dB x c u x d x

Tangent derivative of double layer potential:

'( , ) ( ) ( ) ' ( ) ( )

t

BM x x u x dB x c u x d x

Boundary term

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

H PV M x x u x dB x C PV M x x u x dB x

Discussions on the Laplace and Helmholtz equations at a corner

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2 2 ( )u k u Q x ⋯ Helmholtz equation

2 0u ⋯ Laplace equation

“k” is very small and can be negligible

22

2 2

1 ( , )( , ) ( , )

u x tu x t Q x t

c t

⋯ Wave equation

kc

: frequency : c wave velocity

Conclusions

• The free terms of the six kernel functions in the dual integral equation for the Helmholtz equation at a corner have been examined

• It is discovered that employing the contour appr-oach the jump term comes half and half from the free terms in the L and M kernel integrations, re-spectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only.

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• Laplace equation is a special case of the Helmholtz equation when the value of w-ave number approaches zero.

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THANKS FOR YOUR KIND OF ATTENTION

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