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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