邊界元素法專題進度報告

指導老師：陳正宗 終身特聘教授學生：河工 3A蔡育仁學號： B97520034

目錄 1. H. L. Huan, Two-Dimensional Green’s

Function and Kaplquist’s Equation, Chinese Journal of Physics , Vol , 20, No,3-4.

2. J.A.F. Santiagoa and L.C. Wrobelb, Modified Green’s Functions for Shallow Water Acoustic Wave Propagation

Two-Dimensional Green’s Function and Kaplquist’s Equation

2 2

2

2

( ) ( )

. .

( , ) 0

( , ) ( ) ( )

. .

( , ) ( , )

( , ) 0

R c

d d dsn n

G E

x y

BC

V

Two-Dimensional Green’s Function and Kaplquist’s Equation

2 2

2 2

2 2

2 2

0

0

0

( , ) ( , )

1 ( ) ( )( , ; , ) ln

2 ( ) ( )

1(0, 1)

( )

1 1( , ) ( ,0)

( )

2 2

( , ) 2 2

2 2

c

Gx y V ds

n

x yG x y

x y

G G G y

n x y

yx y V d

x y

V g

g gV H

V g

2 2

1 10

2 2

( )2 2 2( , ) {( ) tan ( ) ( ) tan ( ) }2 2 2 ( )2

g g gx x x yH yg gx y x xgy y x y

Modified Green’s Functions for Shallow Water Acoustic Wave Propagation

2 2

1

G.E.

( , )

B C.

At free surface

( ) 0

At the bottom

( )0

( )( )

Nes

k B E s

x

x

n

xik x

n

.

Sommerfeld radiation condition at infinity

1

( ) ( , )( ) ( , ) ( ) ( , )

Nesx G s xs G s x d x d G E s

n n

Modified Green’s Functions for Shallow Water Acoustic Wave Propagation

(1) (1) (1 )0 0 F m

1

1 (1) (2 ) (1) (3 ) (1) (4 ) (1) (5 )F m 0 0 0 0

( )

1

First Image method

At free surface

1( , ) ( ) ( ) G ( , )

4

G ( , ) ( 1) [ ( ) ( ) ( ) ( )]

Moran

2

FF

m

m F F F Fm m m m

b

F

b

G s x i H kr H kr s x

s x H kr H kr H kr H kr

m m me

r r F

與 作比較1

2 2 2 2

(1) (1) (1 )0 0

1

(1) (2 ) (1) (3 ) (1) (4 ) (1) (5 ) 0 0 0 0

{ ( ) }

At the bottom

1( , ) { ( ) ( ( , )}

4

( , ) ( 1) [ ( ) ( ) ( ) ( )]

Second Ei1gnfuction expansion

1( , )

BB B m

m

m B B B BB m m m m m

M

x y z d

G s x i H kr H kr G s x

G s x H kr H kr H kr H kr

G s x GH

1

2 2

( , )

( , ) sin[ ( )]sin[ ( )]

1( )

2

xm

M mm

k x u

M m ym F ym F

xm

ym

xm ym

s x

eG s x k Y v k Y y

k

k mH

k k k

2 2

( ) 2 ( ) 2

(1 )

(2 )

(3 )

(4 )

(5 )

( ) ( )

( ) ( )

2

2( 1) 2

2 ( )

2 ( )

2( 1) 2

jF jF jF

Fm F

Fm F B

Fm F B

Fm F B

Fm F B

r x s x u y v

r x s x u y v

v Y v

v m Y Y v

v m Y Y v

v m Y Y v

v m Y mY v

The End

Thanks for your attention.