Null-field BIEM for solving a scattering problem from a point

source to a two-layer prolate spheroid

Null-field BIEM for solving a scattering problem from a point

source to a two-layer prolate spheroid Jia-Wei Lee (李家瑋 ), Department of Harbor and River Engineering, National Taiwan Ocean University, Taiwan (29952008@mail.ntou.edu.tw)Advisor: Prof. Jeng-Tzong Chen (陳正宗 )

Providence University, Taichung City, Taiwan, June 1-2, 2013

第 13屆計算數學研討會暨台灣工業與應用數學會年會(海報論文：博士生組 )

第 13屆計算數學研討會暨台灣工業與應用數學會年會(海報論文：博士生組 )

[1] J. W. Lee and J. T. Chen, A semi-analytical approach for a nonlocal suspended strip in an elliptical waveguide, IEEE Trans. Microw. Theory Tech., 60(12) (2012), pp. 3642-3655. [2] J. T. Chen, Y. T. Lee and Y. J. Lin, Analysis of multiple-spheres radiation and scattering problems by using a null-field integral equation approach, Appl. Acoust., 71 (2010), pp. 690-700.[3] A. Charalambopoulos, D. I. Fotiadis and C. V. Massalas, Scattering of a point generated field by kidney stones, Acta Mech., 153, (2002), pp. 63-77.[4] P. Morse and H. Feshbach, Method of Theoretical Physics, McGraw-Hill, New York (1953)[5] S. Zhang and J. Jin, Computation of Special Functions, John Wiley & Sons, New York (1996)

Results and discussion

Problem description

Abstract In this work, the acoustic scattering problem from a point source to a two-layer prolate spheroid is solved by using the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels. To fully utilize the spheroid geometry, the fundamental solutions and the boundary densities are expanded by using the addition theorem and spheroidal harmonics in the prolate spheroidal coordinates, respectively. For the confocal structure, the analytical solution can be analytically derived by using the null-field BIEM. Besides, it is interesting that the kidney-stone system can be simulated by a two-layer spheroid structure. Finally, an example is considered for the parameter study. Also, a special case of the acoustic scattering problem of a point source by a rigid scatterer is also done by setting the density of inner medium to be infinity.

Conclusions1. Based on the addition theorem, the closed-form fundamental solution is expanded into the degenerate kernel in the prolate spheroidal coordinates.2. The size of kidney-stone can be predicted by the parameter study. 3. Numerical results can be used as a reference for clinical medical treatment.

References

The boundary condition on the surface of the igid scatterer is

The governing equation of the scattering problem of a point source is the non-homogeneous three-dimensional Helmholtz equation as follows:

2 2( ) ( ) 4 ( ) k u x x r

The interface conditions on the surface are

1

( )0,

ku

Snx

xx

0

0

( ) ( ) ( ),

( ( ) ( )) ( )1 1,

in sc k

in sc k

ext k

u u u S

u u uS

n n x x

x x x x

x x xx

Null-field boundary integral formulations-the present version

4 ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),

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

S S

c

S S

u T u dS U t dS V S

T u dS U t dS V S

x s x s s s x s s x

s x s s s x s s x

Expansion for boundary densities 0 0

0 0

( , ) 2 ( , ) ( , )

( , ) ( , ) cos[ ( )], ,( , )

( , ) 2 ( , ) ( , )

( , ) ( , ) cos[ ( )], ,

N nE mN mn mn

n m mn

mn mn

N nI mN mn mn

n m mn

mn mn

U ik je q he q

S q S q mU

U ik je q he q

S q S q m

s x

s x s x x s

x s

s x s x x s

s x

s x

s x

The closed-form fundamental solution

(1)0( , ) ( ),

ikreU ikh kr r

rs x x s

s s0 0 1 1

2s

s s2 20 0 1 1s s

( ) ( , ) cos( ) ( , )sin( ),

1( ) ( , ) cos( ) ( , )sin( ) , ,

v v

wv wv wv wvv w v w

v v

wv wv wv wvv w v w

u g S q w h S q w S

t p S q w q S q w Sc

s s

s s

s s

s sFig. 1 The sketch of the taking free body

Degenerate kernel

Fig. 2 Scattering field using different number of truncation terms

0 0.5 1 1.5 2 2.5 3 3.5

0.1

0.11

0.12

0.13

0.14

0.15

0.16

u sc (x )

N = 2N = 3N = 5N = 1 0N = 2 0

0 0.5 1 1.5 2 2.5 3 3.5

0.1

0.12

0.14

0.16

u sc (x )

a 1/b 1= 1 0

a 1/b 1= 5

a 1/b 1= 4

a 1/b 1= 2

0 0.5 1 1.5 2 2.5 3 3.5

0.08

0.12

0.16

0.2

0.24

0.28

0.32

u sc (x )

r 0= 0 .0 6 m

r 0= 0 .0 8 m

r 0= 0 .1 m

0 0.5 1 1.5 2 2.5 3 3.5

0

0.04

0.08

0.12

0.16

u sc (x )

0= 2 1

0= 3 1

0= 4 1

0 0.5 1 1.5 2 2.5 3 3.5

0

1

2

3

4

5

u sc (x )

r 0= 0 .1 mA n a ly tic a l so lu tio n (c o n fo c a l ca se )

A n a ly tic a l so lu tio n (s in g le r ig id sc a tte re r)

A B A Q U S

0 0.5 1 1.5 2 2.5 3 3.5

0.4

0.8

1.2

1.6

2

2.4

u sc (x )

r 0= 0 .2 mA n a ly tica l so lu tio n (c o n fo c a l ca se )

A n a ly tica l so lu tio n (s in g le r ig id sc a tte re r)

A B A Q U S

3

3

0

1

1

0 1

1022

1533 sec

1000

1493 sec

0.1

0.05

0.01

0.005

4

k

k

ext

ext

Kg m

c m

Kg m

c m

r m

r m

a m

b m

0 0.5 1 1.5 2 2.5 3 3.5

0.05

0.1

0.15

0.2

0.25

0.3

u sc (x )

a 0= 0 .0 0 8 ma 0= 0 .0 0 9 ma 0= 0 .0 1 ma 0= 0 .0 1 1 ma 0= 0 .0 1 2 m

Fig. 3 Scattering field versus by changing the ratio of kidney-stone radii

Fig. 4 Scattering field versus by varying the position of the point source

Fig. 5 Scattering field versus by changing the radial parameter of the kidney surface

Fig. 6 Scattering field versus by changing the length of semi-major axis

Fig. 8 Scattering field versus for a case of a single rigid scatterer (r0=0.2m)

Fig. 7 Scattering field versus for a case of a single rigid scatterer (r0=0.1m)

(a)