analysis of two-spheres radiation problems by using the null-field integral equation approach

Analysis of two-spheres radiation problems by using the

null-field integral equation approach

The 32nd Conference of Theoretical and Applied Mechanics

Ying-Te Lee ( 李應德 ) and Jeng-Tzong Chen ( 陳正宗 )

學校 : 國立臺灣海洋大學科系 : 河海工程學系時間 : 2008 年 11 月 28-29日地點 : 國立中正大學

The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi

2

Outline

Introduction1.

2.

3.

Problem statement

Method of solution

Numerical examples4.

5. Concluding remarks


3

Motivation

Numerical methods for engineering problems

FDM / FEM / BEM / BIEM / Meshless method

BEM / BIEM

Treatment of siTreatment of singularity and hyngularity and hy

persingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate

1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks


4

Advantages of BEM

1. Mesh reduction

2. Solve infinite problem


FEM BEM

Area (2D) Line (1D)2-D problem: Volume (3D) Surface (2D)3-D problem:

Only boundary discretization is needed and without the DtN map.


5

BEM FEM

5

DtN interface

BEM and FEM



6

Singular and hypersingular integrals


x

sx

s

Conventional approach to calculate singular and hypersingular integral (Bump contour)

Present approach x

s

x

s


7

Fictitious frequency

(1) CHIEF method (Schenck, JASA , 1968)

(2) Burton and Miller method (Burton and Miller, PRS , 1971)

(3) SVD updating term technique (Chen et al., JSV, 2002)


tL

Uu

M

T

uMk

iTtL

k

iU

uTt

U

Additional constraint (CHIEF point)

Non-unique solution:

0 2 4 6 8

-2

-1

0

1

2UT m ethod

LM method

Burton & Miller method

t(a,0)

1),( au0),( au

Drruk ),( ,0),()( 22

9

1),( au0),( au

Drruk ),( ,0),()( 22

9


8

Successful experiences in 2-D problems using the present approach

)()()( sdBsxB ),( xsK

),( xsK e

Fundamental solutionFundamental solution

Advantages of present approach:1. No principal value2. Well-posed model3. Exponential convergence4. Free of mesh generation

Degenerate kernelDegenerate kernel

xsxsK

xsxsKe

i

),,(

),,(

),( xsK i

sx ln


The proposed approach will be extended to deal with 3-D problem.

)()1(0 sxkH

(Laplace)

(Helmholtz)


9

3-D radiation problem


Dxxuk ,0)()( 22


10

Interior case Exterior case

cD

D D

x

xx

xcD

x

Degenerate (separate) formDegenerate (separate) form

DxsdBstxsUsdBsuxsTxuBB

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

BxsdBstxsUVPRsdBsuxsTVPCxuBB

),()(),(...)()(),(...)(2

1

Bc

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

B

Boundary integral equation and null-field integral equation

s

s

ikr

n

sust

n

xsUxsT

r

exsU

)()(

),(),(

4),(


x


11

),,( UE

x),,( UI x

Expansions


Expand fundamental solution by the using degenerate kernel

Expand boundary densities by using the spherical harmonics

,),()(),(

,),()(),(

),(

0

0

sxsBxAxsU

sxxBsAxsU

xsU

jjj

E

jjj

I

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

kv

v

w

wv

kvwk BswPBst

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

kv

v

w

wv

kvwk BswPAsu

M term in the real implementation

),,( s


12

Degenerate kernels

0 0

)2( )(cos)(cos)(cos)()()!(

)!()12(

4

cossin

44),(

n m

mn

mnnnm

ikr

mPPkhkjmn

mnn

ik

kr

krikrik

r

exsU

0 0

)2( )(cos)(cos)(cos)()()!(

)!()12(

4),(

n m

mn

mnnnm mPPkhkj

mn

mnn

ikxsU


,)(cos)()()12(sin

0nnnn Pkjkjn

kr

kr

n

m

mn

mnnnn mPP

mn

mnPPP

1

)(cos)(cos)(cos)!(

)!(2)(cos)(cos)(cos

,)(cos)()()12(cos

0nnnn Pkykjn

kr

krAddition theorem

),,(

),,(

x

s


13

U(s,x)

T(s,x)

L(s,x) M(s,x)

sn

sn xn

xn

Relationship of kernel functions


DxsdBstxsUsdBsuxsTxuBB

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

DxsdBstxsLsdBsuxsMxtBB

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


14

Degenerate kernels


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

)!()12(

4

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

)!()12(

4),()2(

0 0

)2(

0 0

mkhkjPPmn

mnn

ikU

mkhkjPPmn

mnn

ikU

xsU

nnmn

mn

n

n

mm

e

nnmn

mn

n

n

mm

i

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

)!()12(

4

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

)!()12(

4),()2(

0 0

2

)2(

0 0

2

mkhkjPPmn

mnn

ikT

mkhkjPPmn

mnn

ikT

xsT

nnmn

mn

n

n

mm

e

nnmn

mn

n

n

mm

i

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

)!()12(

4

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

)!()12(

4),()2(

0 0

2

)2(

0 0

2

mkhkjPPmn

mnn

ikL

mkhkjPPmn

mnn

ikL

xsL

nnmn

mn

n

n

mm

e

nnmn

mn

n

n

mm

i

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

)!()12(

4

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

)!()12(

4),()2(

0 0

3

)2(

0 0

3

mkhkjPPmn

mnn

ikM

mkhkjPPmn

mnn

ikM

xsM

nnmn

mn

n

n

mm

e

nnmn

mn

n

n

mm

i


15

Adaptive observer system


),,( 111 ),,( 222


16

Problem with spherical boundaries

Null-field BIE

Expansion

Degenerate kernel for the fundamental

solution

Spherical harmonics for

boundary density

Collocating the collocation point and matching the boundary conditions

Boundary integration in adaptive observer system

Linear algebraic system

Obtain the unknown spherical harmonics coefficients

Velocity potential

Flowchart


Sound pressure


17

Case 1: A sphere pulsating with uniform radial velocity Case 2: A sphere oscillating with non-uniform radial velocity Case 3: Two spheres vibrating from uniform radial velocity

Numerical examples



18

a

Case 1: A sphere pulsating with uniform radial velocity


Uniform radial velocity U0 (Seybert et al., JASA, 1985)

Oy

z

x

)(0

0

1aikeU

ika

kaizap

Analytical solution:


19

Results

0 0

)cos()(cos)(v

v

w

wvvw wPBst

0

)()( U

n

sust

s

000 UB


Uniform radial velocity U0

S

i

S

i sdSstxsUsdSsuxsT )()(),()()(),(0 0)2(0

)2(0

00 )(

)(1U

kah

kah

kA

)(

)()(

)2(0

)2(0

00 kah

khUizp

S

e

S

e sdSstxsUsdSsuxsTxu )()(),()()(),()(0)2(

0

)2(0

)(

)(1U

kah

kh

ku

ukizuip 00

000 UB


20

Comparison


)(0

0

1aikeU

ika

kaizap

Exact solution (Seybert et al.)

Spherical Hankel function of series form:

n

k

kizn

n izknk

kne

z

izh

0

1)2( )2(

)1(!

)!()(

n

k

kkizn

n

k

kiznn

k

kizn

n

zkiknk

kne

z

i

izknk

kne

z

iiz

knk

kne

z

izh

0

)1(1

0

2

02

1)2(

)()2()1(!

)!(

)2()1(!

)!()2(

)1(!

)!()(

)(

)()2(

0

)2(0

00 kah

khUizp

Present approach


21

Distribution of collocation points



22

1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarksSound pressure

0 1 2 3 4 5k a

0

0.2

0.4

0.6

0.8

1

Re(

p/z 0

U0)

P resen t a p p ra o chE x a ct so lu tio n

0 1 2 3 4 5

k a

0

0.2

0.4

0.6

Im(p

/z0U

0)

P resen t a p p ro a chE x a ct so lu tio n

Real part of non-dimensional pressure on the surface

Imaginary part of non-dimensional pressure on the surface

ka=π exist a fictitious frequency in the result of Seybert et al.


23

a


Non-uniform radial velocity U0cosθ (Seybert et al., JASA, 1985)

Oy

z

x

Analytical solution:

Case 2: A sphere oscillating with non-uniform radial velocity

)(022

0

2

cos)1(2

)1( aikeUakika

ikkaizap


24

Result


0 0

)cos()(cos)(v

v

w

wvvw wPBst

cos)(

)( 0Un

sust

s

000 UB

Uniform radial velocity U0

S

i

S

i sdSstxsUsdSsuxsT )()(),()()(),(0 0)2(1

)2(1

10 )(

)(1U

kah

kah

kA

cos)(

)()( 0)2(

1

)2(1

0 Ukah

khizp

S

e

S

e sdSstxsUsdSsuxsTxu )()(),()()(),()(0)2(

1

)2(1

)(

)(1U

kah

kh

ku

ukizuip 00

010 UB


25

Exact solution


cos

)(

)(0)2(

1

)2(1

0 Ukah

khizp

)(

0220

2

cos)1(2

)1( aikeUakika

ikkaizap

Exact solution (Seybert et al.)

Spherical Hankel function of series form:

n

k

kizn

n izknk

kne

z

izh

0

1)2( )2(

)1(!

)!()(

n

k

kkizn

n

k

kiznn

k

kizn

n

zkiknk

kne

z

i

izknk

kne

z

iiz

knk

kne

z

izh

0

)1(1

0

2

02

1)2(

)()2()1(!

)!(

)2()1(!

)!()2(

)1(!

)!()(

Present approach


26

Case 3: Two spheres vibrating from uniform radial velocity


Uniform radial velocity U0 (Dokumaci, JSV, 1995)

2a 2a

Oy

z

x

aa


27

Distribution of collocation points



28

Contour of sound pressure


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

-6

-4

-2

0

2

4

6

8

z=0 and ka=1

Present approach

(Dokumaci and Sarigül, JSV, 1995)

Surface Helmholtz Integral Equation (SHIE)


29

1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarksContour of sound pressure

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

-6

-4

-2

0

2

4

6

8

Present approach

SHIE (Dokumaci and Sarigül)

z=0 and ka=2


30

Contour of sound pressure

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

-6

-4

-2

0

2

4

6

8


Present approach

SHIE (Dokumaci and Sarigül)

z=0 and ka=0.1


31

Potential of the nearest point


SHIE SHIE+CHIEF


32


0 2 4 6 8 10k a

- 6

- 4

- 2

0

2

4

Re(

p/z 0U

0)

N ea res t p o in tR ea l p a rt (M = 1 0 )


Present approach Burton & Miller approach0 2 4 6 8 10

k a

- 2

- 1

0

1

2

Re(

p/z 0

U0)

N ea rest p o in tR ea l p a rt (M = 1 0 )


33

Potential of the furthest point

SHIE SHIE+CHIEF



34

Potential of the furthest point

0 2 4 6 8 10k a

- 6

- 4

- 2

0

2

4

6

Re(

p/z 0U

0)

F u rth es t p o in tR ea l p a rt (M = 1 0 )

Present approach Burton & Miller approach

0 2 4 6 8 10k a

- 2

- 1

0

1

2

Re(

p/z 0U

0)

F u rth es t p o in tR ea l p a rt (M = 1 0 )



35

Concluding remarks

A systematic approach, null-field integral equation in conjunction with degenerate kernel and spherical harmonics, was successfully proposed to deal with the three-dimensional radiation problem.

1.

3.

The present approach can be seen as one kind of semi-analytical approach, since error comes from the number of truncated term of spherical harmonics.


4.

A general-purpose program for multiple radiators of various number, radii, and arbitrary positions was developed.

2. Only boundary nodes were needed in the present approach.

5.

The Burton and Miller approach was successfully used to remedy the fictitious frequency.


36

The End

Thanks for your kind attention

http://ind.ntou.edu.tw/~msvlabWelcome to visit the web site of MSVLAB/NTOU


37


0 2 4 6 8 10k a

- 1

0

1

2

3

4

5

Re(

p/z 0U

0)

N ea res t p o in tR ea l p a rt (M = 1 6 )

Im ag in a ry p a rt (M = 1 6 )

4 .49 5 .7 6 6 .9 8

7 .7 3

8 .2 2 9 .0 9

9 .3 6


38

Elliptic coordinates

01

000

01

0000

],sinsinsinhcoscos[cosh4

)4

ln(2

],sinsinsinhcoscos[cosh4

)4

ln(2),(

0

n

n

n

n

nnnnnnen

a

nnnnnnen

a

xsU

Degenerate kernel in the 2-D Laplace problems

Degenerate kernel in the 2-D Helmholtz problems

01

00

00

0

01

00

00

0

,)cosh,()cosh,()cos,()(

)cos,(

)cosh,()cosh,()cos,()(

)cos,(

,)cosh,()cosh,()cos,()(

)cos,(

)cosh,()cosh,()cos,()(

)cos,(

),(

nmmmo

m

m

nmmme

m

m

nmmmo

m

m

nmmme

m

m

hHohJohSohM

hSo

hHehJehSehM

hSe

hHohJohSohM

hSo

hHehJehSehM

hSe

xsU

P. M. Morse and H. Feshbach, Methods of theoretical physics, 1953.


39

Motivation

BEM / BIEMBEM / BIEM

Improper integralImproper integral

Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity

Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary

Collocation Collocation pointpoint

Fictitious BEMFictitious BEM

Null-field approachNull-field approach

CPV and HPVCPV and HPVIll-posedIll-posed

Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)

Waterman (1965)Waterman (1965)

Achenbach Achenbach et al.et al. (1988) (1988)


40

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

BEM (64 elements) FEM (2791 elements)

Non-uniform radiation problem (2D)– Dirichlet BC)2( ka

Successful experience


41

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

BEM (63 elements) FEM (7816 elements))4( ka

Successful experience

Non-uniform radiation problem (2D)– Neumann BC


42

3-D radiation problem


Dxxuk ,0)()( 22

2a 2a

Oy

z

x

aa

analysis of two-spheres radiation problems by using the null-field integral equation approach

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