The 32 nd National Conference on Theoretical and Applied Mechanics

Scattering of sound from axisymetric sources by multiple circular cylinders using addition

theorem and superposition technique

The 32nd National Conference on Theoreticaland Applied Mechanics

Authors : Yi-Jhou Lin, Ying-Te Lee , I-Lin Chen and Jeng-Tzong ChenDate: November 28-29, 2008 Place: National Chung Cheng University, Chia-Yi

Reporter : Yi-Jhou Lin

National Taiwan Ocean UniversityDepartment of Harbor and River Engineering

The 32nd National Conference on Theoretical and Applied Mechanics

2

Outlines

Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third

identity and that of superposition technique

Numerical examples Concluding remarks

Introduction


3

Motivation

Numerical methods for engineering problems

FDM / FEM / BEM / BIEM / Meshless method

BEM / BIEM

Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate


4

MotivationBEM / 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)


5

Present approach)()()( sdBsx

B ),( xsK

),( xsK e

Fundamental solutionFundamental solution

No principal valueNo principal value

Advantages of present approach1. mesh-free generation2. well-posed model3. principal value free4. elimination of boundary-layer effect5. exponential convergence

Degenerate kernelDegenerate kernel

CPV and HPVCPV and HPV

xsxsK

xsxsKe

i

),,(

),,(

),( xsK i

4)()1(

0 kriH


6

Green’s third identityGreen’s third identity

BIE for Green’s functionBIE for Green’s function


7

Outlines




Problem statement


8

Problem statementOriginal Problem

Free field

Radiation field (typical BVP)

DxxxGk ),(),()( 22

.1

H

jjBB

BxxG ,0),( (soft)


9

FlowchartOriginal problem

Decompose two parts

Free field Radiation field

Expansion

Fourier series of boundary densities

Degenerate kemelFor fundamental solution

Collocate of the real boundary

Linear algebraic system

Calculation of the unknown Fourier

BIE for the domain point

Superposing the solution of two parts

Total field


10

Outlines




Method of solution


11

Method of solutionBoundary integral equation and null-field integral equation

Interior case Exterior case

cD

D D

x

xx

xcD

x x

Degenerate (separate) formDegenerate (separate) form

DxsdBstxsUsdBsuxsTxuBB

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

BxsdBstxsUVPRsdBsuxsTVPCxuBB

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

Bc

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

B

s

s

n

ss

n

xsUxsT

kriHxsU

)()(

),(),(

4

)(),(

)1(0


12

Degenerate kernel and Fourier series

,,,2,1,,)sincos()(1

0 NkBsnbnaas kkn

kn

kn

kk

,,,2,1,,)sincos()(1

0 NkBsnqnpps kkn

kn

kn

kk

s

Ox

R

kth circularboundary

cosnθ, sinnθboundary distributions

eU

x

iU

Expand fundamental solution by using degenerate kernel

Expand boundary densities by using Fourier series

,),()(),(

,),()(),(

),(

0

0

sxsBxAxsU

sxxBsAxsU

xsU

jjj

E

jjj

I


13

Degenerate kernels

xn

xn

sn

U(s,x)U(s,x) T(s,x)T(s,x)

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


14

Degenerate kernels

,)],(cos[)()(4

),(

,)],(cos[)()(4

),(),(

0

)1(

0

)1(

RmkHkRJki

xsL

RmkRHkJki

xsLxsL

mmmm

e

mmmm

i

,)],(cos[)()(4

),(

,)],(cos[)()(4

),(),(

0

)1(

2

0

)1(

2

RmkHkRJik

xsM

RmkRHkJik

xsMxsM

mmmm

e

mmmm

i

,)],(cos[)()(4

),(

,)],(cos[)()(4

),(),(

0

)1(

0

)1(

RmkHkRJki

xsT

RmkRHkJki

xsTxsT

mmmm

e

mmmm

i


15

Adaptive observer system

Source pointSource point

Collocation pointCollocation point


16

Linear algebraic system

B¥

1B2B

NB

x

y


17

Outlines




Mathematical Equivalence


18

Mathematical equivalence between the solution of Green’s third identity and that of superposition technique

+=( , )rG x

( , )G x

( , )fG x

Green’s third identity

Superposition technique


19

Outlines




Numerical examples


20

An infinite plane with two equal circular

cylinders subject to a point sound source.

2 0 313

2

.; ;b a

k k

( )(1)0 ( )

, , where4

iH krU s x r s x

-= º -

2 2 k G x x x D ,,

Governing equation:

Dirichlet Boundary condition:

(soft)

Fundamental solution:

0,Bx

G x


21

Distribution potential on the artificial

boundaries in the free field


22

Case 1 parameter use size and cylinder

B1

B2

b

by

Probe(soft)

(soft)

parameter Probe

Case 1-1 2

b

2

k

0 313.

ak

240 y

Case 1-2 2

b

2

k

1 253.

ak

180 y

Case 1-3 2

b

2

k

2

ak

150 y


23

Distribution potential on the artificial boundaries in the free field versus polar angle.

0 60 120 180 240 300 360

angle

-0 .015

-0.01

-0.005

0

0.005

0.01

uM = 2 0

rea l p a rt (se rie s -fo rm )

im ag . p a rt (se rie s -fo rm )

rea l p a rt ( c lo sed -fo rm )

im ag . p a rt ( c lo sed -fo rm )


24

Relative amplitude of total field versus the probe location y (M=20).

0 4 8 12 16 20 24

P r o b e p o s itio n y (c m )

0

0.2

0.4

0.6

0.8

1

1.2

Rel

ativ

e to

tal s

catt

ered

fie

ld

k a = 0 .3 1 3T H E O

E X P

P resen t m eth o d

Total field

Free field

B1

B2

b

bProbe

(soft)

(soft)

B1

B2

b

bProbe

(soft)

(soft)

Versus


25

Relative amplitude of total field versus the probe location (M=20).

0 2 4 6 8 10 12 14 16 18


0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rel

ativ

e to

tal s

catt

ered

fie

ldk a = 1 .2 5 3

E X P E R IM E N T A L

G A U S S -S E ID E L A P P R O X

E X A C T S O L U T IO N

IN D E P E N D E N T

P R E S E N T M E T H O D

Total field

Free field

B1

B2

b

b

(soft)

(soft)

B1

B2

b

b

(soft)

(soft)

Probe

Probe

Versus


26

Relative amplitude of total field versus

(M=20).

0 4 8 12 16


0

0.4

0.8

1.2

1.6

Rel

ativ

e to

tal s

catt

ered

fie

ld

k a = 2 .0C O U P L E D


T H E O R


B1

B2

b

bProbe

(soft)

(soft)

B1

B2

b

bProbe

(soft)

(soft)

Versus

Total field

Free field


27

Convergence test of Parseval’s sum for (real part).

0 2 4 6 8 10 12 14 16 18 20

T erm s o f F o u rie r se rie s (M )

0.00264

0.00272

0.0028

0.00288

0.00296

Pars

eval

's s

um o

f r

eal p

art s

olut

ion

xnxG ),(


28

Convergence test of Parseval’s sum for

(imaginary part).

0 2 4 6 8 10 12 14 16 18 20

T erm s o f F o u rie r se rie s (M )

0.0093

0.009302

0.009304

0.009306

0.009308

0.00931P

arse

val's

sum

of

imag

inar

y pa

rt s

olut

ion

xnxG ),(


29

parameter

Case 2-1 2

k

42

~b

1 253.a

k

Case 2-2 2

k

42

~b

1 5.a

k

Case 2-3 2

k

42

~b

2a

k

Case 2-4 2

k

42

~b

3a

k

Case 2 parameter use cylinder center-to-center

B1

B2

b

bProbe

(soft)

(soft)


30


(M=20).

0 2 4 6 8

S p a c in g b etw een cen ters o f cy lin d ers 2 b /

0

0.4

0.8

1.2

1.6

2

Rel

ativ

e to

tal s

catt

ered

fie

ld a

t p

rob

e

k a = 1 .2 53C O U P L E D



B1

B2

b

bProbe

(soft)

(soft)


31


(M=20).

0 1 2 3 4 5 6 7 8


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Rel

ativ

e to

tal s

catt

ered

fie

ld a

t p

rob

e

k a = 1 .5 0IN D E P E N D E N T

C O U P L E D


D IA G O N A L T E E M S O N L Y

B1

B2

b

bProbe

(soft)

(soft)


32


(M=20).

0 1 2 3 4 5 6 7 8


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Rel

ativ

e to

tal s

catt

ered

fie

ld a

t p

rob

e

k a = 2 .0IN D E P E N D E N T

C O U P L E D



B1

B2

b

bProbe

(soft)

(soft)


33


(M=20).

0 1 2 3 4 5 6 7 8

S pa c in g b e tw e e n c e nter s o f c y lin d e rs 2 b /

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Rel

ativ

e to

tal s

catt

ered

fie

ld a

t p

rob

e

k a = 3 .0C O U P L E D




b

bProbe

(soft)

(soft)

B1

B2


34

Outlines



Numerical examples Concluding remarksConcluding remarks


35

Concluding remarks

A general-purpose program for solving the problems with arbitrary number, size and various locations of circular cavities was developed.

We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.


36

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The 32 nd National Conference on Theoretical and Applied Mechanics

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