1 the method of fundamental solutions for exterior acoustic problems 報告人：陳義麟...

1

The Method of Fundamental solutions for exterior acoustic problems

報告人：陳義麟國立高雄海洋科技大學造船系

高雄 2006/3/19

NSC-92-2611-E-022-004

Outline

Introduction

Method of fundamental solutions

Governing equation

Mathematical analysis

Numerical examples

Conclusions

2

Introduction

Method of Fundamental Solutions

Governing Equation


Numerical examples

Conclusions

3

Numerical method Numerical method

Finite Difference Method

Finite Difference Method

Finite Element Method

Finite Element Method

Boundary Element Method

Boundary Element Method

Numerical method

Meshless Method Meshless Method

Mesh Mesh MeshSingular integral

X X XX 4

Introduction


Governing Equation


Numerical examples

Conclusions

5

Method of fundamental solutions

j

ijji xscxu ),()(

),(),( ijij rxs

Field representation:

ijij xsr

xi

sj

rijsj+1

Interior problem Exterior problem

xi

sjrij

sj+1

Source pointObservation point

6

Acoustic problem

Dxxukxu ,0)()( 22

Governing Equation:

is the acoustic pressure u

k is the wave number

2 is the Laplacian operator

D is the domain of the interest

7

Fundamental solution

)(2),(),( 22 sxxsUkxsU

)(2

),( )1(0 krH

ixsU

The fundamental solution satisfy

8

is the Dirac delta function

Kernel functions

),( xsU Fundamental solution

9

xn

UxsL

),(

xs nn

UxsM

2

),(

r

nykrHki

n

UxsT ii

s

)(

2),(

)1(1

Acoustic pressure and flux

N

jjj sxsUxu

2

1

)(),()(

N

jjj sxsLxt

2

1

)(),()(

Single-layer potential approach

10

N

jjj sxsTxu

2

1

)(),()(

N

jjj sxsMxt

2

1

)(),()(

Double-layer potential approach

Using MFS to solve the acoustic field

11

}{][

}{][}{

}]{[}{

1

1

uUwu

uU

Uu

For the Dirichlet B.C., uu single-layer potential approach


12

uu

}{][

}{][}{

}]{[}{

1

1

tLwu

tL

Lt

For the Neumann B.C., tt single-layer potential approach


13

}{][

}{][}{

}]{[}{

1

1

uTvu

uT

Tu

For the Dirichlet B.C., uu double-layer potential approach


14

uu

}{][

}{][}{

}]{[}{

1

1

tMvu

tM

Mt

For the Neumann B.C., tt double-layer potential approach

Introduction


Governing Equation


Numerical examples

Conclusions

15

Degenerate kernels for circular case

(0,0) X1

Y1

R1

EUIU

Y2

(0,0)X2

R22.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

2.80

IU

EU

),( x

),( x

1

2

16

RnkRJkYkiJRU

RnkJkRYkRiJRUxsU

nnn

ne

nnn

ni

)),(cos()()]()([2

),;,(

)),(cos()()]()([2

),;,(),(

012321

10432

324201222

22321012

1222210

][

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

U

N

NNNN

NNN

NN

CirculantDiscritization into 2N nodes on the circular boundary

12212

222210 )()(][

NNNNN CaCaCaIaU

17

NN

NC

22

2

00001

10000

00100

00010

Circulant

)2

2sin()

2

2cos(2

2

Ni

Ne N

i

: eigenvalue of C2N

18

Eigenvalue of U kernel 12

122

210

NNaaaa

Riemann sum reduces to integral:

),( aRa

)()()1(2 kRJkHi lll

dU

mUmN

mN

)0,()cos(1

)0,()cos(lim

2

0

12

0

19

NNl ),1(,...2,1,0

)()(' )1(2 kRJkHi lll

)(')()1(2 kRJkHi lll

)(')(' )1(2 kRJkHi lll

Eigenvalues of L, T and M influence matrices

NNl ),1(,...2,1,0

20

Fictitious frequency (exterior problem)

0 2 4 6 8

-0.8

-0.4

0 .0

0 .4

0 .8

1 .2UT method

LM method

Burton & Miller method

Analytical solution

ka

u(a,0)

1),( at0),( at

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

9

1),( at0),( at

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

9

-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

21

tt

0)(

0)('

)()('2

kRJ

kaH

kRJkaHi

l

l

lll

Radiation problem with Neumann B. C.,

22

single-layer potential approach

}{][}{

}]{[}{1 tL

Lt

uu

0)(

0)(

)()()1(

)1(2

kRJ

kaH

kRJkaHi

l

l

lll

Radiation problem with Dirichlet B. C.,

23

single-layer potential approach

}{][}{

}]{[}{1 uU

Uu

tt

0)('

0)(

)(')()'1(

)'1(2

kRJ

kaH

kRJkaHi

l

l

lll

Radiation problem with Neumann B. C.,

24

double-layer potential approach

}{][}{

}]{[}{1 tM

Mt

uu

0)('

0)(

)(')()1(

)1(2

kRJ

kaH

kRJkaHi

l

l

lll

Radiation problem with Dirichlet B. C.,

25

double-layer potential approach

}{][}{

}]{[}{1 uT

Tu

Introduction


Governing Equation


Numerical examples

Conclusions

26

Radiation problem (Dirichlet type) for a cylinder

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

4cos),( au

Γ

Radiatora

27

Collocation pointSource point

R x

s

Ra

The located position of source and collocation points

28

The contour plot for the real-part solutions

29

0 2 4 6 8 10

-12

-8

-4

0

4

8

BEM

MFS

k

t

)58.7(

)(4 kaJ

)43.8(

)(4 kRJ

t(a,0) versus k for by using the MFS and BEM.

30

}{]][[}{

}{][}{

}]{[}{

1

1

uULt

uU

Uu

32

5

32

5

1),( at

The nonuniform radiation problem for a cylinder.

31

Numerical solution for the nonuniform radiation problem

32

0 2 4 6 8

-0.4

0

0.4

0.8

1.2

UL method

TM method

Burton & Miller

Analytical solution

k

u

u versus k using the MFS for the nonuniform radiation by a circular cylinder.

33

Introduction


Governing Equation


Numerical examples

Conclusions

34

Conclusions

We have verified that the fictitious frquency depends on the location of the source point.

For the MFS, the sources can be distributed on the real boundary without any difficulty. However, the sources must be distributed outside the domain to avoid the singularity when the MFS are utilized.

Fictitious frequencies for the exterior acoustic problems were analytically derived by using the degenerate kernels and circulants.

1.

2.

3.

35

The EndThanks for your

attention

36

1 the method of fundamental solutions for exterior acoustic problems 報告人：陳義麟...

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