M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 1...

MMSS

1

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海洋大學力學聲響振動實驗室MSV LAB HRE NTOU

國立台灣海洋大學河海工程學系陳正宗教授

河海工程概論

結構工程

國立台灣海洋大學河海工程學系

Analysis of acoustic eigenfrequencies and eigenmodes by using t

he meshless method

指導教授 : 陳正宗教授學生 : 張銘翰

15, June, 2001

MMSS

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Outlines

I. Numerical Methods.

II. The developments of meshless methods and radial basis functions.

III. The approaching methods of the diagonal elements.

IV. The technique for extracting true eigenvalues.

V. The techniques for filtering out the spurious eigenvalues

VI. Conclusions.

MMSS

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Numerical Methods Numerical Methods

Mesh MethodsMesh Methods

Finite Difference Method

Finite Difference Method

Meshless Method Meshless Method

Finite Element Method

Finite Element Method

Boundary Element Method

Boundary Element Method

Numerical methods

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VV


Meshless methodsMeshless methods

Variational methods(Galerkin methods)

Variational methods(Galerkin methods) BIEMBIEM BEMBEM

Continuous moving least square

Belyschko et al. (1994)

Continuous moving least square

Belyschko et al. (1994)

Continuous KernelMonagh (1982)Liu et al. (1995)

Continuous KernelMonagh (1982)Liu et al. (1995)

Moving least squareLancaster & Salkauskas Belyschko et al. (1994)Nayroles et al. (1992)

Moving least squareLancaster & Salkauskas Belyschko et al. (1994)Nayroles et al. (1992)

Discrete Kernel

Monaghan (1982)

Liu et al. (1995)

Discrete Kernel

Monaghan (1982)

Liu et al. (1995)

Partitious of UnityBabuska and Melenk

Duarte and Oden (1995)

Partitious of UnityBabuska and Melenk

Duarte and Oden (1995)

Boundary node method

Mukherjee, Huang,

Chen & Kang

Boundary node method

Mukherjee, Huang,

Chen & Kang

Local BIE

(unsymmetric)

Atluri, Zhu, and Sladek

Local BIE

(unsymmetric)

Atluri, Zhu, and Sladek

Local Petrov-Galerkin a

pproach

(symmetric)

Atluri, Zhu, and Liu

Local Petrov-Galerkin a

pproach

(symmetric)

Atluri, Zhu, and Liu

Complete solution+

Particular solution

Complete solution+

Particular solution

Completesolution

Completesolution

Particularsolution

Particularsolution

Volumepotential

Volumepotential

RBFsolution

RBFsolution

Chen & KangChen & Kang

The developments of meshless methods

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The developments of radial basis functions

Radial basis function (RBFs)

Radial basis function (RBFs)

Mesh methodMesh method

Globally-supported RBFsGlobally-supported RBFs

DRBEMNardiniBrebbia

DRBEMNardiniBrebbia

Method of particular integral

Ahmad & Banerjee

Method of particular integral

Ahmad & Banerjee

rr 1)( rCr )(

Meshless methodMeshless method

Globally-supported RBFsGlobally-supported RBFsCompactly-supported RBFs

Wu 、 Buhmann & Wendland

Compactly-supported RBFs

Wu 、 Buhmann & Wendland

Volume potential

Volume potential

Method of fundamental

solution

Method of fundamental

solution

)322581()1()(

)35183()1()(

ln26

1

15

163

3

16

6

19

15

1)(

ln23

4

3

1)(

)530

7282366()1()(

)312164()1()(

328

26

26

5432

232

54

326

324

rrrrr

rrrr

rrr

rrrrr

rrrrr

rr

rrrrr

rrrrr

)(

)(

),( )(

rU

sxU

xsUr

c

c

c

(Potential theory)

This thesisThis thesis

r

kr

xsUrD

krJ

xsUrDk

kr

xsUrD

c

c

c

)sin(

)},( Im{)(3

)(

)},( Im{)(22

)cos(

)},( Im{)( 1

0

：

：

：

MMSS

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Complex-valued BEMComplex-valued BEM

Half-effort computation


Imaginary-partformulations

Imaginary-partformulations

Multiple Reciprocity Method(MRM)

Multiple Reciprocity Method(MRM)

Real-part BEMReal-part BEM

Differed by a complementary solution


Singular integralsSingular integrals Avoid singular integralsAvoid singular integrals

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Governing Eq.: Governing Eq.: 0)()( 22 xuk

Two-dimension

Two-dimension

One-dimension

One-dimension

Three-dimension

Three-dimension

)Im(r

eikr

r

kr)sin(

)2

)(Im( 0 krikH

)2

Im(k

ieikr

k

kr

2

)cos()(0 krJ

Imaginary–part formulation

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Double-layer

potential approach

Single-layer

Potential approach

* NDIF method by Kang is the special case

Dirichlet problem

Neumann problem

Dirichlet problem

Neumann problem

Distributed-type

B

I sdBsxsUxu )()(),()(

B

I sdBsxsLxt )()(),()(

B

I sdBsxsTxu )()(),()(

B

I sdBsxsMxt )()(),()(

The distribute and concentrate-type

Dirichlet problem

Neumann problem

Dirichlet problem

Neumann problem

Concentrated-type*

jijj

jijI

i ASMAxsUxu )(),()(

jijj

jijI

i ASMAxsLxt )(),()(

jijj

jijI

i BSMBxsTxu )(),()(

jijj

jijI

i BSMBxsMxt )(),()(

MMSS

9

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Influence matricesInfluence matrices

L’Hôpital’s ruleL’Hôpital’s rule

2),(lim

0),(lim

0),(lim

1),(lim

2kxsM

xsL

xsT

xsU

xs

xs

xs

xs

Invariant methodInvariant method

Indeterminate forms ( )Indeterminate forms ( )0

0

The derivation of indeterminate forms

RBF for RBF for )(0 krJ

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)],([ xsU

.)()(22

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

0

kJkJNNa

NNkJkNJ

.)(2)(1 220

m

m kJkJ Addition theorem ofBessel function

)]),(([ xsUdiag .10 a

The diagonal elements of U kernel

MMSS

11

VV


)],([ xsT

.)()( 22

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

0

kJkJkNNb

NNkJkJN

.)()(2)()(0 00

m

mm kJkJkJkJ Addition theorem ofBessel function

)]),(([ xsTdiag .00 b

The diagonal elements of T kernel

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Two-dimensional cavity

Influence matricesU(s,x), L(s,x), T(s,x), M(s,x)

+Indirect method

Influence matricesU(s,x), L(s,x), T(s,x), M(s,x)

+Indirect method

RBF for RBF for )(0 krJ

EigenvaluesEigenvalues

EigenmodesEigenmodes

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Single-layerpotential approach


0 2 4 6

k

-40

-30

-20

-10

0

Log

T : T ru e e ig en v a lu eS : S p u rio u s e ig en v a lu e(): A n a ly tica l so lu tio n

T

T T T

K 1

2.405

J0(1)

(2 .405)

K 2

3.830

J1(1)

(3 .832)

K 3

5.130

J2(1)

(5 .136) K 4

5.520

J0(2)

(5 .520)

N u m b er o f n o d es : 1 0k : 0 .0 0 5

T

K 5

6.215

J3(1)

(6 .380)

0 2 4 6

k

-30

-20

-10

0

10

Log


ST S

K3.055

J'2(1)

(3 .054)

K 2

3.830

J1(1)

(3 .832)

K4.205

J'3(1)

(4 .201)

K 4

5.375

J0(2)

(5 .520)

K 3

5.135

J2(1)

(5 .136)

K 5

6.295

J3(1)

(6 .380)

N u m b er o f n o d es : 1 0k : 0 .0 0 5

T

K 1

2.405

J0(1)

(2 .405)

S ST T S

T

Eigenfrequencies for Dirichlet B.C.

Double-layerpotential approach


True eigenvaluesSpurious eigenvalues

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0 2 4 6

k

-50

-40

-30

-20

-10

0

10

Log


TS

T

K 1

3.055

J'2(1)

(3 .054)

K 3

4.205

J'3(1)

(4 .201)

K5.135

J 2(1)

(5 .136)

K 5

6.295

J'5(1)

(6 .416)

K 4

5.340

J'4(1)

(5 .318)

S

K 2

3.830

J'0(1)

(3 .832)

TT

K2.405

J0(1)

(2 .404)

S S S T

N u m b er o f n o d es : 1 0k : 0 .0 0 5

0 2 4 6

k

-50

-40

-30

-20

-10

0

10

Log


TT

T T

K 1

1.840

J'1(1)

(1 .841)

K 2

3.055

J'2(1)

(3 .054)

K 3

3.830

J'0(1)

(3 .831)

K 4

4.180

J'3(1)

(4 .201)

T

K 5

5.330

J'1(2)

(5 .331)

T

K 6

6.415

J'5(1)

(6 .415)

N u m b er o f n o d es : 1 0k : 0 .0 0 5

Eigenfrequencies for Neumann B.C.





True eigenvaluesSpurious eigenvalues

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To extract the true eigenvalues

Deficient constraintsDeficient constraints

Spurious eigensolutionsSpurious eigensolutions

SVDupdating terms

SVDupdating terms

To extract the true eigensolutionsTo extract the true eigensolutions

Additional constraints

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Direct method for Dirichlet B. C. :

SVD updating terms

SVD updating terms

SVD updating terms

Singular equation(UT method)

Hypersingular equation(LM method)

}{~

jt

0}{

jE

E

L

U

MMSS

17

VV


SVD updating terms

SVD updating terms

To extract the true eigenfrequencies

}0{}{ }0{}{}}{{ U

jUjψψi

TUj

Uj

j

Uj

ijjUj

}0{}{ }0{}{}}{{ L

jLjψi

TLj

Lj

j

Lj

ijjLj

0 Li

Ui

}0{}]{[

}0{}]{[

iE

iE

L

U

MMSS

18

VV


The degenerate kernels for interior and exterior problems:

,))(cos()()(

2),,(

,))(cos()()(2

),,(),(

RmkRJkJRU

RmkJkRJRUxsU

nmm

E

nmm

I

；

；

The degenerate kernels

X

Y

R

S

X

Interior problem

X

Y

R

S

X

Exterior problem

Blue: field pointsRed: source points

IE UU

MMSS

19

VV


I

I

TL

UU

E

E

T

U

L

UIE

IE

IE TL

The degenerate kernels

,))(cos()()(

2),,(

,))(cos()()(2

),,(),(

RmkJkRJk

RT

RmkRJkJk

RTxsT

nmm

E

nmm

I

；

；

,))(cos()()(

2),,(

,))(cos()()(2

),,(),(

RmkJkRJk

RL

RmkRJkJk

RLxsL

nmm

E

nmm

I

；

；

MMSS

20

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Indirect method for Dirichlet B. C. :

SVD updating terms

SVD updating terms

0}{

jI

I

T

U

}0{}{ }0{}{}}{{ U

jUjψψi

TUj

Uj

j

Uj

ijjUj

}0{}{ }0{}{}}{{ T

jTjψi

TTj

Tj

j

Tj

ijjTj

Singular-layer approach:

Double-layer approach:

sum) no ( ,0)()( ikJkJ ii sum) no ( ,0)()( ikJkJ ii

To extract the true eigenvalues by the indirect method

MMSS

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VV



SVDupdating-terms

0 2 4 6

k

-30

-20

-10

0

10

Log


ST S

K3.055

J'2(1)

(3 .054)

K 2

3.830

J1(1)

(3 .832)

K4.205

J'3(1)

(4 .201)

K 4

5.375

J0(2)

(5 .520)

K 3

5.135

J2(1)

(5 .136)

K 5

6.295

J3(1)

(6 .380)

N u m b er o f n o d es : 1 0k : 0 .0 0 5

T

K 1

2.405

J0(1)

(2 .405)

S ST T S

T

Examples for the SVD updating termsFor Dirichlet B.C.

0 2 4 6

k

-40

-30

-20

-10

0

10

Log

T

K 1

2.405

T

K 2

3.832

J1(1)

(3 .832)

T

J0(1)

(2 .405)

K 3

5.136

J2(1)

(5 .136)

K 4

5.520

J0(2)

(5 .520)

K 5

6.368

J3(1)

(6 .380)

T T


N u m b er o f n o d es : 1 2k : 0 .0 0 1

I

I

T

UTrue eigenvalues

MMSS

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VV



SVDupdating-terms

For Neumann B.C.

Examples for the SVD updating terms

0 2 4 6

k

-50

-40

-30

-20

-10

0

10

Log


TS

T

K 1

3.055

J'2(1)

(3 .054)

K 3

4.205

J'3(1)

(4 .201)

K5.135

J 2(1)

(5 .136)

K 5

6.295

J'5(1)

(6 .416)

K 4

5.340

J'4(1)

(5 .318)

S

K 2

3.830

J'0(1)

(3 .832)

TT

K2.405

J0(1)

(2 .404)

S S S T

N u m b er o f n o d es : 1 0k : 0 .0 0 5

0 2 4 6

k

-40

-30

-20

-10

0

10

Log


T

K 1

3.054

J'2(1)

(3 .054)

K 2

3.832

J'0(1)

(3 .832)

K 3

4.201

J'3(1)

(4 .201)

K 4

5.331

J'1(2)

(5 .331)

T T T

N u m b er o f n o d es : 1 2k : 0 .0 0 1

I

I

M

LTrue eigenvalues

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23

VV


Deficient constraintsDeficient constraints

Spurious eigensolutionsSpurious eigensolutions

Mathematically explainingMathematically explaining

Fredholm alternative theoremFredholm alternative theorem

Numerical techniqueNumerical technique

SVD updating documentsSVD updating documents

To filter out the spurious solutions

MMSS

24

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Fredholm’s alternative theorem:

For solving an algebraic system:

Fredholm alternative theorem

Adjoint homogenous sol.

0}{ * b

bAx

bAx

Alternative theoremAlternative theorem

0 det A

0}{

}{}{*

***

A

bAx

bAx 1 Solutionx

: the transpose conjugate matrix ofA*Areal is if * AAA T

MMSS

25

VV


For double-layer potential approach:

}{ )],([)(

}{ )],([)(

xsMxt

xsTxu

0][][}{ or 0}{][

][

MT

M

T T

T

T

SVD updating documents

Ab x

0}{or 0}{ AA TT

MMSS

26

VV




Dirichlet problem:

Neumann problem: sum) no ( ,0)()( ikJkJ ii sum) no ( ,0)()( ikJkJ ii

}0{}{ }0{}{}}{{}{

T

jTji

TTj

Tj

j

Tj

ijjTT

j

}0{}{ }0{}{}}{{}{

M

jMji

TMj

Mj

j

Mj

ijjTM

j

For double-layer potential approach:


0 Mi

Ti

Spurious modes

T

MMSS

27

VV



SVDupdating documents

0 2 4 6

k

-30

-20

-10

0

10

Log


ST S

K3.055

J'2(1)

(3 .054)

K 2

3.830

J1(1)

(3 .832)

K4.205

J'3(1)

(4 .201)

K 4

5.375

J0(2)

(5 .520)

K 3

5.135

J2(1)

(5 .136)

K 5

6.295

J3(1)

(6 .380)

N u m b er o f n o d es : 1 0k : 0 .0 0 5

T

K 1

2.405

J0(1)

(2 .405)

S ST T S

T

0 2 4 6

k

-40

-30

-20

-10

0

10

Log


SS S S

K 1

3.054

J'2(1)

(3 .054)

K 2

3.832

J'0(1)

(3 .832)

K 3

4.201

J'3(1)

(4 .201)

K 4

5.331

J'1(2)

(5 .331)

N u m b er o f n o d es : 1 2k : 0 .0 0 1

For Dirichlet B.C.

][ II MT

Examples for the SVD updating documents

Spurious eigenvalues

MMSS

28

VV




For Dirichlet B.C.

Examples for the SVD updating document

0 2 4 6

k

-40

-30

-20

-10

0

Log

S

K 1

2.405

K 2

3.830

J1(1)

(3 .832)

J0(1)

(2 .405)

K 3

5.135

J2(1)

(5 .136) K 4

5.520

J0(2)

(5 .520)

K 5

6.367

J3(1)

(6 .380)


S S S S

N u m b er o f n o d es : 1 2k : 0 .0 0 1

0 2 4 6

k

-50

-40

-30

-20

-10

0

10

Log


TS

T

K 1

3.055

J'2(1)

(3 .054)

K 3

4.205

J'3(1)

(4 .201)

K5.135

J 2(1)

(5 .136)

K 5

6.295

J'5(1)

(6 .416)

K 4

5.340

J'4(1)

(5 .318)

S

K 2

3.830

J'0(1)

(3 .832)

TT

K2.405

J0(1)

(2 .404)

S S S T

N u m b er o f n o d es : 1 0k : 0 .0 0 5

Spurious eigenvalues][ II LU

MMSS

29

VV


The interior modes of a circular cavity

Single-layer protential approach


Double-layer protential approach

Double-layer protential approach Analytical solutionAnalytical solution

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

-0 .80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

MMSS

30

VV




0 2 4 6 8 10

k

-40

-30

-20

-10

0

Log


K 1

4.440

K 2

7.030

K 11

(4 .4429)

K 21

(7 .0248)

K 3

8.860

K 22

(8 .8858)

K 4

9.600

K 31

(9 .9346)

T

T TT

N u m b er o f n o d es : 1 2k : 0 .0 0 1



0 2 4 6 8 10

k

-25

-20

-15

-10

-5

0

5

Log

T

K 1

4.440

T

K 2

7.060

T

K 11

(4 .4429)

T


K 21

(7 .0248)

K 3

8.610

K 22

(8 .8858)

K 4

9.870

K 31

(9 .934)S

S

S

S

S

N u m b er o f n o d es : 1 2k : 0 .0 0 1

SVD techniques for a square cavity

SVDupdating terms

SVDupdating terms

0 2 4 6 8 10

k

-40

-30

-20

-10

0

10

Log

T

K 1

4.440

T

K 2

6.990

T

K 11

(4 .4429)

T


K 21

(7 .0248)

K 3

8.610

K 22

(8 .8858)

K 4

9.960

K 31

(9 .9346)

N u m b er o f n o d es : 1 2k : 0 .0 1

True eigenvalues

0 2 4 6 8 10

k

-30

-20

-10

0

10

Log

S

K 1

3.140

K 2

4.470

K 11

(3 .1416)


K 21

(4 .4429)

K 3

6.280 K 22

(6 .2831)

K 4

7.050

K 31

(7 .0248)

K 5

9.070

K 32

(8 .8858)

S

S

SS

N u m b er o f n o d es : 1 2k : 0 .0 1




For Dirichlet B.C.

MMSS

31

VV


0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00




Double-layer protential approach Analytical solutionAnalytical solution

The interior modes of square cavity

MMSS

32

VV


Three-dimensional spherical cavity

Indirect methodIndirect method

RBF for RBF for r

kr)sin( Continuous systemContinuous system

Degenerate kernelDegenerate kernel

X

Z

Y

S

X

X

Z

Y

S

X

Blue： source pointsRed： field points

EigenvaluesEigenvalues

EigenmodesEigenmodes

EigensolutionsEigensolutions

MMSS

33

VV


Eigenfrequencies for Dirichlet B.C.

0 2 4 6

k

-25

-20

-15

-10

-5

0

5

Log


T

K 2

4.478

j1(1)

(4 .4934)

N u m b er o f n o d es : 2 0

k : 0 .0 0 1

T

K 1

3.141

j1(1)

(3 .1416)

T

K 3

5.871 j2

(1)

(5 .7635)





0 2 4 6

k

-25

-20

-15

-10

-5

0

5

Log


TT

K 3

5.746

j2(1)

(5 .7635)


k : 0 .0 0 1

K 1

3.141 j0

(1)

(3 .1416)

T

K 2

4.489

j1(1)

(4 .4934)

S

T

SS S S

True eigenvalues

Supurious eigenvalues

MMSS

34

VV


0 2 4 6

k

-25

-20

-15

-10

-5

0

5

Log


T T

K 3

4.489

j '0(1)

(4 .4934)

K 4

4.519

j '3(1)

(4 .5141)

N u m ber o f n o d es : 2 0

k : 0 .0 0 1

T

K 5

5.611

j '4(1)

(5 .6467)

K 2

3.345

j '2(1)

(3 .3421)

T

T

K 1

2.082

j '1(1)

(2 .0816)

S

S SS

0 2 4 6

k

-20

-15

-10

-5

0

5

Log


T

T

K 3

4.494

j '0(1)

(4 .4934)


k : 0 .0 0 1

T

K 1

2.082

j '1(1)

(2 .0816)

K 2

3.323

j '2(1)

(3 .3421)

T

K 4

5.552

j '4(1)

(5 .6467)

T

K 5

5.912

j '1(2)

(5 .9404)





True eigenvalues


Eigenfrequencies for Neumann B.C.

MMSS

35

VV




0 2 4 6

k

-25

-20

-15

-10

-5

0

5

Log


T T

K 3

4.489

j '0(1)

(4 .4934)

K 4

4.519

j '3(1)

(4 .5141)

N u m ber o f n o d es : 2 0

k : 0 .0 0 1

T

K 5

5.611

j '4(1)

(5 .6467)

K 2

3.345

j '2(1)

(3 .3421)

T

T

K 1

2.082

j '1(1)

(2 .0816)

S

S SS



0 2 4 6

k

-20

-15

-10

-5

0

5

Log


T

T

K 3

4.494

j '0(1)

(4 .4934)

N u m b er o f n od es : 2 0

k : 0 .0 0 1

T

K 1

2.082

j '1(1)

(2 .0816)

K 2

3.323

j '2(1)

(3 .3421)

T

K 4

5.552

j '4(1)

(5 .6467)

T

K 5

5.912

j '1(2)

(5 .9404)

SVD techniques for a spherical cavity

Neumann B.C.

SVDupdating terms

SVDupdating terms

0 2 4 6

k

-20

-15

-10

-5

0

5

Log


T

T

K 3

4.517

j '3(1)

(4.4934)

N u m b er o f n od es : 2 0

k : 0 .0 0 1

T

K 1

2.082

j '1(1)

(2.0816)

K 2

3.344

j '2(1)

(3.3421)

T

K 4

5.623

j '4(1)

(5.6467)

True eigenvalues



0 2 4 6

k

-25

-20

-15

-10

-5

0

5

Log


S

K 2

4.482

j1(1)

(4 .4934)


k : 0 .0 0 1

S

K 1

3.141

j0(1)

(3 .1416)

S

K 3

5.611

j2(1)

(5 .7635)


MMSS

36

VV


n =0, k=0.0

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Analyticalmodes

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Single-layerapproach

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Double-layerapproach

m =0

otherwise,)(cos

0 ,1),,1(

immn ep

mnu

Contours for the boundary modes

MMSS

37

VV


1.8 1.9 2 2.1 2.2

k

- 7

- 6

- 5

- 4

- 3

- 2

- 1

Log

T h e 1 s t m in im u m e ig en v a lu e

T h e 2 n d m in im u m e ig en v a lu eT h e 3 rd m in im u m e ig e n v a lu e

T h e 4 th m in im u m e ig e n v a lu e

ith

minimumeigenvalue

Analytical eigenvalue

Numerical eigenvalue

Test

1st

2.082

2.081 ˇ

2nd 2.079 ˇ

3rd 2.079 ˇ

4th 2.035

Multiplicity of eigenvalues

4th: 2.035

Multiplicity=3

MMSS

38

VV


otherwise,)(cos

0 ,1),,1(

immn ep

mnu

m =0m =-1 m =1

n =0

n =1

Spherical harmonics

P. A. NELSON Y. KAHANA

MMSS

39

VV


0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Single-layerapproach

n =1, k=2.082

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Analyticalmodes

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.000.00

50.00

100.00

150.00

Double-layerapproach

m =-1 m =0 m =-1

Contours of the boundary modes

MMSS

40

VV


I. Only the boundary nodes are required such that the influence matrices can be determinated by the two-point function.

II. The diagonal elements can be derived by invariant method.

III. Based on the imaginary-part formulations, the deficient constraints cause the spurious eigensolutions.

IV. The SVD techniques can solve the true or spurious eigensolutions well.

Conclusions

MMSS

41

VV


• To overcome the ill-posed problem, the generalized singular value decomposition (GSVD) could be applied.

• The extension to several problems, degenerate boundary, crack and solid corner may be considered.

• The meshless method based on the imaginary-part formulation can be extented to structure vibration problems.

Further research

MMSS

42

VV


The EndThank you for your kind attention

感謝委員們辛勤指導

MMSS

43

VV


0)( kJm

0)( kJm

E

E

L

U

I

I

T

U0)( kJm

0)( kJm EE ML II MT

0)( kJm0)( kJm

E

E

M

T

I

I

M

L

0)( kJm

0)( kJm EE TU II LU

Boundary Value Problem

Eigen-solution

Density function True and spurious eigenvalues

Single-layer

potential

Double layer

potentialDirect method Indirect method

Dirichlet

Problem

TrueSVD

updating term

SVD updating

term

Spurious

SVDupdating documen

t

SVDupdating documen

t

Neumann

Problem

TrueSVD

Updatingterm

SVD updating

term

Spurious

SVDupdating documen

t

SVDupdating documen

t

SVD techniques for eigensolutions

MMSS

44

VV


To overcome the ill-posed problems

克服病態問題分析步驟

三維球形聲場二維圓形聲場

病態指標 - 條件數病態指標 - 條件數

病態問題

SVD 補充行SVD 補充列

Tikhonov 正規化法

GSVD

克服病態

Preconditioner

MMSS

45

VV


Distributed & concentrated type

Distributed-type Concentrated-type*

Single-layer

potential approach

Double-layer

potential approach

B

I sdBsxsUxu )()(),()(

B

I sdBsxsTxu )()(),()(

B

I sdBsxsMxt )()(),()(

B

I sdBsxsLxt )()(),()(

Dirichlet problem

Neumann problem

Dirichlet problrm

Dirichlet problrm

Dirichlet problrm

Neumann problrm

Neumann problrm

Neumann problrm

jijjj

ijI

i ASMAxsUxu )(),()(

jijxjj

ijI

i BSMBxsLxt )(),()(

jijsjj

ijI

i ASMAxsTxu )(),()(

jijsxjj

ijI

i BSMBxsMxt )(),()(

* NDIF method by Kang is the special case

MMSS

46

VV


One-dimensional duct ---Dirichlet problem


0 4 8 12

k

-24

-20

-16

-12

-8

-4

0

Log

3.145

6.285

9.425

12.565


N u m b er o f n o d es : 2k : 0 .0 0 1

T TT T

(3 .1416)

(6.2832)

(9.4248)

(12.5663)


0 4 8 12

k

-10

-8

-6

-4

-2

0

Log

3.145

6.285

9.425

12.565


N u m b er o f n o d es : 2k : 0 .0 01

(3.1416)

(6.2832)

(9.4248)

(12.5663)

T T TT

MMSS

47

VV


One-dimensional duct ---Dirichlet problem

0 0.2 0.4 0.6 0.8 1

a

0.48

0.52

0.56

0.6

0.64

0.68

u(a )

Mode 1k=3.145

0 0.2 0.4 0.6 0.8 1

a

-0 .4

0

0.4

0.8

1.2

sin(ka)

Mode 1k=3.1416

0 0.2 0.4 0.6 0.8 1

a

-0.2

0

0.2

0.4

0.6

u(a )

Mode 1k=3.145



Analytical mode

MMSS

48

VV


One-dimensional duct ---



Neumann problem

0 4 8 12

k

-10

-8

-6

-4

-2

0

Log

3.145

6.285

9.425

12.565


N u m b er o f n o d es : 2k : 0 .0 0 1

(3.1416)

(6.2832)

(9.4248)

(12.5663)

T T T T

0 4 8 12

k

-20

-16

-12

-8

-4

0

4

Log

3.145 6.285

9.425

12.565


N u m ber o f n o d es : 2k : 0 .0 0 1

(3 .1416) (6.2832)

(9.4248)

(12.5663)

TT

T T

MMSS

49

VV


One-dimensional duct ---Neumann problem



Analytical mode

0 0.2 0 .4 0 .6 0 .8 1

a

-0 .4

0

0 .4

0 .8

1 .2

u(a )

Mode 1k=3.145

0 0.2 0 .4 0 .6 0 .8 1

a

-1 .5

-1

-0 .5

0

0 .5

1

1.5

u(a )

Mode 1k=3.145

0 0.2 0 .4 0 .6 0 .8 1

a

- 4

- 2

0

2

4

sin(ka)

Mode 1k=3.1416

MMSS

50

VV


張銘翰口試問題記錄 Analysis of acoustic eigenfrequencies and eigenmodes

by using the meshless method

1. 無網格可否做大空間問題 ?2. 此種方法有何限制 ? 病態問題是否加重 ?3. RBF 取法是否唯一 ?RBF 隱含於 kernel?4. 與李慶鋒有何不同 ?5. 為何取虛部 ? 若 kernel 為非奇異時，則 free term 能產生

嗎 ? 跳還是 u?6. Eq. (2-37) & Eq. (2-38) 為何為零 ? 7. 將 mode 相差調正 ?8. 模態可否疊加實驗之 data 結果 ?9. GSRBF & CSRBF 何者適合大空間問題 ?

何者適合病態問題 ?10. RBF 的定義問題 ?

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 1...

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