spurious eigensolutions and fictitious frequencies for acoustic problems with the mixed-type...

Spurious Eigensolutions and Fictitious Frequencies for ASpurious Eigensolutions and Fictitious Frequencies for Acoustic Problems with the Mixed-type Boundary coustic Problems with the Mixed-type Boundary

Conditions by using BEMConditions by using BEM

邊界元素法於混合型邊界條件問題之邊界元素法於混合型邊界條件問題之假根及虛擬頻率探討假根及虛擬頻率探討

國立台灣海洋大學河工二館 307 教室中華民國九十二年七月十五日

研究生：林宗衛指導老師：陳正宗教授

海洋大學力學聲響振動實驗室

OutlinesOutlines

Motivation

Boundary integral equations

Examples

Conclusions

Semi-analytical approach

Further research

02

Techniques and treatments


OutlinesOutlines

Motivation


Examples

Conclusions


Further research

03



MotivationMotivation

04

Dirichlet Neumann Mixed-type

spurious

Eigenvalue

fictitious

frequency

spurious

Eigenvalue

fictitious

frequency

spurious

Eigenvalue

fictitious

frequency

Complex-valued

BEM N Y N Y ? ?Real-part

BEM Y - Y - ? ?Imaginary-part

BEM Y - Y - ? ?

MRM Y - Y - ? ?


OutlinesOutlines

05

Motivation


Examples

Conclusions


Further research

05



Governing equationGoverning equation

Duk x ,0x 22

where

D : the domain of interest

k : the wave number

x : the domain point

u(x): the acoustic potential 06


Types of boundary conditionsTypes of boundary conditions

Mixed-type

u=u

Neumann Dirichlet

t= =un

u

tt

uu

07


The null-field integral formulationsThe null-field integral formulations

c

BB

c

BB

D),(dB)(t),(L)(dB)(u),(M0

D),(dB)(t),(U)(dB)(u),(T0

x ss xsss xs

x ss xsss xs

D Dc

8

DDc


s

xs

n

),(U

x

xs

n

),(U

sx

xs

nn

),(U2

Dc: the complementary domain of D,x,s: the field and source point,u(s): the potential on the boundaryt(s): the normal derivative of potential on the boundaryU (s,x): the kernel function,

T(s,x) M(s,x)

L (s,x)9


UU ( (ss,,xx)) kernel in different methods kernel in different methods

2

1

0krHi

U

x,s

2

0krY

U

xs,

2

0krJ

U

xs,

krJk

krYU00

]2

[ln2

)sx,(

Complex-valued BEM:

Real-part BEM:

Imaginary-part BEM:

MRM:

10


Discretization (singular formulatioDiscretization (singular formulation)n)

11 NNNNNN uTtU

11

N

NNRL

N

NNRL u

uTT

t

tUU

ut

11


Rearrangement of the Rearrangement of the influences matricesinfluences matrices

11

N

NNRL

N

NNRLt

uUT

u

tTU

1NNN1NNN qBpA

12


DiscretizationDiscretization(hypersingular formulation)(hypersingular formulation)

11 NNNNNN uMtL

11 NNNNNN qDpC

13


OutlinesOutlines

14

Motivation


Examples

Conclusions


Further research



0

)(

)(

t

t

kC

kA

0)( pkC

0)( pkA

kt: the true eigenvalues

15

Extraction of the true Extraction of the true eigensolutions by using SVD updatieigensolutions by using SVD updating termsng terms

0q


Detection of the spurious eigensolDetection of the spurious eigensolutions by using utions by using SVD updating documentsSVD updating documents

by using the Fredhohm’s alternative theorem

16

0)(

)(

T

s

T

s

kB

kA

0)(

0)(

T

T

kB

kA

T: transposeks: the spurious eigenvalues


Burton & Miller methodBurton & Miller method

[ikU(s, x)+L(s, x)]u(s) = [ikT(s, x)+M(s, x)]t(s)

1NNN1NNN uMtL

1NNN1NNN uTtU

+)

ik

17


CHIEF methodCHIEF method

11 NNNNNN uTtU

1N

N)aN(c

Na

BNN

1N

N)aN(c

Na

BNN u

T

Tt

U

U

18


OutlinesOutlines

19

Motivation


Examples

Conclusions


Further research



Interior problemsInterior problems

20


A finite rodA finite rod

s0=0x x

0 1

s1=1

Type I : u(0)=0; t(1)=0Type II: t(0)=0; u(1)=0

21

B.C.:


Comparison of the eigenfuctionsComparison of the eigenfuctions

Interior problem

Dirichlet

u(0)=0, u(1)=0

Neumann

t(0)=0, t(1)=0

Mixed-type

(u(0)=0, t(1)=0)

True Spurious True Spurious True Spurious

Complex –valued BEM

UT Sink - Sink - Cosk -

LM Sink - Sink - Cosk -

Real-part

BEM

UT Sink Sink Sink Sink Cosk Sink

LM Sink Sink Sink Sink Cosk Sink

Imaginary-part BEM

UT Sink Sink Sink Sink Cosk Sink

LM Sink Sink Sink Sink Cosk Sink

22

海洋大學力學聲響振動實驗室 23

Real-part BEM

Imaginary-part BEM

Complex-valued BEM

0 4 8 12

k

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

S ingular form ula tion

H ypersingu lar form ulationS ingular form ula tion

H ypersingu lar fo rm ulation

0 4 8 12

k

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

Singular form ula tion

H ypersingular form ulation

0 4 8 12

k

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

T

T

T

TTT T

T T

T

T

T

S

S

SSS S


A circular cavityA circular cavity

u2=0 t1=0R=1 m

24

Elements: N=30


Detection the eigenvalues using thDetection the eigenvalues using the complex-valued BEMe complex-valued BEM

0 1 2 3 4 5

k

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5

k

0.01

0.1

1

10

C

A

0 1 2 3 4 5

k

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5

k

0.01

0.1

1

10

B

D

25


True

Spurious

kYn

of Zeros kYn

of Zeros

0 1 2 3 4 5

k

1E-005

0.0001

0.001

0.01

0.1

1

NNA

0 1 2 3 4 5

k

0.01

0.1

1

10

NN2C

A

Updating term

0 1 2 3 4 5

k

0.0001

0.001

0.01

0.1

1

10

N2NBA

Updating document

Detection of true and spurious Detection of true and spurious eigenvalues using eigenvalues using the real-part BEMthe real-part BEM

26


Detection of true and spurious Detection of true and spurious eigenvalues using eigenvalues using the real-part BEMthe real-part BEM

0 1 2 3 4 5

k

0.01

0.1

1

10

NN2C

A

0 1 2 3 4 5

k

0 . 1

1

1 0

N2NDC

0 1 2 3 4 5

k

0.0001

0.001

0.01

0.1

1

10

NNC

True

Spurious

kYn

' of Zeros kYn

' of Zeros Updating term

Updating document

27


The comparison of the spurious The comparison of the spurious eigenvalues with different B.C.eigenvalues with different B.C.

0 1 2 3 4 5

k

0.0001

0.001

0.01

0.1

1

10

0 1 2 3 4 5

k

0 . 1

1

1 0

0 1 2 3 4 5

k

0.001

0.01

0.1

1

10

0 1 2 3 4 5

k

0.1

1

10

kYn

of Zeros

kYn

' of Zeros

Neumann or Dirichlet Mixed-typed

28


The spurious eigenvalues using diThe spurious eigenvalues using different methodsfferent methods

Complex-valued

BEM

Real-part

BEM

Imaginary-part

BEMMRM

Singular

formulation -

Hyperingular

formulation -'

nY'

nY '

nJ

nYnJ

nY

.5772.0krJ]2

k[ln

2krYY nnn

and where

29


The comparison for the former fivThe comparison for the former five eigenmodese eigenmodes

Real –part BEM

FEM

30


The comparison for the former five The comparison for the former five eigenvalueseigenvalues

k1 k2 k3 k4 k5

Real –part BEM 1.222 2.544 2.954 3.802 4.231

FEM(ABAQUS)

1.254 2.593 2.934 3.842 4.194

31


Exterior problemsExterior problems

32


A semi-infinite rodA semi-infinite rod

s0=1x

0 1

s1=

33

B.C.: mt(1)+u(1)=n


Comparison of Comparison of the fictitious frequenciesthe fictitious frequencies

Interior problem

Dirichlet

u(1)=0

Neumann

t (1)=0

Mixed-type

mt(1)+u(1)=n

Fictitious eigenfunctions

T(s,x0)=0UT cosk cosk cosk

LM sink sink sink

U(s,x0)=0UT sink sink sink

LM cosk cosk cosk

34

海洋大學力學聲響振動實驗室T(s,x)=0

0 4 8 12

k

-8000

-4000

0

4000

Pot

entia

l

Singular form ulation


Exect so lu tion: u(k)=1

0 4 8 12

k

-8000

-4000

0

4000

Pot

entia

lExect so lu tion: u(k)=1



U(s,x)=035

The fictitious frequency of 1-D The fictitious frequency of 1-D problemproblem


A circular radiatorA circular radiator

θ4)k(H

)kR(H)(u

)1(4

)1(4

2 cos

θ4)R

4-

)(kρH

(kR)H(k)(t

)1(4

)1(5

1 cos

36

R=1 m

Elements: N=30

海洋大學力學聲響振動實驗室0 2 4 6 8

k

- 2

0

2

4

u(1,

0)


Hypersingular form ulation

Analytica l solution: u(k)=1

J'12

J41J13J02

J'51J'31J'21

CHIEF m ethod (X)

Burton & M iller m ethod

Analytical solution: u(k)=1

0 2 4 6 8

k

- 2

0

2

4

u(1,

0)

The positions of irregular for The positions of irregular for u(1,0) u(1,0) and their treatmentsand their treatments

x

37

海洋大學力學聲響振動實驗室0 2 4 6 8

k

- 2

0

2

4

t(1,

)


Hypersingular form ulation

Analytica l solution: t(k)=1

J'12

J'51

J'31

J'21

The positions of irregular for The positions of irregular for t(1,t(1,) ) and their treatmentsand their treatments

CHIEF m ethod (X)

Burton & M iller m ethod

Analytical solution: t(k)=1

0 2 4 6 8

k

- 2

0

2

4

t(1,

)x

38


Comparison of the BEM and exact Comparison of the BEM and exact solutions for radiation problemsolutions for radiation problem

BEM solution Analytical solution

39


OutlinesOutlines

40

Motivation


Examples

Conclusions


Further research



The null-field integral formulationsThe null-field integral formulations

c

BB

c

BB

D),(dB)(t),(L)(dB)(u),(M0

D),(dB)(t),(U)(dB)(u),(T0

xss xsss xs

xss xsss xs

41


Semi-analytical approachSemi-analytical approach

42

12n42n42n4 vW0

12n4H


0 1 2 3 4 5

k

1E-024

1E-023

1E-022

1E-021

1E-020

1E-019

1E-018

1E-017

1E-016

1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

n=5m =20

S

T

T TT

T

0 1 2 3 4 5

k

1E-016

1E-015

1E-014

1E-013

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

n=5m =20

TT T

T T

Real-part BEM

Imaginary-part BEM

Complex-valued BEM

Interior problemInterior problem

0 1 2 3 4 5

k

100000

1000000

10000000

100000000

1000000000

10000000000

n=5m =20

S

SST

TT

T TS


J01

J12

J21

J02J31

J13J41

0 2 4 6 8

k

0.001

0.01

0.1

1

n=5m =20

n=5m =20

0 2 4 6 8

k

1E-012

1E-011

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

10

J '11 J '21 J'02

J'31J'41 J'51

J'22J'03

J'61

Exterior problemExterior problem

Singular Formulation

Hypersingular Formulation


OutlinesOutlines

45

Motivation


Examples

Conclusions


Further research



ConclusionsConclusions

The results in the numerical experiment match well with those in our semi-analytical results.

The eigensolutions and fictitious frequencies

were solved successfully by using semi-analytical approach and BEMs

46


ConclusionsConclusions

The spurious eigenvalues and fictitious frequencies depend on the representation no matter what the given types of B.C. for the problem are specified.

47


OutlinesOutlines

48

Motivation


Examples

Conclusions


Further research



Further researchFurther research

49

It will be interesting to know that if spurious eigenvalues and fictitious frequencies locate at the zeros of the Mathieu functions in the shape of ellipse.

The quantity of the fictitious frequencies depend on the modal participation factor.



50

To derive the fictitious frequencies analytically by using the complex-valued BEM for mixed-type boundary condition problem.

The extension to multiple radiators & scatters and half-plane problems using the similar algorithm can be conducted to examine the occurrence of the fictitious frequencies.



51

The extension to 3-D problems deserves further study to reconfirm the conclusion proposed in the thesis.


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spurious eigensolutions and fictitious frequencies for acoustic problems with the mixed-type...

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