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Recent development of BEM/BIEM in vibration and

acoustics陳正宗

海洋大學 特聘教授河海工程學系

Nov. 19, 2004, NSYSU, 14:10~16:00

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VOutlines

Introduction Exterior acoustics - adaptive BEM Interior acoustics - multiply-connected

eigenproblems Conclusions

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VGrowth of BEM papers

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VIntroduction

Finite difference method (FDM) Finite element method (FEM) Boundary element method (BEM) Meshless method (MM) Boundary integral equation method (BIEM)

FDMFEM

Domain discretization

BEM

Boundary discretization

MM BIEM

No meshNo discretization for circular boun

daries

No meshNo node

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Adaptive BEM for exterior radiation

and scattering problems

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VProblem statement

Non-uniform radiator problem Scattering problem

1),( au0),( au

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

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

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VAdaptive scheme

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

u x C PV T s x u s dB s R PV U s x t s dB s Solver

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

t x H PV M s x u s dB s C PV L s x t s dB s Error indicator

R.P.V. is Riemann Principal Value

H.P.V. is Hadamard Principal Value

C.P.V. is Cauchy Principal Value

Singular formulation

Hypersingular formulation

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VAdaptive mesh

Uniform mesh Adaptive mesh

-1.00 -0.50 0.00 0.50 1.00

-1.00

-0.50

0.00

0.50

1.00

-1 .00 -0.50 0.00 0.50 1.00

-1.00

-0.50

0.00

0.50

1.00

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VRefinement scheme

h-version p-version r-version

1. Element number increasing2. Interpolation function order increasing3. Optimum nodal collocation

1 2 3

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VMesh

BEM FEM(DtN)

- 1 - 1 0 1 1

- 1

- 1

0

1

1

Taiwan, NTOU

US Navy. Stanford Univ.

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Non-uniform radiation ： Dirichlet problem

-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

Numerical solution: BEM Numerical solution: FEM(DtN) 64 ELEMENTS 2791 ELEMENTS

2ka

Taiwan, NTOU

US Navy. Stanford Univ.

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Non-uniform radiation ： Dirichlet problem

-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

Analytical solution: n=20 2ka

nkaH

krH

n

nru

n

n

ncos

)(

)(sin2),(

)1(

)1(

0

1),( au0),( au

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

32

5

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VSuperposition principle

),(),(),( )()( rururu si

+

),()( ru s),( ru i cosikre∥

0),(

n

ru

n

u

n

u is

)()(

n

u i

)(

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Scattering ： Neumann problem

-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

Numerical solution: BEM Numerical solution: FEM(DtN) 63 ELEMENTS 7816 ELEMENTS

4ka

Taiwan, NTOU

US Navy. Stanford Univ.

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Scattering ：Neumann problemAnalytical solution: n=20

-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

4ka

nkrHkaH

kaJikrH

kaH

kaJru n

n

nn

ncos)(

)(

)(2 )(

)(

)(),( )1(

)1(1

)1(0

)1(0

0

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

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Fictitious frequency ： Non-uniform radiation problem

ka0 2 4 6 8

-0.8

-0.4

0.0

0.4

0.8

1.2UT method

LM method

Burton & Miller method

Analytical solution

u(a,0)

1),( at0),( at

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

9

1),( at0),( at

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

9

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Fictitious frequency ： The scattering Dirichlet problem

t(a, 0)

ka

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

•

0 2 4 6 8

-12

-8

-4

0

4

8

UT m ethod

LM m ethod

Burton & Miller m ethod

Analytical solution

1

1J

1

2J

2

0J

1

3J

1

4J

1

5J

2

2J

3

0J

1

6J

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VSummary

Fictitious frequency depends on the formulation (singular or hypersingular) instead of B.C. (Dirichlet or Neumann).

Burton & Miller method and CHIEEF method can overcome the problem of fictitious frequency.

Fictitious frequency happens to be the true eigenvalues of the interior problem

(SingularDirichlet, HypersingularNeumann).

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Spurious eigenvalues for multiply-connected pro

blems

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VProblem domain

Doubly-connected domain

Multiply-connected domain

Simply-connected domain

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VBEM&BIEM

BEM

BIEM

.

.

.

.

.

.

.

.

.

..

.

Boundary discretization Fourier series

0 2 0 2

u(θ) or t(θ)

θ θ

u(θ) or t(θ)

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The flowchart to determine the eigenvalues and mode shape by BEM

Given G.E.and B.C.

Solve thefundamental solution

BIE for domain point

Moving to the boundary

BIE for boundary point

Boundary element discretization

Linear algebraic system

Solve boundary

dataPotential

SVD

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The flowchart to determine the eigenvalues and mode shape by BIEM

Degenerate kernel

Fourier series

Null-field equation

Algebraic equation Fourier Coefficients

Potential

Analytical

Numerical

SVD

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VIntegral Formulation

Null-field integral equations:

0 ( , ) ( ) ( ) ( , ) ( ) ( ), .eB BT s x u s dB s U s x t s dB s x D= - Îò ò%% % % %% % % %

0 ( , ) ( ) ( ) ( , ) ( ) ( ), .eB BM s x u s dB s L s x t s dB s x D= - Îò ò%% % % %% % % %

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Vs

x

EU

rO

RIU

x ： source point ； s ： field point

s

r

x

Degenerate kernels

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VDegenerate kernels

Degenerate kernels:

(1)

0

(1)

0

( , ) ( | |) ( | |) cos( ),| | | |2

( , )

( , ) ( | |) ( | |) cos( ),| | | |2

Im m j m j j j

m

Em m j m j j j

m

iU s x J k x c H k s c m s c x c

U s xi

U s x H k x c J k s c m x c s c

1, 0,.

2, 0,m

m

m

(1)

0

(1)

0

( | |)( , ) ( | |){ }cos( ), | | | |

2( , )

( | |)( , ) ( | |){ }cos( ), | | | |

2

m jIm m j j j

m j

m jEm m j j j

m j

H k s ciT s x J k x c m s c x c

RT s x

J k s ciT s x H k x c m x c s c

R

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VDegenerate kernels

(1)

0

(1)

0

( | |)( , ) ( | |){ cos( ) cos( )

2

1 cos( )( | |) cos( )}, | | | |

2( , )

( | |)( , ) ( | |){ cos(

2

m jI

m m jm

m j j j

m jE

m m jm

c jj

c jj j

j

J k x cis x H k s c m

mJ k x c s c x c

L s xH k x ci

s x J k s c m

L

L

(1)

) cos( )

1 cos( )( | |) cos( )}, | | | |

2m j j j

c j

c jj j

mH k x c x c s c

(1)

0

(1)

0

( | |)( , ) { cos( ) cos( )

2

cos( )( | |) cos( )}, | | | |

2( , )

( , )2

( | |)

1

( | |) ( | |){

m j m jI

mm

m j j j

m j m jE

mm

c jj j

c jj

j

j

J k x cis x m

mJ k x c s c x c

M s xi

s x

H k s cM

R

J k s c H k x cM

R

(1)

cos( ) cos( )

( | |) cos( ) | | | |2

1 cos( )},m j j j

c jj

c jjj

m

H k s c x c s cm

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Fourier series for boundary densities

Fourier series:

01

( ) ( cos sin ),j nj j nj j jn

u s a a n b n s Bq q¥

=

= + + Îå% %

01

( ) ( cos sin ),j nj j nj j jn

t s p p n q n s Bq q¥

=

= + + Îå% %

a),( Rs

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VCollocation points

By choosing M terms of Fourier series, we select 2M+1 collocation points on the circle.

01

( ) ( cos sin )M

n nn

u x a a n b n

2M+1 terms

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VIntegral representation

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

u x T s x u s dB s U s x t s dB s x Dp = - Îò ò% %% % % %% % % %2 ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),

B Bt x M s x u s dB s L s x t s dB s x Dp = - Îò ò% %% % % %% % % %

Integral equation formulation:

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VNumerical examples

Example 1

2 2( ) ( ) 0,k u x x D

1 0.5r

2 2.0r

1B

0u 2B

0u

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The eigenfrequenies by using singular equation

More accurate

( ) Numerical

[ ] exact

0.0 1.0 2.0 3.0 4.0 5.0

1E-004

1E-003

1E-002

1E-001

1E+000

0.5 2.0

P resent m ethod[U c] kerne l

BEM[U c] kerne l

J 1 0

(4 .83)[4.81]

[4.81]

Contaminated by spurious eigenvalues

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Relation of spurious eigenvalue and true eigenvalue

( 0.5) 0 4.81nJ k k

0.5 True

Spurious eigenvalue using singular formulation happens to be the true eigenvalue of the associated interior Dirichlet problem.

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The eigenfrequenies by using hyper-singular equation

( ) Numerical

[ ] exact

More accurate

More accurate

0.0 1.0 2.0 3.0 4.0 5.0

1.0E-003

1.0E-002

1.0E-001

1.0E+000

1.0E+001

J '1

0

(0.35)[0 .00]

J '2 0

(3 .68)[3.68]

J '2 0

(3.77)[3 .68]

[0.00][3.68]

0.5 2.0

Present m ethod[L c] kerne l

BEM[L c] kerne l

Contaminated by spurious eigenvalues

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Relation of spurious eigenvalue and true eigenvalue

( 0.5) 0 0, 3.68nJ k k

0.5 True

Spurious eigenvalue using hypersingular formulation happens to be the true eigenvalue of the associated interior Neumann problem.

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The spurious eigenvalues are filtered by Burton&Miller method

Only true eigenvalues appear

0.0 1.0 2.0 3.0 4.0 5.0

1E-002

1E-001

1E+000

1E+001

0.5 2 .0

B urton&M iller m ethod(P resent m ethod)

B urton&M iller m ethod

(BE M )

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The former five eigenvalues of Helmholtz eigenproblem with an eccentric d

omain

1 2 3 4 5

FEM [Chen et. al.] 1.73 2.13 2.45 2.76 2.95

BEM[Chen and Zhou] 1.75 2.14 2.47 2.78 2.97

BEM [Chen et. al.] 1.74 2.14 2.47 2.78 2.98

Present method 1.74 2.14 2.46 2.78 2.96

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The former five eigenmodes for eccentric case using present method, FEM a

nd BEM

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VNumerical examples

Example 2

R=1

c1=0.3c2=0.4

e=0.5

2 2( ) ( ) 0,k u x x D

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Extraction of the spurious eigenvalues by using SVD updating docu

ment

0 1 2 3 4 5 6 7 8 9

0.001

0.01

0.1

1

10

100

1000

The m in im um singular va lues 1 for the updating docum ent by BEM (90 e lem ents)

The m in im um singular va lues 1 for the updating docum ent by B IEM (63 co llocation po ints)

( ) : e x ac t so lu tio n

6.01 8.02More accurate

More accurate

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The former five eigenvalues for a multiply-connected problem with two unequal holes using different approaches

Method ki

k1 k2 k3 k4 k5

Burton & Miller method 4.82 4.82 6.72 6.72 7.82

Direct BEM + SVD Updating 4.81 4.81 6.73 6.73 7.81

Null-field BEM + SVD Updating 4.81 4.81 6.73 6.73 7.82

Fictitious BEM + SVD Updating 4.80 4.80 6.72 6.72 7.79

Direct BEM + CHIEF method 4.81 4.81 6.73 6.73 7.82

Null-field BEM + CHIEF method 4.83 4.83 6.74 6.74 7.84

Fictitious BEM + CHIEF method 4.77 4.77 6.68 6.68 7.88

FEM 4.790 4.801 6.619 6.634 7.797

Present method 4.85 4.85 6.77 6.77 7.91

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The former five modes for a circle domain with two unequal holes using present method, BEM and FEM

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VSummary

Spurious eigenvalues depend on formulation (singular or hyper-singular).

Spurious eigenvalues are independent of B.C. (Dirichlet or Neumann).

Spurious eigenvalues happens to be the true eigenvalues of the interior problem

(Dirichletsingular, Neumannhypersingular).

To overcome the spurious eigenvalues Burton&Miller, SVD updating term, SVD upd

ating document…….

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VConclusions

Exterior acoustic problems (radiation and scattering) were solved by using adaptive BEM.

Good accuracy and efficiency of the present method were obtained in comparison with those with FEM.

Spurious eigenvalues embedded in the BIEM/BEM were examined and filtered out in this study.

Both the fictitious frequency and spurious eigenvalue depend on the formulation instead of B.C. .

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歡迎參觀海洋大學力學聲響振動實驗室 烘培雞及捎來伊妹兒

URL: http://ind.ntou.edu.tw/~msvlab/ Email: jtchen@mail.ntou.edu.tw

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