A Case Study for a Coupled System of Acoustics and Structures

A Case Study for a Coupled A Case Study for a Coupled System of Acoustics and System of Acoustics and

StructuresStructuresDeng Li Deng Li

((Japan Research Institute, Tokyo Institute of TechnologyJapan Research Institute, Tokyo Institute of Technology))Craig C. Douglas (Craig C. Douglas (UK, Yale UniversityUK, Yale University))

Takashi, Kako Takashi, Kako (University of Electro-Communications)(University of Electro-Communications)

Ichiro, Hagiwara (Ichiro, Hagiwara (Tokyo Institute of TechnologyTokyo Institute of Technology))

March 23, 2006 March 23, 2006

CS521 in UKCS521 in UK

This research was supported in part by National Science Foundation grants This research was supported in part by National Science Foundation grants

EIA-0219627, ACI-0324876, ACI-0305466, and OISE-0405349EIA-0219627, ACI-0324876, ACI-0305466, and OISE-0405349

OUTLINEOUTLINEBasic IdeaBasic IdeaBackground Background Mathematical AnalysisMathematical AnalysisDiscretizationDiscretization by FEM by FEMPerturbation MethodPerturbation MethodError EstimationError EstimationApplication on Nastran SoftwareApplication on Nastran SoftwareNumerical ResultsNumerical ResultsFuture WorkFuture Work

This research was supported in part by National Science Foundation grantsThis research was supported in part by National Science Foundation grants

Basic IdeaBasic IdeaUsing uncoupled eigen-pairs (eigenvalue Using uncoupled eigen-pairs (eigenvalue and eigenvector) to calculate coupled and eigenvector) to calculate coupled eigen-pairs.eigen-pairs.

coupled eigen-pairs:coupled eigen-pairs:Acoustic and Structure coupled systemAcoustic and Structure coupled system

Uncoupled eigen-pairs:Uncoupled eigen-pairs:Acoustic systemAcoustic system

Structure systemStructure system


BackgroundBackgroundWe study a numerical method to calculate the eigen-frequencies of We study a numerical method to calculate the eigen-frequencies of the coupled vibration between an acoustic field and a structure. A the coupled vibration between an acoustic field and a structure. A typical example of the structure in our present study is a plate which typical example of the structure in our present study is a plate which forms a part of the boundary of the acoustic region, its application of forms a part of the boundary of the acoustic region, its application of this research is a problem to reduce a noise inside a car which is this research is a problem to reduce a noise inside a car which is caused by an engine or other sources of the sound. More in detail, caused by an engine or other sources of the sound. More in detail, the interior car noises such as a booming noise or a road noise are the interior car noises such as a booming noise or a road noise are structural-acoustic coupling phenomena.structural-acoustic coupling phenomena. Our present study was motivated by the work Our present study was motivated by the work

of Hagiwara, where they developed intensively of Hagiwara, where they developed intensively the sensitivity analysis based on the eigenvalue the sensitivity analysis based on the eigenvalue calculation and applied the results to the design calculation and applied the results to the design of motor vehicles with a lower inside noise.of motor vehicles with a lower inside noise.

This research was supported in part by This research was supported in part by National Science Foundation grantsNational Science Foundation grants

y

z

x

Fig.2

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ž S

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x

y

š

š

š

š

š

( a )

( b )

P1

P2

u

Px1 Px2

S

S’

( c )

š

ž ?

ž

š

P1

P2

Fig??

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Mathematical Analysis (1)Mathematical Analysis (1)3D Coupled Problem3D Coupled Problem

wherewhereΩ0Ω0 : a three-dimensional acoustic region, : a three-dimensional acoustic region,S0S0 : a plate region, : a plate region,Γ0=∂Ω0Γ0=∂Ω0 ＼＼ S0S0 : a part of the boundary : a part of the boundary of the acoustic field,of the acoustic field,∂∂S0S0 : the boundary of the plate, : the boundary of the plate,P0P0 : the acoustic pressure in : the acoustic pressure in Ω0Ω0,,U0U0 : the vertical plate displacement, : the vertical plate displacement,cc : the sound velocity, : the sound velocity,ρ0ρ0 : the air mass density, : the air mass density,DD : the flexural rigidity of plate, : the flexural rigidity of plate,ρ1ρ1 : the plate mass density, : the plate mass density,nn : the outward normal vector on : the outward normal vector on ∂Ω∂Ω from from Ω0Ω0, , σσ : the outward normal vector on : the outward normal vector on ∂S0∂S0 from from S0S0..


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SonUn

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Mathematical Analysis (2)Mathematical Analysis (2)2D Coupled Problem2D Coupled Problem

apply Fourier mode decomposition to apply Fourier mode decomposition to PP and and UU

in the in the zz direction: direction:


.0)(

)()0(

)0(

,|)2(

,|

,0|

,0)/(

2

2

2

2

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2

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Mathematical Analysis (3)Mathematical Analysis (3)

Introduce Parameter Introduce Parameter : :


.0)(

)()0(

)0(

,|)2(

,|

,0|

,0)/(

2

2

2

2

12

2

22

4

4

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2222,

Sondy

udu

dy

udu

Sonpudy

udm

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Sonux

ponp

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S

S

yx

.0|

,0|

,0)/( 2222,

Sonx

ponp

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.0)(

)()0(

)0(

,0)2(

2

2

2

2

12

2

22

4

4

Sondy

udu

dy

udu

Sonudy

udm

dy

udD

DiscretizationDiscretization by FEM (1) by FEM (1)

Ka and Ma: the stiffness and mass matrices for the Ka and Ma: the stiffness and mass matrices for the acoustic field, acoustic field,

Kp and Mp: the stiffness and mass matrices for the plate, Kp and Mp: the stiffness and mass matrices for the plate, L and LT :the coupling matrices. L and LT :the coupling matrices. The precise definitions of The precise definitions of KaKa, , MaMa, , KpKp, , MpMp, , LL and and LTLT are as are as

follows: follows:


2hCi

,0)(

,0)(2

22

hphphT

hhhaha

uMKpL

LupMK

h

h

p

ah

h

h

pT

a

u

p

M

LM

u

p

KL

K

0

02

DiscretizationDiscretization by FEM (2) by FEM (2)


).,()(

),,()(

),,()(

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

)),,((1

)(

,),(,1

)(

1

4''2""

20

2

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hishjij

hihjijp

hihjhihjhihjijp

hihjija

hihjhihjija

upL

puL

uuM

uumuumuuDK

ppc

M

ppmppK

Perturbation Method (1)Perturbation Method (1)

Introduce Parameter Introduce Parameter : :


.)(

),()( 2

h

hh

hh

u

p

,0

0

h

h

p

a

h

h

pT

ah u

p

M

LM

u

p

KL

K


There are two orthonormality conditions for There are two orthonormality conditions for the eigenvector:the eigenvector:


ih

(0)K

a0

0 KP

jh(0)

ij and

ih( )

Ka

0

0 KP

ih(0) 1.


Perturbation SeriesPerturbation Series


...,...)0()(

......,...)0()()(3)3(2)2()1(

)(3)3(2)2()1(

nnihihihihihih

nnihihihihihih

),2(}),...)((),(1

),1(),)(

),1(),,...)((

),1(,0

1

1

)1()2()3()2()1()()()(

)1(

)1()22()32()2()12()2(

)12(

nLLLL

nLL

nLLLL

n

ij

n

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nih

nihih

nihih

Tjh

kih

knih

ihjh

nih

jij ji

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ih

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ihn

ihihn

ihihTn

ih

nih

2hCiihi

Error EstimationError EstimationWe give the order of convergence for the error between exact and approximate We give the order of convergence for the error between exact and approximate eigenvalues using the standard result of Babuska and Osborn, where we assume a eigenvalues using the standard result of Babuska and Osborn, where we assume a certain regularity condition for the corresponding inhomogeneous problem. For 2D certain regularity condition for the corresponding inhomogeneous problem. For 2D coupled eigenvalue problem, we obtain the order estimatecoupled eigenvalue problem, we obtain the order estimate

After a few calculation, we can get the similar order estimation:After a few calculation, we can get the similar order estimation:


)()( ihi2hC

i

Application on Nastran Software(1)Application on Nastran Software(1)

Ortho-normality Condition for EigenvectorOrtho-normality Condition for Eigenvector


ijnastranjh

p

anastranih M

M

)0(

0

0)0(

ihp

aih

p

aih M

M

K

K

0

0

0

0

)0()0( nastranihihih

Application on Nastran Software (2)Application on Nastran Software (2)

How to Get the CoefficientHow to Get the Coefficient


)()0()( 4)2(2 Oiii

ij ji

iT

ijjiiT

iT

ii

iiT

ii

LLLL

LL

LL

))(,)(,)((

))(,(

),)(()1(

)1()2(

Numerical Results(1)Numerical Results(1)

Exact SolutionExact Solution

Coupled Eigenvalue ProblemCoupled Eigenvalue Problem


)2(,0))(/(1,cos)(

sin/cos

)1(,0))(/(1,cosh)(

sinh/cosh

)0(,0))(/(1,)/(

)(

2222222

01

2222222

01

2224

10

TypegcnmwithgnmDg

ggg

TypegcnmwithgnmDg

ggg

Typecnmwithc

D


Exact SolutionExact Solution

Acoustic EigenvalueAcoustic Eigenvalue

Structure EigenvalueStructure Eigenvalue


,)2/1((

12222, kjmc

skj .

))2/1((

sin)2/1sin(2),(

2/12220

, kjm

kyxjyxp kj

2221

)( jmDpj

)(

sin222 jmD

jyu j

Parameters in ExampleParameters in Example


10,...3,2,1,5.2,1,2,50,5,1 10 ncmD


Example 1 of Perturbation Analysis: Example 1 of Perturbation Analysis:

The First Eigenvalue of Type 1The First Eigenvalue of Type 1

the first eigenvalue the first eigenvalue error= error= 0.0 6.2500000000.0 6.2500000000.001 6.25000044470.001 6.2500004447 6.2500004453 -6.0E-106.2500004453 -6.0E-100.01 6.2500444660 6.2500444652 8.0E-100.01 6.2500444660 6.2500444652 8.0E-100.1 6.2544465978 6.2544466536 -5.58E-80.1 6.2544465978 6.2544466536 -5.58E-80.5 6.3611649457 6.3611542771 1.06686E-50.5 6.3611649457 6.3611542771 1.06686E-50.8 6.53458226090.8 6.5345822609 6.5345070074 7.52535E-56.5345070074 7.52535E-51.0 6.69447657561.0 6.6944765756 6.6946597826 -1.832070E-46.6946597826 -1.832070E-4


)2(1

21 )(1 )(1 )2(

12

1


Example 2 of Perturbation Analysis: Example 2 of Perturbation Analysis: The First Eigenvalue of Type 2The First Eigenvalue of Type 2 the first eigenvalue the first eigenvalue error= error=

0.0 0.071111110.0 0.07111111 0.0 7111111 0.00.0 7111111 0.00.001 0.0711111090.001 0.071111109 0.071111109 0.00.071111109 0.00.01 0.071110909 0.071110907 2E-9 0.01 0.071110909 0.071110907 2E-9 0.1 0.071090951 0.071090759 1.92E-7 0.1 0.071090951 0.071090759 1.92E-7 0.5 0.07061816 0.070602315 1.5846E-50.5 0.07061816 0.070602315 1.5846E-50.8 0.06989274 0.069808592 8.4151E-50.8 0.06989274 0.069808592 8.4151E-51.0 0.069251598 0.069085726 1.65871E-41.0 0.069251598 0.069085726 1.65871E-4


)2(1

21 )(1 )(1 )2(

12

1

Numerical Results (5)Numerical Results (5)

Relationship between Error and a Number of Relationship between Error and a Number of Used Eigen-pairsUsed Eigen-pairs

A number of used eigen-pairs jA number of used eigen-pairs j Error Error 11 6.261615E-46.261615E-433 -9.130154E-5-9.130154E-555 -1.621518E-4-1.621518E-41010 -1.83207E-4-1.83207E-4100100 -1.865037E-4-1.865037E-410001000 -1.865071E-4-1.865071E-41000010000-1.865071E-4-1.865071E-4This research was supported in part by National Science Foundation grantsThis research was supported in part by National Science Foundation grants

))0(()1( )2(iii

Numerical Results (6)Numerical Results (6)

A Special CaseA Special Case

An approaching phenomenon of An approaching phenomenon of eigenvalues which cannot be described by eigenvalues which cannot be described by FEM but can be described by the FEM but can be described by the perturbation method.perturbation method.


Numerical Results(7)Numerical Results(7)Exact ResultExact Result


Numerical Results(8)Numerical Results(8)FEM ResultFEM Result


Numerical Results(9)Numerical Results(9)compare the resultscompare the results


λ vs. ε

0. 0366

0. 0368

0. 037

0. 0372

0. 0374

0. 0376

0. 0378

0. 038

0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

ε

λλ 6(FEM)λ 7(FEM)λ 6λ 7

Future WorkFuture Work

We expect to obtain a mathematically We expect to obtain a mathematically rigorous estimation of the magnitude of the rigorous estimation of the magnitude of the convergence radius of the perturbation convergence radius of the perturbation series. series.

We need consider how to modify the We need consider how to modify the perturbation series in the case of perturbation series in the case of eigenvalue is not simple.eigenvalue is not simple.


A Case Study for a Coupled System of Acoustics and Structures

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