1

SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND

MODAL ANALYSIS

Ahlem ALIA

presented by Nicolas AQUELET

Laboratoire de Mécanique de Lille

Université des Sciences et Technologies de Lille

Avenue Paul Langevin, Cité Scientifique

59655 Villeneuve d’Ascq, France

2

Introduction

Structure Fluid

The vibrations generate sound

The sound engenders vibrations

Main industrial concern in Vibroacoustics: Reduction of NOISE

Actually, noise constitutes an important indicator of quality in many industrial products such as vehicles, machinery…

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Introduction

Analytical technique

Simple geometry

Under very restrictive hypothesis

Numerical methods

FEM / FEM

FEM / BEM

4

IntroductionClassical FEM / FEM

Six Nodes / Wavelength Application domain: Low

Frequency Range

DOF f

5

FEM / FEM with Modal analysis

Modal analysis solves the vibroacoustic problem with some modes

Reduction of the problem size

The modal analysis is applied with a Lumped mass representation

Lumped Mass matrix consists of Zero-off diagonal terms

Advantage of this approach:

Reduction of the computational cost

Introduction

(100 modes in our problem versus 432 physical unknowns)

6

Introduction Model the vibroacoustic behavior of an acoustic cavity with one flexible wall boundary by using FEM/FEM with:

Modal analysis

Lumped mass representation

Simply supported elastic plate

Mechanical load

Rigid wall

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Governing equations Structure

Fluid

i

2

j

ij wx

0pkp 22 Fluid (f)

(f)

(sf)

Structure(s)

Vibroacoustic problem

0pnn ijij

0wnp

n

2

Pressure Continuity

Normal Displacement Continuity

BC at Coupling InterfaceP: pressure

k=/c wave number

w: displacement

: stress

n: interface normal

(1)

(2)

(3)

(4)

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The application of the FEM to the variational formulation of structure cavity system yields to the following linear system :

Coupling system

Ks, Ms: structural stiffness and mass matrices

Kf, Mf: fluid matrices

B: coupling matrix

Fs, Ff: mechanical load, acoustical sources

c: sound velocity, f: fluid density

f

s

fT

s

f

s

F

F

p

w

MBc

M

K

BK2

20

0

MK F

p

wMK

2

,dnNNB

,dNNM,dNNM

,dNNcK,dNDNK

sfsTf

ffTf

ffssTs

ss

ffTf2

fssTs

s

Ns, Nf : structural and fluid shape function

(5)

(6)

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The problem (6) can be seen as an eigenvalue problem:

(K - 2 M ) = F

p

w

Coupling system

For a great number of DOF, solving the system directly is always hard in term of CPU time.

(6)

10

Purpose of the approach

LFIRp

w 12

LKRILMR We search two matrices L and R verifying:

- L and R contain the LEFT and RIGHT eigenvectors, respectively

MRKR - is a diagonal matrix containing the eigenvalues of:

We obtain the physical unknowns of

MLKL HH

(K - 2 M ) = F

p

w

by this relation:

(7)

(10)

(8)

(9)

(11)(6)

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Efficient eigenvalue algorithms can’t be used

A symmetric form of eigenvalue problem is required

(K - 2 M ) = F

p

w

Since

is non-symmetric,

Sandberg’s method enables us to make it symmetric by using Modal analysis

Purpose of the approach

(6)

12

Classical modal analysis

fffff XMXK sssss XMXK

s , f : the structural and the fluid eigenvalues.

Xs , Xf : the structural and the fluid eigenvectors.

Cavity with stiff boundaries Structure in vacuum

fXsX

Solved

Independently

(12) (13)

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Classical modal analysis

f

s

f

s

p

w

0

0

fffTffff

Tf

sssTssss

Ts

DK,IM

DK,IM

Ds , Df are diagonal matrices containing the structural and the fluid eigenvalues.s, f represent the modal structural displacement and the modal fluid pressure.

fXsX

s, f are matrices containing some eigenvectors of structure and fluid

s, f verify the following properties:

(14)

(15)

(50 modes) (50 modes)

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Classical modal analysisHence, the coupling system (5) can be rewritten as the reduced system (17):

f

s

f

s

p

w

0

0 s, fw , p

f

s

fT

s

f

s

F

F

p

w

MBc

M

K

BK2

20

0

fff

sTs

f

s

f2

fsTT

f22

fTss

2s

F

F

IDBc

BID

The system is reduced (the problem size is divided by 4) but it remains non symmetric

(5)

(14)

(17)

432 physical unknowns 100 modal unknowns

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Modal analysis (Sandberg Method)

fff

sTs

f

s

f2

fsTT

f22

fTss

2s

F

F

IDBc

BID

f

s

f

s

f

s

f

sS

I

Dc

0

012

sTss

TTff

Tf

sTss

f

s

fTss

TTffss

TTf

fTsss

FBcF

FDc

BBcDDBc

BDcDI

2

2

22

22

Symmetric system

Non Symmetric system

Transition matrix

(17)

(18)

(19)

16

sTss

TTff

Tf

sTss

f

s

fTss

TTffss

TTf

fTsss

FBcF

FDc

BBcDDBc

BDcDI

2

2

22

22

Symmetric generalized eigenvalue problem

FAIF

S

2

VAV

Modal analysis (Sandberg Method)

V : right eigenvector matrix of the symmetric system

(19)

(20)

(21)

17

V

I

DcRf

s

f

s

0

00

01

2

f

s

f

s

p

w

0

0s, f

w , p

f

s

f

sS s, f

Modal analysis (Sandberg Method)

R: right eigenvector matrix of the original system

The left eigenvector matrix “ L” is obtained in the same manner

MRKR

(18)(14)

(22)

(9)

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Modal analysis (Sandberg Method)

(K - 2 M ) = F

p

w

LFIRp

w 12

LKR

ILMR &

is a diagonal matrix containing the eigenvalues of:

(6)

(11)

MRKR MLKL TT (10)

(9)

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Numerical results

Coupling

InterfaceFluidRigid

Structure

Elastic Structure

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Numerical results

Cavity

(c)

Plate

(b)

Discrete Kirchhoff Quadrilateral (DKQ) plate element thin plate

Kirchhoff theory

8-node brick isoparametric acoustic element

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Structure ( Frequency response) Simply supported plate (0.5m 0.5m)

Unit punctual force (0.125m 0.125m)

Variation of the displacement with the frequency at the load point

Results given by Migeot et al (1)Numerical results

(1) 2nd Worldwide Automotive Conference Papers,1-7

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Structure ( Natural frequencies)Structure: Simply supported plate (0.2m 0.2m) made of brass

0 20 40 60 80 1000

1000

2000

3000

4000

5000

6000

Mode number

Fre

que

ncy

(Hz)

Analytical Consistent mass Lumped mass

Natural frequencies of the plate

Consistent and lumped mass matrices are in good agreement with analytical ones as long as low frequencies are considered (<50th mode).

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Cavity ( Natural frequencies)

0 20 40 60 80 1000

2500

5000

Mode number

Fre

que

ncy

(Hz)

Analytical FEM

Rigid cavity (0.2m 0.2m 0.2m)

FEM leads to good results below the 50th mode

Natural frequencies of the rigid cavity

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Coupling problem

1 0 0 0- 8 0

- 6 0

- 4 0

- 2 0

0

2 0

F r e q u e n c y ( H z )

Pr

es

su

re

( d

B )

D i r e c t F E M M o d a l F E M w i t h c o n s i s t e n t m a s s M o d a l F E M w i t h l u m p e d m a s s

10 100 1000-100

-80

-60

-40

-20

0

20

40

Frequency ( Hz )

Pre

ssure

(dB

)

Direct FEM Modal FEM with consistent mass Modal FEM with lumped mass

Mesh (1266)Direct Method Modal analysis32mn 54s 56s

Results Given by Lee et al (2)

CPU Time

Simply supported elastic plate

Field point

Pressure at the point (0.1,0.1,0.2)

Numerical results

(2) Engineering Analysis with Boundary Elements, 16 (1995) 305-315

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Structure ( Frequency response)

200 400 600 800 1000-200

-150

-100

-50

0

50

Frequency (HZ)

Dis

plac

emen

t (dB

)200 400 600 800 1000

-150

-100

-50

0

50

Frequency ( Hz )

Dis

plac

emen

t (dB

)

Plate quadratic displacement of the structure

In vaccum Plate-cavity (air)

Coupling effect

854Hz

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Mea

n sq

uare

pre

ssur

e

Frequency (Hz)

Mea

n sq

uare

vel

ocity

Frequency (Hz)

Coupling problem (air)

Comparison between the direct and the modal results

Mean square pressure: cavity Mean square velocity:structure

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Coupling problem (water)

Comparison between the direct and the modal results

Mean square pressure: cavity Mean square velocity: structure

Mea

n sq

uare

vel

ocity

Frequency (Hz)

Mea

n sq

uare

pre

ssur

e

Frequency (Hz)

Mea

n sq

uare

vel

ocity

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FEM / FEM -- FEM / BEM

FEM-FEM FEM-BEM comparison

Frequency (Hz)

Pre

ssu

re (

dB

)

Simply supported elastic plate

Field point

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Conclusion FEM / FEM with modal analysis and lumped mass representation has been used to model a simple vibroacoustic problem.

A good representation of the mass is very essential to achieve accurate results.

Modal FEM / FEM with only small number of modes is less efficient for strong coupling.

More modes must be taken into account ( disadvantage)

Solution: Improve the numerical results by using Modal correction for diagonal system