Transcript FINITE DIFFERENCE MODELLING OF THE ULTRASONIC …

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
1
Introduction
The vibrations generate sound
Structure
Fluid
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…
2
Introduction
Analytical technique
 Simple geometry
 Under very restrictive hypothesis
Numerical methods
FEM / FEM
FEM / BEM
3
Introduction
Classical FEM / FEM
Six Nodes / Wavelength  Application domain:
Low Frequency Range
f
DOF
4
Introduction
 FEM / FEM with Modal analysis
 Modal analysis solves the vibroacoustic problem with some
modes

Reduction of the problem size
(100 modes in our problem versus 432 physical unknowns)
 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
5
Introduction
 Model the vibroacoustic behavior of an acoustic
cavity with one flexible wall boundary by using
FEM/FEM with:
 Modal analysis
 Lumped mass representation
Mechanical load
Simply supported
elastic plate
Rigid wall
6
Governing equations
 Structure
Structure
(s)
 ij
  2 w i (1)
x j
 Fluid
(sf)
Fluid
(f)
 p  k p  0 (2)
2
(f)
2
BC at Coupling Interface
Vibroacoustic problem
Pressure Continuity (3)  ijn j  pn i  0
Normal Displacement (4)
Continuity
p
 2 w n  0
n
P: pressure
k=/c wave number
w: displacement
: stress
n: interface normal
7
Coupling system
The application of the FEM to the variational formulation of structure
cavity system yields to the following linear system :
0   w   Fs 
  K s  B 2  M s

     (5)


2
T



 0 K
 p

c
B
M
F
f
f





 f

w
K   MM 
  F
K
 p

Ks, Ms: structural stiffness and mass matrices
Kf, Mf: fluid matrices
B: coupling matrix
2




(6)

   N  N d ,
K s   N s DN s ds , K f  c 2  N f N f d f ,
T
 
M s  s  N s N s ds , M f
T
 
T
f T
f
f
f
B   N f nNs dsf ,
T
Fs, Ff: mechanical load, acoustical sources
c: sound velocity, f: fluid density
Ns, Nf : structural and fluid shape function
8
Coupling system
 For a great number of DOF, solving the system directly is
always hard in term of CPU time.
The problem (6) can be seen as an eigenvalue problem:
(K -
2
M )
w
 = F
 p
(6)
9
Purpose of the approach
We search two matrices L and R verifying:
LMR  I
LKR  
(7)
(8)
- L and R contain the LEFT and RIGHT eigenvectors, respectively
(9) KR   MR
-  is a diagonal matrix containing the eigenvalues of:
(10) LH K  LH M
We obtain the physical unknowns of
by this relation:
(K -
w
2

R



I
 
 p


1
2
M )
w
 = F
 p  (6)
LF (11)
10
Purpose of the approach
Since
(K -
2
M )
w
 = F
 p
(6)
is non-symmetric,
Efficient eigenvalue algorithms can’t be used
A symmetric form of eigenvalue problem is required
Sandberg’s method enables us to make it symmetric by using
Modal analysis
11
Classical modal analysis
Solved
Independently
Structure in vacuum
K s X s   s M s X s (12)
Xs
Cavity with stiff
boundaries
K f X f   f M f X f (13)
Xf
s , f : the structural and the fluid eigenvalues.
Xs , Xf : the structural and the fluid eigenvectors.
12
Classical modal analysis
Xs
X f (50 modes)
(50 modes)
s, f are matrices containing some eigenvectors of structure and fluid
 w   s 0    s 
 
   



0


p
f  f 
  
(14)
s, f verify the following properties:
sT M s s  I s ,sT K s s  Ds
(15)
fT M f f  I f ,fT K f f  Df
Ds , Df are diagonal matrices containing the structural and the fluid
eigenvalues.
s, f represent the modal structural displacement and the modal fluid pressure.
13
Classical modal analysis
Hence, the coupling system (5) can be rewritten as the reduced system (17):
0   w   Fs 
  K s  B 2  M s







 
2
T



 0 K

f 
c B M f   p   F f 

w,p
432 physical unknowns
 w   s 0    s 
  (14)
   

 
0

p
f
  
 f 
(5)
 s,  f
100 modal unknowns
 Ds   I s
  B f    s    F 







2 2 T T
2
  c  f B s Df   I f  f    F 
2
T
s
T
s s
f
(17)
f f
The system is reduced (the problem size is divided by 4) but it remains non symmetric
14
Modal analysis (Sandberg Method)
 sT Bf    s 
   
2
Df   I f   f 
Non
 Ds   2 I s
Symmetric

system (17)   c 2  2  fT B T  s
Transition   s
matrix (18)   f
 


 


c
2
Ds

1
0


Ds
  2 I  
2 T
T
Symmetric 



c

B
 s Ds
f


system
(19)
0   s


If 
 f
sT Fs 
 f 
 f Ff 

 s

   S 


 f




   s
 
D f  c 2 Tf B T  s Ts B f    f
  c 2 D T F

s
s
s


 T F   c 2 T B T   T F 
f
s s
s 
 f f
 c 2 Ds Ts B f
15




Modal analysis (Sandberg Method)


Ds
  2 I  
2 T
T




c

B
 s Ds
f


   s
 
D f  c 2 Tf B T  s Ts B f    f
  c 2 D T F

s
s
s


 T F   c 2 T B T   T F 
f
s s
s 
 f f
 c 2 Ds Ts B f

 S 
  I  A    F  (20)
 F 

2

Symmetric generalized eigenvalue problem
AV  V
(21)
V : right eigenvector matrix of the symmetric system
16




(19)
Modal analysis (Sandberg Method)
w,p
 w   s 0    s 
 
   

 
0

p
f
  
 f 
 s
R
0
(14)
 s 
 s 
   S  
 
 
 f
 f
 s,  f

0   c 2 D
s


 f 
0

KR  MR

1
0 
V
I f 
(18)
 s,  f
(22)
(9)
R: right eigenvector matrix of the original system
The left eigenvector matrix “ L” is obtained in the same manner
17
Modal analysis (Sandberg Method)
(K -
LMR  I
&
LKR  
2

M )
w
 
 p
= F

(6)
w
2

R



I
 
 p
 is a diagonal matrix containing the eigenvalues of:

1
LF
(11)
KR  MR
T
T
L K  L M
18
(9)
(10)
Numerical results
Elastic Structure
Coupling
Rigid
Structure
Fluid
Interface
19
Numerical results
Plate
Discrete Kirchhoff Quadrilateral
(DKQ) plate element thin plate
(b)
Kirchhoff theory
Cavity
8-node brick isoparametric
acoustic element
(c)
20
Structure ( Frequency response)
 Simply supported plate (0.5m 0.5m)
 Unit punctual force (0.125m 0.125m)
Numerical results
Results given by Migeot et al (1)
Variation of the displacement with the frequency
at the load point
(1) 2nd
Worldwide Automotive Conference Papers,1-7
21
Structure ( Natural frequencies)
Structure: Simply supported plate (0.2m 0.2m) made of brass
6000
Analytical
Consistent mass
Lumped mass
Frequency (Hz)
5000
4000
3000
2000
1000
0
0
20
40
60
80
100
Mode number
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).
22
Cavity ( Natural frequencies)
Rigid cavity (0.2m 0.2m 0.2m)
5000
Frequency (Hz)
Analytical
FEM
2500
0
0
20
40
60
80
100
Mode number
Natural frequencies of the rigid cavity
FEM leads to good results below the 50th mode
23
Coupling problem
Results Given by Lee et al (2)
20
P re s s u re
( d B
40
)
Numerical results
2 0
0
-2 0
-4 0
D ir e c t F E M
M o d a l F E M
M o d a l F E M
-6 0
w ith
w ith
c o n s is te n t m a s s
lu m p e d m a s s
-8 0
1 0 0 0
Pressure (dB)
0
F re q u e n c y
( H z )
-20
-40
-60
-80
Direct FEM
Modal FEM with consistent mass
Modal FEM with lumped mass
-100
10
100
Frequency ( Hz )
1000
Pressure at the point (0.1,0.1,0.2)
CPU Time
Mesh (1266)
Direct Method
Modal analysis
32mn 54s
56s
Simply supported
elastic plate
Field point
(2) Engineering Analysis
with Boundary Elements, 16 (1995) 305-315
24
Structure ( Frequency response)
In vaccum
Plate-cavity (air)
50
50
Coupling effect
Displacement (dB)
Displacement (dB)
0
0
-50
-100
-50
-100
-150
-150
200
400
600
Frequency ( Hz )
800
1000
-200
200
400
600
800
1000
Frequency (HZ)
Plate quadratic displacement of the structure
854Hz
25
Coupling problem (air)
Mean square velocity:structure
Mean square velocity
Mean square pressure
Mean square pressure: cavity
Frequency (Hz)
Frequency (Hz)
Comparison between the direct and the modal results
26
Coupling problem (water)
Frequency (Hz)
Mean square velocity
Mean square velocity: structure
Mean square velocity
Mean square pressure
Mean square pressure: cavity
Frequency (Hz)
Comparison between the direct and the modal results
27
FEM / FEM -- FEM / BEM
Pressure (dB)
FEM-FEM FEM-BEM comparison
Simply supported
elastic plate
Frequency (Hz)
Field point
28
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
29