SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE
FORMULATION AND MODAL ANALYSIS
ALIA Ahlem
Laboratoire de Mécanique de Lille
Université des Sciences et Technologies de Lille
1, Boulevard Paul Langevin, Cité Scientifique
59655 Lille, France
Ahlem.Alia@ed.univ-lille1.fr
ABSTRACT
Vibroacoustic consists of the interaction between
elastic and acoustic waves. This interaction causes the
acoustic pressure to exert a force to the structure and the
structural motion produces an effective fluid load. Hence,
studying vibroacoustic problems requires that both elastic
structure and fluid to be modeled.
In this paper, a simple vibroacoustic problem is
modeled by a coupled FE/FE modal analysis method in
which a lumped mass representation is considered. The
numerical results are represented for an elastic plate backed
cavity. In this method, the coupling system is projected on
some structural modes in vacuum and some rigid cavity
modes. The obtained system is put in a diagonal form by
solving left and right eigenvalues problem. Some results are
presented to illustrate the effect of the modal analysis and
the lumped mass formulation. Some numerical results were
compared to analytical solutions.
I. INTRODUCTION
Actually, the design of comfortable products such as
vehicles, aircraft, machinery… draws attention to engineers
because noise constitutes one of the important indicators of
their quality. It is desirable to reduce and control products
noise at the design process in order to minimize the costs.
Hence, using noise prediction methods is very attractive in
the design process since they provide accurate information
about structural–acoustic performance.
Analytical techniques can be only used for solving a
very simple class of vibroacoustic problems under very
restrictive hypothesis. For real and more complex systems,
numerical methods are often indispensable. This class of
method is based on the computer hardware and software
environments. Today, the growing advance of the modeling
of noise and vibration has been furthered by the recent
development of fast computers. The actual powerful
machines allow solving more and more complex problems
without spend excessive time. In the other hand, the
development of the visualization tools of the software
environment helps the interpretation of results and makes
easier to create and solve models. Many numerical methods
have been developed to simulate the vibroacoustic
problems. Each one can be used according to the domain of
interest.
FE formulation has been commonly used for predicting
the vibroacoustic behavior of complex systems. Since the
spatial FE increment is dimensioned to the wavelength,
typically six nodes per wavelength, this limits its application
for only low frequency range. In addition, when the problem
size gets bigger, FE discretized coupled problem becomes
very large. In this situation, to being computationally
efficient, modal coupling analysis constitutes a very
convenient alternative [1].
Sometimes, modal techniques are not sufficient for
dynamic analysis of complex structures. In fact, consistent
mass matrix formulation, which has non zero-off diagonal
terms, involves large number of DOF and consequently
requires higher computational effort. Using diagonal lumped
mass formulations reduces the computational time and
simplifies the program coding [2]. To be accurate, the mass
representation must preserve the total mass in the element.
The BEM is a numerical method based on a Boundary
Integral Equations (BIE) form of the governing equations of
the field in a given domain to solve the boundary value
problems. The name of Boundary Element Method (BEM)
appeared in the late 70’s in an attempt to make analogy with
the FEM.
FEM has provided solutions for a wide range of engineering
problems. To this day, it remains the dominant method of
computational engineering analysis. However, in recent
years the BEM is used in many technological fields,
particularly, in many problems of radiation and diffusion in
acoustics. It is employed in automobile industry to reduce
the engines noise, in audio industry to optimize the
loudspeakers, microphones…etc. However, its practical
application, limited for some range of domains, is relatively
recent, not as well known as FEM which came first, and
therefore, received the most research attention.
Comparing to the FEM, the important advantage of the
BEM is the discretization. While in FEM the complete
domain has to be discretized, the BEM discretization is
restricted to the boundary [3]. Therefore for a given
problem, BEM generates a smaller full matrix than FEM,
which its system is characteristically sparse. Consequently,
to solve matrices of the same size, FEM requires less time
and memory than BEM. Although BEM model has fewer
elements than the FEM one, it doesn’t mean that FEM
results in a higher computational efficiency.
The BEM, which satisfies implicitly the Somerfeld radiation
condition, addressed the difficulty of the modeling of the
open boundary problems by the FEM in which absorbing
boundary conditions must be used. In fact, FEM is
conventionally limited to interior problems whereas BEM is
considered as the best method for the analysis of unbounded
problems since they are treated in the same way as interior
problems without any additional effort.
Another feature of this method is its employment for certain
class of Partial Differential Equations (PDE) (here the
Helmholtz equation) having fundamental solutions, for
instance, the Green’s functions “G” in acoustics which are
analytically the free space solutions under the action of a
point source [3]. However, this feature is regarded by many
as a drawback in the BEM because treatment of complex
problems requires the knowledge, a priori, of suitable
fundamental solutions. Besides, singularities in the BIE arise
from the use of the singular fundamental solutions [3]. Both
FE and BE acoustics formulations can be successfully
coupled to FE structure formulation in order to model a
vibroacoustic problem in the frequency domain.
The purpose of the present paper is to model the
vibroacoustic behavior of an acoustic cavity with one
flexible wall boundary using FEM. Both modal analysis and
lumped mass representation are applied. It is shown that
such techniques lead to satisfactory results as long as the
used frequency is relatively low.
In the same manner, we can write the governing elastic
wave motion:
(3)
 ij
   s 2ui  f i
( s )
x j
where  is the stress, s is the structure density, ui is the
displacement in the ith direction and fi is the mechanical
load.
We consider that the surface (f) is rigid, so:
(4)
p
At the interface fluid–structure (sf), the continuity of
the normal displacement and the stress gives:

p
  f  2u .n ( sf )
n
 ij n j   pni (  sf )
    
  2 M f   p  F f 

 N  DN d ,
K  c N  N d ,

M   N  N d ,

M   N  N d ,

B  N  nN d ,

s T
Ks 
s
s
f T
2
f
f
f
f
s
s
s
f
f
(7)
f
f T
s
sf
where D is a symmetric matrix containing the structural
elastic coefficients, N designs the shape function and “n”
is the normal direction at the wet surface.
Or in a condensed form as:
(8)
u 
2
(K -  M )   =F
 p
Structure
(s)
(sf)
(f)
III. MODAL ANALYSIS [1]
(f)
Figure(1): Fluid-structure problem
For the time harmonic acoustical pressure
P  p eit
f
where
Ks is the structural stiffness matrix
Ms is the structural mass matrix
( Kf /fc2 ) is the cavity kinetic energy matrix
( Mf /fc2 ) is the fluid compressibility matrix
B is the coupling matrix
f T
Fluid
K

2 2 T
  f  c B
s T
The problem we wish to solve is dealing with the
interaction of an elastic structure (s) with an ideal fluid
(f) [Figure (1)].
(5)
The application of the FEM method to the variationnal
formulation yields to the following system:
 Ks   2M s
 u   Fs 
B
(6)
s
II. PROBLEM STATEMENT
 0 ( s )
n
(1)
in a compressible and non-viscid fluid, the wave equation
reduces to the following Helmholtz equation:
(2)
p  k 2 p  0 ( f )
where p is the pressure amplitude, k=/c is the wave
number, c is the sound velocity in the fluid and  is the
circular frequency of the pressure oscillations.
Solving system (6) directly is always hard in term of
computational costs because of the great number of DOF
and due to the non symmetric character of this system. As
motioned above, modal analysis reduces significantly the
discretized problem size. In the other hand, it makes
possible to diagonalize each block independently. To do
this, we solve independently the following generalized
eigenvalues problems corresponding , respectively, to the
structure in vacuum and to the fluid with stiff boundaries:
K s X s  s M s X s
K f X f  f M f X f
(9)
where s and f are, respectively, the structural and the fluid
eigenvalues.
Assuming s and f the matrices consisting of the
eigenvectors in the structural and the fluid domains
respectively, such as:
 u   s 0   s 
   
 
 p   0  f   f 
 sT M s s  I s ,  sT K s s  Ds
Mechanical load
(a)
x
y
z
(10)
 Tf M f  f  I f ,  Tf K f  f  D f
Simply supported
elastic plate
where Ds and Df are diagonal matrices containing,
respectively, the structural and the fluid eigenvalues and s,
f represent, respectively, the modal structural displacement
and the modal fluid pressure.
Hence, the coupling system can be rewritten as
follows:
 Ds   2 I s

2 2 T T
 c   f B  s
  sT B f    s    sT Fs
    f
D f   2 I f   f   f F f



Rigid wall
x
(11)
The resolution of the last algebraic linear system
[equation (11)] for each frequency allows obtaining the
modal unknowns s, f and consequently u and p can be
deduced from equation (10).
Although the system dimension is reduced it stills non
symmetric. Therefore, it requires expensive algorithms.
Another alternative proposed by Sandberg [1] consists of
putting the system matrices in a diagonal form by solving
left and right eigenvalues problem.
In fact, if we assume that L and R matrices consisting,
respectively, of the left and right eigenvectors corresponding
to all modes up to 1.5fmax where fmax is the highest frequency
of interest, then
(12)
LKR  
designs a diagonal matrix containing the eigenvalues of the
following coupled problem:
(13)
KR  MR
By putting
(14)
LMR I
the initial system becomes
u 
(15)
2
Plate
1
LF
(16)
So , the solution is obtained by simple matrices
multiplication.
The most important advantage of this method is that
the resonance frequencies and the mode shapes of the
coupled system can be deduced, what gives the
vibroacoustic behavior of this system in its totality.
IV. FEM
In this paper, a flexible simply supported plate backed
cavity is considered. The plate is subjected to a uniformly
distributed harmonic pressure load [Figure(2.a)].
Node 3 (w,x,y)
z,w
Node 1 (w,x,y)
Node 2 (w,x,y)
(b)
(d.2)
(d.3)
Cavity
(d.4)
(c)
Figure (2): Cavity–plate interaction
a) Coupling problem
b) Plate mesh
c) Cavity mesh
d) DKQ plate element and brick fluid element
 


ns
y
and consequently, the unknowns are given by:

(d.1)
x
Node 4 (w,x,y)
   I L p  LF
u 
2
   R   I
 p
y
IV.1. Thin plate
The plate is divided into quadrilateral elements
[Figure(2.b)]. In this paper, the used Discrete Kirchhoff
Quadrilateral (DKQ) plate element [Figure(2.d.1)] is based
on thin plate Kirchhoff theory in which the shear
deformation is neglected. At each node of this 4 nodded–
element correspond three DOF: the normal displacement
“w“ and its derivatives corresponding to two rotations
 x  w / y ,  y  w / x at the nodes. The Kirchhoff
assumptions must be satisfied along the boundaries of the
element. The stiffness matrix of the DKQ element is defined
as [4]:
(17)
K se  Bs T D Bs dSe

Se
where
(18)
 x x



Bs   y y



 x y   y x
with  x and y are rotations expressed in an incomplete
cubic polynomial form and
0
(19)
1 

3
Et
 1
12 1   2 
0 0




1   / 2
0
When the element is rectangular, using a non complete
fourth order polynomial expression allows to define the
displacement function as [5]:
(20)
N  PC 1
where C is a 1212 matrix depending on the nodal
coordinates and P is given by:
P  1 x y x 2 xy y 2 x 3 x 2 y xy 2 y 3 x 3 y
xy 3  (21)

U  u1
w  NU
 x1  y1 u 2
 x2  y2
u 3  x3  y3
B
 N 
f T
 N 
f T
case : fig d .2 
N s d sf
(24)
case : fig d .3 
N s d sf
V. NUMERICAL RESULTS
in which E,  and t represent, respectively, the Young
modulus, the Poisson ratio and the plate thickness.
with
B
u 4  x4
 y4

(22)
Thus, the consistent mass matrix for a rectangular plate
element of a constant thickness “t” becomes:
(23)


T
 
M se   s t  C 1 PT PC 1dS e 
 e

S

Generally, this matrix is populated. For many
problems, the lumped mass, which is formed by lumping of
mass at the element nodes, is more economical. However, it
results in a potential loss of accuracy compared to the
consistent mass matrix. Although this, it still be used widely
since it results in admissible physical results. There are
many methods employed for the construction of the lumped
mass matrices.
In the typical lumped mass called equal lumped mass [2],
only the transnational mass is lumped to the nodes. The
mass of an element is preserved by dividing the total mass
amongst the diagonal entries of the mass matrix and setting
all other entries to zero [6]. This matrix is singular since it
has got some pivots equal to zero. Thus, it may not be
inverted.
In row-sum technique, the lumped mass matrix is
constructed by adding the off-diagonal entries in each row to
the diagonal entry [5].
A more sophisticated lumped mass is of HZR (Hilton, Rock
& Zienkiewicz) diagonalization method [6]. The essential
idea is to compute only the diagonal terms of the consistent
mass matrix and to lump it by scaling the diagonal entries to
preserve the total transnational mass of the element.
IV.2. Cavity and cavity-plate interface
The element stiffness and mass matrix are computed
for 8-node brick isoparametric acoustic element [Figure
(2.d.4)]. The fluid domain mesh is shown on figure (2.c).
The accuracy of the response analysis is significantly
affected by errors in natural frequencies and mode shapes,
which are dependent on the quality of both the mass and the
stiffness matrices. For this reason, the accuracy of the
lumped mass formulation is investigated by computing the
natural frequencies of a plate ( 0.2  0.2 ) made of brass
with the following properties: Young modulus E=103 GPa,
s =8500 kg/m3 and Poisson ratio =0.34.
In this paper, the mass is lumped as done in LS-DYNA
in which linear interpolations are used [ 7]. The translational
masses are computed from the consistent mass matrix by
row summing whereas the rotational masses are obtained by
scaling the translational mass at the node by the factor
V
where the V is the element volume.

8t
Figure (3) shows that FE results with consistent or
lumped mass are in good agreement with analytical natural
frequencies [ 8] as long as low frequencies are considered (
 50 modes). Usually, only few structural and acoustic
modes are involved in the analysis. For the frequency band
of interest, keeping 50 modes for the structure and the fluid
seems sufficient.
6000
Analytical
Consistent mass
Lumped mass
5000
Frequency (Hz)
D
The element coupling matrix "B" [eq. 7] between 8-node
isoparametric acoustic element and a 4-node quadrilateral
plate element is calculated as following:
4000
3000
2000
1000
0
0
20
40
60
80
100
Mode number
Figure(3): Natural frequencies of the structure
The natural frequencies of the rigid cavity are shown in
figure (4). As in the structure, FEM leads to good results
below the 50th mode.
5000
Figure (6.a) and (6.b) represent, respectively, the
quadratic displacement of the plate in vacuum and structurecavity coupled system. We notice the weak effect of the
cavity on the first plate mode ( a weak shift towards the low
frequencies) and the appearance of a new resonance at 854
Hz.
Frequency (Hz)
Analytical
FEM
2500
0
0
20
40
60
80
100
Mode number
50
Figure(4): Natural frequencies of the fluid
Displacement (dB)
Consider a rigid cavity ( 0.2  0.2  0.2 ) with one
simply supported flexible plate which is the same as used
above. The cavity contains air of density f =1.21 kg/m3 and
sound velocity c=343m/s. The plate is subjected to a
uniformly distributed harmonic pressure load ( 1psi) .
Figure(5.a) represents the numerical pressure at the point
situated at (0.1,0.1,0.2). For the frequency band [10,1000]
Hz, the three methods lead to practically the same results
except for high frequencies. These results are compared to
the analytical results [Figure(5.b)] given by Lee et al [9], a
good agreement is observed.
0
-50
-100
-150
200
400
600
800
1000
Frequency ( Hz )
50
40
0
Displacement (dB)
20
20
Pressure ( dB )
0
-20
-40
Direct FEM
Modal FEM with consistent mass
Modal FEM with lumped mass
-60
-80
1000
Frequency ( Hz)
-20
-60
-80
-100
10
-50
-100
-150
-40
-200
Direct FEM
Modal FEM with consistent mass
Modal FEM with lumped mass
100
200
400
600
800
1000
Frequency (HZ)
1000
-40
Frequency ( Hz )
-60
-80
Displacement (dB)
Pressure (dB)
0
-100
-120
-140
-160
-180
-200
200
400
600
800
Freqeuncy ( HZ)
Figure (5): Pressure at the point situated at (0.1,0.1,0.2)
a) numerically obtained
b) given by Lee et al
Figure(6): Plate quadratic displacement of the structure
a) in vacuum
b) in the plate-cavity(air) system
c) in the plate-cavity(water) system
1000
The 5th first modes of the plate in the coupled system
are shown in figure (6.b). Because of the weak coupling
between the plate and the cavity this later does not affect
greatly the plate response.
If the cavity contains water ( c=1500 m/s, =1000
kg/m3), the coupling becomes strong and consequently the
frequency response is totally changed. [ Figure(6.c)].This
can be clearly seen in the plate response when it is coupled
to a cavity containing water [ Figure(6.c)]. This is because
water is a heavy fluid compared to air.
V. CONCLUSION
A simple vibroacoustic problem is modeled using FE
modal analysis and lumped mass formulation. It is shown
that this method results in less computational effort.
However, the modal analysis would be less efficient if
more higher frequency range is to be considered because of
the modal truncation in the two subsystems. In order to
handle the vibroacoustic behavior for such frequencies,
modal corrections need to be used [ 10].
In the FEM, a good representation of the mass is very
essential to achieve accurate results. When frequency gets
higher, the inertia contribution in FEM becomes important
which amplifies any errors made in the lumped mass matrix.
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