1 component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models

Component mode synthesis and polynomial chaos expansions for stochastic
frequency functions of large linear FE models
D. Sarsri
a
, L. Azrar
b,
⇑
, A. Jebbouri
c
, A. El Hami
d
a
LTI, ENSA, University Abdelmalek Essaadi, Tangier, Morocco
b
MMC, FST, University Abdelmalek Essaadi, Tangier, Morocco
c
FST, University Abdelmalek Essaadi, Tangier, Morocco
d
Lab. de Mécanique – UMR 6138, Pôle Technologique du Madrillet, Avenue de l’Université – BP 8, 76801 Saint Etienne du Rouvray Cedex, France
article info
Article history:
Received 2 February 2010
Accepted 9 November 2010
Available online 8 December 2010
Keywords:
Frequency transfer function
Component mode synthesis
Random
Polynomial chaos
First two moments
abstract

This paper presents a methodological approach for the numerical investigation of frequency transfer
functions for large FE systems with linear and nonlinear stochastic parameters. The component mode
synthesis methods are used to reduce the size of the model and are extended to stochastic structural
vibrations. The statistical first two moments of frequency transfer functions are obtained by an adaptive
polynomial chaos expansion. Free and fixed interface methods with and without reduction of interface
dof are used. The coupling with the first and second order polynomial chaos expansion is elaborated
for beams and assembled plates with linear and nonlinear stochastic parameters.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
In the dynamic analysis of complex industrial structures using
the finite element method (FEM), a very large number of degrees
of freedom is usually required. This leads to large numerical
problems to solve. Therefore, it is often necessary to reduce the size
of the system before proceeding to numerical computation. To this
end, component mode synthesis (CMS) methods are well-
established methods for efficiently constructing models that are
often described by separate substructure (or component) models.
Typically, each substructure is approximated by a set of basis vec-
tors (Ritz vectors), where the number of vectors is substantially
smaller than the number of the physical degrees of freedom
(dof). The substructure approximations are then assembled to
provide a global approximation of the structure. Substructuring
techniques differ from Ritz representation basis. The latter includes
the vibration normal modes, the rigid body modes, the static
modes, the attachment modes, etc. Since their first introduction
in 1965 by Hurty [1], the CMS methods have been extensively
developed. Depending on the boundary conditions applied to the
substructure interfaces, the CMS methods can be classified into
four groups: fixed interface methods [2]; free interface methods
[3,4]; hybrid interface methods [5] and loaded interface methods

[6].
The aforementioned approaches have been extensively applied
to analyze large structural systems. However, CMS methods are
commonly accomplished assuming deterministic behavior of loads
and model parameters. Although modern computational facilities
allow a very sophisticated and numerically accurate structural
analysis with very detailed deterministic models, quite often the
predicted results do not accurately coincide with experimental
tests. Furthermore, even test results of technically identical models
subjected to identical loading conditions may vary randomly.
Hence, it would be necessary to take account of the model param-
eters uncertainties, if highly reliable structures are to be designed.
In the framework of simulations destined to qualify the
response or the reliability of a structure, it is important first to
identify all sources of uncertainties involved in the modelling of
the structural characteristics. Probabilistic methods provide a
powerful tool for incorporating structural modelling uncertainties
in the analysis of structures by describing the uncertainties as ran-
dom variables. The first and second-order statistics of the response
are commonly investigated once those of the random variables
modelling the structural uncertainties are known. The stochastic
dynamic behaviour of structures is commonly handled by well
established random eigenvalue approaches [7–9].
Furthermore, the finite element method (FEM) represents the
most important tool for structural analysis and design, its applica-
tions are increasing and its progress offers solutions to a wide
variety of problems. Standard deterministic FEM has been ex-
tended to stochastic finite element method (SFEM) to analyze the
0045-7949/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2010.11.009

⇑
Corresponding author. Tel.: +212 62 88 71 48; fax: +212 39 39 39 53.
E-mail addresses: , (L. Azrar).
Computers and Structures 89 (2011) 346–356
Contents lists available at ScienceDirect
Computers and Structures
journal homepage: www.elsevier.com/locate/compstruc
statistical nature of loads and material properties. Substantial
developments of SFEM have been noticed and detailed reviews
on its overall aspects are well documented in the literature
[10,11]. Monte Carlo Simulation (MCS) is often used to obtain ref-
erence results [12]. Although, simulation techniques can be used
for a wide range of structural dynamics problems, it is in general
quite inefficient due to the large number of samples required to
guarantee accurate statistical results.
Alternatively, the SFEM based on perturbation techniques be-
gun to be used [13,14]. The perturbed components of the response
are obtained through the perturbed components of the uncertain
parameters. Therefore, only the low-order perturbation technique
is practically implementable, high-order perturbations is extre-
mely time-consuming. The accuracy of low-order perturbation
method is good enough for the problems with small deviations of
uncertain parameters.
Another alternative approach is based on the expansion of the
response in terms of a series of polynomials that are orthogonal
with respect to mean value operations [15,16]. More precisely;
the Karhunen–Loeve expansion is used to discretize the stochastic
parameters into a denumerable set of random variables, thus pro-
viding a de numerable function space in which the problem is cast.
The polynomial chaos expansion is then used to represent the solu-

tion in this space and the expansion coefficients are evaluated via a
Galerkin procedure in the Hilbert space of random variables. For
large structural vibration systems subjected to stochastic loading
the time domain or the frequency domain can be used. For station-
ary solutions of linear structures, the spectral density of the re-
sponse can be computed from the spectral density of the
excitation using frequency transfer functions in the frequency
domain.
In this paper, a methodological approach for computing the fre-
quency transfer functions of stochastic structures, modelled by
large FE models is presented. A CMS approach is used in order to
reduce the size of the model before proceeding to numerical com-
putations. The second moment characteristics, i.e. mean and
covariance of the frequency transfer function are computed by
combining the CMS and polynomial chaos expansions at first and
second orders. The approach may be construed as an extension
of deterministic computational analysis to the stochastic case with
an appropriate extension to the concepts of projection, orthogonal-
ity and weak convergence. The model parameters are random, give
arise to stochastic static and dynamic Ritz vectors for each sub-
structure. The only assumption involved in the proposed approach,
is that these vectors are defined assuming that the model is deter-
ministic. Different approaches based on the CMS and polynomial
chaos expansions are elaborated. Stochastic beams and assembled
plates with linear and nonlinear random parameters are analysed.
The efficiency of the proposed approach is demonstrated and an
impressive CPU time reduction is resulted.
2. Polynomial chaos expansion
Let us consider a multi-degrees of freedom linear structural sys-
tem with mass, damping and stiffness matrices M, C, and K respec-

tively. The equations of motion describing the forced vibration of a
linear and damped discrete system are:
M
€
yðtÞþC
_
yðtÞþKyðtÞ¼fðt Þ; ð1Þ
where y(t) is the nodal displacement vector and f(t) is the external
excitation. In the frequency domain and with a harmonic excitation,
Eq. (1) can be written in the following form:
Dð
x
Þy ¼ f; ð2Þ
where D(
x
) is the dynamic stiffness matrix defined by:
Dð
x
Þ¼K þ i
x
C À
x
2
M: ð3Þ
In this paper, a hysteretic damping of coefficient
g
is considered and
the dynamic stiffness matrix is rewritten in the form:
Dð
x

Þ¼ð1 þ i
g
ÞK À
x
2
M: ð4Þ
In this analysis the matrices K and M are constant and frequency
independent. The transfer matrix H is defined by:
Dð
x
ÞÁH ¼ I; ð5Þ
where H(i, j) is the frequency response at the ith node with applied
force at the jth node.
In order to reduce the computation, the following vector nota-
tions are used:
Dð
x
ÞH
j
¼ f
j
; ð6Þ
where H
j
and f
j
are the jth column vectors of H and I.
The physical properties of the structural system described by
the mass, damping and stiffness matrices are assumed to be uncer-
tain. Then, M, C and K are random matrices. The issue of represent-

ing stochastic processes is crucial to the SFEM. It involves replacing
a complicated random quantity by a collection of simpler random
quantities that are easier to manage. The random material proper-
ties are then represented by the random processes. These proper-
ties are assumed to be known through their second-order
statistics and vary continuously over the space. The value of these
processes at each spatial location is therefore a random variable,
and the issue is then to replace this uncountable set of random
variables by a countable set that can be truncated at a certain level
and is commensurate with specified representation accuracy. The
Karhunen–Loeve expansion is used for this purpose. The matrices
M and K are represented in the form [16]:
Nomenclature
M, C, K mass, damping and stiffness matrices
D dynamic stiffness matrix
Z transformation matrix
H transfer matrix
H(i, j) frequency response at the ith node with applied force at
the jth node
f vector of force
Q matrix of Ritz vectors
Y truncated undamped normal modes
w
c
matrix of constrained mode
w
r
matrix of rigid body modes
w
a

matrix of attachment modes
G residual flexibility matrix
w
ar
residual attachment modes
M
0
,K
0
average of mass and stiffness matrices
dof degree of freedom
n
i
(i – 0) random variables
w
n
(n
i
) multidimensional orthogonal polynomials chaos
hÁÁi inner product defined by the mathematical expectation
operator
hi mean value
r
standard deviation
D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
347
M ¼
X
Q
1

q
1
¼0
M
q
1
n
q
1
; ð7aÞ
K ¼
X
Q
2
q
2
¼0
K
q
2
n
q
2
; ð7bÞ
where n
0
= 1, the matrices M
0
and K
0

are the average matrices and
M
q
1
and K
q
2
are deterministic while n
q
i
(q
i
– 0) are Gaussian random
variables. The dynamic stiffness matrix D can be similarly repre-
sented in the form:
Dð
x
Þ¼ð1 þ i
g
Þ
X
Q
2
q
2
¼0
K
q
2
n

q
2
À
x
2
X
Q
1
q
1
¼0
M
q
1
n
q
1
: ð8Þ
The real and imaginary parts of the frequency response functions
with random properties must be, obviously, random too. A vector
random process representing the random solution at the nodes of
the finite element mesh is used. This solution is not known a priori,
and should therefore be discretized in a generic way that is inde-
pendent of its unknown properties. This is the reason for what in-
stead of the Karhunen–Loeve expansion; the polynomial chaos
expansion is used. The resulting vector H
j
is expanded along a poly-
nomial chaos basis [15]:
H

j
¼
X
N
n¼0
ðH
j
Þ
n
w
n
ðn
i
Þ; ð9aÞ
where
w
n
(n
i
) are multidimensional orthogonal polynomials in the
random variables n
i
describing the material properties, defined by:
w
n
ðn
i
; ; n
p
Þ¼ðÀ1Þ

p
exp
1
2
T
fngfng

@
p
À
1
2
T
fngfng
@n
i
; ;@n
p
: ð9bÞ
(H
j
)
n
denotes an n-dimensional vector of deterministic coefficients.
In this context, orthogonality is construed to be in the Hilbert space
of random variables with respect to the inner product defined by
the mathematical expectation operator.
Substituting Eqs. (8) and (9) into Eq. (6) and forcing the residual
to be orthogonal to the space spanned by the polynomial chaos
w

n
yield the following system of linear equations:
X
N
n¼0
ð1 þ i
g
Þ
X
Q
2
q
2
¼0
n
q
2
w
n
w
m

K
q
2
À
x
2
X
Q

1
q
1
¼0
n
q
1
w
n
w
m

M
q
1
"#
ðH
j
Þ
n
¼ w
m
hif
j
m ¼ 0; 1; ; N;
ð10Þ
where hÁÁi denotes the inner product defined by the mathematical
expectation operator.
This algebraic equation can be rewritten in a more compact ma-
trix form as:

Dð
x
Þ
ð00Þ
ÁÁÁ Dð
x
Þ
ð0NÞ
: ÁÁÁ :
: Dð
x
Þ
ðnmÞ
:
: ÁÁÁ :
Dð
x
Þ
ðN0Þ
ÁÁÁ Dð
x
Þ
ðNNÞ
2
6
6
6
6
6
6

4
3
7
7
7
7
7
7
5
ðH
j
Þ
0
:
ðH
j
Þ
m
:
ðH
j
Þ
N
8
>
>
>
>
>
>

<
>
>
>
>
>
>
:
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
¼
f
j
0
.
.
.

.
.
.
0
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
=

>
>
>
>
>
>
>
;
; ð11aÞ
where:
Dð
x
Þ
ðnmÞ
¼ð1 þ i
g
Þ
X
Q
2
q
2
¼0
n
q
2
w
n
w
m


K
q
2
À
x
2
X
Q
1
q
1
¼0
n
q
1
w
n
w
m

M
q
1
:
ð11bÞ
The deterministic coefficients of (H
j
)
m

(m =0, 1, , N) can be ob-
tained by solving the algebraic system (11). Once these coefficients
are computed, the mean values and the standard deviations of the
imaginary and real parts of H
ij
are given by the following
relationships:
realðH
ij
Þ

¼ realð½H
ij

0
Þ; ð12aÞ
imagðH
ij
Þ

¼ imagð½H
ij

0
Þ: ð12bÞ
r
realðH
ij
Þ
¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
N
n¼1
realð½H
ij

n
Þ
2
hw
2
n
i
v
u
u
t
; ð13aÞ
r
imagðH
ij
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
N
n¼1
imagð½H
ij


n
Þ
2
hw
2
n
i
v
u
u
t
: ð13bÞ
Note that the equation giving the frequency response function mag-
nitude is nonlinear. Monte Carlo Simulation of the vector of random
variable {n
1
, , n
q
, } is also used in this paper to compute this
magnitude and the obtained results are considered as reference
results
The previous methodological approach is similar to that fol-
lowed by Guedri et al. [17] in which the system (11) is numerically
solved. For large number of dof, the algebraic system (11) becomes
very large and its inversion requires a large amount of CPU time
particularly when standard numerical procedures are used. New
developments of iterative methods for linear systems ended with
the availability of large toolbox of specialized algorithms for solv-
ing the very large problems. The main research developments in

this area during the 20th century are described in the review paper
[18].
Even if new efficient algorithms are available, it is desirable to
avoid solving such large problems. For this reason, a reduction pro-
cedure based on deterministic modal basis is developed here. The
displacement vector y can be written in deterministic modal basis:
y ¼
X
P
p¼1
k
p
/
p
ð14Þ
where k
p
are unknown random coefficients and /
p
are the vectors of
deterministic modal basis. k
p
are also expanded along a polynomial
chaos basis:
k
p
¼
X
N
n¼0

k
n
p
w
n
ðn
i
Þ: ð15Þ
Inserting Eq. (14) into Eq. (6) and using the M and K-orthogonality
conditions, the following equation is obtained:
½ð1 þ i
g
Þ
x
2
j
À
x
2
k
j
þ
X
P
p¼1
k
p
ð1 þ i
g
Þ

X
Q
2
q
2
¼1
n
q
2
T
/
j
K
q
2
/
p
"
À
x
2
X
Q
1
q
1
¼0
n
q
1

T
/
j
M
q
1
/
p
#
¼
T
/
j
f: ð16Þ
Forcing equation (16) to be orthogonal to the approximating space
spanned by the polynomial chaos
w
n
the following algebraic linear
system is obtained:
½ð1 þ i
g
Þ
x
2
j
À
x
2
k

m
j
w
2
m

þð1 þ i
g
Þ
X
P
p¼1
X
N
n¼0
X
Q
2
q
2
¼1
n
q
2
w
n
w
m

k

m
p
T
/
j
K
q
2
/
p
À
x
2
X
P
p¼1
X
N
n¼0
X
Q
1
q
1
¼1
n
q
1
w
n

w
m

k
m
p
T
/
j
M
q
1
/
p
¼
T
/
j
f: ð17Þ
The solution of this system allows one to get the coefficients k
n
p
and
therefore the random vector displacement y:
y ¼
X
P
p¼1
X
N

n¼0
ðk
n
p
/
p
Þw
n
ðn
i
Þ: ð18Þ
348 D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
For
f ¼
0
.
.
.
0
1
0
.
.
.
0
8
>
>
>
>

>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>

>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
;
j; ð19Þ
y corresponds to the j’th column of the frequency transfer matrix H.
It is often necessary to reduce the size of the system before pro-
ceeding to numerical computation. To this end, component mode
synthesis (CMS) methods are used. The same concept used by Gue-
dri et al. [17] is followed here and the following main ideas will be
exploited:
Explicit deterministic transformation matrix is developed for
the used CMS methods and particularly for reducing the num-
ber of interface dof.
Automatic procedure is developed based on this defined trans-

formation matrix allowing a straightforward computation of
the needed condensed random matrices.
Based on the given forms of the condensed matrices, the sto-
chastic finite element approach is easily elaborated for fixed
and free methods with and without reduction of interface dof.
3. Component mode synthesis
Component mode synthesis (CMS) techniques are well
established in the field of response analysis of large and complex
structures. CMS techniques have an advantage of enhancing com-
putational efficiency by reducing the number of degrees of free-
dom of a structure and have been widely developed and used for
larger structural systems [1–6,19,20].
Let us consider a structure, which is decomposed into n
s
susb-
structures SS
(k)
(k =1, , n
s
) which do not overlap. For each sub-
structure k the displacement vector y
(k)
is partitioned into a
vector y
ðkÞ
j
, called interface dof and y
ðkÞ
i
which is the vector of inter-

nal dof. The force vector f
(k)
is composed into vectors f
ðkÞ
j
and f
ðkÞ
e
,
called interface force and external applied force.
In the component mode synthesis methods, the physical dis-
placements of the substructure SS
(k)
are expressed as linear combi-
nation of the substructure modes. After some algebraic
transformations, a set of Ritz vectors Q is obtained and the dis-
placements of SS
(k)
are expressed as [21]:
y
ðkÞ
¼ Q
ðkÞ
y
ðkÞ
j
l
ðkÞ
()
¼ Q

ðkÞ
g
ðkÞ
; ð20Þ
where
l
(k)
are the generalised coordinates. In order to simplify the
writing superscript k is omitted in the following formulations.
3.1. Fixed interface method
In the fixed interface method, the displacements of each sub-
structure are expressed:
y ¼ Y
g
þ w
c
y
j
: ð21aÞ
The matrix Q is given by
Q ¼½w
c
Y ð21bÞ
in which Y is a matrix containing the first eigenmodes of the
undamped substructure SS with a fixed interface as boundary
condition.
w
c
is the matrix of constrained mode associated with
the interface, which is the static deformation shapes of SS obtained

by imposing successively a unit displacement on one interface,
while holding the remaining interface coordinates fixed.
3.2. Free interface method
In the free interface method, the displacements of each sub-
structure are expressed as:
y ¼ Y
g
þ w
r
n
r
þ w
a
n
a
: ð22Þ
Y is a matrix containing the first eigenmodes of the undamped sub-
structure SS with a free interface as boundary condition.
w
r
is the
matrix of rigid body modes for an unconstrained substructure with
a free interface.
w
a
is the matrix of attachment modes associated
with the interface, which are the static deformation shapes of SS ob-
tained by applying successively a unit force to one coordinate of the
interface.
w

a
¼ GF
j
; ð23aÞ
F
j
¼
I
j
0

; ð23bÞ
where G is the residual flexibility matrix. The expression of G de-
pends on the nature of the problem.
If the substructure is statically determined (i.e. no rigid body
modes) then:
G ¼ K
À1
: ð24aÞ
Else,
G ¼
T
AK
À1
ðcÞ
A; ð24bÞ
where: A = I À
u
(r)T
u

(r)
M and
T
u
(r)
M
u
(r)
= I, I: unit matrix and
u
(r)
:
matrix of rigid modes. K
(c)
: stiffness matrix obtained by fixing arbi-
trary dof to make the structure isostatic and replacing the corre-
sponding part of the initial stiffness matrix by zero.
To preserve the interface dof, we use the following partition:
Y ¼
Y
j
Y
i

w
r
¼
w
rj
w

ri

w
a
¼
w
aj
w
ai

: ð25Þ
Using this partition in Eq. (23), one obtains:
n
a
¼ w
À1
aj
y
j
À w
À1
aj
Y
j
g
þ w
À1
aj
w
rj

n
r
: ð26Þ
The matrix Q is then given by:
Q ¼ w
a
w
À1
aj
w
r
À w
a
w
À1
aj
w
rj
Y À w
a
w
À1
aj
Y
j
jk
: ð27Þ
The residual attachment modes
w
ar

, obtained by removing in the
attachment modes the components of the normal mode already re-
tained in Y, can be used to get:
y ¼ Y
g
þ w
r
n
r
þ w
ar
n
ar
; ð28Þ
w
ar
is the residual attachment modes obtained by:
w
ar
¼ RF
j
ð29aÞ
and
R ¼ G À YK
À1
T
Y; ð29bÞ
where
K
is the matrix of the retained eigenvalues. The matrix Q can

be written as:
Q ¼
w
ar
w
À1
arj
w
r
À w
ar
w
À1
arj
w
rj
Y À w
ar
w
À1
arj
Y
j
jk
: ð30Þ
D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
349
3.3. Equation of motion for assembled system
In order to assemble the components, the force and displace-
ment continuity at the interface will be used. That is to say for n

s
substructures coupled at a common boundary:
- Displacement continuity:
y
1
j
¼ y
2
j
¼ÁÁÁ¼y
n
j
¼ y
j
: ð31aÞ
- Equilibrium of coupling forces:
X
n
s
k¼1
f
k
j
¼ 0: ð31bÞ
The conservation of interface dof allows assembling these matrices
as in the classical finite element method. The vector of independent
displacements of the assembled structure
g
is expressed by:
g

¼
l
ð1Þ
.
.
.
l
ðn
s
Þ
y
j
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>

>
=
>
>
>
>
>
;
: ð32Þ
The compatibility of interface displacements of the assembled
structure is obtained by writing, for each substructure SS
(k)
, the fol-
lowing relation:
g
ðkÞ
¼ b
ðkÞ
g
; ð33Þ
where b
(k)
is the matrix of localization or of geometrical connectiv-
ity of the SS
(k)
substructure. It makes possible to locate the dof of
each substructure SS
(k)
in the global ddl of the assembled structure.
They are the Boolean matrices whose elements are 0 or 1.

A transformation matrix can be defined for each substructure
SS
(k)
by:
Z
ðkÞ
¼ Q
ðkÞ
b
ðkÞ
: ð34Þ
The kinetic energy T, the strain energy U and the work of the exter-
nal forces
s
are given by:
T ¼
1
2
T
_
g
M
c
_
g
; ð35aÞ
U ¼
1
2
T

g
K
c
g
; ð35bÞ
s
¼
T
g
f
c
; ð35cÞ
where:
M
c
¼
X
n
s
k¼1
T
Z
ðkÞ
M
ðkÞ
Z
ðkÞ
; ð36aÞ
K
c

¼
X
n
s
k¼1
T
Z
ðkÞ
K
ðkÞ
Z
ðkÞ
; ð36bÞ
f
c
¼
X
n
s
k¼1
T
Z
ðkÞ
ðf
ðkÞ
j
þ f
ðkÞ
e
Þ: ð36cÞ

Using the compatibility of displacements of interface dof, it can be
easily shown that:
X
n
s
k¼1
T
Z
ðkÞ
f
ðkÞ
j
¼ 0: ð37Þ
Thus, the work of the applied forces becomes:
f
c
¼
X
n
s
k¼1
T
Z
ðkÞ
f
ðkÞ
e
: ð38Þ
The reduced equation can be written in the form:
½ð1 þ i

g
ÞK
c
À
x
2
M
c

g
¼ f
c
: ð39Þ
The physical displacements of each substructure are obtained by:
y
ðkÞ
¼ Z
ðkÞ
g
: ð40Þ
This concept may lead to many interface dofs. For the sake of CPU
time reduction, a reduction procedure is also used in this paper.
3.4. Reduction of interface degrees of freedom
In most of the CMS methods, the coupling of the substructures
is performed through the interface displacements, especially when
the size of the coupled system is still large due to great number of
degrees of freedom at the interface. In order to reduce the number
of interface coordinates and therefore the size of the coupled sys-
tem, a procedure based on the interface modes is used.
The interface modes

u
are defined as the first eigenmodes of the
reduced eigenproblem:
ðK
cj
À kM
cj
Þy
j
¼ 0: ð41Þ
This results from the Guyan condensation [22] of the whole struc-
ture to the interface. The displacements of the interface dof are ex-
pressed as:
y
j
¼
ul
j
: ð42Þ
For the assembled structure, the vector of independent displace-
ment is rewritten as:
g
¼
l
ð1Þ
.
.
.
l
ðn

s
Þ
y
j
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>
>
=
>
>
>
>
>
;

¼
I
ð1Þ
.
.
.
I
ðn
s
Þ
u
2
6
6
6
6
4
3
7
7
7
7
5
l
ð1Þ
.
.
.
l
ðn

s
Þ
l
j
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>
>
=
>
>
>
>
>
;

¼ T

g
: ð43Þ
In this case, the transformation matrix becomes:
Z
ðkÞ
¼ Q
ðkÞ
b
ðkÞ
T: ð44Þ
4. Application of the CMS to the frequency transfer matrix
The physical properties of each substructure SS
(k)
described by
the mass, damping and stiffness matrices are assumed to be uncer-
tain and then, M
(k)
, and K
(k)
are random matrices. In stochastic fi-
nite element method (SFEM) the matrices M
(k)
and K
(k)
can be
represented in the form [16]:
M
ðkÞ

¼
X
Q
1
q
1
¼0
M
ðkÞ
q
1
n
q
1
; ð45aÞ
K
ðkÞ
¼
X
Q
2
q
2
¼0
K
ðkÞ
q
2
n
q

2
: ð45bÞ
The transformation matrix Z
(k)
is assumed to be deterministic. The
condensed mass and stiffness matrices are given by:
M
c
¼
X
Q
1
q
1
¼0
M
c
q
1
n
q
1
; ð46aÞ
K
c
¼
X
Q
2
q

2
¼0
K
c
q
2
n
q
2
; ð46bÞ
where:
M
c
q
1
¼
X
n
s
k¼1
T
Z
ðkÞ
M
ðkÞ
q
1
Z
ðkÞ
; ð46cÞ

K
c
q
2
¼
X
n
s
k¼1
T
Z
ðkÞ
K
ðkÞ
q
2
Z
ðkÞ
: ð46dÞ
350 D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
To obtain the column vector H
j
, the external force is:
f
ðk
i
Þ
¼
0
.

.
.
0
1
0
.
.
.
0
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>

>
>
>
>
>
:
9
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>

>
>
;
j; ð47Þ
where j corresponds to a dof of the substructure SS
(ki)
. The con-
densed vector force f
c
is obtained from f
ðk
i
Þ
by:
f
c
¼
T
Z
ðk
i
Þ
f
ðk
i
Þ
ð48Þ
and the condensed displacement vector y
c
is expressed in determin-

istic modal basis as:
y
c
¼
X
P
p¼1
k
p
/
c
p
; ð49Þ
where k
p
are random coefficients which are expanded along a poly-
nomial chaos basis as giving in Eq. (17) and the considered con-
densed vector y
c
is expressed by:
y
c
¼
X
P
p¼1
X
N
n¼0
ðk

n
p
/
c
p
Þw
n
ðn
i
Þ: ð50Þ
The column vector H
j
of the transfer matrix H corresponding to the
substructure SS
(k)
is then given by:
y
ðkÞ
¼
X
P
p¼1
X
N
n¼0
k
n
p
ðZ
ðkÞ

/
c
p
Þw
n
ðn
i
Þ; ð51Þ
where Z
(k)
is the transformation matrix of the substructure SS
(k)
.
Note that to obtain the deterministic modal basis /
c
p
in the last
equation, four component mode synthesis methods are used here:
fixed interface (CB), free interface (FI), fixed interface with reduc-
tion of interface dof (CBR) and free interface with reduction of
interface dof (FIR).
5. Numerical results
In order to demonstrate the efficiency of this method, some
benchmark tests are analyzed with linear and nonlinear parame-
ters. For the sake of accuracy and comparison four methodological
approaches are used. The whole structure discretisation combined
with the MCS (WS + MCS) as well as with polynomial chaos
(WS + chaos) are elaborated. The results obtained by (WS + MCS)
are considered as reference results. The component mode synthe-
sis with fixed interface (CB) and free interface (FI) combined with

polynomial chaos (CB + chaos) and (FI + chaos), with and without
reduction of interface dof, are elaborated and considered to be
the main results of this paper.
5.1. Example 1: Frequency responses of beams
For this simple structure, two cases are studied. First, the ran-
dom parameters intervene linearly in the stiffness and mass matri-
ces of the structure. To this end, the mass density
q
and the Young
modulus E are assumed to be independent random variables. Sec-
ond, the beam’s radius r is assumed to be a random parameter
which intervenes non-linearly in the stiffness and mass matrices.
The frequency responses are computed based on the reduced
model obtained by CMS methods. The fixed interface method CB
(Craig Bampton) and the free interface method (FI) are used. The
pulsation range is [0,
x
u
= 2000 rd/s] and eleven eigenmodes are
considered in this study.
Let us consider the transverse vibration of an Euler beam discre-
tised by 100 simple FE. Each node has 2 dof in-plane rotation and a
transverse displacement. The beam is of length L and of circular
cross-section with radius r. In order to use the presented CMS
methods, the beam is assumed to be composed of two substruc-
tures SS
(1)
and SS
(2)
as presented in Fig. 1. The first substructure

consists of 60 finite elements and the second substructure consists
of 40 ones. The beam is assumed to be clamped at both ends and
the assembled structure has a total of 198 dof. The substructure
SS
(1)
has 120 dof in which 2 are the interface dof and the substruc-
ture SS
(2)
has 80 dof in which 2 are the interface dof. Let E,
q
,
g
and
l denote element Young modulus, mass density, hysteretic damp-
ing coefficient and length. The element stiffness and mass matrices
are defined by:
M ¼
m
420
156 22:l 54 À13:l
22:l 4:l
2
13:l À3:l
2
54 13:l 156 À22:l
À13:l À3:l
2
À22:l 4:l
2
2

6
6
6
4
3
7
7
7
5
; ð52aÞ
K ¼
E:I
l
3
12 6:l À12 6:l
6:l 4:l
2
À6:l 2:l
2
À12 À6:l 12 À6:l
6:l 2:l
2
À6:l 4:l
2
2
6
6
6
4
3

7
7
7
5
; ð52bÞ
where:
m ¼
q
:S:l ¼
q
:
p
r
2
4
:l; I ¼
p
r
4
4
: ð53Þ
For the CMS (CB and FI), the substructure modes whose pulsations
are smaller than a cut-out pulsation defined by
x
cp
=2.
x
u
are se-
lected. For (CB) method, the size of the reduced system is 17, 9 nor-

mal modes are retained for the substructure SS
(1)
, 6 modes for SS
(2)
and 2 interface dof. For (FI) method, 10 normal modes for the sub-
structures SS
(1)
, 7 modes for SS
(2)
, and 2 interface dof are retained.
The size of reduced system is thus 19.
5.1.1. Linear random effect
The mass density
q
and the Young modulus E are supposed
independent random variables and defined as follows:
q
¼
q
0
þ
r
q
n E ¼ E
0
þ
r
E
n;
where n is a zero mean value Gaussian random variable,

q
0
= 7800 kg/m
3
and E
0
=21Â 10
10
N/m
2
are the mean values and
r
q
and
r
E
are the associated standard deviations.
The coefficient of hysteretic damping is assumed to be deter-
ministic and given by
g
= 5%. For this linear random effect only
the first order polynomial chaos approximation is used.
The mean and standard deviation of the magnitude of localized
frequency response H(99, 99) have been investigated by the pro-
posed approaches for
r
E
=
r
q

= 10%. The results obtained by the di-
rect Monte Carlo 500 simulations (WS + MCS) are presented and
considered as reference results. The first order chaos expansion
combined with the fixed interface method (CB) and the free
interface method (FI) are used and the obtained results are well
SS
(1)
SS
(2)
x
y
O
Fig. 1. Example 1: Sub structured clamped beam.
D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
351
compared. Good accuracy is observed for the mean value and stan-
dard deviation of H(99, 99) as clearly shown in Fig. 2.
5.1.2. Nonlinear random effect
The second considered case is a random radius parameter given
by:
r ¼ r
0
þ
r
r
n
where n is a zero mean value Gaussian random variable, r
0
= 0.01 m
is the mean value and

r
r
is the standard deviation of this parameter.
In this nonlinear case, the first and second order polynomial chaos
expansions combined with the fixed interface method (CB) and the
free interface method (FI) are developed.
The mean and standard deviation of the magnitude of localized
frequency responses H(49, 99), H(99, 99) and H(99, 49) have been
calculated by the proposed approach. The obtained results are
compared with those given by the direct Monte Carlo Simulation
1000 simulations. The results are plotted in Fig. 3-4 for
r
r
=2%
and in Fig. 5-6 for
r
r
= 5%. These figures show that the obtained
solutions oscillate around the MCS reference solution. It can be
seen that for small variance range the proposed method, expanded
solutions in first and second order polynomial, provides a very
good accuracy as compared with the direct MCS. When the vari-
ance increases the error increases. This error decreases by increas-
ing the polynomial chaos order. The proposed method with the
whole system and the CMS methods requires much smaller CPU
time than the direct MCS. This is due to the fact that in direct
MCS method, matrix inversions for each pulsation
x
require a large
amount of CPU time.

5.2. Example 2: frequency responses of assembled plates
In order to use the CMS methods of reduction interface dof, let
us consider the structure of an assembly of plane plates as pre-
sented in Fig. 7. The finite element model of the complete structure
is generated with thin shell elements Q4 (quadrilateral element
with six dof per node). The used discretization leads to 3120 active
dof. The structure is divided into three substructures (see Fig. 7).
Fig. 2.1. Mean value of transfer function H(99, 99), the mass density and the Young
modulus are independent random variables.
Fig. 2.2. Standard deviation of transfer function H(99, 99), the mass density and the
Young modulus are independent random variables.
Fig. 3.1. Mean value of transfer function H(49, 99) where the radius is a random
variable,
r
r
= 2%.
Fig. 3.2. Standard deviation of transfer function H(49, 99) where the radius is a
random variable,
r
r
= 2%.
352 D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
Each substructure is a plane plate defined by two junctions lines
with adjacent plates and the substructures SS
(1)
and SS
(2)
have
1320 dof in which 180 are the interface ones. The substructure
SS

(3)
has 726 dof in which 126 are the interface ones.
The following data is considered:
Plate 1: dimensions 1 m à 2 m, thickness 0.02 m
Plate 2: dimensions 1 m à 2 m, thickness 0.02 m
Plate 3: dimensions 1 m à 2 m, thickness 0.05 m
For the three plates, the mass density
q
, and the Young modulus
E are independent random variables, while the deterministic
damping coefficient
g
= 5% is considered:
The frequency responses are investigated based on the reduced
model obtained by CMS methods, fixed interface method CB, free
interface method (FI), fixed interface with reduction of interface
dof (CBR) and free interface with reduction of interface dof (FIR).
The considered pulsation range is fixed between
x
= 0 and
x
u
= 1200 rd/s. The equilibrium equation of the whole structure
is projected on the first 16 eigenmodes.
For the CMS (CB and FI), all the substructure modes whose pul-
sations are smaller than a cut-out pulsation defined by
x
cp
=2.
x

u
are selected. For the (CB) method, the size of the reduced system
is 254 in which we retain respectively six normal modes for sub-
structures SS
(1)
and SS
(2)
, two modes for SS
(3)
and 240 interface
dof. For the (FI) method we retain respectively 15 normal modes
for the substructures SS
(1)
and SS
(2)
, nine modes for SS
(3)
, six rigid
body for SS
(3)
and 240 interface dof, the size of the reduced system
is thus 285. For the CMS with reduction of interface dof, the choice
of the substructure normal modes is the same as in the classical
CMS methods. The interface modes are selected by using similar
criterion with a cut-out pulsation defined by
x
cp
=4.
x
u

, thus we
retain 12 interface modes. The size of the reduced system (total
number of substructure modes and interface modes) varies from
26 for the (CBR) method to 57 for the (FIR) method.
In order to validate the assumption that the transformation ma-
trix Z
(k)
for each substructure can be defined assuming that the
model is deterministic, the first two moments (mean and variance)
of the frequency responses are computed numerically within the
framework of Monte Carlo Simulations from the reduced model
Fig. 4.1. Mean value of transfer function H(99, 99), where the radius is a random
variable,
r
r
= 2%.
Fig. 4.2. Standard deviation of transfer function H(99, 99), where the radius is a
random variable,
r
r
= 2%.
Fig. 5.1. Mean value of transfer function H(99, 99), where the radius is a random
variable,
r
r
= 5%.
Fig. 5.2. Standard deviation of transfer function H(99, 99), where the radius is a
random variable,
r
r

= 5%.
D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
353
Fig. 6.2. Standard deviation of transfer function H(99, 49), where the radius is a
random variable,
r
r
= 5%.
SS
(2)
SS
(3)
SS
(1)
Fig. 7. Example 2: Assembly of plane plates.
Fig. 8.1. The mean value of the transfer function H(243, 3116) where the mass
density and the Young modulus are independent random variables.
Fig. 8.2. The standard deviation of the transfer function H(243, 3116) where the
mass density and the Young modulus are independent random variables.
Fig. 9.1. The mean value of the transfer function H(3116, 3116) where the mass
density and the Young modulus are independent random variables.
Fig. 6.1. Mean value of transfer function H(99, 49), where the radius is a random
variable,
r
r
= 5%.
354 D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
obtained by CMS methods. The results are compared with those
obtained using the whole structure (WS).
Based on the direct Monte Carlo Simulation (MCS) 500 samples,

the obtained mean and standard deviation of the localized fre-
quency responses H(243, 3116) and H(3116, 3116) magnitude are
plotted in Figs. 8 and 9, for
r
q
=
r
E
= 10%. These figures show that
the condensed model obtained by CMS methods yields a good rep-
resentation of the dynamic behavior of the coupled structure with-
in the pulsation range [0–1200 rd/s]. The CPU time is given in Table
1. Compared to the reference case (WS), the gains obtained with
the CMS methods are impressive. The reduced time is 6.4% for
Fig. 9.2. The standard deviation of the transfer function H(3116, 3116) where the
mass density and the Young modulus are independent random variables.
Table 1
CPU time (s): Monte Carlo Simulation 500 samples.
WS CB CB FI FIR
19822 1850 1269 2077 1565
Fig. 10.1. The mean value of the dof 3116 amplitude where the mass density and
the Young modulus are independent random variables.
Fig. 10.2. The standard deviation of the dof 3116 amplitude where the mass
density and the Young modulus are independent random variables.
Fig. 11.1. The mean value of the dof 243 amplitude, where the mass density and
the Young modulus are independent random variables.
Fig. 11.2. The standard deviation of the dof 243 amplitude, where the mass density
and the Young modulus are independent random variables.
D. Sarsri et al. /Computers and Structures 89 (2011) 346–356
355

(CBR), 7.7% for (FIR), 9.33% for (CB), from 9.54 and 10.47% for (FI)
method.
In order to assess the efficiency of the method based on the
polynomial chaos expansions applied to compute the frequency
transfer functions using the CMS methods, we assume that the
assembled plane plates are subjected to three harmonic excitations
f
1
= f
2
= f
3
= 400 (sin
x
t + cos
x
t). f
1
is applied within u
z
at the node
41 (SS
(1)
), f
2
within u
x
at the node 420 (SS
(2)
) and f

3
within u
y
of
node 520 (SS
(3)
). The mean value and the standard deviation of
the displacement amplitude at the dof 3116 and 243 have been ob-
tained by the proposed approach, when the solution is expanded in
the second order polynomial chaos using the whole model and the
condensed model. As the equation giving the amplitude is nonlin-
ear, a Monte Carlo Simulation 500 samples of random variable vec-
tor (n
q
, n
E
) are used to compute this value. The obtained results are
compared with those obtained by direct Monte Carlo Simulation
500 samples using the whole structure (WS, MCS). These results
are plotted in Figs. 10 and 11 and a good agreement between these
results is clearly observed.
The CPU time, needed for the different proposed approaches, is
presented in Table 2. It is clearly observed that the proposed meth-
ods using the whole structure and the condensed approaches lead
to impressive CPU time reductions.
6. Conclusion
A methodological approach based on the use of component
mode synthesis methods and of the polynomial chaos basis is
developed and used to investigate the frequency transfer functions
for large linear FE models of beams and assembled plates with lin-

ear and nonlinear stochastic parameters. The random frequency
transfer function is expanded along a polynomial chaos basis in or-
der to compute the statistical first two moments (mean and vari-
ance). When the random parameters intervene linearly in the
model, the first-order polynomial chaos is sufficient to obtain a
good accuracy. However, when the parameter intervenes non-
linearly, the accuracy in the results could be improved significantly
by increasing the polynomial order. The obtained results based on
the proposed approaches are in good agreement with those ob-
tained by the Monte Carlo Simulation. The presented approaches
are efficient, accurate and fast computational ones for the fre-
quency transfer functions within the MCS for large structural sys-
tems with linear and nonlinear random parameters. The proposed
methodological approach proves to be of particular advantage and
can be improved by higher polynomial chaos for strong nonlinear
stochastic parameters.
References
[1] Hurty WC. Dynamic analysis of structural systems using component modes.
AIAA J 1965;3(4):678–85.
[2] Craig RR, Bampton MCC. Coupling of substructures for dynamics analysis. AIAA
J 1968;6(7):1313–9.
[3] Rubin S. Improved component mode representation for structural dynamic
analysis. AIAA J 1975;13(8):995–1006.
[4] Bouhaddi N, Lombard JP. Improved free interface substructures representation
method. Comput Struct 2000:269–83.
[5] MacNeal RH. A hybrid method of component mode synthesis. Comput Struct
1971;1:581–601.
[6] Benflied WA, Hruda RF. Vibration analysis of structures by component mode
substitution. AIAA J 1971;9(7):1255–61.
[7] Scheidt JV, Purkert W. Random eigenvalue problems. New York: North-

Holland; 1983.
[8] Benaroya H. Random eigenvalues, algebraic methods and structural dynamic
models. Appl Math Comput 1992;52:37–66.
[9] Adhikari S, Friswell MI. Random matrix eigenvalue problems in structural
dynamics. Int J Numer Methods Eng 2007;69(3):562–91.
[10] Schuëller GI. Computational stochastic mechanics – recent advances. Comput
Struct 2001:2225–34.
[11] Stefanou G. The stochastic finite element method: past, present and future.
Comput Methods Appl Mech Eng 2009;198:1031–51.
[12] Shinozuka M. Monte Carlo solution of structural dynamics. Int J Comput Struct
1972;2:855–74.
[13] Kleiber M, Hien TD. The stochastic finite element method. Ed. John Wiley;
1992.
[14] Singh R, Lee C. Frequency response of linear systems with parameters
uncertainties. J Sound Vib 1993;168(1):71–92.
[15] Adhikari S, Manohar CS. Dynamic analysis of framed structures with statistical
uncertainties. Int J Numer Methods Eng 1999;44:1157–78.
[16] Li R, Ghanem R. Adaptive polynomial chaos expansions applied to statistics of
extremes in nonlinear random vibration. Probablist Eng Mech 1998:125–36.
[17] Guedri M, Bouhaddi N, Majed R. Reduction of the stochastic finite element
models using a robust dynamic condensation method. J Sound Vib
2006;297:123–45.
[18] Saad Y, van der Vorst HA. Iterative solution of linear systems in the 20th
century. J Comput Appl Math 2000;123(1-2):1–33.
[19] El Hami A, Radi B. Some decomposition methods in the analysis of repetitive
structures. Comput Struct 1996;58(5):973–80.
[20] Bourquin F. Component mode synthesis and eigenvalues of second-order
operators, discretization and algorithm. Math Model Number Anal
1992;26(3):385–423.
[21] Tran DM. Component mode synthesis methods using interface modes.

application to structures with cyclic symmetry. Comput Struct 2001:209–22.
[22] Guyan RJ. Reduction of stiffness and mass matrices. AIAA J 1965;3(2).
Table 2
CPU time(s): polynomial chaos expansion at order 2.
WS Monte
Carlo
WS chaos
order2
CB chaos
order2
CBR chaos
order2
FI chaos
order2
FIR chaos
order2
19787 59.95 10.93 8.75 10.65 8.90
356 D. Sarsri et al. /Computers and Structures 89 (2011) 346–356