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

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 Hi, j

Component mode synthesis and polynomial chaos expansions for stochastic

frequency functions of large linear FE models

D Sarsria, L Azrarb,⇑, A Jebbouric, A El Hamid

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

a r t i c l e i n f o

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

a b s t r a c t

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 vecvec-tors), where the number of vecvec-tors 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 soluapplica-tions 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.

⇑Corresponding author Tel.: +212 62 88 71 48; fax: +212 39 39 39 53.

E-mail addresses: L.azrar@uae.ma , azrarlahcen@yahoo.fr (L Azrar).

Contents lists available atScienceDirect Computers and Structures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c

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 €yðtÞ þ C _yðtÞ þ K yðtÞ ¼ fðtÞ; 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:

where D(x) is the dynamic stiffness matrix defined by:

DðxÞ ¼ K þ ixC x2M: ð3Þ

In this paper, a hysteretic damping of coefficientgis considered and the dynamic stiffness matrix is rewritten in the form:

DðxÞ ¼ ð1 þ igÞK x2M: ð4Þ

In this analysis the matrices K and M are constant and frequency independent The transfer matrix H is defined by:

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:

where Hjand fjare 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 replacrepresent-ing

a complicated random quantity by a collection of simpler random quantities that are easier to manage The random material 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

wc matrix of constrained mode

wr matrix of rigid body modes

wa matrix of attachment modes

G residual flexibility matrix

war residual attachment modes

M0, K0 average of mass and stiffness matrices dof degree of freedom

ni(i – 0) random variables

wn(ni) multidimensional orthogonal polynomials chaos hi inner product defined by the mathematical expectation

operator

hi mean value

r standard deviation

M ỬXQ1

q1Ử0

K ỬXQ2

q2Ử0

where n0= 1, the matrices M0and K0are the average matrices and

Mq1and Kq2are deterministic while nqi(qiỜ0) are Gaussian random

variables The dynamic stiffness matrix D can be similarly

repre-sented in the form:

Dđxỡ Ử đ1 ợ igỡXQ 2

q 2 Ử0

Kq 2nq2x2XQ 1

q 1 Ử0

Mq 1nq1: đ8ỡ 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 KarhunenỜLoeve expansion; the polynomial chaos

expansion is used The resulting vector Hjis expanded along a

poly-nomial chaos basis[15]:

HjỬXN

nỬ0

wherewn(ni) are multidimensional orthogonal polynomials in the

random variables nidescribing the material properties, defined by:

wnđni; ;npỡ Ử đ1ỡpexp 1

2

Tfngfng

 @p

1 fngfng

@ni; ; @np

: đ9bỡ (Hj)ndenotes 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

SubstitutingEqs (8) and (9)into Eq.(6)and forcing the residual

to be orthogonal to the space spanned by the polynomial chaoswn

yield the following system of linear equations:

XN

nỬ0

đ1 ợ igỡXQ 2

q 2 Ử0

nq 2wnwm

Kq2x2XQ 1

q 1 Ử0

nq 1wnwm

Mq1

đHjỡn

Ử wh mifj m Ử 0; 1; ; N;

đ10ỡ where hi denotes the inner product defined by the mathematical

expectation operator

This algebraic equation can be rewritten in a more compact

ma-trix form as:

Dđxỡđ00ỡ    Dđxỡđ0Nỡ

: Dđxỡđnmỡ :

DđxỡđN0ỡ    DđxỡđNNỡ

2

6

6

6

4

3 7 7 7 5

đHjỡ0 :

đHjỡm :

đHjỡN

8

>

>

<

>

>

:

9

>

>

=

>

>

;

Ử

fj 0

0

8

>

>

>

>

>

>

9

>

>

>

>

>

>

; đ11aỡ

where:

DđxỡđnmỡỬ đ1 ợ igỡXQ 2

q2Ử0

nq 2 wnwm

Kq2x2XQ 1

q1Ử0

nq 1 wnwm

Mq1: đ11bỡ The deterministic coefficients of (Hj)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 Hij are given by the following

relationships:

realđHijỡ

Ử realđơHij0ỡ; đ12aỡ imagđHijỡ

Ử imagđơHij0ỡ: đ12bỡ

rrealđHijỡỬ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

XN nỬ1 realđơHijnỡ2hw2i

v u

rimagđHijỡỬ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

XN nỬ1 imagđơHijnỡ2hw2i

v u

Note that the equation giving the frequency response function mag-nitude is nonlinear Monte Carlo Simulation of the vector of random variable {n1, , nq, } 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 ỬXP pỬ1

where kpare unknown random coefficients and /pare the vectors of deterministic modal basis kpare also expanded along a polynomial chaos basis:

kpỬXN nỬ0

Inserting Eq.(14)into Eq.(6)and using the M and K-orthogonality conditions, the following equation is obtained:

ơđ1 ợ igỡx2

j x2

kjợXP pỬ1

kp đ1 ợ igỡXQ 2

q 2 Ử1

nq2T/jKq 2/p

"

x2XQ1

q1Ử0

nq 1

T/jMq1/p

#

ỬT

Forcing equation(16)to be orthogonal to the approximating space spanned by the polynomial chaoswnthe following algebraic linear system is obtained:

ơđ1 ợ igỡx2

j x2

kmj w2m

ợ đ1 ợ igỡXP

pỬ1

XN nỬ0

XQ2

q2Ử1

nq 2 wnwm

kmp /jKq2/p

x2XP pỬ1

XN nỬ0

XQ 1

q1Ử1

nq1 wnwm

kmpT/jMq1/pỬT/jf: đ17ỡ

The solution of this system allows one to get the coefficients kn

pand therefore the random vector displacement y:

y ỬXP pỬ1

XN nỬ0

đknp/pỡwnđniỡ: đ18ỡ

f ¼

0

0

1

0

0

8

>

>

>

>

<

>

>

>

>

:

9

>

>

>

>

=

>

>

>

>

;

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 ns

susb-structures SS(k)(k = 1, , ns) which do not overlap For each

sub-structure k the displacement vector y(k) is partitioned into a

vector yðkÞ

j , called interface dof and yðkÞ

i which is the vector of inter-nal dof The force vector f(k)is composed into vectors fðkÞj and fðkÞe ,

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]:

yðkÞ¼ QðkÞ yðkÞj

lðkÞ

wherel(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:

The matrix Q is given by

in which Y is a matrix containing the first eigenmodes of the

undamped substructure SS with a fixed interface as boundary

condition.w 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 Ygþ wrnrþ wana: ð22Þ

Y is a matrix containing the first eigenmodes of the undamped sub-structure SS with a free interface as boundary condition.wris the matrix of rigid body modes for an unconstrained substructure with

a free interface.wais 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

Fj¼ Ij 0

 

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:

Else,

where: A = I u(r)Tu(r)M andTu(r)Mu(r)= I, I: unit matrix andu(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 Yj

Yi

 

wr¼ wrj

wri

 

wa¼ waj

wai

 

Using this partition inEq (23), one obtains:

na¼ w1ajyj w1ajYjgþ w1ajwrjnr: ð26Þ The matrix Q is then given by:

Q ¼ waw1

aj wr waw1

ajwrj Y  waw1

ajYj

The residual attachment modeswar, 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 Ygþ wrnrþ warnar; ð28Þ

waris the residual attachment modes obtained by:

and

whereKis the matrix of the retained eigenvalues The matrix Q can

be written as:

Q ¼ warw1arj wr warw1arjwrj Y  warw1arjYj

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 ns

substructures coupled at a common boundary:

- Displacement continuity:

y1

j ¼ y2j ¼    ¼ ynj ¼ yj: ð31aÞ

- Equilibrium of coupling forces:

Xn s

k¼1

The conservation of interface dof allows assembling these matrices

as in the classical finite element method The vector of independent

displacements of the assembled structuregis expressed by:

g¼

lð1Þ

lðn s Þ

yj

8

>

>

>

>

9

>

>

>

>

The compatibility of interface displacements of the assembled

structure is obtained by writing, for each substructure SS(k), the

fol-lowing relation:

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:

ZðkÞ

The kinetic energy T, the strain energy U and the work of the

exter-nal forcessare given by:

T ¼1

2

U ¼1

2

where:

Mc¼Xn s

k¼1

TZðkÞMðkÞZðkÞ; ð36aÞ

Kc¼Xn s

k¼1

TZðkÞKðkÞZðkÞ; ð36bÞ

fc¼Xn s

k¼1

TZðkÞ

ðfðkÞj þ fðkÞe Þ: ð36cÞ

Using the compatibility of displacements of interface dof, it can be

easily shown that:

Xn s

k¼1

Thus, the work of the applied forces becomes:

fc¼Xn s

k¼1

The reduced equation can be written in the form:

½ð1 þ igÞK x2Mg¼ f: ð39Þ

The physical displacements of each substructure are obtained by:

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 modesuare defined as the first eigenmodes of the reduced eigenproblem:

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:

For the assembled structure, the vector of independent displace-ment is rewritten as:

g¼

lð1Þ

lðn s Þ

yj

8

>

>

>

>

9

>

>

>

>

¼

Ið1Þ

Iðn s Þ

u

2 6 6 4

3 7 7 5

lð1Þ

lðn s Þ

lj

8

>

>

>

>

9

>

>

>

>

¼ Tg: ð43Þ

In this case, the transformation matrix becomes:

ZðkÞ

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]:

MðkÞ

¼XQ 1

q 1 ¼0

MðkÞ

KðkÞ¼XQ2

q2¼0

KðkÞ

The transformation matrix Z(k)is assumed to be deterministic The condensed mass and stiffness matrices are given by:

Mc¼XQ1

q1¼0

Mc

Kc¼XQ2

q2¼0

Kcq

where:

Mcq1¼Xn s

k¼1

TZðkÞMðkÞ

Kcq2¼Xn s

TZðkÞKðkÞq2ZðkÞ: ð46dÞ

To obtain the column vector Hj, the external force is:

fðki Þ

¼

0

0

1

0

0

8

>

>

>

>

<

>

>

>

>

:

9

>

>

>

>

=

>

>

>

>

;

where j corresponds to a dof of the substructure SS(ki) The

con-densed vector force fcis obtained from fðk i Þby:

fc¼TZðk i Þfðki Þ

ð48Þ and the condensed displacement vector ycis expressed in

determin-istic modal basis as:

yc¼XP

p¼1

where kpare random coefficients which are expanded along a

poly-nomial chaos basis as giving in Eq.(17)and the considered

con-densed vector ycis expressed by:

yc¼XP

p¼1

XN

n¼0

ðkn

The column vector Hjof the transfer matrix H corresponding to the

substructure SS(k)is then given by:

yðkÞ¼XP

p¼1

XN

n¼0

knpðZðkÞ/cÞwnðniÞ; ð51Þ

where Z(k)is the transformation matrix of the substructure SS(k)

Note that to obtain the deterministic modal basis /cin 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 reducreduc-tion 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 densityqand 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,xu= 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 inFig 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,gand

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 13:l 22:l 4:l2 13:l 3:l2

54 13:l 156 22:l

13:l 3:l2 22:l 4:l2

2 6 6

3 7

K ¼E:I

l3

12 6:l 12 6:l 6:l 4:l2 6:l 2:l2

12 6:l 12 6:l 6:l 2:l2 6:l 4:l2

2 6 6

3 7

where:

m ¼q:S:l ¼q:pr2

4 :l; I ¼

pr4

For the CMS (CB and FI), the substructure modes whose pulsations are smaller than a cut-out pulsation defined byxcp= 2.xuare 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:

q¼q0þrqn E ¼ E0þrEn;

where n is a zero mean value Gaussian random variable,

q0= 7800 kg/m3and E0= 21  1010N/m2are the mean values and

rqandrEare 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 forrE=rq= 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

x y O

compared Good accuracy is observed for the mean value and

stan-dard deviation of H(99, 99) as clearly shown inFig 2

5.1.2 Nonlinear random effect

The second considered case is a random radius parameter given

by:

r ¼ r0þrrn

where n is a zero mean value Gaussian random variable, r0= 0.01 m

is the mean value andrris 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 3-4forrr= 2%

and inFig 5-6forrr= 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 pulsationxrequire 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 inFig 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 (seeFig 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

Fig 3.1 Mean value of transfer function H(49, 99) where the radius is a random variable,rr = 2%.

Fig 3.2 Standard deviation of transfer function H(49, 99) where the radius is a

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  2 m, thickness 0.02 m

Plate 2: dimensions 1 m  2 m, thickness 0.02 m

Plate 3: dimensions 1 m  2 m, thickness 0.05 m

For the three plates, the mass densityq, and the Young modulus

E are independent random variables, while the deterministic

damping coefficientg= 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

xu= 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 byxcp= 2.xu

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 byxcp= 4.xu, 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,rr = 2%.

Fig 4.2 Standard deviation of transfer function H(99, 99), where the radius is a

Fig 5.1 Mean value of transfer function H(99, 99), where the radius is a random variable,rr = 5%.

Fig 5.2 Standard deviation of transfer function H(99, 99), where the radius is a

Fig 6.2 Standard deviation of transfer function H(99, 49), where the radius is a

random variable,rr = 5%.

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 Fig 6.1 Mean value of transfer function H(99, 49), where the radius is a random

variable,rr = 5%.

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 inFigs 8 and 9, forrq=rE= 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 with-inTable

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.

Fig 10.1 The mean value of the dof 3116 amplitude where the mass density and

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