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 vectors (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 parameters 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 random 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 extended 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@uae.ma, azrarlahcen@yahoo.fr (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 reference 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 solution 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 response 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 frequency 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, orthogonality and weak convergence. The model parameters are random, give arise to stochastic static and dynamic Ritz vectors for each substructure. The only assumption involved in the proposed approach, is that these vectors are defined assuming that the model is deterministic. 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 system with mass, damping and stiffness matrices M, C, and K respectively. 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 uncertain. Then, M, C and K are random matrices. The issue of representing 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 properties are then represented by the random processes. These properties 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 represented 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 independent 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 polynomial 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 matrix 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 obtained 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 magnitude 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 followed 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 solving 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 procedure 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 proceeding 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 number of interface dof. Automatic procedure is developed based on this defined transformation matrix allowing a straightforward computation of the needed condensed random matrices. Based on the given forms of the condensed matrices, the stochastic 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 computational efficiency by reducing the number of degrees of freedom 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 substructure 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 displacements 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 displacements 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 substructure 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 substructure 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 substructure 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 obtained 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 corresponding 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 retained 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 displacement 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 following 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 external 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 system, 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 structure to the interface. The displacements of the interface dof are expressed as: y j ¼ ul j : ð42Þ For the assembled structure, the vector of independent displacement 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 uncertain and then, M (k) , and K (k) are random matrices. In stochastic finite 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 condensed 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 deterministic 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 polynomial chaos basis as giving in Eq. (17) and the considered condensed 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 reduction 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 parameters. 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 synthesis 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 random parameters intervene linearly in the stiffness and mass matrices 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 substructures 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 substructure 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 damping 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 selected. For (CB) method, the size of the reduced system is 17, 9 normal 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 substructures 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 deterministic 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 proposed approaches for r E = r q = 10%. The results obtained by the direct 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 standard 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 variance increases the error increases. This error decreases by increasing 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 presented 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 Ã 2 m, thickness 0.02 m Plate 2: dimensions 1 m Ã 2 m, thickness 0.02 m Plate 3: dimensions 1 m Ã 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 pulsations 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 substructures 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 matrix 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 frequency 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 representation of the dynamic behavior of the coupled structure within 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 [...]... 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 linear and nonlinear stochastic parameters The random frequency transfer function is expanded along a polynomial chaos basis in order to compute the statistical first two moments (mean and variance) When the random parameters intervene linearly... computational ones for the frequency transfer functions within the MCS for large structural systems 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 19 65;3(4):678–85... / Computers and Structures 89 (2 011 ) 346–356 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 19 787 59.95 10 .93 8.75 10 .65 8.90 (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... Frequency response of linear systems with parameters uncertainties J Sound Vib 19 93 ;16 8 (1) : 71 92 [15 ] Adhikari S, Manohar CS Dynamic analysis of framed structures with statistical uncertainties Int J Numer Methods Eng 19 99;44 :11 57–78 [16 ] Li R, Ghanem R Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration Probablist Eng Mech 19 98 :12 5–36 [17 ] Guedri M, Bouhaddi... Bampton MCC Coupling of substructures for dynamics analysis AIAA J 19 68;6(7) :13 13–9 [3] Rubin S Improved component mode representation for structural dynamic analysis AIAA J 19 75 ;13 (8):995 10 06 [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 19 71; 1:5 81 6 01 [6] Benflied WA,... Computational stochastic mechanics – recent advances Comput Struct 20 01: 2225–34 [11 ] Stefanou G The stochastic finite element method: past, present and future Comput Methods Appl Mech Eng 2009 ;19 8 :10 31 51 [12 ] Shinozuka M Monte Carlo solution of structural dynamics Int J Comput Struct 19 72;2:855–74 [13 ] Kleiber M, Hien TD The stochastic finite element method Ed John Wiley; 19 92 [14 ] Singh R, Lee C Frequency. .. analysis of structures by component mode substitution AIAA J 19 71; 9(7) :12 55– 61 [7] Scheidt JV, Purkert W Random eigenvalue problems New York: NorthHolland; 19 83 [8] Benaroya H Random eigenvalues, algebraic methods and structural dynamic models Appl Math Comput 19 92;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 ]... Reduction of the stochastic finite element models using a robust dynamic condensation method J Sound Vib 2006;297 :12 3–45 [18 ] Saad Y, van der Vorst HA Iterative solution of linear systems in the 20th century J Comput Appl Math 2000 ;12 3 (1- 2) :1 33 [19 ] El Hami A, Radi B Some decomposition methods in the analysis of repetitive structures Comput Struct 19 96;58(5):973–80 [20] Bourquin F Component mode synthesis and. .. compute the frequency transfer functions using the CMS methods, we assume that the assembled plane plates are subjected to three harmonic excitations f1 = f2 = f3 = 400 (sinxt + cosxt) f1 is applied within uz at the node 41 (SS (1) ), f2 within ux at the node 420 (SS(2)) and f3 within uy of node 520 (SS(3)) The mean value and the standard deviation of the displacement amplitude at the dof 311 6 and 243 have... 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 methods 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 . Component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models D. Sarsri a , L. Azrar b, ⇑ , A is developed and used to investigate the frequency transfer functions for large linear FE models of beams and assembled plates with linear and nonlinear stochastic parameters. The random frequency transfer. substructure SS (1) , 6 modes for SS (2) and 2 interface dof. For (FI) method, 10 normal modes for the substructures SS (1) , 7 modes for SS (2) , and 2 interface dof are retained. The size of reduced

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