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Generalized Wishart Distribution for Probabilistic Structural Dynamics

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Generalized Wishart Distribution for Probabilistic Structural Dynamics Comput Mech (2010) 45:495–511 DOI 10.1007/s00466-010-0467-3 ORIGINAL PAPER Generalized Wishart distribution for probabilistic structural dynamics Sondipon Adhikari Received: 9 April 2009 / Accepted: 3 January 2010 / Published online: 23 January 2010 © Springer-Verlag 2010 Abstract An accurate and efficient uncertainty quantifi- Keywords Unified uncertainty quantification · Random cation of the dynamic response of complex structural sys- matrix theory · Wishart distribution · Model validation · tems is crucial for their design and analysis. Among the Parameter identification many approaches proposed, the random matrix approach has received significant attention over the past decade. In this List of symbols paper two new random matrix models, namely (1) gener- f(t) Forcing vector alized scalar Wishart distribution and (2) generalized diag- M, C and K Mass, damping and stiffness matrices, onal Wishart distribution have been proposed. The central respectively aims behind the proposition of the new models are to (1) φ Undamped eigenvectors improve the accuracy of the statistical predictions, (2) sim- j q(t) Response vector plify the analytical formulations and (3) improve computa- δ Dispersion parameter of G tional efficiency. Identification of the parameters of the newly G n Number of degrees of freedom proposed random matrix models has been discussed. Closed- (•)T Matrix transposition form expressions have been derived using rigorous analyti- R Space of real numbers cal approaches. It is considered that the dynamical system is R+ Space n × n real positive definite matrices proportionally damped and the mass and stiffness properties n Rn×m Space n × m real matrices of the system are random. The newly proposed approaches |•| Determinant of a matrix are compared with the existing Wishart random matrix model etr {•} exp {Trace (•)} using numerical case studies. Results from the random matrix • Frobenius norm of a matrix, approaches have been validated using an experiment on a F • = (•)(•)T 1/2 vibrating plate with randomly attached spring-mass oscilla- F Trace ∼ tors. One hundred nominally identical samples have been Distributed as (•) created and separately tested within a laboratory framework. Trace Sum of the diagonal elements of a matrix Relative merits and demerits of different random matrix for- pdf Probably density function mulations are discussed and based on the numerical and experimental studies the recommendation for the best model has been given. A simple step-by-step method for imple- 1 Introduction menting the new computational approach in conjunction with general purpose finite element software has been outlined. Finite element codes implementing physics based models are used extensively for the dynamic analysis of complex sys- tems. Laboratory based controlled tests are often performed B to gain insight into some specific physics of a problem. Such S. Adhikari ( ) tests can indeed lead to new physical laws improving the School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK computational models. Test data can also be used to calibrate e-mail: [email protected] a known model. However, neither of these activities may 123 496 Comput Mech (2010) 45:495–511 be enough to produce a credible numerical tool because of a central Wishart distribution for the system matrices. In a several types of uncertainties which exist in the physics based recent paper [7] it was shown that Wishart matrix with prop- computational framework. Such uncertainties include, but erly selected parameters can be used for systems with both are not limited to (1) parameter uncertainty (e.g, uncertainty parametric uncertainty and nonparametric uncertainty. The in geometric parameters, friction coefficient, strength of the main conclusions arising from this study were: materials involved); (2) model uncertainty (arising from the lack of scientific knowledge about the model which is a priori – Wishart random matrix distribution can be obtained either unknown); (3) experimental error (uncertain and unknown using the maximum entropy approach or using the matrix errors percolate into the model when they are calibrated factorization approach. against experimental results). These uncertainties must be – The maximum entropy approach gives a natural selection assessed and managed for credible computational predic- for the parameters. Through numerical examples it was tions. shown that the parameters obtained using the maximum The predictions from high resolution numerical models entropy approach may yield non-physical results. In that may sometimes exhibit significant differences with the the ‘mean of the inverse’ and the ‘inverse of the mean’ of results from physical experiments due to uncertainty. When the random matrices can be significantly different. substantial statistical information exists, the theory of prob- – The parameters of the pdf of a Wishart random matrix ability and stochastic processes offer a rich mathematical obtained using the maximum entropy approach are not framework to represent such uncertainties. In a probabilis- unique since they depend on what constraints are used in tic setting, the data (parameter) uncertainty associated with the optimization approach. the system parameters, such as the geometric properties and – Considering that the available ‘data’ is the mean and (nor- constitutive relations (i.e. Young’s modulus, mass density, malized) standard deviation of a system matrix, it was Poisson’s ratio, damping coefficients), can be modeled as shown that when the mean of the inverse equals to the random variables or stochastic processes using the so-called inverse of the mean of the system matrices, the predictive parametric approach. These uncertainties can be quantified accuracy is maximum. and propagated, for example, using the stochastic finite element method [8,9,15,17,20,23,25,26,28,34,40–42,49]. Recently, the uncertainty due to modelling error has received The aim of this paper is to further explore the idea of proper attention as this is crucial for model validation [21,22,35]. parameter selection of Wishart matrices. Specifically, we ask The model uncertainty problem poses serious challenges as the question ‘what is the simplest possible Wishart random the parameters contributing to the modelling errors are not matrix model which can be used without compromising the available a priori and therefore precludes the application of predictive accuracy?’. The definition of ‘simplest’ and ‘accu- a parametric approach to address such issues. Model uncer- racy’ will be explained later in the paper. The outline of the tainties do not explicitly depend on the system parameters. paper is as follows. In Sect. 2 dynamic response of linear For example, there can be unquantified errors associated with stochastic systems is discussed. A brief overview of random the equation of motion (linear or non-linear), in the damping matrix models in probabilistic structural dynamics is given model (viscous or non-viscous [3,11]), in the model of struc- in Sect. 3. The identification method of the two new random tural joints. The model uncertainty may be tackled by the so- matrix model is proposed in Sect. 4. In Sect. 5 the accu- called non-parametric method pioneered by Soize [44–46] racy and efficiency the proposed random matrix models are and adopted by others [5,6,31,32]. numerically verified. An experiment conducted on a vibrat- The equation of motion of a damped n-degree-of-freedom ing plate with randomly attached sprung-mass oscillators is linear dynamic system can be expressed as used to validate the proposed approach in Sect. 6. Based on the numerical and experimental studies, a step-by-step ¨( ) + ˙( ) + ( ) = ( ) Mq t Cq t Kq t f t (1) approach is outlined in Sect. 7 to apply the proposed uncer- where f(t) ∈ Rn is the forcing vector, q(t) ∈ Rn is the tainty quantification approach. response vector and M ∈ Rn×n, C ∈ Rn×n and K ∈ Rn×n are the mass, damping and stiffness matrices respectively. In order to completely quantify the uncertainties associated 2 Dynamic response of discrete stochastic systems with system (1) we need to obtain the probability density functions of the random matrices M, C and K.Usingthe Modal analysis, originally due to Lord rayleigh [39], is the parametric approach, such as the stochastic finite element most widely used approach for linear structural dynamics. method, one usually obtains a problem specific covariance Using this approach the dynamic response of a system can structure for the elements of system matrices. The nonpara- be obtained by linear superposition of vibration modes. The metric approach [5,6,44–46] on the other hand results in extension of classical modal analysis to stochastic systems 123 Comput Mech (2010) 45:495–511 497 requires the calculation of natural frequencies and mode- Since is a symmetric and positive definite matrix, it can shapes which are random in nature. The efficiency and be diagonalized by a orthogonal matrix r such that accuracy of a computational method therefore crucially T = 2 dependent on the efficiency and accuracy of the calcula- r r r (9) tion of random natural frequencies and mode-shapes. In this Here the subscript r denotes the random nature of the eigen- section first we transform the coupled equations of motion values and eigenvectors of the random matrix . Recalling in the modal modal domain and then introduce uncertainty T = that r r In, from Eq. (6) we obtain directly in natural frequencies and mode-shapes using ran- −1 dom matrix theory. The parameter estimation and sampling 2 2 ¯ q¯ = s In + sC + f (10) of this random matrix distribution are discussed in details in −1 later sections. = 2 + ζ + 2 T ¯ r s In 2s r r r f (11) Assuming all the initial conditions are zero and taking the Laplace transform of the equation of motion (1)wehave where s2M + sC + K q¯(s) = f¯(s) (2) ζ = diag [ζ1,ζ2,...,ζn] (12) The response in the original coordinate can be obtained as where (•¯) denotes the Laplace transform of the respective − quantities. The aim here is to obtain the statistical properties 1 q¯(s) = q¯ (s)= s2I + 2sζ +2 ( )T f¯(s) of q¯(s) ∈ Cn when the system matrices are random matrices.
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