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Articles Published in Computer Methods in Applied Mechanics and Engineering

  1. Xiu, D. and Karniadakis, G.E.. "Modeling uncertainty in steady state diffusion problems via generalized chaos" Computer Methods in Applied Mechanics and Engineering. 191 (43). 2002. pp. 4927--4948.

    We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [Amer. J. Math. 60 (1938) 897]. A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations. (C) 2002 Elsevier Science B.V. All rights reserved.


  2. Ghanem, R.G.. "Ingredients for a general purpose stochastic finite element formulation" Computer Methods in Applied Mechanics and Engineering. vol. 168. 1999. pp. 19--34.


  3. Ghanem, R.G., Doostan, A., and Red-Horse, J.R. "A probabilistic construction of model validation" Computer Methods in Applied Mechanics and Engineering. 197 (29-32). 2008. pp. 2585--2595.

    We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise. (C) 2007 Elsevier B.V. All rights reserved.


  4. Ghanem, R.G.. "Probabilistic characterization of transport in heterogeneous media" Computer Methods in Applied Mechanics and Engineering. 158 (3-4). JUN 8 1998. pp. 199--220.

    The mechanics of transport and flow in a random porous medium are addressed in this paper. The hydraulic properties of the porous medium are modeled as spatial random processes. The random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented. Thus, the hydraulic processes are discretized using the spectral Karhunen - Loeve expansion. This expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes. These amplitudes form a basis in the manifold associated with the random processes. The concentrations over the whole domain are also random processes, with unknown probabilistic structure. These processes are represented using the Polynomial Chaos basis. This is a basis in the functional space described by all second order random variables. The deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator. Once the spatio-temporal variation of the concentrations has been specified in terms of the Polynomial Chaos expansion, individual realizations can be readily computed. (C) 1998 Elsevier Science S.A.


  5. Matthies, H.G. and Keese, A.. "Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations" Computer Methods in Applied Mechanics and Engineering. 194 (12-16). 2005. pp. 1295--1331.

    Stationary systems modelled by elliptic partial differential equations-linear as well as nonlinear-with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations-in particular with respect to the stochastic part-are given and investigated with regard to stability. Different and increasingly sophisticated computational approaches involving both Wiener's polynomial chaos as well as the Karhunen-Loeve expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed. The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration. (C) 2004 Elsevier B.V. All rights reserved.


  6. Sachdeva, S.K., Nair, P.B., and Keane, A.J.. "Comparative study of projection schemes for stochastic finite element analysis" Computer Methods in Applied Mechanics and Engineering. 195 (19-22). 2006. pp. 2371--2392.

    We present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency. More specifically, we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic Krylov subspace. Numerical studies are presented for two problems: (1) static analysis of a plate with random Young's modulus and (2) settlement of a foundation supported on a randomly heterogeneous soil. Monte Carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared. We show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach, particularly for large-scale problems with many random variables. For the class of problems considered, we find that stochastic reduced basis methods can be up to orders of magnitude faster, while providing results of comparable or better accuracy.


  7. Doostan, A., Ghanem, R.G., and Red-Horse, J.R. "Stochastic model reduction for chaos representations" Computer Methods in Applied Mechanics and Engineering. 196 (37-40). 2007. pp. 3951--3966.

    This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galerkin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) in the Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms at hand, the solution of SPDE's becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided for the adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics.


  8. Wan, X. and Karniadakis, G.E.. "Long-term behavior of polynomial chaos in stochastic flow simulations" Computer Methods in Applied Mechanics and Engineering. 195 (41-43). 2006. pp. 5582--5596.

    In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend these results to other random distributions and high-dimensional random inputs with numerical verification using the algebraic convergence rate of ME-gPC. Finally, we apply our results to noisy flow past a stationary circular cylinder. Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior. (c) 2005 Elsevier B.V. All rights reserved.


  9. Jardak, M. and Ghanem, R.G.. "Spectral stochastic homogenization of divergence-type PDEs" Computer Methods in Applied Mechanics and Engineering. 193 (6--8). 2004. pp. 429-447.

    This paper presents a formulation and numerical analysis of the homogenization of stochastic PDEs. The framework of homogenization is adopted to describe an effective medium that is equivalent in some sense to a heterogeneous medium of interest. The parameters of the resulting homogeneous medium are described as stochastic processes characterized by their polynomial chaos decomposition. The formulation yields a chaos decomposition for the predicted behavior of the homogeneous medium that captures, in addition to the effect of heterogeneity, the effect of variability. Once this description has been computed, various statistics of the solution can be efficiently evaluated. (C) 2003 Elsevier B.V. All rights reserved.


  10. Ghanem, R.G., Masri, S., Pellissetti, M., and Wolfe, R.. "Identification and prediction of stochastic dynamical systems in a polynomial chaos basis" Computer Methods in Applied Mechanics and Engineering. 194 (12-16). 2005. pp. 1641--1654.

    Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust. (C) 2004 Elsevier B.V. All rights reserved.


  11. Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation" Computer Methods in Applied Mechanics and Engineering. vol. 194. 2005. pp. 1251--1294.