Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms

Nouy, Anthony

doi:10.1016/j.cma.2008.06.012

Cited by 129 publications

(153 citation statements)

References 35 publications

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“…[66,139,29,95,43,22]. Recently, a generalization of the classical spectral decomposition (truncated K-L expansion) for the solution of the problem interpreted as an ''extended" eigenproblem has been proposed together with ad hoc iterative solution techniques inspired by classical techniques for solving the eigenproblem [122,123]. This method leads to further computational savings and reduction of memory requirements compared to Krylov-type techniques in the solution of linear problems.…”

Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

862

407

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a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

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Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

862

407

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“…Several strategies have been proposed for the extraction of reduced bases from approximate Karhunen-Loeve expansions (or classical spectral decompositions) of the solution [113,103]. Another method, called Generalized Spectral Decomposition method, has been recently proposed for the construction of such representations without knowing a priori the solution nor an approximation of it [69,114,70]. A major advantage of these algorithms is that they allow a separation of deterministic problems for the computation of deterministic functions and stochastic algebraic equations for the computation of stochastic functions.…”

Section: Propagation Of Uncertainties or What Are The Methods To Solvmentioning

confidence: 99%

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

Soize

2013

Journal of Sound and Vibration

139

106

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To cite this version:Christian Soize. Stochastic modeling of uncertainties in computational structural dynamics -Recent theoretical advances. Journal of Sound and Vibration, Elsevier, 2013, 332 (10) AbstractThis paper deals with a short overview on stochastic modeling of uncertainties. We introduce the types of uncertainties, the variability of real systems, the types of probabilistic approaches, the representations for the stochastic models of uncertainties, the construction of the stochastic models using the maximum entropy principle, the propagation of uncertainties, the methods to solve the stochastic dynamical equations, the identification of the prior and the posterior stochastic models, the robust updating of the computational models and the robust design with uncertain computational models. We present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties, and for their identification with experimental data. We also present the construction of the posterior stochastic model of uncertainties using the Bayesian method when experimental data are available.

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“…Detailed discussion on the efficient solution of the linear system in Equation (39) is beyond the scope of the current work, and hence, the reader is referred to a review of the important literature in this domain [38,39].…”

Section: Solution Methodologymentioning

confidence: 99%

Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty

Kundu

Adhikari

Friswell

2014

Numerical Meth Engineering

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SUMMARYThe problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method. A randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered. The discretized parametric fields are interpolated within each element with multidimensional Lagrange polynomials and integrated into the weak formulation. The proposed discretized random-field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables. The description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain. A method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral Galerkin framework is studied in detail. The treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems. The convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme.

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Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms

Cited by 129 publications

References 35 publications

The stochastic finite element method: Past, present and future

The stochastic finite element method: Past, present and future

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty

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