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Efficient uncertainty quantification in stochastic finite element analysis based on functional principal components. (English) Zbl 1326.65158

Summary: The great influence of uncertainties on the behavior of physical systems has always drawn attention to the importance of a stochastic approach to engineering problems. Accordingly, in this paper, we address the problem of solving a finite element analysis in the presence of uncertain parameters. We consider an approach in which several solutions of the problem are obtained in correspondence of parameters samples, and propose a novel non-intrusive method, which exploits the functional principal component analysis, to get acceptable computational efforts. Indeed, the proposed approach allows constructing an optimal basis of the solutions space and projecting the full finite element problem into a smaller space spanned by this basis. Even if solving the problem in this reduced space is computationally convenient, very good approximations are obtained by upper bounding the error between the full finite element solution and the reduced one. Finally, we assess the applicability of the proposed approach through different test cases, obtaining satisfactory results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
62H25 Factor analysis and principal components; correspondence analysis
74S05 Finite element methods applied to problems in solid mechanics

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References:

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