An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis

• Published in: Probabilistic engineering mechanics Vol. 25; no. 2; pp. 183 - 197
• Main Authors: Blatman, Géraud, Sudret, Bruno
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.04.2010; Elsevier
• Subjects: Adaptive stochastic finite elements; Algorithms; Applied sciences; Building structure; Buildings. Public works; Chaos theory; Computational efficiency; Construction (buildings and works); Exact sciences and technology; Experimental design; Finite element method; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematical models; Mathematics; Personal computers; Physics; Probability and statistics; Probability theory and stochastic processes; Regression; Response surfaces; Sciences and techniques of general use; Sequential experimental design; Solid mechanics; Sparse polynomial chaos expansion; Statistics; Stochastic processes; Stochasticity; Structural and continuum mechanics; Structural reliability; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Regression; Structural reliability; Adaptive stochastic finite elements; Sparse polynomial chaos expansion; Sequential experimental design; Response surfaces; Chaos; Adaptive algorithm; Cross validation; Random vector; Stochastic process; Modeling; Adaptive method; Finite element method; Sparse matrix; Probability learning; Random variable; Statistical analysis; Probabilistic approach; Sequential design; Frame construction; Well posed problem; Orthogonal polynomial; Program verification; Stochastic analysis; Truss structure; Experimental design; Deterministic model; Galerkin method
• DOI: 10.1016/j.probengmech.2009.10.003
• ISSN: 0266-8920

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e.of Galerkin type) or non-intrusive) unaffordable when the deterministic finite element model is expensive to evaluate. To address such problems, this paper describes a non-intrusive method that builds a sparse PC expansion. An adaptive regression-based algorithm is proposed for automatically detecting the significant coefficients of the PC expansion. Besides the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to ensure the well-posedness of the various regression problems. The accuracy of the PC model is checked using classical tools of statistical learning theory (e.g.  leave-one-out cross-validation). As a consequence, a rather small number of PC terms is eventually retained ( sparse representation), which may be obtained at a reduced computational cost compared to the classical “full” PC approximation. The convergence of the algorithm is shown on an academic example. Then the method is illustrated on two stochastic finite element problems, namely a truss and a frame structure involving 10 and 21 input random variables, respectively.