Efficient uncertainty quantification in stochastic finite element analysis based on functional principal components

Abstract

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

Applying functional principal components to structural topology optimization

“…The methodological content of this section is based on the FPCA reduction approach developed in the work of Bianchini et al; the interested reader can refer to such a reference for further details. However, as mentioned, the objective of the aforementioned work was to investigate the variability of a certain output quantity of interest given that some mechanical parameters are random. Here, on the contrary, the optimization problem does not include any randomness/variability, since we are looking for the (deterministic) configuration of parameters that leads to the optimal output.…”

“…All of the works in which the decomposition exploits the covariance matrix are based on the standard PCA, where data are treated as multivariate vectors, thus merely considering the displacements of the nodes of the mesh. On the contrary, the FPCA has been only adopted in the work of Bianchini et al, to build the reduced basis in the context of stochastic FEA. The approach in the aforementioned work relies on the idea of evaluating the distance between the complete FEA solution and the solution obtained in the reduced space to assess whether to keep or to discard a reduced solution (as in the work of Florentin and Diéz); however, a data‐driven basis is built according to the FPCA.…”

“…To conclude, PCA has been already adopted in the literature to address optimization problems (an overview of several optimization problems related to PCA under various geometric perspectives can be found in the work of Reris and Brooks). However, to the best of our knowledge, the functional counterpart (FPCA) that we are here proposing has never been coupled with FEA in the context of structural optimization, and only once in the context of stochastic FEA …”

“…However, to the best of our knowledge, the functional counterpart (FPCA) that we are here proposing has never been coupled with FEA in the context of structural optimization, and only once in the context of stochastic FEA. 10…”

Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

“…An adaptative RB strategy, based on local error estimator, was introduced in , and a sensibility analysis on a quantity of interest (QoI) was performed. Recently, a different solution has been presented in to obtain a reliable and uncostly solution in the same framework.…”