On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties

Pivovarov D, Willner K, Steinmann P (2019)

Publication Type: Journal article

Publication year: 2019


Book Volume: 350

Pages Range: 432-461

DOI: 10.1016/j.cma.2019.02.024


In this work we review a unified formulation for spectral fuzzy, spectral stochastic, and spectral fuzzy–stochastic FEM. We propose some modifications for the fuzzy and fuzzy–stochastic FEM involving local bases in the non-deterministic dimensions and the incorporation of extended FEM into the non-deterministic FEM. These modifications were previously used for stochastic FEM only. We discuss advantages of the proposed techniques for problems with uncertainties in the geometry and demonstrate their application to computational homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. We address also some theoretical aspects of fuzzy FEM, that have not been tackled in the literature, namely the inner product of fuzzy variables. We address also another important problem, which becomes often the target of criticism of fuzzy approach. The membership function of the fuzzy input has often some degree of arbitrariness: it is known only approximately or it is constructed based on some assumptions. Here we propose a fuzzy FEM representation, which requires only the modal value and support of the fuzzy input parameters and is completely independent of the membership function's shape. We also discuss the case of imprecise probabilities which allows for severe dimension reduction of the non-deterministic problem if the accurate spectral simulation technique is used. Application of dimension reduction is also demonstrated on the example of computational homogenization.

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Pivovarov, D., Willner, K., & Steinmann, P. (2019). On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties. Computer Methods in Applied Mechanics and Engineering, 350, 432-461. https://doi.org/10.1016/j.cma.2019.02.024


Pivovarov, Dmytro, Kai Willner, and Paul Steinmann. "On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties." Computer Methods in Applied Mechanics and Engineering 350 (2019): 432-461.

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