Three Versions of the Finite Element Method Based on Concepts of Either Stochasticity, Fuzziness, or Anti-Optimization

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Abstract
In this review article, we describe in some detail three versions of the FEM pertinent to the uncertainty triangle, introduced by the present writer; in particular, to describe various types of uncertainties in the structure, namely a) non-perturbational finite element method for stochastic structures, b) finite element method associated with determination of least favorable responses through the method of anti-optimization in interval setting of parameters, and c) finite element method based on the fuzzy sets. The comparison between the probabilistic and convex modeling of uncertainty is conducted. The combined optimization and anti-optimization due to presence uncertainty is advocated. The author trusts that hybrid approaches, rather than a) genuine probabilistic treatment, b) anti-optimization, or c) fuzzy-sets-based modeling are called for to rigorously treat the realistic structures. In the end, the issue of relative robustness of these approaches is discussed. This review should be viewed as a continuation of our previous essays about non-probabilistic modeling of uncertainty, random vibrations, and introduction of initial imperfection-sensitivity concept into design, respectively (Comput Struct, 1995; Appl Mech Rev, 1995; NASA SP, 1998; Comput Meth in Appl Mech Eng, with Ren YJ, 1998), devoted to various aspects of uncertainty modeling in engineering. As its predecessors, it contains personal perspectives on the subject. It is envisioned, that at a later stage, these reviews, combined, will be published as a book, Thoughts on Uncertainty. This review article contains 126 references.

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