Towards error bounds of the failure probability of elastic structures using reduced basis models

Abstract

International audienceStructural reliability methods aim at computing the probability of failure of systems with respect to prescribed limit state functions. A common practice to evaluate these limit state functions is using Monte Carlo simulations. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. Surrogate models, which are built from a limited number of runs of the original model, have been developed, as substitute of the original model, to reduce the computational cost. However, these surrogate models, while decreasing drastically the computational cost, may fail in computing an accurate failure probability. In this paper, we focus on the control of the error introduced by a reduced basis surrogate model on the computation of the failure probability obtained by a Monte Carlo simulation. We propose a technique to determine bounds of this failure probability, as well as a strategy of enrichment of the reduced basis, based on limiting the bounds of the error of the failure probability for a multi-material elastic structur

Adaptive reduced basis strategy for rare‐event simulations

“…To construct the stress RB, we follow the method presented by Gallimard et al The first step consists in building a stress σ neu ( θ ), which verifies the FE equilibrium (see Equation ) for all θ in (we refer the reader to the work of Gallimard et al for more details about the construction of σ neu ( θ )). Let us consider the set of stress fields computed from the snapshot solutions (see Equation ) and the set of stress fields defined by It follows that is a set of stress fields equilibrated to zero in the FE sense.…”

“…Following the work of Ladevèze et al, we consider the following auxiliary problem: Find such that and its solution in the RB It can be shown that (we refer the reader to previous studies for more details). where …”

“… where each is a fourth‐order tensor defined on Ω. These hypotheses are required by the adaptive RB method, as shown in Section 4.2.…”

“…To evaluate a failure probability P f =10 − n with a relative deviation δ =10 % , the MC method necessitates 10 n +2 FOM computations. To reduce the computational effort, it is interesting to introduce ROMs based on the RB method . As shown in Section 4, this method involves the construction of a basis (the so‐called RB).…”

“…where eachS q (x) is a fourth-order tensor defined on Ω. These hypotheses are required by the adaptive RB method, 21 as shown in Section 4.2.…”