Laurent Gallimard

Summary: Monte Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, these approaches do not lead to accurate results. Indeed, the number of samples is often insufficient to estimate such low probabilities (at least 10ⁿ⁺² samples are needed to estimate a probability of 10⁻ⁿ with 10% relative deviation of the Monte Carlo estimator). Even within the framework of reduced order methods, such as a reduced basis approach, it seems difficult to predict accurately low‐probability events. In this paper, we propose to combine a cross‐entropy method with a reduced basis algorithm to compute rare‐event (failure) probabilities.

Summary: Structural 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 structure. Copyright © 2017 John Wiley & Sons, Ltd.

An entropy approach has been recently developed, based on the irreversible thermodynamics and entropy production. It has been observed that, for low cycle fatigue, metals undergoing cyclic load reach fracture, at a certain level of entropy production called fracture fatigue entropy (FFE) (Naderi and Khonsari, 2010), solely dependent on the material. Until now, we only used the two principles of thermodynamics separately to follow the behaviour of a solid continuum problem as low cycle fatigue. We propose here to use the two laws of thermodynamics together via the Gouy-Stodola equation and the concept of exergy to provide a new description for metals undergoing low cycle fatigue.