Error bounds for the reliability index in finite element reliability analysis

Abstract

SUMMARYThis work presents an extension of the goal-oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first-order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index . The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented.

Error control in FORM reliability analysis

“…As shown in Gallimard (2011a), the precision of the finite element analyses has a great influence on the computed failure probability's error. The control of the quantity e FORM;h can be achieved by using the techniques developped for goal-oriented mesh adaptivity (Diez & Calderon, 2007) applied to the control on the error on the loading effect SðuðyÞÞ.…”

“…As far as we know, only the first point has been addressed in the literature (see Mitteau, 1999). In Gallimard (2011aGallimard ( , 2011b we have shown that the discretisation error may have an important impact on the computed failure probability. In this paper, we also combine both errors in order to find a finite element discretisation adapted to the problem.…”

“…Recently, numerous works have been published which provide error estimates and bounds for several quantities of interest (Gallimard, 2006;Gallimard & Panetier, 2006;Ladevèze, Rougeot, Blanchard, & Moreau, 1999;Peraire & Patera, 1998;Prudhomme & Oden, 1999). However, these works have been developed for deterministic FEM and few researches have been carried out on the verification of probabilistic models (Deb, Babuška, & Oden, 2001;Gallimard, 2011a;Ladevèze & Florentin, 2006).…”

Adaptive reduced basis strategy for rare‐event simulations

“…A first approach is the first-order reliability method and the second-order reliability method, 1,2 which consist in building a simple analytical approximation of the limit state function around the so-called design point followed by a direct estimation of the failure probability. [3][4][5][6][7][8] A second approach consists in building a surface response as a surrogate model of the limit state function (quadratic response surfaces, polynomial chaos expansions, Kriging surrogates, etc). [9][10][11][12][13] The MC algorithm can be then applied on this surrogate model.…”

Post optimization for accurate and efficient reliability‐based design optimization using second‐order reliability method based on importance sampling and its stochastic sensitivity analysis

“…In reliability analysis, such uncertainties are usually expressed as random variables, and a reliability or probability of failure is computed in order to quantify safety of the engineering system under influence of the uncertainties. It is quite difficult to estimate the probability of failure defined as a multi-dimensional integration over a nonlinear domain in a real engineering problem especially including finite element analysis, so reliability methods based on function approximation are commonly used such as first-order reliability method (FORM) [1][2][3][4], second-order reliability method (SORM) [5][6][7][8][9][10][11], dimension reduction method [12][13][14][15][16], response surface method [17][18][19], and polynomial chaos expansion [20]. FORM and SORM approximate the performance function at the most probable point (MPP), which has the highest probability density on a limit-state surface and can be obtained by searching the minimum distance from the origin to the limit-state surface in the standard normal space (U-space).…”