Directional methods for structural reliability analysis

Abstract

Directional simulation reduces the dimension of the limit state probability integral by identifying a set of directions for integration, integrating either in closed-form or by approximation in those directions, and estimating the probability as a weighted average of the directional integrals. Most existing methods identify these directions by a set of points distributed on the unit hypersphere. The accuracy of the directional simulation depends on how the points are identified. When the limit state is highly nonlinear, or the inherent failure probability is small, a very large number of points may be required, and the method can become inefficient. This paper introduces several new approaches for identifying directions for evaluating the probability integral — Spherical t-design, Spiral Points, and Fekete Points — and compares the failure probabilities with those determined in a number of examples in previously published work. Once these points have been identified for a probability integral of given dimension, they can be used repeatedly for other probability integrals of the same dimension in a fashion analogous to Gauss Quadrature.