Augmented Single Loop Single Vector Algorithm Using Nonlinear Approximations of Constraints in Reliability-Based Design Optimization

Abstract

Reliability-based design optimization (RBDO) aims at minimizing a function of probabilistic design variables, given a maximum allowed probability of failure. The most efficient methods available for solving moderately nonlinear problems are single loop single vector (SLSV) algorithms that use a first-order approximation of the probability of failure in order to rewrite the inherently nested structure of the loop into a more efficient single loop algorithm. The research presented in this paper takes off from the fundamental idea of this algorithm. An augmented SLSV algorithm is proposed that increases the rate of convergence by making nonlinear approximations of the constraints. The nonlinear approximations are constructed in the following way: first, the SLSV experiments are performed. The gradient of the performance function is known, as well as an estimate of the most probable failure point (MPP). Then, one extra experiment, a probe point, per performance function is conducted at the first estimate of the MPP. The gradient of each performance function is not updated but the probe point facilitates the use of a natural cubic spline as an approximation of an augmented MPP estimate. The SLSV algorithm using probing (SLSVP) also incorporates a simple and effective move limit (ML) strategy that also minimizes the heuristics needed for initiating the optimization algorithm. The size of the forward finite difference design of experiment (DOE) is scaled proportionally with the change of the ML and so is the relative position of the MPP estimate at the current iteration. Benchmark comparisons against results taken from the literature show that the SLSVP algorithm is more efficient than other established RBDO algorithms and converge in situations where the SLSV algorithm fails.