A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization

Abstract

The evaluation of probabilistic constraints in the probabilistic structural design optimization (PSDO) problem can be carried out using either the conventional reliability index approach (RIA) or the more recently proposed performance measure approach (PMA). The latter is sometimes regarded as more efficient and stable with less dependence on probabilistic distribution types of random variables. Herein we apply PMA to evaluate probabilistic constraints and solve the PSDO using the sequential approximate programming (SAP) strategy. The sequential linear programming (SLP) approach in structural optimization achieves optimum design by solving a sequence of sub-programming problems iteratively. Each sub-programming consists of a linear objective subjected to a set of linear constraints, all based on approximation of the original objective and constraints at the current design. Implementing this approach to PSDO in a straightforward manner will require linear approximation of probabilistic performance measure and its sensitivities, thus implying large number of iterations and huge computational cost. In our new approach, rather than using the linear expansion of the probabilistic performance measure, we propose a formulation for an approximate probabilistic performance measure and its linearization. Obtained based on optimality conditions in the vicinity of the minimum performance target point (MPTP), the approximate measure and its sensitivity enables efficient sub-programming step. We update MPTP simultaneously at each step using iterative formula from the advanced mean-value (AMV) method and apply it as the initial estimate for the next step. As the sub-programming steps are no longer the linear approximation of the original problem, this is essentially a sequential approximate programming (SAP) approach. Through the application of this method to the most relevant examples frequently cited in similar studies, we compare its efficiency to other existing approaches and illustrate the concurrent convergence of both optimization and probabilistic performance measure calculation.