Evaluation and Maximization of Robustness of Trusses by using Semidefinite Programming

3. Introduction In structural and mechanical design, deterministic design optimization models have been successfully developed. Recently, the robust structural design has received increasing attention. Based on stochastic uncertainty models of mechanical parameters, various techniques were proposed for evaluation and estimation of failure probabilities [7, 11], that can be utilized in the reliability-based structural design methods. Besides stochastic uncertainty models, non-probabilistic uncertainty models have also been developed, where so-called unknownbut-bounded uncertain parameters are included in a system. Ben-Haim and Elishakoff [2] developed the so-called convex model, with which Pantelides and Ganzerli [14] proposed a robust truss optimization method. For various classes of convex optimization problems, a unified methodology of robust optimization, or robust counterpart scheme, was developed by Ben-Tal and Nemirovski [4], where the data in optimization problems are assumed to be unknown but bounded. Calafiore and El Ghaoui [6] proposed a method for finding the ellipsoidal bounds of the solution set of uncertain linear equations by using the semidefinite program (SDP) [17]. This paper deals with trusses consisting of the members with uncertain stiffness and/or suffering the uncertain external forces. The uncertain parameters are assumed to be unknown-but-bounded and non-probabilistic. We evaluate the robustness performance of a truss by using the concept of robustness function [1], which expresses the greatest level of uncertainty at which any constraint on mechanical performance cannot be violated. As a first contribution of this paper, we propose a numerically tractable formulation for computing the robustness functions of trusses. It is difficult to compute the robustness functions in general, because we have to solve an optimization problem with infinitely many constraints. Instead of this intractable problem, we propose a quasiconvex optimization problem [5, Section 4.2.5] that is regarded as an extension of the SDP problem. Secondly, we attempt to find a set of cross-sectional areas maximizing the robustness function, which is referred to as the maximization problem of robustness function. A sequential SDP method is proposed for solving the maximization problem of robustness function, in which we successively solve some SDP problems. At each iteration, an SDP problem can be solved easily by using the primal-dual interior-point method [3, 17]. Numerical experiments are presented for trusses under various uncertainty circumstances by using the proposed sequential SDP methods.