The use of probabilistic optimization in structural design applications is hindered by the huge computational cost associated with evaluating probabilistic characteristics, where the computationally expensive finite element method (FEM) is often used for simulating design performance. In this paper, a Sequential Optimization and Reliability Assessment (SORA) method with analytical derivatives is applied to improve the efficiency of probabilistic structural optimization. With the SORA method, a single loop strategy that decouples the optimization and the reliability assessment is used to significantly reduce the computational demand of probabilistic optimization. Analytical sensitivities of displacement and stress functionals derived from finite element formulations are incorporated into the probability analysis without recurring excessive cost. The benefits of our proposed methods are demonstrated through two truss design problems by comparing the results with using conventional approaches. Results show that the SORA method with analytical derivatives is the most efficient with satisfactory accuracy. * KEY WORDSprobabilistic optimization, structural design, Sequential Optimization and Reliability Assessment, finite element method, analytical derivative

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In robust design, it is important not only to achieve robust design objectives but also to maintain the robustness of design feasibility under the effect of variations (or uncertainties). However, the evaluation of feasibility robustness is often a computationally intensive process. Simplified approaches in existing robust design applications may lead to either over-conservative or infeasible design solutions. In this paper, several feasibility-modeling techniques for robust optimization are examined. These methods are classified into two categories: methods that require probability and statistical analyses and methods that do not. Using illustrative examples, the effectiveness of each method is compared in terms of its efficiency and accuracy. Constructive recommendations are made to employ different techniques under different circumstances. Under the framework of probabilistic optimization, we propose to use a most probable point (MPP) based importance sampling method, a method rooted in the field of reliability analysis, for evaluating the feasibility robustness. The advantages of this approach are discussed. Though our discussions are centered on robust design, the principles presented are also applicable for general probabilistic optimization problems. The practical significance of this work also lies in the development of efficient feasibility evaluation methods that can support quality engineering practice, such as the Six Sigma approach that is being widely used in American industry. [S1050-0472(00)00904-1]

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Two types of uncertainty exist in engineering. Aleatory uncertainty comes from inherent variations while epistemic uncertainty derives from ignorance or incomplete information. The former is usually modeled by the probability theory and has been widely researched. The latter can be modeled by the probability theory or nonprobability theories and is much more difficult to deal with. In this work, the effects of both types of uncertainty are quantified with belief and plausibility measures (lower and upper probabilities) in the context of the evidence theory. Input parameters with aleatory uncertainty are modeled with probability distributions by the probability theory. Input parameters with epistemic uncertainty are modeled with basic probability assignments by the evidence theory. A computational method is developed to compute belief and plausibility measures for black-box performance functions. The proposed method involves the nested probabilistic analysis and interval analysis. To handle black-box functions, we employ the first order reliability method for probabilistic analysis and nonlinear optimization for interval analysis. Two example problems are presented to demonstrate the proposed method.