SUMMARYRobust optimization problems are newly formulated and an e cient computational scheme is proposed. Both design variables and design parameters are considered as random variables about their nominal values. To ensure the robustness of objective performance, we introduce a new performance index bounding the performance together with a constraint limiting the performance variation. The constraint variations are regulated by considering the probability of feasibility. Each probability constraint is transformed into a sub-optimization problem by the advanced ÿrst-order second moment (AFOSM) method for computational e ciency. The proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second-order sensitivity information is not required even in case of gradient-based optimization process. The suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature.

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An equivalence between the enhanced assumed strain (EAS) method based on the Hu-Washizu principle, recently proposed by Simo and Rifai, and assumed stress hybrid (hybrid) method based on the Hellinger-Reissner principle is investigated. It is proved that not only the displacements but also the stresses of the EAS-elements calculated from the strains are identical to those of the corresponding hybrid-elements at least at the Gauss integration points provided the spaces of the trial functions for enhanced assumed strains and for assumed stresses satisfy the orthogonality and the inclusion or the invertibility condition. By virtue of this equivalence, a stress recovery procedure of the EAS-elements is devised. This procedure is variationally consistent and more efficient than those proposed by Simo and Rifai and Andelfinger and Ramm. Since the classical method of incompatible displacement modes is a special case of the EAS-method, this procedure also can be used to evaluate variationally consistent stresses for the non-conforming elements.

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SUMMARYFirst‐order reliability method (FORM) has been mostly utilized for solving reliability‐based design optimization (RBDO) problems efficiently. However, second‐order reliability method (SORM) is required in order to estimate a probability of failure accurately in highly nonlinear performance functions. Despite accuracy of SORM, its application to RBDO is quite challenging due to unaffordable numerical burden incurred by a Hessian calculation. For reducing the numerical efforts, a quasi‐Newton approach to approximate the Hessian is introduced in this study instead of calculating the true Hessian. The proposed SORM with the approximated Hessian requires computations only used in FORM, leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes a generalized chi‐squared distribution in order to achieve better accuracy. Furthermore, SORM‐based inverse reliability method is proposed in this study. An accurate reliability index corresponding to a target probability of failure is updated using the proposed SORM. Two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed. The proposed SORM‐based inverse analysis is then extended to RBDO in order to obtain a reliability‐based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system. The numerical study results show that the proposed reliability analysis and RBDO achieve efficiency of FORM and accuracy of SORM at the same time. Copyright © 2014 John Wiley & Sons, Ltd.