This paper presents a general approach for the probabilistic constraint evaluation in the reliability-based design optimization (RBDO). Different perspectives of the general approach are consistent in prescribing the probabilistic constraint, where the conventional reliability index approach (RIA) and the proposed performance measure approach (PMA) are identified as two special cases. PMA is shown to be inherently robust and more efficient in evaluating inactive probabilistic constraints, while RIA is more efficient for violated probabilistic constraints. Moreover, RBDO often yields a higher rate of convergence by using PMA, while RIA yields singularity in some cases.

In this paper, a continuum-based shape design sensitivity formulation for a frictional contact problem with a rigid body is proposed using a meshless method. The contact condition is imposed using the penalty method that regularizes the solution of variational inequality. The shape dependency of the contact variational form with respect to the design velocity ®eld is obtained. The dependency of the response with respect to the shape of the rigid body is also considered. It is shown that the sensitivity equation needs to be solved at the ®nal converged load step for the frictionless contact problem, whereas for the frictional contact case the sensitivity solution is needed at the converged con®guration of each load step because the sensitivity of the current load step depends on that of the previous load step. The continuumbased contact formulation and consistent linearization is critical for accurate shape design sensitivity results. The accuracy of the proposed method is compared with the ®nite difference result and excellent agreement is obtained for a door seal contact example. A design optimization problem is formulated and solved to reduce the contact gap opening successfully in a demonstration of the proposed method. IntroductionThe contact problem in linear elasticity can be categorized as a free boundary value problem. In general, the region where contact occurs is unknown until the solution is obtained. Stampacchia and Lions [1, 2], pioneers in this ®eld, formulated the free boundary value problem as a variational inequality (VI). The VI in linear elasticity can be considered as the projection of a solution in Hilbert space into a convex subset. Since the projection is a type of constraint, VI can be reformulated as a constrained minimization problem. This problem can then be solved by either the Lagrange multiplier method or the penalty method. Hughes et al. [3], proposed a ®nite element analysis (FEA) method for the contact problem, with a small deformation assumption. Kikuchi and Oden [4] studied contact VI theoretically from an engineering standpoint and formulated it as a constrained minimization problem.As the structure experiences large deformation, the small deformation assumption is no longer valid. New algorithms for general large deformation contact problems were proposed in several papers. For example, Wriggers and Simo [5] suggested an algorithm for fully nonlinear contact problems by introducing a contact tangent stiffness matrix. Parisch [6] proposed a three-dimensional contact algorithm and used a linear master segment for discretization of contact surface. Saleeb et al.[7] developed a contact algorithm for arbitrary curved geometry. Laursen and Simo [8] proposed a continuum-based contact algorithm. The contact condition is imposed on both the continuum and discrete domains. The continuumbased formulation is critical for the continuum-based shape design sensitivity analysis (DSA) method proposed in this paper.Since the mechanism of frictional phenomena is quite complicated, a represent...

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