  This paper addresses the computational challenges in reliability-based
topology optimization (RBTO) of structures associated with the estimation of
statistics of the objective and constraints using standard sampling methods,
and overcomes the accuracy issues of traditional methods that rely on
approximating the limit state function. Herein, we present a stochastic
gradient-based approach, where we estimate the probability of failure at every
few iterations using an efficient sampling strategy. To estimate the gradients
of the failure probability with respect to the design parameters, we apply
Bayes' rule wherein we assume a parametric exponential model for the
probability density function of the design parameters conditioned on the
failure. The design parameters as well as the parameters of this probability
density function are updated using a stochastic gradient descent approach
requiring only a small, e.g., O(1), number of random samples per iteration,
thus leading to a considerable reduction of the computational cost as compared
to standard RBTO techniques. We illustrate the proposed approach with a
benchmark example that has an analytical solution as well as two widely used
problems in structural topology optimization. These examples illustrate the
efficacy of the approach in producing reliable designs.
