Robust and Reliability-Based Structural Topology Optimization Using a Continuous Adjoint Method

Abstract

A unified framework for robust and reliability-based structural topology optimization, considering structural model and loading uncertainties, is presented using a continuous adjoint formulation. The topology optimization is formulated for objective functions related to uncertainty measures of the compliance or displacements of the structure. Uncertainty measures, such as mean, standard deviation, and level exceedance probability or failure probability, involve the estimation of multidimensional integrals over the uncertain parameter space. These integrals are evaluated using (1) sparse grid quadrature techniques for the mean and standard deviation and (2) the approximate first-order reliability method (FORM) for failure probability. A unified continuous adjoint formulation is presented for the different objective function formulations. Simplifications are proposed that result in increased computational efficiency and accuracy for special cases of uncertainties and structural performance measures. The proposed formulation is demonstrated by computing the optimal distribution of material in a two-dimensional structure under loading and material uncertainties. For the approximate FORM-based topology optimization, computational difficulties are pointed out because of the existence of multiple design points in the uncertain parameter space.