Robust Topology Optimization: A New Algorithm for Volume-constrained Expected Compliance Minimization with Probabilistic Loading Directions using Exact Analytical Objective and Gradient

Abstract

This paper presents a new algorithm for volume-constrainedexpected compliance minimization of continuum structureswith probabilistic loading directions using analytically determinedexact objective and gradient functions. The algorithmis based upon the finding that for a particular set of statisticalparameters the integration in the expected compliance functioncan be done symbolically and automatically using symbolicmanipulation software. In this study, Mathematica wasused to integrate and simplify the highly nonlinear expectedcompliance function. It will be demonstrated by examples thatthe result of the symbolic pre-processing step is a simple linearfunction defined on a particular subset of the inverse stiffnessentries which is needed in the compliance computation. Thecoefficient set of this linear function forms the base of the exactanalytical gradient computation used in the optimal solutionsearching optimality criteria (OC) method to define the steepestdescent direction. Naturally, the applied OC method canbe replaced by any other appropriate nonlinear solver. Matlabcodes of the algorithm for 2D and 3D structures with exactanalytical sensitivities have been developed using the topologyoptimization codes presented by Andreassen et al. [1] for2D and Liu and Tovar [2] for 3D structures as starting points.Illustrative examples with Mathematica and Matlab codes arepresented to demonstrate the essence and viability of the proposedapproach and highlight the potential of the automaticsymbolic computation in structure optimization.