Momentum-based Accelerated Mirror Descent Stochastic Approximation for Robust Topology Optimization under Stochastic Loads

TL;DR

This paper introduces an accelerated stochastic optimization method for robust topology design under uncertain loads, significantly reducing computational costs while maintaining high-quality solutions.

Contribution

It develops a momentum-based accelerated mirror descent stochastic approximation approach that efficiently handles stochastic gradients with minimal sample size for robust topology optimization.

Findings

Reduces sample size to two for unbiased gradient estimation.

Demonstrates effectiveness in 2D and 3D topology optimization examples.

Accelerates convergence and stabilizes the optimization process.

Abstract

Robust topology optimization (RTO) improves the robustness of designs with respect to random sources in real-world structures, yet an accurate sensitivity analysis requires the solution of many systems of equations at each optimization step, leading to a high computational cost. To open up the full potential of RTO under a variety of random sources, this paper presents a momentum-based accelerated mirror descent stochastic approximation (AC-MDSA) approach to efficiently solve RTO problems involving various types of load uncertainties. The proposed framework can perform high-quality design updates with highly noisy stochastic gradients. We reduce the sample size to two (minimum for unbiased variance estimation) and show only two samples are sufficient for evaluating stochastic gradients to obtain robust designs, thus drastically reducing the computational cost. We derive the AC-MDSA update formula based on $\ell_1$-norm with entropy function, which is tailored to the geometry of the feasible domain. To accelerate and stabilize the algorithm, we integrate a momentum-based acceleration scheme, which also alleviates the step size sensitivity. Several 2D and 3D examples with various sizes are presented to demonstrate the effectiveness and efficiency of the proposed AC-MDSA framework to handle RTO involving various types of loading uncertainties.