Momentum‐based accelerated mirror descent stochastic approximation for robust topology optimization under stochastic loads

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 article 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 performs high‐quality design updates with highly noisy and biased stochastic gradients. The sample size is reduced to two (minimum for unbiased variance estimation) and is shown to be sufficient for evaluating stochastic gradients to obtain robust designs, thus drastically reducing the computational cost. The AC‐MDSA update formula based on entropic ℓ1‐norm is derived, which mimics the feasible space geometry. A momentum‐based acceleration scheme is integrated to accelerate the convergence, stabilize the design evolution, and alleviate step size sensitivity. Several 2D and 3D examples are presented to demonstrate the effectiveness and efficiency of the proposed AC‐MDSA to handle RTO involving various loading uncertainties. Comparison with other methods shows that the proposed AC‐MDSA is superior in computational cost, stability, and convergence speed.

An Eigenfrequency-Constrained Topology Optimization Method with Design Variable Reduction

“…Joo and Jang [31] proposed a deep neural network topology optimization algorithm, which can improve the convergence speed by obtaining the history of intermediate designs. Li and Zhang [32] used high noise and unbiased random gradients to update design variables and expedite the convergence process. Du et al [33] shared a set of efficient topology optimization Matlab codes, which resulted in faster convergence speeds by removing the freedom not belonging to the transmission path in the finite element analysis.…”

Quasi-static frictional contact analysis free from solutions of linear equations: an approach based on primal-dual algorithm

“…Accelerated first-order methods for convex optimization have received substantial interests, especially as efficient methods for large-scale problems [1]. Efficiency of these methods has recently been validated also for several problems in applied mechanics, including elastoplastic analysis [2][3][4] and bi-modulus elasticity [5], as well as continuum-based topology optimization [6][7][8]. Unlike second-order optimization methods, e.g., interiorpoint methods, these accelerated first-order methods do not involve solution of linear equations system.…”

Accelerated projected gradient method with adaptive step size for compliance minimization problem

“…In contrast, very few applications to topology optimization are found in the literature. For example, Li and Zhang [8] applied the accelerated mirror descent method to a robust topology optimization problem. They reduce the computational cost of the robust topology optimization problem by the stochastic gradient method.…”