PGD-TO: A Scalable Alternative to MMA Using Projected Gradient Descent for Multi-Constraint Topology Optimization

TL;DR

This work tackles the scalability bottlenecks of projected gradient descent in topology optimization under nonlinear and multi-constraint settings. It proposes PGD-TO, which replaces the projection step with a regularized convex quadratic program and solves it via a semismooth Newton method for general constraints or binary-search projection for independent constraints, augmented by spectral step-size and nonlinear conjugate-gradient directions. The approach achieves convergence and comparable objective values to MMA and OC across four benchmark TO problems, while delivering 10–43x faster iterations in general and 115–312x when constraints are independent, enabling large-scale, multi-constraint TO. The method is supported by theory (sublinear convergence under L-smoothness and robust regularization) and open-source GPU/CPU-accelerated code, underscoring its practical impact for scalable TO design.

Abstract

Projected Gradient Descent (PGD) methods offer a simple and scalable approach to topology optimization (TO), yet they often struggle with nonlinear and multi-constraint problems due to the complexity of active-set detection. This paper introduces PGD-TO, a framework that reformulates the projection step into a regularized convex quadratic problem, eliminating the need for active-set search and ensuring well-posedness even when constraints are infeasible. The framework employs a semismooth Newton solver for general multi-constraint cases and a binary search projection for single or independent constraints, achieving fast and reliable convergence. It further integrates spectral step-size adaptation and nonlinear conjugate-gradient directions for improved stability and efficiency. We evaluate PGD-TO on four benchmark families representing the breadth of TO problems: (i) minimum compliance with a linear volume constraint, (ii) minimum volume under a nonlinear compliance constraint, (iii) multi-material minimum compliance with four independent volume constraints, and (iv) minimum compliance with coupled volume and center-of-mass constraints. Across these single- and multi-constraint, linear and nonlinear cases, PGD-TO achieves convergence and final compliance comparable to the Method of Moving Asymptotes (MMA) and Optimality Criteria (OC), while reducing per-iteration computation time by 10-43x on general problems and 115-312x when constraints are independent. Overall, PGD-TO establishes a fast, robust, and scalable alternative to MMA, advancing topology optimization toward practical large-scale, multi-constraint, and nonlinear design problems. Public code available at: https://github.com/ahnobari/pyFANTOM

Figures (76)

• Figure 1: Overview of the four TO benchmark problems used to evaluate convergence and robustness across solvers. Each problem is solved at three mesh resolutions, with detailed results provided in Appendix \ref{['app:results']}.
• Figure 2: Scaling of iteration time with problem size and number of constraints. PGD achieves roughly an order-of-magnitude improvement in iteration time across all constraint counts and scales almost log-linearly with the number of elements. Notably, PGD with four constraints matches the iteration time of MMA with a single constraint, while MMA’s cost rises sharply as the number of constraints increases from two to four—highlighting the superior scalability of the proposed approach.
• Figure 3: Convergence behavior at fine mesh resolution for each benchmark problem. Each column compares objective evolution (left) and relative design-variable change (right) for single- and multi-constraint problems with both linear and nonlinear constraints. Across all four scenarios, PGD, MMA, and OC exhibit nearly identical convergence trajectories, confirming that the proposed PGD framework is broadly applicable and numerically stable across diverse constraint types. PGD$^*$ denotes PGD using the Newton-based projection instead of binary search in cases with independent constraints; the default PGD employs the faster binary-search projection, and PGD$^*$ is shown for completeness.
• Figure 4: An example of linear approximation of constraint leading to step size relaxation in our algorithm, keeping the problem stable and convergence behavior robust-- as seen visually and in convergence plots-- without the need to adjust the global step size relaxation from the start.
• Figure 5: The designs each optimizer produces at five log-spaced iterations. Here, we visualize the solutions each optimizer produces for the volume-constrained minimum compliance problem on the cantilever beam problem with a volume fraction target of $0.2$.

Theorems & Definitions (6)

• Theorem 1: Solution to the Linearized Projection Problem
• proof
• Theorem 2: Solution to the Regularized Projection Problem
• proof
• Theorem 3: Convergence of Projected Gradient Descent with Linearized Constraints
• proof