Investigating the Interplay of Parameterization and Optimizer in Gradient-Free Topology Optimization: A Cantilever Beam Case Study

TL;DR

The paper addresses how parameterization of the design space and the choice of gradient-free optimizer affect topology optimization under expensive simulations. It systematically compares three parameterizations—Honeycomb Tilings, MMC, and Curved MMC—with three optimizers—DE, CMA-ES, and HEBO—across 10D, 20D, and 50D cantilever problems, incorporating a volume constraint and a connectivity constraint via a Minimum Spanning Tree formulation. The unconstrained objective is obtained by penalizing violations with $f_{obj}(\mathbf{x})$, and a budget of $20D$ evaluations is used to assess performance across 27 configurations and 15 seeds. The main finding is that parameterization quality dominates optimizer choice: well-crafted geometries yield robust performance across optimizers, while poor representations lead to strong optimizer dependence and degraded results, highlighting the central role of problem formulation in gradient-free TO. The work suggests focusing on geometry design as a primary driver of practical TO performance and points to extending the study with more optimizers and parameterizations across varied TO problems to validate generality.

Abstract

Gradient-free black-box optimization (BBO) is widely used in engineering design and provides a flexible framework for topology optimization (TO), enabling the discovery of high-performing structural designs without requiring gradient information from simulations. Yet, its success depends on two key choices: the geometric parameterization defining the search space and the optimizer exploring it. This study investigates this interplay through a compliance minimization problem for a cantilever beam subject to a connectivity constraint. We benchmark three geometric parameterizations, each combined with three representative BBO algorithms: differential evolution, covariance matrix adaptation evolution strategy, and heteroscedastic evolutionary Bayesian optimization, across 10D, 20D, and 50D design spaces. Results reveal that parameterization quality has a stronger influence on optimization performance than optimizer choice: a well-structured parameterization enables robust and competitive performance across algorithms, whereas weaker representations increase optimizer dependency. Overall, this study highlights the dominant role of geometric parameterization in practical BBO-based TO and shows that algorithm performance and selection cannot be fairly assessed without accounting for the induced design space.

Figures (6)

• Figure 1: Overview of the parameterizations considered in this work: (A) HT, (B) MMCs, and (C) Curved MMCs. Dark-blue regions indicate material directly generated by the parameterization, while light-blue regions show the mirrored material domain enforced by symmetry about the red midline. (D–F) illustrate examples of 20D parameterizations for the full cantilever: (D) HT, (E) MMCs, and (F) Curved MMCs.
• Figure 2: (A) A schematic overview of the studied problem, showing the design domain $D$ and application of load $\mathbf{F}$. (B) The optimal topology obtained by SIMP for the same mesh using andreassen2011efficient (adapted from bujny2020level).
• Figure 3: An example for calculating the least distance to connect four connected components of geometry to each other and to edge- and load-point boundary geometry.
• Figure 4: The HTs for dimensions: (A) 10D, (B) 20D, and (C) 50D respectively, showing the deletion of tiles: one for 20D and two for 50D.
• Figure 5: Mean convergence curves for all experiment configurations (10D, 20D, and 50D), with standard error shown as shaded regions. The top row plots compliance against the total evaluation budget (including both feasible and infeasible evaluations). Here, we plot the objective value only for feasible points, as the infeasible ones are not simulated. The bottom row reports compliance with respect to the number of feasible evaluations only, i.e., simulation calls.