Achieving binary topology optimization solutions via automatic projection parameter increase

TL;DR

The paper tackles the difficulty of obtaining near-binary designs in three-field density-based topology optimization, where traditional continuation schemes for the projection parameter $\beta$ require problem-specific tuning. It introduces an automatic, iteration-wise update rule for $\beta$ that links its increase to the observed progress of the objective via $\Delta \beta^k = -\frac{\gamma}{2}\frac{f^k+f^{k-1}}{f^k-f^{k-1}}$, with updates $\beta^{k+1} = \beta^k + \min(\Delta \beta^k, \Delta \beta_{max})$, starting from $\beta=1$, and a stopping condition based on the grayness measure $G(\bar{x})$ with threshold $\epsilon=0.01$. The approach removes much of the problem-specific tuning and is validated on 2D benchmarks including linear buckling and geometrically nonlinear analyses (e.g., compressed column and cantilever), showing faster convergence to near-binary designs while maintaining or improving objective values compared to standard continuation schemes. The results indicate the method’s robustness across problems and mesh refinements, offering a practical pathway to more broadly applicable, efficient binary topology optimization. Overall, the work provides a simple, effective mechanism to automate a key hyperparameter, facilitating broader deployment of density-based topology optimization in engineering design.

Abstract

A method is created to automatically increase the threshold projection parameter in three-field density-based topology optimization to achieve a near binary design. The parameter increase each iteration is based on an exponential growth function, where the growth rate is dynamically changed during optimization by linking it to the change in objective function. This results in a method that does not need to be tuned for specific problems, or optimizers, and the same set of hyper-parameters can be used for a wide range of problems. The effectiveness of the method is demonstrated on several 2D benchmark problems, including linear buckling and geometrically nonlinear problems.

Figures (9)

• Figure 1: Example problems, a) compressed column, b) cantilever.
• Figure 2: Compressed column results for maximization of buckling load factor. Solutions with filter radius = $4h$: a) Default continuation scheme, b) modified scheme, c) automatic scheme. Solutions with filter radius = $8h$: d) Default continuation scheme, e) modified scheme, f) automatic scheme.
• Figure 3: Compressed column convergence history for maximization of buckling load factor. Convergence of volume objective for: a) filter radius = $4h$, c) filter radius = $8h$. Convergence of gray level indicator function, $G(x)$ , and threshold projection parameter, $\beta$, for b) filter radius = $4h$, d) filter radius = $8h$.
• Figure 4: Compressed column results for minimization of volume. Solutions with filter radius = $4h$: a) Default continuation scheme, b) modified scheme, c) automatic scheme. Solutions with filter radius = $8h$: d) Default continuation scheme, e) modified scheme, f) automatic scheme.
• Figure 5: Compressed column convergence history for minimization of volume. Convergence of volume objective for: a) filter radius = $4h$, c) filter radius = $8h$. Convergence of gray level indicator function, $G(x)$, and threshold projection parameter, $\beta$, for b) filter radius = $4h$, d) filter radius = $8h$.