Adaptive beta update scheme in heaviside projection method of topology optimization

Abstract

The Heaviside projection method is widely used to obtain binary solutions in topology optimization, and the projection steepness parameter beta is typically increased by doubling at fixed update intervals. However, such interval-based schemes often lead to excessive iterations and numerical oscillations during the optimization process. In this study, we propose an adaptive beta update strategy that extends the role of the gray-level indicator, a measure of non-discreteness, to an adaptive parameter governing the progression of beta throughout the optimization. The proposed method consists of two phases: a stability-based Phase 1 that guides a gradual reduction of intermediate densities, and a prediction-based Phase 2 that adjusts beta when beta-update congestion is detected to ensure continuous and stable projection progression. Numerical experiments across various physical problems and parameter settings demonstrate that the proposed approach significantly reduces the number of iterations required to reach convergence while maintaining or improving the final objective performance. These results indicate that the adaptive beta update strategy can serve as a consistent and effective beta update framework for the Heaviside projection in topology optimization.