Application of grayscale analysis in heat transfer topology optimization: A study on the impact of filter radius on numerical stability and thermodynamic performance

Abstract

Topology optimization, as an efficient material distribution design method, faces challenges of numerical instability (such as checkerboard patterns, grayscale phenomena, and mesh dependency) in the field of thermal management. These issues directly affect the manufacturability and thermodynamic performance of the optimization results. This study proposes a quantitative method based on grayscale analysis, conducting frequency-domain analysis of the grayscale distribution in the optimization results through two-dimensional discrete Fourier transform. It systematically investigates the influence mechanism of the filtering radius on numerical instability issues. The research results show that the choice of filtering radius has a significant impact on the numerical stability and thermodynamic performance of the optimization results. When the filtering radius is within the range of 1 to 2 mm, it effectively reduces numerical instability issues while significantly improving the stability and accuracy of the structural performance. By defining a comprehensive metric S and conducting thermal analysis of the topology structure for different filtering radii, the effectiveness of the proposed method is validated. The numerical results indicate that when the filtering radius is 1.5 mm, the optimization results achieve an ideal balance between multiple optimization indicators, including objective function value, computational cost, discretization, and grayscale rate. The objective function value is significantly improved, and heat transfer efficiency is optimized; computational cost is reasonably reduced, and the number of iterations is decreased; optimization of discretization and grayscale rate ensures the uniformity of the structure and the effectiveness of the thermal flow channels, significantly enhancing the accuracy and stability of the topology optimization results. The innovation of this study lies in quantifying the impact of the filtering radius on the optimization results through frequency-domain analysis, providing a theoretical basis for scientifically selecting the filtering radius and avoiding the subjectivity and uncertainty in the traditional methods of selecting the filtering radius. Compared with existing research, this paper not only systematically analyzes the quantitative impact of the filtering radius on numerical stability and thermodynamic performance but also validates the effectiveness of the proposed method through numerical experiments. It provides clear guidance for selecting the filtering radius in topology optimization, contributing to the application of topology optimization techniques in complex thermal management problems and improving engineering design efficiency and manufacturing quality.