Automatic projection parameter increase for three-field density-based topology optimization

A method is proposed to automatically increase the threshold projection parameter in three-field density-based topology optimization to achieve near binary designs. The three-field method is composed of an element-wise design density field that is filtered and then passed through a smooth threshold projection function to compute the projected density field, which is then used to compute element properties, e.g., using a power law for stiffness. The sharpness of the threshold projection function is controlled by a parameter beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. In this paper, a method is introduced to automatically increase this parameter during optimization by linking it to the change in objective function. Furthermore, the gray value indicator is added as a stopping criterion to guarantee the solution is near binary. This results in a method that does not need to be tuned for specific problems, or optimizers, and the same set of user-defined parameters can be used for a wide range of problems. However, a high value of the threshold projection parameter may cause convergence issues for some optimizers, such as the optimally criteria method, and an adaptive move limit strategy is introduced to overcome this problem. It is also shown that some problems require length-scale control to achieve a near binary design. The effectiveness of the method is demonstrated on several benchmark problems, including linear compliance, linear buckling, compliant mechanism, heat conduction, and geometrically nonlinear problems.