Non-Differentiable Functions in Topology Optimization: Application to Additive Manufacturing

This paper was produced for the 2019 NAFEMS World Congress in Quebec Canada

Resource Abstract

It is well known that non-differentiable functions cannot be used for gradient-based optimization because the optimization algorithms assume a smooth change in gradients. In order to handle non-differentiable functions in optimization, engineers often approximate the non-differentiable function using a smooth differentiable function. However, the approximation process makes the function highly nonlinear, which makes the optimization difficult to handle. The main contribution of the paper is to show that in topology optimization, the difficulty associated with non-differentiable functions is diminished due to a fine discretization of the domain and a large number of design variables. We show the efficacy of non-differentiable function in topology optimization using additive manufacturing constraint, where maximum or minimum functions are used to determine the topological density of an element. In this paper, we compare the performance of smooth functions against directly imposing the maximum and minimum functions in topology optimization. We also compared the difference in gradients between the two approaches and showed that the effect of the non-differentiability issue is not noticeable in topology optimization.