Finite Element Approximation of a Time-Dependent Topology Optimization Problem

In most additive manufacturing technologies, support structures are required to sustain overhanging surfaces. These additional structures have negative effects on both processing time and material consumption. Moreover, additional post-processing effort is required for their removal. Therefore, reducing the use of the supports would have a beneficial impact on the overall manufacturing process. An optimization procedure aiming at the optimal placement (and design) of the supports in additive manufacturing should include the intrinsic time-dependent nature of the process. More precisely, it should take into account not only the final configuration, but all the intermediate shapes that are obtained during the additive process. As a consequence, we believe that it is necessary to go beyond standard topology optimization methods, where typically only the final shape is optimized. The model proposed in this work relies on the solution of a time-dependent minimal compliance problem based on the classical Solid Isotropic Material with Penalization (SIMP) method. In particular, we first introduce a continuous optimization problem with the state equation de- fined as the time-integral of a linear elasticity problem on a space-time domain. The objective functional is given by the mean compliance over a time interval. The optimality conditions for this optimization problem are then derived and a fixed-point algorithm is introduced for the iterative computation of the optimal solution. Numerical examples showing the differences between a standard SIMP method, which only optimizes the shape at the final time, and the proposed time-dependent approach are presented and discussed.