Topology optimization with a time-integral cost functional

We present a topology optimization based procedure aiming at the optimal placement (and design) of the supports in problems characterized by a time dependent construction process. More precisely, we focus on the solution of a time-dependent minimal compliance problem based on the classical Solid Isotropic Material with Penalization (SIMP) method. In particular, a continuous optimization problem with the state equation defined as the time-integral of a linear elasticity problem on a space-time domain is firstly introduced and the mean compliance over a time interval objective functional is then selected as objective function. The optimality conditions are 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.