On-the-fly dual reduction for time-dependent topology optimization

Abstract

A new model reduction approach is presented for time-dependent topology optimization of elastodynamic problems. Two salient features of the model reduction approach are on-the-fly and dual reduction. That is, the snapshots and reduced basis are obtained during the optimization rather than through off-line training; primal and adjoint equations are reduced independently. For the on-the-fly reduction, we use two complementary dynamic sampling strategies for snapshots: nearest nf full-order solutions and rolling transient ensemble of nt solutions. For the dual reduction, we apply the proper orthogonal decomposition and Galerkin procedure independently to the primal equations and to the adjoint equations for reduction. Central to our dual reduction strategy is the use of continuous adjoint to obtain the adjoint equations so that the adjoint equations can be reduced independently and along side the primal equation reduction. The residuals of the governing primal and adjoint equations are used to determine whether full-order model (FOM) based solutions should be used. Two-dimensional and three-dimensional numerical examples on topology optimization of elastodynamics problems are presented. Our numerical results demonstrate that the on-the-fly dual-reduction approach leads to two orders of magnitude improvement in FOM reduction (in terms of the number of FOM solution procedures being invoked) and data storage (in terms of storage of primal solutions required to solve the backward time-dependent adjoint equations and compute the sensitivity).