Topology optimization for linear thermo-mechanical transient problems: Modal reduction and adjoint sensitivities

This paper focuses on topology optimization for linear transient thermo-mechanical problems. The latter are, for example, encountered for extreme precision tools and instrumentation. Due to the transient nature, a standard adjoint sensitivity analysis will result in a backward transient analysis for the adjoint variables, leading to both storage and computational inefficiencies. A method is proposed that rigorously eliminates the backward transient integration for the adjoint sensitivity analysis. At the basis is a model-order reduction technique, which relies on a reduced thermal modal basis combined with static correction. The modal amplitudes can be readily obtained semi-analytically using simple convolutions. This accurate but reduced-order model is the starting point for an adjoint sensitivity analysis. Via a tactic selection of adjoint variables, the backward transient analysis for the adjoints is completely eliminated, whereas computational efficiency and consistency are maintained. The effectiveness of the resulting adjoint sensitivities and their application in topology optimization are demonstrated on the basis of several test examples.