Adjoint sensitivities of time-periodic nonlinear structural dynamics via model reduction

This work details a comparative analysis of six methods for computing the transient system response and adjoint design derivatives of a nonlinear structure under a periodic external actuation. Time marching via implicit integration, a time-periodic spectral element method, and a time-periodic finite difference method are all considered, each with or without POD-based model reduction of the system of equations. Details of each method, and the concomitant adjoint sensitivities, are provided in terms of accuracy, stability, and Jacobian sparsity patterns. Each method is used for a gradient-based optimization of a nonlinear planar beam, periodically actuated at its root, subject to a large number of structural design parameters. The method with the lowest computational design cost is found to be a strong function of the harmonic content and the nonlinearity of the transient response.