An adaptive hybrid expansion method (AHEM) for efficient structural topology optimization under harmonic excitation

One challenge of solving topology optimization problems under harmonic excitation is that usually a large number of displacement and adjoint displacement vectors need to be computed at each iteration step. This work thus proposes an adaptive hybrid expansion method (AHEM) for efficient frequency response analysis even when a large number of excitation frequencies are involved. Assuming Rayleigh damping, a hybrid expansion for the displacement vector is developed, where the contributions of the lower-order modes and higher-order modes are given by the modal superposition and Neumann expansion, respectively. In addition, a simple (yet accurate) expression is derived for the residual error of the approximate displacement vector provided by the truncated hybrid expansion. The key factors affecting the convergence rate of the truncated hybrid expansion series are uncovered. Based on the Strum sequence, the AHEM can adaptively determine the number of lower-order eigenfrequencies and eigenmodes that need to be computed, while the number of terms that need to be preserved in the truncated Neumann expansion can be determined according to the given error tolerance. The performance of the proposed AHEM and its effectiveness for solving topology optimization problems under harmonic excitation are demonstrated by examining several 2D and 3D numerical examples. The non-symmetry of the optimum topologies for frequency response problems is also presented and discussed.