Reduced-order methods for dynamic problems in topology optimization: A comparative study

The dynamics of engineering structures are of great importance for topology optimization problems in both academia and industry. However, for design problems where broadband frequency responses are required, the computational burden becomes enormous, especially for large-scale applications. To remedy this numerical bottleneck, using the Reduced-Order Methods (ROMs) is an efficient approach by recasting the original problem into a subspace with a much smaller dimensionality than the full model. In this paper, a systematic comparative study of some typical and potential ROMs for solving the broadband frequency response optimization problems is provided, including the Quasi-Static Ritz Vector (QSRV), the Padé expansion and the second-order Krylov subspace method. Furthermore, the effects of the orthonormalization processes are discussed. Two representative test problems, a vibration problem and a wave propagation problem, are solved, analyzed, and compared based on the ROMs’ accuracy, their stability in approximating the state and adjoint equations and the applicability to topology optimization problems. From the extensive numerical results, we find that the second-order Krylov subspace with moment-matching Gram–Schmidt orthonormalization (SOMMG) and the Second-Order Arnoldi method (SOAR) provides superior accuracy and stability. Moreover, the results verify that the basis vectors computed for the state equation cannot be reused for solving the adjoint equation, and hence, that new basis vectors should be constructed. Analysis of the computational cost for the 3D test problems shows an improvement in numerical performance in the order of 100–10000 for the ROMs compared to the full approach.