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.

工程结构的动力学对于学术界和工业界的拓扑优化问题都非常重要。然而，对于需要宽带频率响应的设计问题，计算负担变得巨大，特别是对于大规模应用。为了解决这个数值瓶颈，使用降阶方法 (ROM) 是一种有效的方法，它可以将原始问题重铸到维数比完整模型小得多的子空间中。在本文中，对解决宽带频率响应优化问题的一些典型和潜在的 ROM 进行了系统的比较研究，包括准静态 Ritz 向量 (QSRV)、Padé 展开和二阶Krylov 子空间方法. 此外，还讨论了正交归一化过程的影响。基于 ROM 的精度、它们在逼近状态和伴随方程方面的稳定性以及对拓扑优化问题的适用性，解决、分析和比较了两个具有代表性的测试问题，即振动问题和波传播问题。从广泛的数值结果中，我们发现具有矩匹配 Gram-Schmidt 正交归一化 (SOMMG) 和二阶阿诺尔迪方法 (SOAR)的二阶Krylov 子空间提供了卓越的精度和稳定性。此外，结果证实为状态方程计算的基向量不能重复用于求解伴随方程，因此，应该构建新的基向量。3D 测试问题的计算成本分析表明，与完整方法相比，ROM 的数值性能提高了 100-10000。