Lattice-type structures can provide a combination of stiffness with light weight that is desirable in a variety of applications. Design optimization of these structures must rely on approximations of the governing physics to render solution of a mathematical model feasible. In this paper, we propose a topology optimization (TO) formulation that approximates the governing physics using component-wise reduced order modeling as introduced in Huynh et al. (2013); Eftang and Patera (2013), which can reduce solution time by multiple orders of magnitude over a full-order finite element model while providing a relative error in the solution of $<$1%. In addition, the offline training data set from such component-wise models is reusable, allowing its application to many design problems for only the cost of a single offline training phase, and the component-wise method is nearly embarrassingly parallel. We also show how the parameterization chosen in our optimization allows a simplification of the component-wise reduced order model (CWROM) not noted in previous literature, for further speedup of the optimization process. The sensitivity of the compliance with respect to the particular parameterization is derived solely at the component level. In numerical examples, we demonstrate a 1000x speedup over a full-order FEM model with relative error of $<$1% and show minimum compliance designs for two different cantilever beam examples, one smaller and one larger. Finally, error bounds for the displacement field, compliance, and compliance sensitivity of the CWROM are derived.

格子型结构可以提供刚度和轻量的组合，这在各种应用中都是理想的。这些结构的设计优化必须依赖于控制物理的近似，才能使数学模型的求解可行。在本文中，我们提出了一种拓扑优化（TO）公式，该公式使用Huynh等人介绍的基于组件的降阶建模来近似控制物理。（2013）；Eftang and Patera（2013），它可以在一个全阶有限元模型上将求解时间减少多个数量级，同时在以下条件下提供相对误差：$<$1％。此外，这种基于组件的模型的脱机训练数据集是可重用的，从而仅以单个脱机训练阶段的成本就可以将其应用于许多设计问题，而基于组件的方法几乎令人尴尬地是并行的。我们还展示了在优化中选择的参数化方式如何简化以前文献中未提到的组件级降序模型（CWROM），以进一步加快优化过程。关于特定参数设置的顺应性敏感性仅在组件级别上得出。在数值示例中，我们展示了相对误差为的全阶FEM模型的1000倍加速$<$1％，并显示了两个不同的悬臂梁示例（一个较小而另一个较大）的最小顺应性设计。最后，得出了CWROM的位移场，顺应性和顺应性敏感性的误差范围。