  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, 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 less than one percent. 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 in the component level. In
numerical examples, we demonstrate a 1000x speedup over a full-order FEM model
with relative error of less than one percent and show minimum compliance
designs for two different cantilever beam examples, one smaller and one larger.
Finally, error bounds for displacement field, compliance, and compliance
sensitivity of the CWROM are derived.
