Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two -- at a first sight -- very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline phase. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

翻译：多材料设计优化问题经离散化后，可通过迭代求解在展开点处对原问题进行一阶近似的简单子问题来解决。其中，基于凸可分离一阶近似构建的模型在设计优化领域有着悠久而成功的传统，并催生了诸如广泛应用的运动渐近线法（MMA）等强大优化工具。本文针对一个模型问题提出了若干新型可分离近似方法，并从精度与快速求解两个维度予以评估。这些模型通常可以是非凸的，分别基于Sherman-Morrison-Woodbury矩阵恒等式与拓扑导数的数学概念构建。我们揭示了这两种初看迥异的概念所导出的两个模型之间令人意外的关联。数值实验表明，我们提出的两个模型具有高精度水平，且一旦在离线阶段完成足够数据预计算，其求解过程便可高效执行。此外，实验结果还证明，采用我们最精确的模型可避免次优决策。