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Multi-material design optimization problems can, after discretization, besolved by the iterative solution of simpler sub-problems which approximate theoriginal problem at an expansion point to first order. In particular, modelsconstructed from convex separable first order approximations have a long andsuccessful tradition in the design optimization community and have led topowerful optimization tools like the prominently used method of movingasymptotes (MMA). In this paper, we introduce several new separableapproximations to a model problem and examine them in terms of accuracy andfast evaluation. The models can, in general, be nonconvex and are based on theSherman-Morrison-Woodbury matrix identity on the one hand, and on themathematical concept of topological derivatives on the other hand. We show asurprising relation between two models originating from these two -- at a firstsight -- very different concepts.  Numerical experiments show a high level of accuracy for two of our proposedmodels while also their evaluation can be performed efficiently once enoughdata has been precomputed in an offline phase. Additionally it is demonstratedthat suboptimal decisions can be avoided using our most accurate models.

Efficient and accurate separable models for discrete material optimization: A continuous perspective