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The a posteriori analysis of the discretization error and the modeling erroris studied for a compliance cost functional in the context of the optimizationof composite elastic materials and a two-scale linearized elasticity model. Amechanically simple, parametrized microscopic supporting structure is chosenand the parameters describing the structure are determined minimizing thecompliance objective. An a posteriori error estimate is derived which includesthe modeling error caused by the replacement of a nested laminatemicrostructure by this considerably simpler microstructure. Indeed, nestedlaminates are known to realize the minimal compliance and provide a benchmarkfor the quality of the microstructures. To estimate the local difference in thecompliance functional the dual weighted residual approach is used. Differentnumerical experiments show that the resulting adaptive scheme leads to simpleparametrized microscopic supporting structures that can compete with theoptimal nested laminate construction. The derived a posteriori error indicatorsallow to verify that the suggested simplified microstructures achieve theoptimal value of the compliance up to a few percent. Furthermore, it is shownhow discretization error and modeling error can be balanced by choosing anoptimal level of grid refinement. Our two scale results with a single scalemicrostructure can provide guidance towards the design of a produciblemacroscopic fine scale pattern.

A posteriori modeling error estimates in the optimization of two-scale elastic composite materials