  The a posteriori analysis of the discretization error and the modeling error
is studied for a compliance cost functional in the context of the optimization
of composite elastic materials and a two-scale linearized elasticity model. A
mechanically simple, parametrized microscopic supporting structure is chosen
and the parameters describing the structure are determined minimizing the
compliance objective. An a posteriori error estimate is derived which includes
the modeling error caused by the replacement of a nested laminate
microstructure by this considerably simpler microstructure. Indeed, nested
laminates are known to realize the minimal compliance and provide a benchmark
for the quality of the microstructures. To estimate the local difference in the
compliance functional the dual weighted residual approach is used. Different
numerical experiments show that the resulting adaptive scheme leads to simple
parametrized microscopic supporting structures that can compete with the
optimal nested laminate construction. The derived a posteriori error indicators
allow to verify that the suggested simplified microstructures achieve the
optimal value of the compliance up to a few percent. Furthermore, it is shown
how discretization error and modeling error can be balanced by choosing an
optimal level of grid refinement. Our two scale results with a single scale
microstructure can provide guidance towards the design of a producible
macroscopic fine scale pattern.
