# A Posteriori Error Estimates for Nested Laminates in Shape Optimization

Benedict Geihe^{*}

Structural elastic shape optimization amounts to the problem of distributing a given solid material over a working domain exposed to volume and surface loads. The resulting structure should be as rigid as possible while meeting a constraint on the volume spent. Without any further regularization of the perimeter length this renders the problem ill-posed and on a minimizing sequence the onset of microstructures can be observed. \newline To describe resulting relaxed optimal shapes several constructions are known. The most general one are nested laminates which have been adopted in a shape optimization algorithm by Allaire and coworkers. They consist of iteratively stacked layers of hard and soft material whose effective elastic properties are determined by means of homogenization. Thereby at any point the locally optimal microstructure is given by a small set of lamination parameters that can explicitly be computed from the local stress. \newline We are interested in approximating these optimal microstructures by simple geometries within a two-scale model. It turns out that it is crucial to sharply separate regions in which a trivial structure like bulk or void is sufficient from regions where a complex structure is needed. \newline As a first step we consider the nested laminates model and apply the dual weighted residual approach to derive goal-oriented a posteriori error estimates. These will be used to steer a mesh adaptive algorithm and numerical results will be presented.

Mathematics Subject Classification: 49M29 65K10

Keywords: shape optimization; nested laminates; microstructures; a posteriori error estimates; adaptivity

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