As suggested by Sigmund, material tailoring can be formulated as a structural optimization problem. However microstructural geometries can be rather complex while optimization process may introduce large shape modifications of boundaries and material interfaces. To circumvent the technical difficulties of classical shape optimization relying on numerous and tedious CAD model manipulations and remesh- ing, the Level Set description combined with a non-conforming XFEM analysis is a promising approach. In the present work, this approach that has been developed in work by Van Miegroet is extended to the investigation of periodic microstructures subject to compliance or stress constraints. The approach is illustrated on the famous problem of the Vigdergauz microstructure.
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