Designing materials with prescribed elastic properties using polygonal cells

An extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional e ort. In particular, the periodic homogenization problem de ned on polygonal base cells that tile the plane can be replaced and analysed more e ciently by an equivalent problem that uses simple parallelograms as base cells. Di erent material layouts can be obtained by varying just two parameters that a ect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an e cient way to organize the search of the design space for all possible single-scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results.