We present here a topology optimization method based on a homogenization approach to design oriented and parametrized cellular structures. The present work deals with 2-D square cells featuring a rectangular hole, because their structure is close to that of rank-2 sequential laminates, which are optimal for compliance optimization. For several cells, the value and the parametric sensitivities of their effective elastic tensor can easily be computed, by the resolution of a cell problem. The obtained results can be used to build a surrogate model for the homogenized constitutive law. Moreover, we add the local orientation of the cells to our problem. Then, an optimal composite shape is computed.
The crucial ingredient of the methodology is the extraction of a quasi-periodic and additive manufacturable structure from the previously obtained composite shape, thanks to a space transformation. Indeed, after a light regularization of the optimal orientation field, a conformous map can be computed. It distorts a regular grid in order to align the cells with the given orientation, and to preserve the isometry of each cell.