Optimization of bodies with locally periodic microstructure by varying the shape, the topology and the periodicity pattern

Mimicking nature, an optimization method that makes the link between microstructure and macrostructure is considered. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body.   The already known alternate optimization of shape and topology of the model cell is a procedure that gives a limited flexibility to the microstructure for adapting to the macroscopic loads. Beyond that, one may vary the periodicity cell itself during the optimization process, thus allowing the microstructure to adapt more freely to the given loads.   What we propose is a method that combines the three optimization techniques : the shape, the topology and the periodicity pattern. By combining variations of these three ingredients, the obtained optimal design approaches the homogenized structure of the body, giving one the possibility to obtain a manufacturable design with smooth transition of material properties as in functionally graded materials.   Numerical examples will be presented. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.