Hierarchical Topology Optimization of Structures Subjected to Constraints on Material Design

The hierarchical topology optimization of three-dimensional structures is addressed in this paper. The structure lay-out (macroscale) and material microstructure (microscale) are optimized concurrently to minimize compliance. It is assumed that structure is made up of a periodic cellular material. This two-scale optimization problem is subjected to global/local material resource constraints and to specific local constraints controlling the perimeter on material microstructure design. To handle local constraints a proper algorithmic strategy is used combining two versions of MMA (Method of Moving Asymptotes). Local design problems can be solved independently and one takes advantage of this property by using parallel computing techniques to speed-up the solution task. Concerning applications one might think on man’s made structures but the model is used to gain insight in natural hierarchical materials such as bone. At apparent level (macroscale) bone exhibits two types of tissues in a well-defined arrangement (cortical and trabecular) while going down on length scale (microscale) one distinguish trabeculae from marrow space. Actually, bone adapts to the mechanical environment by changing its microstructure and this natural adaptation is simulated in the present work. Further, among morphometric parameters used to characterize trabecular architecture there is the surface-to-volume ratio, which depends on anatomical site of the trabecular region. This biological feature of bone architecture is taken into account in the present study by means of a local material design constraint involving perimeter control. The results obtained for a proximal femur bone show a fin