Topology optimization for structure with multi-gradient materials

Functionally gradient materials (FGM) widely exist in biological structures, which offer indispensable support and potential protection. In this paper, we expand topology optimization for coated-base materials to multi-gradient structure. In the process of modeling the biological bones, we define three levels of density gradients: a spongy bone as the base in between, wrapped by a layer of compact bone, and multi-layers of periosteum side by side. By defining the properties and thickness of each layer, the base domain is derived by the gradient norm while a modified non-monotone filter function is imposed to separate periosteum and compact bone. Meanwhile, periosteum is separated into multi-gradient layers by Heaviside projection function in order to further mimic the skeletal of bone structure. The method of robust control, using geometric constraints, is applied to the design domain in both solid and void phases to avoid local singularity during the optimal process. Finally, the validity of the algorithm is illustrated by two benchmark problems with a variety of layer combinations.