An unified material interpolation for topology optimization of multi-materials

Topology optimization is one of the engineering tools for finding efficient design. For the material interpolation scheme, it is usual to employ the SIMP (Solid Isotropic Material with Penalization) or the homogenization based interpolation function for the parameterization of the material properties with respect to the design variables assigned to each finite element. For topology optimization with single material design, i.e., solid or void, the parameterization with 1 for solid and 0 for void becomes relatively straight forward using a polynomial function. For the case of multiple materials, some issues of the equality modeling of each material and \textcolor{red}{the clear 0, 1 result of each element for the topology optimization} issues become serious because of the curse of the dimension. To relieve these issues, this research proposes a new mapping based interpolation function for multi-material topology optimization. Unlike the polynomial based interpolation, this new interpolation is formulated by the ratio of the $p$-norm of the design variables to the 1-norm of the design variable multiplied by the design variable for a specific material. With this alternative mapping based interpolation function, each material are equally modeled and \textcolor{red}{ the clear 0, 1 result of each material for the multi-material topology optimization model} can be improved. This paper solves several topology optimization problems to prove the validity of the present interpolation function.