  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.
