  This paper provides an extended level set (X-LS) based topology optimiza-
tion method for multi material design. In the proposed method, each zero level
set of a level set function {\phi}ij represents the boundary between materials
i and j. Each increase or decrease of {\phi}ij corresponds to a material change
between the two materials. This approach reduces the dependence of the initial
configuration in the optimization calculation and simplifies the sensitivity
analysis. First, the topology optimization problem is formulated in the X-LS
representation. Next, the reaction-diffusion equation that updates the level
set function is introduced, and an optimization algorithm that solves the
equilibrium equations and the reaction-diffusion equation using the fi- nite
element method is constructed. Finally, the validity and utility of the
proposed topology optimization method are confirmed using two- and three-
dimensional numerical examples.
