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We present a level-set based topology optimization algorithm for designoptimization problems involving an arbitrary number of different materials,where the evolution of a design is solely guided by topological derivatives.Our method can be seen as an extension of the algorithm that was introduced in(Amstutz, Andrae 2006) for two materials to the case of an arbitrary number $M$of materials. We represent a design that consists of multiple materials bymeans of a vector-valued level set function which maps into $\mathbb R^{M-1}$.We divide the space $\mathbb R^{M-1}$ into $M$ sectors, each corresponding toone material, and establish conditions for local optimality of a design basedon certain generalized topological derivatives. The optimization algorithmconsists in a fixed point iteration striving to reach this optimalitycondition. Like the two-material version of the algorithm, also our methodpossesses a nucleation mechanism such that it is not necessary to start with aperforated initial design. We show numerical results obtained by applying thealgorithm to an academic example as well as to the compliance minimization inlinearized elasticity.

A multi-material topology optimization algorithm based on the topological derivative