“Color” level sets: a multi-phase method for structural topology optimization with multiple materials

Abstract

In this paper we address the problem of structural shape and topology optimization in a multi-material domain. A level-set method is employed as an alternative approach to the popular homogenization-based methods of rule of mixtures for multi-material modeling. A multi-phase level-set model is adapted for material and topology representation. This model eliminates the need for a material interpolation or phase mixing scheme. It only requires m level-set functions to represent a structure of n=2 m different material phases, in a principle similar to combining colors from the three primary colors. Therefore, this multi-phase model may be referred to as a “color” level-set representation which has its unique benefits: it is flexible to handle complex topologies; it substantially reduces the number of model functions when n>3; it automatically avoids the problem of overlap between material phases of a conventional partitioning approach. We describe numerical techniques for efficient and robust implementation of the method, by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain. This would separate the issues of accuracy in numerical calculations of the physical equation and in the level-set model propagation. A gradient projection method is described for incorporating multiple constraints in the problem. Finally, the benefits and the advantages of the developed method are illustrated with several 2D examples of mean compliance minimization of multi-material structures.