Optimization of heterogeneous solids with level set methods

A heterogeneous object is referred to as a solid object made of different constituent materials. The object is of a finite collection of regions of a set of prescribed material classes of continuously varying material properties. These properties have a discontinuous change across the interface of the material regions. We present a variational framework of Mumford-Shah model for a well-posed formulation for the design of the heterogeneous solids. We discuss two approaches to the optimization problem of free-discontinuities: a multi-phase level-set model and a multi-material phase-field model, both to represent the discontinuities implicitly. These models yield a computational system of coupled geometric evolution and/or diffusion partial differential equations. Promising features of the proposed method include strong regularity in problem formulation, topological flexibility, and inherent capabilities of geometric, physical and material modeling, incorporating dimension, shape, topology, material properties, and even micro-structures within a common framework for optimization of the heterogeneous solids. The proposed methods are illustrated with several 2D examples of topology optimization of multi-material structures, material design, and compliant mechanism synthesis.