Topological design for mechanical metamaterials using a multiphase level set method

Metamaterials usually refer to artificially engineered composites with unusual properties that cannot be easily found in nature. This paper will develop a topological shape optimization method for design of mechanical metamaterials of thermoelastic micro-structured composites, which integrates numerical homogenization method with a multi-phase level set method (MPLSM). The homogenization method is applied to evaluate the effective macroscopic properties of a periodic microstructure, while the MPLSM will be utilized to implement shape and topology evolutions of the microstructure. A multi-phase level set representation model is established to describe the boundaries of the multi-material microstructure using a combination of all level set functions, without overlaps and empties. The Hamilton-Jacobi partial differential equation-based topological shape optimization problem will be transformed to a generalized size optimization problem. Typical numerical examples are used to demonstrate the effectiveness of the proposed method for designing metamaterials with expected and extreme thermal expansion coefficients.