Topological shape optimization of microstructural metamaterials using a level set method

Metamaterials usually refer to artificial composite materials consisting of an array of periodically arranged microstructures, engineered to provide unusual material properties that may not be easily found in nature. This paper proposes a new topological shape optimization method for systematic computational design of a type of mechanical metamaterials with negative Poisson's ratios (auxetic materials), which integrates the numerical homogenization approach into a powerful parametric level set method (PLSM). The homogenization method is used to obtain the effective properties of the periodic microstructure, while the PLSM is applied to achieve shape evolutions and topological changes of the microstructure, until the desired material properties are achieved. The key concept of the PLSM is the interpolation of the implicit level set surface by using a given set of compactly supported radial basis functions (CSRBF), which are positioned at a number of given and fixed knots inside the design domain. Several typical numerical examples are used to demonstrate the favorable characteristics of the proposed method in the design of micro-structured metamaterials.