Topology Optimization Design of Adjustable Thermal Expansion Metamaterial Based on Independent Continuous Variables

This paper presents a topology optimization method to construct adjustable thermal expansion metamaterials. The negative thermal expansion (NTE) and extreme positive thermal expansion (EPTE) microstructures are designed, respectively. First, the effective elastic modulus and the effective thermal expansion coefficient of microstructure are derived by the multiscale asymptotic homogenization theory based on the periodic characteristics of the metamaterial. Second, a topology optimization model aiming at extreme thermal expansion coefficient is established and solved based on independent continuous topological variables, which is subjected to the effective elastic modulus and structure weight fraction. Then a cross-precision progressive optimization method is provided to save the computation time and make a smoother boundary. In addition, the specific topological feature for the initial configuration is discussed. Five holes are the necessary condition to obtain adjustable thermal expansion metamaterials for four-axisymmetric structures. Finally, a transformation strategy from NTE to EPTE and the change law of thermal expansion coefficient bounds with effective elastic modulus are studied. Numerical examples show that the NTE and EPTE can be designed in the range of effective elastic modulus from 0.01 to 0.05. The proposed method provides a reference for the novel configuration design of metamaterials.