Topology and Shape Optimization of 2-D and 3-D Micro-Architectured Thermoelastic Metamaterials Using a Parametric Level Set Method

2-D and 3-D micro-architectured multiphase thermoelastic metamaterials are designed and analyzed using a parametric level set method for topology optimization and the finite element method. An asymptotic homogenization approach is employed to obtain the effective thermoelastic properties of the multiphase metamaterials. The epsilon-constraint multi-objective optimization method is adopted in the formulation. The coefficient of thermal expansion (CTE) and Poisson's ratio (PR) are chosen as two objective functions, with the CTE optimized and the PR treated as a constraint. The optimization problems are solved by using the method of moving asymptotes. Effective isotropic and anisotropic CTEs and stiffness constants are obtained for the topologically optimized metamaterials with prescribed values of PR under the constraints of specified effective bulk modulus, volume fractions and material symmetry. Two solid materials along with one additional void phase are involved in each of the 2-D and 3-D optimal design examples. The numerical results reveal that the newly proposed approach can integrate shape and topology optimizations and lead to optimal microstructures with distinct topological boundaries. The current method can topologically optimize metamaterials with a positive, negative or zero CTE and a positive, negative or zero Poisson's ratio.