Voxel-Based Asymptotic Homogenization of the Effective Thermal Properties of Lattice Materials with Generic Bravais Lattice Symmetry

In this paper, voxel-based Asymptotic Homogenization (AH) is employed to calculate the thermal expansion and thermal conductivity characteristics of lattice materials that have a Representative Volume Element (RVE) with non-orthogonal periodic bases. The non-orthogonal RVE of the cellular lattice is discretized using voxel elements (iso-parametric hexahedral element, on a cartesian grid). A homogenization framework is developed in python that uses a fast-nearest neighbor algorithm to approximate the (non-orthogonal) periodic boundary conditions of the discretized RVE. Validation studies are performed where results of the homogenized Thermal Expansion Coefficient (TEC) and thermal conduction performed in this paper are compared with results generated by commercially available software. These included comparison with the results for (a) bi-material unidirectional composite with orthogonal RVE cell envelope; (b) bi-material hexagon lattice with orthogonal cell envelope; (c) bi-material hexagon lattice with non-orthogonal cell envelope; and (d) bi-material square lattice. A novel approach of visualizing the contribution of each voxel towards the individual terms within the homogenized thermal conductivity matrix is presented, which is necessary to mitigate any potential errors arising from the numerical model. Additionally, the effect of the thermal expansion and thermal conductivity for bi-material hexagon lattice (orthogonal and non-orthogonal RVE cell envelope) are presented for varying internal cell angles and all permutations of material assignments for a relative density of 0.3. It is found that when comparing the non-orthogonal RVE with the Orthogonal RVE as a reference model, the numerical error due to approximating the periodic boundary condition for the non-orthogonal bi-material hexagon is generally less than 2% as the numerical error is pseudo-cyclically dependent on the discretization along the cartesian axis.