Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties

Abstract

In this work, a topology optimization (TO) based framework for functional grading of triply periodic minimal surfaces (TPMS) based lattice structures is developed, implemented and demonstrated. Material interpolation laws of the gyroid, G-prime and Schwarz-D surfaces are derived by numerical homogenization for transversely isotropic elasticity and are represented as convex combinations of solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP) models. These convex combinations are implemented in the TO-based compliance problem with new upper and lower bounds on the density variables representing the volume fraction limits of the lattices. The lower bound on the density variables is treated by introducing a sigmoid filter in the optimization loop forcing densities below the lower boundary towards zero. The optimal density solution is represented by Shepard interpolations or radial basis function networks, which, in turn, are utilized for the thickness grading of the TPMS-based lattices. In addition, the global boundary of the lattice structure is identified by support vector machines. Finally, a standard triangle language (STL) file is generated from the implicit surfaces by using marching cubes, which is utilized for further studies by nonlinear finite element analysis and to set up 3D printing of the optimal component quickly. The framework is demonstrated for the established L-shaped benchmark and the well-known General Electric engine bracket.