Adaptation and validation of FFT methods for homogenization of lattice based materials

TL;DR

This paper develops and validates FFT-based methods tailored for homogenizing lattice materials, effectively handling large empty regions and surface roughness, with applications to printed lattices and comparisons to FEM.

Contribution

It introduces adapted FFT frameworks for lattice homogenization that improve accuracy and efficiency, especially for domains with large empty spaces and surface irregularities.

Findings

FFT becomes competitive with FEM above 7% density.

The adapted Galerkin FFT with Voigt smoothing yields best accuracy and efficiency.

Simulations quantify stiffness reduction and stress localization in printed lattices.

Abstract

An FFT framework which preserves a good numerical performance in the case of domains with large regions of empty space is proposed and analyzed for its application to lattice based materials. Two spectral solvers specially suited to resolve problem containing phases with zero stiffness are considered (1) a Galerkin approach combined with the MINRES linear solver and a discrete differentiation rule and (2) a modification of a displacement FFT solver which penalizes the indetermination of strains in the empty regions, leading to a fully determined equation. The solvers are combined with several approaches to smooth out the lattice surface, based on modifying the actual stiffness of the voxels not fully embedded in the lattice or empty space. The accuracy of the resulting approaches is assessed for an octet-lattice by comparison with FEM solutions for different relative densities and discretization levels. It is shown that the adapted Galerkin approach combined with a Voigt surface smoothening was the best FFT framework considering accuracy, numerical efficiency and h-convergence. With respect to numerical efficiency it was observed that FFT becomes competitive compared to FEM for cells with relative densities above 7%. Finally, to show the real potential of the approaches presented, the FFT frameworks are used to simulate the behavior of a printed lattice by using direct 3D tomographic data as input. The approaches proposed include explicitly in the simulation the actual surface roughness and internal porosity resulting from the fabrication process. The simulations allowed to quantify the reduction of the lattice stiffness \rev{as well as to resolve the stress localization of 50% near large pores.