[Paper Review] Adaptation and validation of FFT methods for homogenization of lattice based materials

S. Lucarini, Lucia Cobian+2 more|arXiv (Cornell University)|Oct 2, 2021

Composite Material Mechanics52 references44 citations

TL;DR

This paper proposes and validates an optimized FFT homogenization framework for lattice-based materials with large void fractions, combining a modified Galerkin FFT solver with Voigt-based surface smoothening to improve accuracy and efficiency. The approach enables direct simulation from 3D tomographic data, revealing up to 50% higher local stresses near pores and a 10% reduction in effective stiffness due to fabrication defects.

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 ev{as well as to resolve the stress localization of 50% near large pores.

Motivation & Objective

• To develop an FFT-based homogenization framework that maintains high accuracy and numerical efficiency for lattice materials with large regions of empty space.
• To address the challenge of infinite stiffness contrast between solid lattice members and voids in spectral solvers.
• To enable direct use of 3D tomographic data as input, preserving actual surface roughness and internal porosity from additive manufacturing.
• To validate the framework against FEM simulations and assess its performance across varying relative densities and discretization levels.
• To demonstrate the method’s capability in predicting real-world mechanical behavior, including stress concentrations and stiffness reduction due to fabrication defects.

Proposed method

• Adapted a Galerkin FFT approach using MINRES linear solver and modified Fourier frequencies to implement a discrete differentiation rule (rotated forward scheme).
• Proposed a modified displacement-based FFT (MoDBFFT) that penalizes strain indeterminacy in void regions, resulting in a fully determined system.
• Applied Voigt-type analytic smoothening to interpolate stiffness in interfacial voxels based on distance to the true lattice surface.
• Used voxel-based phase maps derived from tomographic images, with modifications to stiffness in partially filled voxels to reduce Gibbs ringing and improve geometric fidelity.
• Combined the FFT solvers with multiple surface smoothening techniques and evaluated their impact on effective properties and local field accuracy.
• Validated results against FEM simulations for octet-lattice RVEs across multiple relative densities (3.6% to 30%) and grid resolutions.

Experimental results

Research questions

• RQ1 Can FFT homogenization be made numerically efficient and accurate for lattice materials with very low relative densities and infinite stiffness contrast?
• RQ2 How do different FFT solvers (Galerkin vs. displacement-based) perform in terms of convergence, accuracy, and efficiency when applied to lattice structures with voids?
• RQ3 What is the optimal surface smoothening technique for minimizing numerical artifacts while preserving the true relative density and geometric features from tomographic data?
• RQ4 To what extent can FFT simulations using direct tomographic input predict real mechanical behavior, including stress concentrations and stiffness reduction, compared to idealized designs?
• RQ5 At what relative density does FFT become computationally competitive with FEM for lattice homogenization?

Key findings

• The adapted Galerkin FFT solver with Voigt-based smoothening achieved the best balance of accuracy, numerical efficiency, and h-convergence across all tested cases.
• FFT simulations showed a 10% reduction in effective Young’s modulus compared to the ideal design, primarily due to internal porosity (3.6%) and surface roughness.
• Local stress concentrations near large pores increased by up to 50% in real microstructures compared to idealized geometries, indicating higher risk of fracture initiation.
• The modified Galerkin FFT framework reduced microfield differences with FEM to below 20% across all tested materials and densities.
• FFT became computationally competitive with FEM for relative densities above approximately 7%, achieving 4–8 times faster simulation times at 30% density.
• Direct simulation using 3D tomographic data revealed that fabrication-induced defects significantly alter both macroscopic stiffness and local stress distributions, validating the method’s predictive capability.