An explicit dynamic FFT method for homogenizing heterogeneous solids under large deformations

We propose an explicit displacement-based FFT method for computational homogenization and multi-scale modeling of heterogeneous solids. Unlike existing FFT methods that solve the static equilibrium equation using implicit iterative techniques, our proposed explicit method solves dynamic equations with the Central Difference method. The voxel-like discretizations of the periodic domain and the direct calculation of gradient and divergence operations using Discrete Fourier Transform and its inverse result in a straightforward algorithm that avoids convergence issues and does not require material consistent tangents. The stability condition of this explicit method is also addressed. Four numerical examples (2D particle-reinforced composites of hyperelasticity and viscoelasticity, a 3D fiber-reinforced composite, and a 3D polycrystalline metal) illustrate that the proposed method calculates as accurately and efficiently as implicit FFT methods for quasi-static problems. Furthermore, the proposed explicit displacement-based FFT method outperforms implicit FFT methods when dealing with microstructures comprising phases with very high contrast. Moreover, since the method does not require material consistent tangents, it extends the applicability of FFT methods to the homogenization of materials with complex or black-boxed constitutive models, where consistent tangents are challenging to compute.