DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior

Abstract

All the FFT-based methods available for homogenization of the mechanical response share a common point: the unknown field is the strain/deformation gradient, second order tensors which compatibility is imposed using Green's function or projection operators. This implies the allocation of redundant information and, when the method is based in a solving a linear system, the rank-deficiency of the resulting systems. In this work we propose a new, fast, robust and memory-efficient FFT approach in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. In the linear case, the method results in a linear equation defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. The method is extended to non-linear problems and strain, stress and mixed control. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.

An FFT-based approach for Bloch wave analysis: application to polycrystals

“…being the 3 × 3 matrix inverse computed only once, at the beginning of the algorithm. Note that the preconditionerP is equivalent to the one derived in [41] for the displacementbased FFT approach, DBFFT, but replacing the Fourier frequency with the sum of that frequency with the Bloch wave vector , ξ ← ξ + k. For the value of C 0 , the volume averaged stiffness provides good numerical performance, as shown in [41] for static simulations. It must be remarked that the use of the preconditioner is essential for the application of the method since it reduces the number of iterations by orders of magnitude.…”

“…[34]). Also, alternative FFTbased formulations have been developed as Fourier-Galerkin approaches [35][36][37][38][39], or related methods based on the use of displacements instead of strains as primary unknown field [40,41]. Although there are potential advantages (i.e.…”

A review of nonlinear FFT-based computational homogenization methods

“…However, additional fields may be stored in displacement form. For instance, the CG method can be implemented on a single strain and three displacement fields [105][106][107]. Thus, a low memory implementation is possible, but not to the same degree as for other discretization schemes.…”

Computing the effective response of heterogeneous materials with thermomechanically coupled constituents by an implicit fast Fourier transform‐based approach

“…Memory efficient versions of the Newton-CG method 23,88,89 operating on the displacement instead of the strain can be implemented, provided a finite-difference 90,91 or a finite-element 92,93 discretization is used. Additional memory can be saved by storing the Hessian in single precision, 26 resulting in a reduced memory footprint of nine displacement fields.…”