An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

Abstract

Summary The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.

A two‐scale FE‐FFT approach to nonlinear magneto‐elasticity

“…At this point, we would like to refer to the works of Miehe et al and Terada and Kikuchi, where this idea has been worked out in the context of computational homogenization and the FE 2 method. For very recent contributions concerned with the consistent linearization of the effective response obtained by DFT‐based homogenization, we refer to the contributions of Göküzüm and Keip and Göküzüm et al While the derivation of the consistent tangent in the latter two works was based on solvers for the Lippman‐Schwinger equation, we later show the consistent linearization of the projection approach discussed earlier. Nevertheless, both approaches yield equivalent results.…”

“…In the context of scale bridging, we derive a consistent macroscopic tangent operator that is a crucial ingredient for multiscale simulations. Göküzü and Keip recently discussed such a tangent operator for classical Fourier‐based schemes. However, the present work is based on an FE perspective of Fourier schemes leading to a derivation more familiar to an audience with a background in finite elements.…”

Consistent macroscopic tangent computation for FFT‐based computational homogenization at finite strains

“…which is a linear equation in ∂ F /∂F and is solved under zero-average constraint [4]. In contrast to finite-difference based approaches, where the tangent is calculated via a set of perturbed strain cases, this is a great advantage.…”

Numerically robust two‐scale full‐field finite strain crystal plasticity simulations of polycrystalline materials

“…[3,5]) and direct tangent computations (e.g. [7]) have been utilized. Numerical tangent computations are correlated with high computational costs, in general.…”