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

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.