Non-convex, ringing-free, FFT-accelerated solver using an incremental approximate energy functional

Abstract

Fourier-accelerated micromechanical homogenization has been developed and applied to a variety of problems, despite being prone to ringing artifacts. In addition, the majority of Fourier-accelerated solvers applied to fast Fourier transform (FFT)-accelerated schemes only apply to convex problems. We here introduce a first order approximation incremental energy functional (FAIEF) that allows to employ modern efficient and non-convex iterative solvers, such as trustregion solvers or Low memory Broydon-Fletcher-Goldfarb-Shanno (LBFGS) in a FFT-accelerated scheme. These solvers need the explicit energy functional of the system in their standard form. We develop a modified trust region solver, capable of handling non-convex micromechanical homogenization problems such as continuum damage employing the FAIEF. We use the developed solver as the solver of a ringing-free FFT-accelerated solution scheme, namely the projection based scheme with finite element discretization.