Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at large deformations

• Published in: Computational mechanics Vol. 54; no. 6; pp. 1497 - 1514
• Main Authors: Kabel, Matthias, Böhlke, Thomas, Schneider, Matti
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Approximation; Classical and Continuum Physics; Computational Science and Engineering; Deformation; Elastic deformation; Elasticity; Engineering; Engineering Sciences; Exact solutions; Fiber reinforced polymers; Fixed points (mathematics); Glass fiber reinforced plastics; Homogenization; Homogenizing; Iterative methods; Laminated materials; Mathematical models; Mechanics; Mechanics of materials; Nonlinearity; Original Paper; Solvers; Theoretical and Applied Mechanics; 45G10; 65T50; Composite materials; Finite deformations; FFT; Lippmann–Schwinger equation; 74B20
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-014-1071-8

In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton–Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly 100 times less iterations than the nonlinear fixed point method. However, the Newton–Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant–Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.