Computational Mechanics

, Volume 54, Issue 6, pp 1497–1514 | Cite as

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

  • Matthias KabelEmail author
  • Thomas Böhlke
  • Matti Schneider
Original Paper


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.


Composite materials Finite deformations Lippmann–Schwinger equation FFT 

Mathematics Subject Classification

74B20 45G10 65T50 



The authors benefited from many fruitful discussions with Heiko Andrä and Andreas Günnel.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthias Kabel
    • 1
    Email author
  • Thomas Böhlke
    • 2
  • Matti Schneider
    • 3
  1. 1.Department of Flow and Material SimulationFraunhofer ITWMKaiserslauternGermany
  2. 2.Intitute of Engineering MechanicsKarlruhe Institute of TechnologyKarlsruheGermany
  3. 3.Faculty of Mechanical EngineeringChemnitz University of TechnologyChemnitzGermany

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