A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromech...

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Bibliographic Details
Published in:Computational mechanics 2020-07, Vol.66 (1), p.239-257
Main Author: Schneider, Matti
Format: Article
Language:English
Subjects:
Classical and Continuum Physics
Computational Science and Engineering
Conjugates
Context
Damping
Engineering
Fast Fourier transformations
Fourier transforms
Micromechanics
Original Paper
Parameters
Search methods
Solvers
Theoretical and Applied Mechanics
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Summary:For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically.
ISSN:0178-7675
1432-0924
DOI:10.1007/s00466-020-01849-7