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A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics. (English) Zbl 07229270
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.

MSC:
74S25 Spectral and related methods applied to problems in solid mechanics
74M25 Micromechanics of solids
Software:
FFTW; L-BFGS
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