Abstract
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.
Highlights
fast Fourier transform (FFT)-based solvers, as pioneered by Moulinec–Suquet [1,2], enjoy great popularity for solving homogenization problems on complex microstructures
FFT-based methods naturally support matrix-free solvers for strongly nonlinear mechanical problems, which can be decisive if memory occupancy is an issue
We demonstrate the usefulness of the proposed method as a general-purpose solver for FFT-based computational micromechanics in Sect. 4, comparing the method to a multitude of state-of-the-art numerical solvers for problems of industrial scale
Summary
FFT-based solvers, as pioneered by Moulinec–Suquet [1,2], enjoy great popularity for solving homogenization problems on complex microstructures. Schneider [39] proposed the Barzilai–Borwein method as a general-purpose method for computational micromechanics It may be interpreted as the basic scheme with adaptive time-stepping (or, equivalently, adaptive reference material). Nonlinear conjugate gradient methods were among the first general-purpose solution schemes for unconstrained optimization problems, and have been studied thoroughly, see Nocedal–Wright [42] for a comprehensive textbook treatment They are based upon an iteratively updated search direction, involving a coefficient (the “β”) to be chosen wisely and a step-size which is typically determined by line search. Quasi-Newton methods of the Broyden class, see chapter 8 in Nocedal–Wright [42], may be interpreted as generalizations of linear CG They typically outperform nonlinear conjugate gradient methods because they asymptotically accept the proposed step-size— in contrast, the most efficient conjugate gradient methods are reported to require about two line-search steps per iteration to be efficient. We demonstrate the usefulness of the proposed method as a general-purpose solver for FFT-based computational micromechanics in Sect. 4, comparing the method to a multitude of state-of-the-art numerical solvers for problems of industrial scale
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