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.

中文翻译：

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非线性共轭梯度方法在基于 FFT 的计算微力学中的应用的动力学视图

对于基于快速傅里叶变换 (FFT) 的计算微观力学，求解器需要快速、高效内存且独立于繁琐的参数校准。在这项工作中，我们研究了非线性共轭梯度 (CG) 方法在基于 FFT 的计算微观力学背景下的好处。传统上，非线性 CG 方法需要专用的线搜索程序才能有效，这使得它们在基于 FFT 的环境中没有竞争力。我们通过利用非线性 CG 方法和加速梯度方法之间的相似性，为没有线搜索的非线性 CG 方法做出贡献。更准确地说，通过让步长变为零，我们将 Fletcher-Reeves 非线性 CG 展示为具有状态相关非线性阻尼的动态系统。我们展示了如何为基于 FFT 的计算微力学实现非线性 CG 方法，