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

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.

Lippmann‐Schwinger solvers for the computational homogenization of materials with pores

“…Of particular interest to us is the strain field within the pores, where it is devoid of any physical significance. Rather, as predicted by the theory of Section 2.2, it corresponds to the elastic extension (12). For all three discretizations, the strain fields within the pores appear similar if we ignore the artifacts for the moment.…”

“…Put differently, vector fields in V may take arbitrary values in Y 1 , and their boundary values on the interface I are extended elastically (12) into the pore space Y 0 .…”

“…3. The interested reader might wonder about the form of the elastic extension operator (12). The results of this section may be extended to the more general inner product…”

“…Here, the key technical argument is based on the (trivial) fact that finite elements admit well-defined boundary values on the interface I h , that is, nodal values, and that the definition of V h (36) involves a discretized variant of the elastic extension (12).…”

“…The first class is derived from the basic scheme, and operates on the displacement 8 (or, equivalently, on compatible strain fields). These solvers include Newton's method, [8][9][10] the linear and nonlinear conjugate gradient methods, 11,12 fast gradient methods, 13,14 the Barzilai-Borwein scheme, 15 the recursive projection method, 16 the Anderson-accelerated basic scheme, 17 and other Quasi-Newton methods. 18 The second class of solution methods are the polarization schemes, which operate on so-called polarization fields which are neither compatible nor equilibrated.…”