Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients

Abstract

a b s t r a c tIn this short note, we present a new technique to accelerate the convergence of a FFT-based solver for numerical homogenization of complex periodic media proposed by Moulinec and Suquet [1]. The approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to Vainikko [2], to give a linear system which can be efficiently solved by conjugate gradient methods. Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration.

A finite element perspective on nonlinear FFT-based micromechanical simulations

“…As follows from earlier developments [13,20], it is convenient to use the fundamental trigonometric polynomials defined on the grid Z 2 N , e.g. [25,Chapter 8],…”

“…where the matrix Γ ref is block-diagonal in the Fourier space; see [13] for a more detailed explanation and Appendix Appendix B for the matrix representations. The remaining steps in the solution of the non-linear system (37) now closely follow those of Sections 2.5 and 2.6, once the projection matrix G is replaced with Γ ref , including the fact that Γ ref enforces nodal equilibrium and strain compatibility.…”

“…For this choice, the number of iterations to reach the given tolerance, η CG in Algorithm 1, grows linearly with the condition number λ max (i) /λ min (i) . On the other hand, when the linear system (38) is solved with, e.g., Conjugate Gradients as proposed by Zemanet al [13] for linear problems and by Gélébart and Mondon-Cancel [34] for non-linear problems, it suffices for the convergence of the algorithm that the condition number is finite. This in turn implies that the CG method works for any choice of reference media, no updates of C ref during the Newton increment are needed, and the number of iterations to reach the given accuracy η CG grows as λ max Even though the collocation and variational approaches become equivalent for specific choices of the reference stiffness tensor C ref and the iterative solver, the variational formulation offers at least two advantages.…”

“…In particular, Zemanet al [13] found the integral formulation to be equivalent to a spectral collocation method resulting in a fully populated system of linear equations with a sparse representation; the convergence of approximate solutions for non-smooth coefficients has been proven by Vondřejc [14, pages 116-117] and by Schneider [15]. The original Moulinec-Suquet scheme is recovered when solving the system by the Richardson iteration [16], but other low-memory iterative solvers, such as conjugate gradients, offer even better performance.…”

Nanoindentation Based Analysis of Heterogeneous Structural Materials

“…The system can be resolved e.g. by the conjugate gradient method as proposed by Zeman et al (Zeman et al, 2010). Elastic constants received from grid nanoindentation have been used as input parameters for this FFT homogenization with the assumption of plane strain conditions.…”