Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

Abstract

Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. T. B. Jönsthövel (B)·

Preconditioning

“…The deflation idea is most effective when the problem naturally presents such a space. For example, see Brandt and Ta'asan (1986), who deflate the low-frequency eigenspaces which correspond to negative eigenvalues for a Helmholtz problem, and Jonsthovel et al (2012), who describe a prototypical application involving composite materials. However, various subspaces which arise from iterations of a previous linear system solve, or other sources, can be used.…”

A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

“…The deflation technique to accelerate the convergence of Krylov subspace methods for the solution of a given linear system has been known for a long time; see, e.g., [14,15,44] and the extensive bibliography proposed in [29]. Applications to structural mechanics have been provided in, e.g., [32,33] in the symmetric positive definite case. In the last decade, deflation has been used and analyzed in combination with multigrid and domain decomposition methods, which results in efficient algorithms [32,62].…”

“…Applications to structural mechanics have been provided in, e.g., [32,33] in the symmetric positive definite case. In the last decade, deflation has been used and analyzed in combination with multigrid and domain decomposition methods, which results in efficient algorithms [32,62]. Extensions to nonsymmetric or non-Hermitian problems have been provided in, e.g., [20][21][22]30].…”

“…Deflated and augmented Krylov subspaces [14,15,44,57] or Krylov subspace methods with recycling [26,36,50,53,54,61,67] have been proposed in this setting. Applications in structural mechanics have been provided in, e.g., [32,33] in the symmetric positive definite case. We refer the reader to [15,21,22,30,60] for a comprehensive theoretical overview on these methods and to references therein for a summary of applications, where the relevance of these methods has been shown.…”

Efficient Two-Level Preconditioned Conjugate Gradient Method on the GPU

“…More recently in [3] also multigrid has been investigated for solving Poisson type problems. In [11] a comparative study is presented between deflation and multigrid. It shows that the former is a competitive technique in comparison with the latter.…”