Performance of Krylov subspace methods In equivariant Branch following

Abstract

The theoretical and practical performance of the Krylov subspace solvers Generalized Minimal Residuals (GMRES), Bi-Conjugate Gradients - Stabilized (BiCGSTAB), and Transpose-Free Quasi-Minimal Residual (TFQMR) used in nonlinear solver routines during branch-following analysis in the presence of symmetry is measured and evaluated. Group theory provides that along branches of symmetric configurations, the solution of the nonlinear system of equations lies in a subspace, called the fixed point space, of the full configuration space. Due to equivariance, the incremental linear system solutions as well as the itermediate approximations of the solution also lie within the fixed point space when using exact arithmetic. Consequently, Krylov subspace methods are expected to converge in a number of iterations commensurate with the dimension of the fixed point space. However, rounding errors introduced by computing in a floating-point number environment cause the behavior in practice. Branch following analyses of both an infinite strip wrinkling under compression and of a loaded structural dome using Krylov solvers are performed. For the compressed strip, a spectral formulation is developed. For this case, Krylov subspace methods converge in many fewer iterations than what is predicted from group theory. In contrast, convergence for the structural dome requires more iterations than theoretically required. In all cases, consistent early convergence of Krylov methods is observed relative to the full dimension of the solution space without explicitly modifying the calculation process to account for symmetry, suggesting that symmetric systems provide a class of applications where Krylov subspace methods offer intrinsic advantages. Row scaling of the linear systems was shown to improve convergence for one of the test cases. Other techniques for adjusting Krylov subspace solver behavior, such as restarting and preconditioning, warrant further investigation owing to the strong performance of these solvers in the set of benchmark activities performed in this work. The spectral formulation developed for the compressed infinite strip is not found elsewhere in literature. In this work it is shown that this formulation provides a feasible alternative to the typical Finite Element Analysis (FEA) approach, especially when employing Jacobian-Free Newton-Krylov approaches that mitigate the expensive task of forming the dense Jacobian. An explanation of the approach and design rationale for the software implemented to support this work is provided, including classes established, approach to parallel processing, and use of symbolic computing.