A robust adaptive algebraic multigrid linear solver for structural mechanics

Abstract

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.

Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

“…To show that it is also convenient in terms of overall efficiency we carried out a comparison with a state-of-the-art parallel preconditioned solver for SPD linear system. It is the solver chronos, available at the webpage https://www.m3eweb.it/chronos/, which makes use of an enhanced AMG solver, partially based on a FSAI smoother with dynamical nonzero pattern selection 20,21 In Table 8, we report the results in solving the FD matrix with nx = 512 for the PCG method accelerated with either the AMG or the FSAI preconditioners, after some trials to select the optimal parameters. Since the setup time to evaluate the preconditioner is rather high for this approach we reported this in the table as T setup while the CPU time for the PCG solution is T solver .…”

A General-Purpose AMG Linear Solver for High Performance Computing

“…with I the identity matrix and ω a relaxation factor necessary whenever ρ(M −1 A) > 2, see e.g. [6] for an explanation. Typilly, the smoother is given by a simple pointwise relaxation method such as (block)…”

“…By distinction to Gauss-Seidel smoother, aFSAI application is perfectly parallel also in the application as, giving an explicit approximation of the system inverse, it can be applied simply by a matrix-vector product. The price to pay for the use of aFSAI is a not always negligible set-up cost that is usually compensated by a faster covergence, especially in ill-conditioned problems [12,6], where standard smoother fail in dumping high frequencies.…”

“…To overcome this issue, Chronos also makes available the so-called BAMG prolongation whose coefficients are computed by fitting a set of test vectors through a least square minimization process. The test vectors must represent, at least approximately, the near null space of the operator and, in the case of elasticity, they can be guessed using the Rigid Body Modes (RBM) of the free body [6].…”

“…In this work, we propose Chronos a massively parallel implementation of a novel AMG framework [12,6] which is able to adapt all of its components to the problem at hand, from the smoother set-up, to the coarse grid hierarchy and prolongation definition. This is achieved by guessing and iteratively improving in a bootstrap fashion the near-null space of the system, which allows for both testing the smoother and the prolongation operator as well as for inferring the connection strengths between system unknowns.…”