Effective multigrid algorithms for algebraic system arising from static peridynamic systems

Abstract

Peridynamics is a nonlocal continuum theory that uses integral equations with no assumption of differentiability of displacement fields. One of the advantages of implicit type schemes for peridynamic systems is that they guarantee the equilibrium of the numerical solution, regardless of existing discontinuities on displacement fields. In this work, we propose multigrid algorithms for implicit discretizations of static peridynamic systems. The considerations are emphasized on the selection of horizon parameters at coarse levels. Among some versions, the multigrid algorithms which fix the horizon size at the coarse level are shown to be most efficient. In such case, the computational complexity of a smoothing (say Jacobi or Gauss-Seidel) at coarse level decreases by 1/16 as the level decreases. To enhance the efficiency of multigrid algorithms, the interpolation operator is modified near the crack on the underlying domain so that the energy-like norm does not blow up near the crack. In the numerical experiments, we report the performance of preconditioned conjugated gradient (PCG) preconditioned by our multigrid algorithms (MG-PCG). The computational complexity of MG-PCG is O(N) where N implies the number of the unknowns. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.