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 \(\mathcal {O}(N)\) where N implies the number of the unknowns.

近场动力学是一种非局部连续介质理论，它使用积分方程，不假设位移场的可微性。近场动力学系统隐式类型方案的优点之一是它们可以保证数值解的平衡，而不管位移场上存在的不连续性如何。在这项工作中，我们提出了用于静态近场动力学系统隐式离散化的多重网格算法。重点是在粗略级别选择水平参数。在一些版本中，在粗略级别固定水平大小的多重网格算法被证明是最有效的。在这种情况下，粗略级别的平滑（例如 Jacobi 或 Gauss-Seidel）的计算复杂度随着级别的降低而降低 1/16。为了提高多重网格算法的效率，插值算子在底层域上的裂缝附近被修改，以便类似能量的范数不会在裂缝附近爆炸。在数值实验中，我们报告了由我们的多重网格算法 (MG-PCG) 预处理的预处理共轭梯度 (PCG) 的性能。MG-PCG 的计算复杂度为\(\mathcal {O}(N)\)其中N表示未知数的数量。