A Multiscale Implementation Based on Adaptive Mesh Refinement for the Nonlocal Peridynamics Model in One Dimension

Abstract

Peridynamics models for solid mechanics feature a horizon parameter δ that specifies the maximum extent of nonlocal interactions. In this paper, a multiscale implementation of peridynamics models is proposed. In regions in which the displacement field is smooth, grid sizes are large relative to δ, leading to a local behavior of the models, whereas in regions containing defects, e.g., cracks, δ is larger than the grid size. Discontinuous (continuous) Galerkin finite element discretizations are used in regions where defects do (do not) occur. Moreover, in regions where no defects occur, the multiscale implementation seamlessly transitions to the use of a standard finite element discretization of a corresponding PDE model. Here, we demonstrate the multiscale implementation in a simple one-dimensional setting. An adaptive strategy is incorporated to detect discontinuities and effect grid refinement, resulting in a highly accurate and efficient implementation of peridynamics.

A Cookbook for Finite Element Methods for Nonlocal Problems, Including Quadrature Rule Choices and the Use of Approximate Balls

“…Finite element methods for nonlocal volume-constrained problems have been studied using continuous and discontinuous piecewise-linear finite element spaces and discontinuous piecewise-constant finite element spaces; see, e.g., Refs. 17,49,50,55,56. These approaches have been tested on manufactured smooth solutions (e.g., polynomial solutions).…”

Peridynamics review

“…In addition, Du et al [99] implemented the two-scale convergence model to study the homogenization of the non-local convection-diffusion equation and the statebased PD for heterogeneous media. Xu et al [100] developed a two-dimensional multiscale implementation of finite-element discretization of non-local models, see also [101]. It is noted that PD has been used for mesoscale fracture analysis in aggregate materials, specifically in multiphase cementitious composites [102].…”

An hp-Galerkin method with fast solution for linear peridynamic models in one dimension

“…The refinement in Step 3 of Algorithm 1 can be based on only h-or only p-refinement. Some discussion on how to decide whether to do h-or p-refinement in Step 3 was considered in [22].…”

“…The above adaptive algorithm allows for hp-refinement in Galerkin finite element discretizations. An adaptive hp-refinement for a finite element discretization of the peridynamic model considered here has been presented in [22]. However, adaptive mesh refinement changes the character of stiffness matrices, resulting in a loss of their block Toeplitz structure.…”

“…Here, to make the fast method effective and to improve the convergence rate, we consider two hp-Galerkin finite element discretizations based on piecewise uniform meshes. As in [22], we assume x is a point of discontinuity. The parameter x ∈ (α, β) allows the location of the discontinuity to be varied for different experiments.…”