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

Peridynamics models for solid mechanics feature a horizon parameter $\delta$ 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 $\delta$, leading to a local behavior of the models, whereas in regions containing defects, e.g., cracks, $\delta$ 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.