Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

TL;DR

This paper develops an asymptotically compatible reproducing kernel collocation method for the peridynamic Navier equation, ensuring convergence to local and nonlocal limits, with stability analysis and numerical validation.

Contribution

It introduces a novel RK collocation scheme for the peridynamic Navier equation that guarantees asymptotic compatibility and convergence to the correct limits.

Findings

The scheme converges to both nonlocal and local limits.

Stability is established via Fourier analysis.

Numerical experiments validate theoretical convergence.

Abstract

In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. We then apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lam\'{e} parameters satisfy $\lambda \geq \mu$ to avoid adding extra constraints on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.