An asymptotically compatible meshfree quadrature rule for non-local problems with applications to peridynamics

TL;DR

This paper introduces a meshfree quadrature rule for non-local equations that ensures asymptotic compatibility and high-order convergence, improving peridynamic simulations without the need for background meshes.

Contribution

The paper develops a novel meshfree quadrature rule that guarantees asymptotic compatibility in peridynamics, handling complex boundary conditions and damage modeling.

Findings

Achieves high-order convergence to local solutions

Handles traction-free conditions and surface effects

Successfully reproduces impact experiment results

Abstract

We present a meshfree quadrature rule for compactly supported non-local integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a strong-form meshfree discretization of a peridynamic solid mechanics model that requires no background mesh. Existing discretizations of peridynamic models have been shown to exhibit a lack of asymptotic compatibility to the corresponding linearly elastic local solution. By posing the quadrature rule as an equality constrained least squares problem, we obtain asymptotically compatible convergence via reproducability constraints. Our approach naturally handles traction-free conditions, surface effects, and damage modeling for both static and dynamic problems. We demonstrate high-order convergence to the local theory by comparing to manufactured solutions and to cases with crack singularities for which an analytic solution is available. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments.