An asymptotically compatible meshfree quadrature rule for non-local problems with applications to peridynamics
This work addresses the lack of asymptotic compatibility in peridynamic discretizations, a key bottleneck for meshfree methods in solid mechanics.
The authors propose a meshfree quadrature rule for non-local integro-differential equations that ensures asymptotic compatibility to local elasticity, achieving high-order convergence. The method is validated on manufactured solutions, crack singularities, and Kalthoff-Winkler impact experiments.
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.