OBMeshfree: An optimization-based meshfree solver for nonlocal diffusion and peridynamics models

TL;DR

OBMeshfree is a novel meshfree solver for nonlocal diffusion and peridynamics models that accurately captures fractures and converges to local solutions, demonstrated on realistic impact problems.

Contribution

It introduces an optimization-based quadrature rule for meshfree discretization that ensures asymptotic compatibility and sharp fracture representation.

Findings

Converges to analytical nonlocal solutions

Achieves asymptotic compatibility with local theory

Successfully models high-velocity impact experiments

Abstract

We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. As numerical examples, we consider the problem of modeling both homogeneous and heterogeneous materials with nonlocal diffusion and peridynamics models. Convergences to the analytical nonlocal solution and to the local theory are demonstrated. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments. Discussions on possible immediate extensions of the code to other nonlocal diffusion and peridynamics problems are provided. OBMeshfree is freely available on GitHub.