On The Convergence of the Discretized Linear Static State-Based Peridynamic Equations

TL;DR

This paper analytically proves the convergence of discretized linear state-based static peridynamic equations to their continuous counterparts, including practical discretization methods like meshfree lattice approaches.

Contribution

It provides the first rigorous convergence proofs for discretized peridynamic models, including practical quadrature and meshfree discretizations.

Findings

Convergence proven for discretized solutions to continuous peridynamic equations.

Inclusion of one-point quadrature in discretization maintains convergence.

Results applicable even with discontinuities in material parameters and external forces.

Abstract

In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply one-point-quadrature to the terms in the discrete equations. The resulting system coincides with the commonly used meshfree discretization using a regular lattice, including the possibility of using partial area algorithms to improve the numerical behavior. We again prove convergence, this time for fixed choices of a weighting function commonly used in literature and stronger assumptions on the input data. We note however, that these assumptions are not significantly restrictive for practical purposes. In particular, they still allow discontinuities in the material parameters and external body forces.