Finite element approximation of nonlocal fracture models

TL;DR

This paper analyzes the convergence of finite element methods for nonlocal fracture models, establishing solution existence, energy stability, and error rates for time-stepping schemes in peridynamics.

Contribution

It introduces a finite element approximation framework for nonlocal nonlinear potentials, providing convergence rates and stability analysis for the first time.

Findings

Convergence rate of $C_t riangle t + C_s h^2/7^2$ for FE approximations.

Existence of $H^2$ solutions over finite time intervals.

Energy stability of semi-discrete time schemes and CFL-like condition for nonlinearity.

Abstract

We consider nonlocal nonlinear potentials and estimate the rate of convergence of time stepping schemes to the peridynamic equation of motion. We begin by establishing the existence of $H^2$ solutions over any finite time interval. Here spatial approximation by finite element interpolations are considered. The energy stability of the associated semi-discrete time stepping scheme is established and the approximation of strong and weak formulations of the evolution using FE interpolations of $H^2$ solutions are investigated. The strong and weak form of approximations are shown to converge to the actual solution in the mean square norm at the rate $C_t\Delta t +C_s h^2/\epsilon^2$ where $h$ is the mesh size, $\epsilon$ is the size of nonlocal interaction and $\Delta t$ is the time step. The constants $C_t$ and $C_s$ are independent of $\Delta t$, and $h$. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed.