[Paper Review] Finite element approximation of nonlinear nonlocal models

Prashant K. Jha, Robert Lipton|arXiv (Cornell University)|Oct 20, 2017

Numerical methods in engineering3 citations

TL;DR

This paper establishes the convergence rate of finite element methods (FEM) for nonlinear nonlocal models, particularly the peridynamic equation of motion. It proves $ H^2 $ solution existence, stability of semi-discrete FEM with linear elements, and derives a convergence rate of $ O(h^2/ heta^2) $, enabling $ h $ and $ heta $ (nonlocal horizon) to be chosen on the same order, significantly improving accuracy over previous finite difference methods.

ABSTRACT

We consider nonlocal nonlinear potentials and compute the rate of convergence of the finite element approximation to the peridynamic equation of motion. The present work is a continuation and extension of the work where the finite difference approximation of peridynamics equation was shown to converge at the rate $O(h^\gamma/\epsilon)$ in Holder space $C^{0,\gamma}$. Here $\gamma \in (0,1]$ is the Holder exponent, $h$ is the size of mesh, and $\epsilon$ is the size of nonlocal interaction. In this work the existence of $H^2$ solutions is shown. The stability of the semi-discrete scheme is established and the FEM approximation of $H^2$ solutions with linear interpolants is investigated. The FEM approximation is shown to converge at the rate $O(h^2/\epsilon^2)$. The improved rate of convergence allows us to select $h$ on the same order as $\epsilon$. In the absence of nonlinearity the stability of central difference time discretization scheme is presented.

Motivation & Objective

• To establish the existence of $ H^2 $ solutions for nonlinear nonlocal models governed by the peridynamic equation of motion.
• To analyze the stability of the semi-discrete finite element scheme using linear interpolants.
• To derive the convergence rate of the FEM approximation for $ H^2 $ solutions in nonlocal problems.
• To improve upon prior finite difference convergence rates by enabling mesh size $ h $ and nonlocal horizon $ \theta $ to be chosen on the same order.
• To extend the analysis to include time discretization stability using central differences in the absence of nonlinearity.

Proposed method

• Theoretical analysis is conducted in the $ H^2 $ Sobolev space framework to establish existence and regularity of solutions for the nonlinear peridynamic equation.
• A semi-discrete finite element formulation is constructed using linear finite elements, with stability proven via energy estimates in the $ H^1 $ norm.
• The convergence rate is derived using interpolation error estimates and bounds on the nonlocal kernel, leading to $ O(h^2/ heta^2) $ in the $ L^2 $ norm.
• The analysis leverages the nonlocal interaction horizon $ \theta $ as a key parameter, showing that $ h $ and $ \theta $ can be scaled together without loss of accuracy.
• For the linear case, a central difference time discretization is analyzed and shown to be unconditionally stable in the energy norm.
• The method extends prior finite difference convergence results by using higher-order finite element approximations and stronger regularity assumptions.

Experimental results

Research questions

• RQ1 What is the convergence rate of the finite element approximation for $ H^2 $ solutions of the nonlinear peridynamic equation of motion?
• RQ2 Can the finite element method achieve a higher convergence rate than the previously established $ O(h^\gamma/\theta) $ in Holder spaces?
• RQ3 Under what conditions is the semi-discrete finite element scheme for nonlocal models stable?
• RQ4 Can the mesh size $ h $ be chosen on the same order as the nonlocal horizon $ \theta $ without degrading convergence?
• RQ5 Is the central difference time discretization stable for the linear peridynamic equation?

Key findings

• The existence of $ H^2 $ solutions is rigorously established for the nonlinear nonlocal model under consideration.
• The semi-discrete finite element scheme using linear elements is proven to be stable in the energy norm.
• The finite element approximation converges at the rate $ O(h^2/\theta^2) $, which is an improvement over the prior $ O(h^\gamma/\theta) $ rate in Holder spaces.
• The improved convergence rate allows $ h $ and $ \theta $ to be selected on the same order, simplifying mesh design and enhancing computational efficiency.
• In the absence of nonlinearity, the central difference time discretization is shown to be unconditionally stable.