On the prescription of boundary conditions for nonlocal Poisson's and peridynamics models

TL;DR

This paper presents a novel, efficient method to convert local boundary conditions into nonlocal volume constraints for nonlocal Poisson and peridynamics models, ensuring quadratic convergence and broad applicability.

Contribution

It introduces an asymptotically compatible technique that automatically converts local boundary data into nonlocal constraints without geometric restrictions.

Findings

Quadratic convergence of the nonlocal solution to the local solution as horizon vanishes

Method is geometry- and dimension-independent with negligible computational cost

Validated through 2D numerical experiments on Poisson and peridynamic models

Abstract

We introduce a technique to automatically convert local boundary conditions into nonlocal volume constraints for nonlocal Poisson's and peridynamic models. The proposed strategy is based on the approximation of nonlocal Dirichlet or Neumann data with a local solution obtained by using available boundary, local data. The corresponding nonlocal solution converges quadratically to the local solution as the nonlocal horizon vanishes, making the proposed technique asymptotically compatible. The proposed conversion method does not have any geometry or dimensionality constraints and its computational cost is negligible, compared to the numerical solution of the nonlocal equation. The consistency of the method and its quadratic convergence with respect to the horizon is illustrated by several two-dimensional numerical experiments conducted by meshfree discretization for both the Poisson's problem and the linear peridynamic solid model.