Convergence, adaptive refinement, and scaling in 1D peridynamics

Authors

Florin Bobaru Ph.D., University of Nebraska at LincolnFollow
Mijia Yabg Ph.D., University of Nebraska at Lincoln
Leonardo F. Alves M.S., University of Nebraska at Lincoln
Stewart A. Silling Ph.D., Sandia National Laboratories
Ebrahim Askari Ph.D., Boeing Co., Bellevue, WA
Jifeng Xu Ph.D., Boeing Co., Bellevue, WA

Date of this Version

2009

Comments

Published in International Journal for Numerical Methods in Engineering 77 (2009), pp. 852–877. Published by Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/nme.2439
This article is a U.S. government work and is not subject to copyright in the United States.

Abstract

We introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed (in J. Mech. Phys. Solids 2000; 48:175–209) as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes.