A Priori and A Posteriori Analysis of the Quasi-Nonlocal Quasicontinuum Method in 1D

TL;DR

This paper provides a comprehensive analysis of the quasi-nonlocal quasicontinuum method in 1D, establishing its accuracy and stability for large deformations through novel a priori and a posteriori estimates.

Contribution

It introduces new stability estimates and error analyses for the QNL-QC method, ensuring a reliable correspondence with atomistic solutions under large deformations.

Findings

Guarantees a one-to-one correspondence between atomistic and QNL-QC solutions

Validates the method's effectiveness for large deformations

Provides novel error and stability estimates

Abstract

For a next-nearest neighbour pair interaction model in a periodic domain, a priori and a posteriori analyses of the quasinonlocal quasicontinuum method (QNL-QC) are presented. The results are valid for large deformations and essentially guarantee a one-to-one correspondence between atomistic solutions and QNL-QC solutions. The analysis is based on truncation error and residual estimates in negative norms and novel a priori and a posteriori stability estimates.