Analysis of a Predictor-Corrector Method for Computationally Efficient Modeling of Surface Effects in 1D

TL;DR

This paper introduces a predictor-corrector approach that enhances the Cauchy--Born method for 1D materials by accurately modeling surface effects, which are crucial at small scales.

Contribution

A novel corrector method is developed to improve surface effect predictions in the Cauchy--Born framework for 1D materials.

Findings

The method accurately captures surface effects within a boundary layer.

Error estimates show the correction improves predictions at small scales.

The approach is validated through theoretical analysis.

Abstract

The regular Cauchy--Born method is a useful and efficient tool for analyzing bulk properties of materials in the absence of defects. However, the method normally fails to capture surface effects, which are essential to determining material properties at small length scales. In this paper, we present a corrector method that improves upon the prediction for material behavior from the Cauchy--Born method over a small boundary layer at the surface of a 1D material by capturing the missed surface effects. We justify the separation of the problem into a bulk response and a localized surface correction by establishing an error estimate, which vanishes in the long wavelength limit.