A Note on Automatic Kernel Carpentry for Atomistic Support of Continuum Stress

TL;DR

This paper introduces an adaptive kernel method using local linear regression for atomistic stress estimation in multiscale modeling, improving accuracy at complex features but with higher computational costs.

Contribution

It applies automatic kernel carpentry with local linear regression to adaptively weight atoms for stress calculation, enhancing accuracy at interfaces and faults.

Findings

Local linear regression improves stress estimation at complex features.

Adaptive weighting offers better accuracy than isotropic kernels.

Computational costs are higher without significant benefits for crystalline solids.

Abstract

Research within the field of multiscale modelling seeks, amongst other questions, to reconcile atomistic scale interactions with thermodynamical quantities (such as stress) on the continuum scale. The estimation of stress at a continuum point on the atomistic scale requires a pre-defined kernel function. This kernel function derives the stress at a continuum point by averaging the contribution from atoms within a region surrounding the continuum point. Commonly the kernel weight assignment is isotropic: an identical weight is assigned to atoms at the same spatial distance, which is tantamount to a local constant regression model. In this paper we employ a local linear regression model and leverage the mechanism of automatic kernel carpentry to allow for spatial averaging adaptive to the local distribution of atoms. As a result, different weights may be assigned to atoms at the same spatial distance. This is of interest for determining atomistic stress at stacking faults, interfaces or surfaces. It is shown in this study that for crystalline solids, although the local linear regression model performs elegantly, the additional computational costs are not justified compared to the local constant regression model.