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

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.