Automatic adaptivity in the fully nonlocal quasicontinuum method for coarse‐grained atomistic simulations

Abstract

The quasicontinuum (QC) method is a concurrent scale-bridging technique that extends atomistic accuracy to significantly larger length scales by reducing the full atomic ensemble to a small set of representative atoms and using interpolation to recover the motion of all lattice sites where full atomistic resolution is not necessary. While traditional QC methods thereby create interfaces between fully-resolved and coarsegrained regions, the recently introduced fully-nonlocal QC framework does not fundamentally differentiate between atomistic and coarsened domains. Adding adaptive refinement enables us to tie atomistic resolution to evolving regions of interest such as moving defects. However, model adaptivity is challenging because large particle motion is described based on a reference mesh (even in the atomistic regions). Unlike in the context of, e.g., finite element meshes, adaptivity here requires that (i) all vertices lie on a discrete point set (the atomic lattice), (ii) model refinement is performed locally and provides sufficient mesh quality, and (iii) Verlet neighborhood updates in the atomistic domain are performed against a Lagrangian mesh. With the suite of adaptivity tools outlined here, the nonlocal QC method is shown to bridge across scales from atomistics to the continuum in a truly seamless fashion, as illustrated for nanoindentation and void growth.

A quasicontinuum theory for the nonlinear mechanical response of general periodic truss lattices

“…A key benefit of the fully-nonlocal QC method over other concurrent scale-bridging techniques is that no conceptual distinction is made between the fully-resolved and coarse-grained regions, allowing for automatic adaptive refinement of the discretization down to the fully-resolved limit. Note that we here constrain ourselves to adaptive refinement without coarsening (see the discussion in Tembhekar et al (2017)). Adaptivity thus requires a refinement criterion and a geometric refinement algorithm.…”

“…It is also used in Section 3.3 as a general measure of the localized deformation. Once all coarse-grained elements have been identified for refinement, we apply a longest-edge bisection algorithm (Tembhekar et al, 2017) to insert new repUCs with the complicating constraint that all repUCs are to be located on valid lattice sites. To this end, we insert a new repUC at that vacant UC location nearest to the mid-point of the longest edge of each element to be refined.…”

A Posteriori Error Estimation and Adaptive Algorithm for Atomistic/Continuum Coupling in Two Dimensions

“…We will choose free parameters α and α K in the subsequent analysis to keep the estimate of the stability constant sharp. Applying (49) and (50) to (47), taking the same α and α K for each bond b P B and using the fact that |ρ b | " 1 , we obtain…”

“…Recently, Kochmann et al [47] proposed an adaptivity strategy for the so-call "fully-nonlocal quasicontinuum" method which apply a discrete model in the entire computational domain without coupling of different models. This approach aims to minimize the ghost force rather than eliminate it as in the consistent a/c coupling method.…”

A refinement indicator for adaptive quasicontinuum approaches for structural lattices

“…Adaptivity of QC approaches is mainly investigated in the context of atomistic lattices for metals 3,18,30‐32 . Error estimators were proposed from a mathematical point of view for the force based atomistic QC by Dobson et al, 33 Arndt and Luskin 34 and for the energy based atomistic QC by Wang 35 and Ortner and Wang 36 .…”