a b s t r a c tThe quasicontinuum (QC) method coarse-grains crystalline atomic ensembles in order to bridge the scales from individual atoms to the micro-and mesoscales. A crucial cornerstone of all QC techniques, summation or quadrature rules efficiently approximate the thermodynamic quantities of interest. Here, we investigate summation rules for a fully nonlocal, energy-based QC method to approximate the total Hamiltonian of a crystalline atomic ensemble by a weighted sum over a small subset of all atoms in the crystal lattice. Our formulation does not conceptually differentiate between atomistic and coarsegrained regions and thus allows for seamless bridging without domain-coupling interfaces. We review traditional summation rules and discuss their strengths and weaknesses with a focus on energy approximation errors and spurious force artifacts. Moreover, we introduce summation rules which produce no residual or spurious force artifacts in centrosymmetric crystals in the large-element limit under arbitrary affine deformations in two dimensions (and marginal force artifacts in three dimensions), while allowing us to seamlessly bridge to full atomistics. Through a comprehensive suite of examples with spatially non-uniform QC discretizations in two and three dimensions, we compare the accuracy of the new scheme to various previous ones. Our results confirm that the new summation rules exhibit significantly smaller force artifacts and energy approximation errors. Our numerical benchmark examples include the calculation of elastic constants from completely random QC meshes and the inhomogeneous deformation of aggressively coarse-grained crystals containing nano-voids. In the elastic regime, we directly compare QC results to those of full atomistics to assess global and local errors in complex QC simulations. Going beyond elasticity, we illustrate the performance of the energy-based QC method with the new second-order summation rule by the help of nanoindentation examples with automatic mesh adaptation. Overall, our findings provide guidelines for the selection of summation rules for the fully nonlocal energy-based QC method.

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.

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a b s t r a c tSurface effects in nanoscale mechanical systems such as nanoporous solids or small-scale structures can have a significant impact on the effective material response which deviates from the material behavior of bulk solids. Understanding such phenomena requires modeling techniques that locally retain atomistic information while transitioning to the relevant macroscopic length scales. We recently introduced a fully-nonlocal energy based quasicontinuum (QC) method equipped with new summation rules. This technique accurately bridges across scales from atomistics to the continuum through a thermodynamicallyconsistent coarse-graining scheme. Beyond minimizing energy approximation errors and spurious force artifacts, the new method also qualifies to describe free surfaces, which is reported here. Surfaces present a major challenge to coarse-grained atomistics, which has oftentimes been circumvented by costly ad hoc extensions of the traditional QC method. We show that our new coarse-graining scheme successfully and automatically reduces spurious force artifacts near free surfaces. After discussing the computational model, we demonstrate its benefits in the presence of free surfaces by several nanomechanical examples including surface energy calculations, elastic size effects in nano-rods and in plates with nano-sized holes. Overall, we demonstrate the importance of surface effects as well as a new strategy to accurately capture those computationally via coarse-grained atomistics.

A new energy-based quasicontinuum formulation is presented which is based on sampling the crystal energy at carefullychosen lattice sites and which allows for efficiently bridging from the atomistic to the continuum length scale. The presented technique is applied to experiments of nanoindentation whose microstructure-induced size effects can now be studied with full atomistic detail at the micrometer scale without the necessity of phenomenological material models.