A Caltech Library Service

Summation rules for a fully nonlocal energy-based quasicontinuum method

Amelang, J. S. and Venturini, G. N. and Kochmann, D. M. (2015) Summation rules for a fully nonlocal energy-based quasicontinuum method. Journal of the Mechanics and Physics of Solids, 82 . pp. 378-413. ISSN 0022-5096.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


The 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 coarse-grained 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Kochmann, D. M.0000-0002-9112-6615
Additional Information:© 2015 Elsevier. Received 17 August 2014; Received in revised form 15 February 2015; Accepted 22 March 2015; Available online 30 March 2015. The authors gratefully acknowledge support from the Department of Energy's National Nuclear Security Administration (NNSA) under Award no. DE-FC52-08NA28613 as well as from the National Science Foundation (NSF) under Grant no. CMMI-123436.
Funding AgencyGrant Number
Department of Energy (DOE) National Nuclear Security AdministrationDE-FC52-08NA28613
Subject Keywords:Atomistics; Quasicontinuum; Multiscale modeling; Error analysis
Record Number:CaltechAUTHORS:20151023-142752873
Persistent URL:
Official Citation:J.S. Amelang, G.N. Venturini, D.M. Kochmann, Summation rules for a fully nonlocal energy-based quasicontinuum method, Journal of the Mechanics and Physics of Solids, Volume 82, September 2015, Pages 378-413, ISSN 0022-5096, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:61506
Deposited By: Tony Diaz
Deposited On:26 Oct 2015 20:59
Last Modified:24 Nov 2020 00:43

Repository Staff Only: item control page