{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T06:16:28Z","timestamp":1771481788123,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2015,11,19]],"date-time":"2015-11-19T00:00:00Z","timestamp":1447891200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2016,10]]},"DOI":"10.1007\/s00211-015-0772-z","type":"journal-article","created":{"date-parts":[[2015,11,19]],"date-time":"2015-11-19T08:34:26Z","timestamp":1447922066000},"page":"275-326","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Analysis of blended atomistic\/continuum hybrid methods"],"prefix":"10.1007","volume":"134","author":[{"given":"Xingjie Helen","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christoph","family":"Ortner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander V.","family":"Shapeev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brian","family":"Van Koten","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,11,19]]},"reference":[{"key":"772_CR1","doi-asserted-by":"crossref","unstructured":"Miller, R., Tadmor, E.: A unified framework and performance benchmark of fourteen multiscale atomistic\/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17 (2009)","DOI":"10.1088\/0965-0393\/17\/5\/053001"},{"key":"772_CR2","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/j.cma.2013.10.007","volume":"268","author":"XH Li","year":"2014","unstructured":"Li, X.H., Luskin, M., Ortner, C., Shapeev, A.V.: Theory-based benchmarking of blended force-based quasicontinuum method. Comput. Methods Appl. Mech. Eng. 268, 763\u2013781 (2014)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"772_CR3","doi-asserted-by":"crossref","unstructured":"Luskin, M., Ortner, C., Van Koten, B.: Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253 (2013)","DOI":"10.1016\/j.cma.2012.09.007"},{"issue":"5","key":"772_CR4","doi-asserted-by":"crossref","first-page":"2182","DOI":"10.1137\/10081071X","volume":"49","author":"B Van Koten","year":"2011","unstructured":"Van Koten, B., Luskin, M.: Analysis of energy-based blended quasi-continuum approximations. SIAM J. Numer. Anal. 49(5), 2182\u20132209 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"772_CR5","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/07069969X","volume":"7","author":"S Badia","year":"2008","unstructured":"Badia, S., Parks, M.L., Bochev, P.B., Gunzburger, M., Lehoucq, R.B.: On atomistic-to-continuum coupling by blending. SIAM J. Multiscale Model. Simul. 7(1), 381\u2013406 (2008)","journal-title":"SIAM J. Multiscale Model. Simul."},{"issue":"4","key":"772_CR6","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s00466-008-0291-1","volume":"42","author":"PT Bauman","year":"2008","unstructured":"Bauman, P.T., Ben Dhia, H., Elkhodja, N., Oden, J.T., Prudhomme, S.: On the application of the Arlequin method to the coupling of particle and continuum models. Comput. Mech. 42(4), 511\u2013530 (2008)","journal-title":"Comput. Mech."},{"key":"772_CR7","doi-asserted-by":"crossref","unstructured":"Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193(17-20), 1645\u20131669 (2004)","DOI":"10.1016\/j.cma.2003.12.053"},{"key":"772_CR8","doi-asserted-by":"crossref","unstructured":"Li, X.H., Luskin, M., Ortner, C.: Positive-definiteness of the blended force-based quasicontinuum method. SIAM J. Multiscale Model. Simul. 10, 1023\u20131045 (2012)","DOI":"10.1137\/110859270"},{"issue":"1","key":"772_CR9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1002\/cpa.21429","volume":"66","author":"J Lu","year":"2013","unstructured":"Lu, J., Ming, P.: Convergence of a force-based hybrid method in three dimensions. Commun. Pure Appl. Math. 66(1), 83\u2013108 (2013)","journal-title":"Commun. Pure Appl. Math."},{"key":"772_CR10","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1615\/IntJMultCompEng.v1.i1.100","volume":"1","author":"T Belytschko","year":"2003","unstructured":"Belytschko, T., Xiao, S.P.: Coupling methods for continuum model with molecular model. Int. J. Multiscale Comp. Eng. 1, 115\u2013126 (2003)","journal-title":"Int. J. Multiscale Comp. Eng."},{"issue":"45\u201348","key":"772_CR11","doi-asserted-by":"crossref","first-page":"4548","DOI":"10.1016\/j.cma.2007.05.020","volume":"196","author":"J Fish","year":"2007","unstructured":"Fish, J., Nuggehally, M.A., Shephard, M.S., Picu, C.R., Badia, S., Parks, M.L., Gunzburger, M.: Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput. Methods Appl. Mech. Eng. 196(45\u201348), 4548\u20134560 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"772_CR12","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1016\/j.cma.2005.05.042","volume":"195","author":"WK Liu","year":"2006","unstructured":"Liu, W.K., Park, H., Qian, D., Karpov, W.G., Kadowaki, H., Wagner, G.J.: Bridging scale methods for nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 195, 1407\u20131421 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"41\u201342","key":"772_CR13","doi-asserted-by":"crossref","first-page":"3399","DOI":"10.1016\/j.cma.2008.03.014","volume":"197","author":"S Prudhomme","year":"2008","unstructured":"Prudhomme, S., Dhia, H.B., Bauman, P.T., Elkhodja, N., Oden, J.T.: Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method. Comput. Methods Appl. Mech. Eng. 197(41\u201342), 3399\u20133409 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"772_CR14","doi-asserted-by":"crossref","first-page":"831","DOI":"10.4208\/cicp.2009.09.053","volume":"7","author":"P Seleson","year":"2010","unstructured":"Seleson, P., Gunzburger, M.: Bridging methods for atomistic-to-continuum coupling and their implementation. Commun. Comp. Phys. 7, 831\u2013876 (2010)","journal-title":"Commun. Comp. Phys."},{"key":"772_CR15","doi-asserted-by":"crossref","unstructured":"Luskin, M., Ortner, C.: Atomistic-to-continuum-coupling. Acta Numerica (2013)","DOI":"10.1017\/S0962492913000068"},{"key":"772_CR16","doi-asserted-by":"crossref","unstructured":"Ortner, C.: The role of the patch test in 2D atomistic-to-continuum coupling methods. ESAIM Math. Model. Numer. Anal. 46 (2012)","DOI":"10.1051\/m2an\/2012005"},{"key":"772_CR17","doi-asserted-by":"crossref","unstructured":"Ortner, C., Shapeev, A.V.: Analysis of an energy-based atomistic\/continuum coupling approximation of a vacancy in the 2D triangular lattice. Math. Comput. 82 (2013)","DOI":"10.1090\/S0025-5718-2013-02687-7"},{"key":"772_CR18","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1137\/100792421","volume":"9","author":"AV Shapeev","year":"2011","unstructured":"Shapeev, A.V.: Consistent energy-based atomistic\/continuum coupling for two-body potentials in one and two dimensions. SIAM J. Multiscale Model. Simul. 9, 905\u2013932 (2011)","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"772_CR19","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1007\/s00205-012-0582-8","volume":"207","author":"C Makridakis","year":"2013","unstructured":"Makridakis, C., S\u00fcli, E.: Finite element analysis of Cauchy\u2013Born approximations to atomistic models. Arch. Ration. Mech. Anal. 207, 813\u2013843 (2013)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"772_CR20","doi-asserted-by":"crossref","unstructured":"Ortner, C., Theil, F.: Justification of the Cauchy\u2013Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207 (2013)","DOI":"10.1007\/s00205-012-0592-6"},{"key":"772_CR21","doi-asserted-by":"crossref","first-page":"1258","DOI":"10.1137\/130933873","volume":"12","author":"C Ortner","year":"2014","unstructured":"Ortner, C., Shapeev, A., Zhang, L.: (In-)stability and stabilisation of QNL-type atomistic-to-continuum coupling methods. SIAM J. Multiscale Model. Simul. 12, 1258\u20131293 (2014)","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"772_CR22","unstructured":"Ortner, C., Shapeev, A.: Interpolants of lattice functions for the analysis of atomistic\/continuum multiscale methods. arXiv:1204.3705 (2012)"},{"issue":"2","key":"772_CR23","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s00211-010-0338-z","volume":"118","author":"L Beirao da Veiga","year":"2011","unstructured":"Beirao da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Num. Math. 118(2), 271\u2013305 (2011)","journal-title":"Num. Math."},{"key":"772_CR24","volume-title":"p-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics","author":"C Schwab","year":"1998","unstructured":"Schwab, C.: p-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford (1998)"},{"key":"772_CR25","unstructured":"Ehrlacher, V., Ortner, C., Shapeev, A.V.: Analysis of boundary conditions for crystal defect atomistic simulations. arXiv:1306.5334v2 (2013)"},{"key":"772_CR26","unstructured":"Ehrlacher, V., Ortner, C., Shapeev, A.V.: Analysis of boundary conditions for crystal defect atomistic simulations. arXiv:1306.5334v1 (2013)"},{"issue":"3","key":"772_CR27","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1007\/s00205-014-0746-9","volume":"213","author":"T Hudson","year":"2014","unstructured":"Hudson, T., Ortner, C.: Existence and stability of a screw dislocation under anti-plane deformation. Arch. Ration. Mech. Anal. 213(3), 887\u2013929 (2014)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"772_CR28","volume-title":"Theory of Dislocations","author":"JP Hirth","year":"1982","unstructured":"Hirth, J.P., Lothe, J.: Theory of Dislocations. Krieger Publishing Company, Malabar (1982)"},{"key":"772_CR29","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1051\/m2an\/2011014","volume":"46","author":"T Hudson","year":"2012","unstructured":"Hudson, T., Ortner, C.: On the stability of Bravais lattices and their Cauchy\u2013Born approximations. ESAIM Math. Model. Numer. Anal. 46, 81\u2013110 (2012)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"772_CR30","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1137\/090767005","volume":"8","author":"M Dobson","year":"2010","unstructured":"Dobson, M., Luskin, M., Ortner, C.: Sharp stability estimates for force-based quasicontinuum methods. SIAM J. Multiscale Model. Simul. 8, 782\u2013802 (2010)","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"772_CR31","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s00205-009-0276-z","volume":"197","author":"M Dobson","year":"2010","unstructured":"Dobson, M., Luskin, M., Ortner, C.: Stability, instability and error of the force-based quasicontinuum approximation. Arch. Ration. Mech. Anal. 197, 179\u2013202 (2010)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3","key":"772_CR32","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1137\/110825704","volume":"10","author":"M Dobson","year":"2012","unstructured":"Dobson, M., Ortner, C., Shapeev, A.V.: The spectrum of the force-based quasicontinuum operator for a homogeneous periodic chain. SIAM J. Multiscale Model. Simul. 10(3), 744\u2013765 (2012)","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"772_CR33","doi-asserted-by":"crossref","unstructured":"Lu, J., Ming, P.: Stability of a force-based hybrid method in three dimension with sharp interface (2012)","DOI":"10.1002\/cpa.21429"},{"key":"772_CR34","doi-asserted-by":"crossref","unstructured":"Olson, D., Bochev, P., Luskin, M., Shapeev, A.V.: Development of an optimization-based atomistic-to-continuum coupling method. Large-Scale Scientific Computing, Springer, Heidelberg, pp 33\u201344 (2014)","DOI":"10.1007\/978-3-662-43880-0_3"},{"key":"772_CR35","doi-asserted-by":"crossref","unstructured":"Hardy, R.J.: Formulas for determining local properties in molecular dynamics simulations: shock waves. J. Chem. Phys. 76(622), 628 (1982)","DOI":"10.1063\/1.442714"},{"key":"772_CR36","volume-title":"A Review of a Posteriori Error Estimation and Adaptive Mesh-refinement Techniques","author":"R Verf\u00fcrth","year":"1996","unstructured":"Verf\u00fcrth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-refinement Techniques. Wiley, New York (1996)"},{"key":"772_CR37","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1051\/m2an:1999158","volume":"33","author":"R\u00fcdiger Verf\u00fcrth","year":"1999","unstructured":"Verf\u00fcrth, R\u00fcdiger: Error estimates for some quasi-interpolation operators. M2AN Math. Model. Numer. Anal. 33, 695\u2013713 (1999)","journal-title":"M2AN Math. Model. Numer. Anal."},{"key":"772_CR38","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics in Applied Mathematics, vol.\u00a040. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002) (Reprint of the 1978 original)","DOI":"10.1137\/1.9780898719208"},{"key":"772_CR39","doi-asserted-by":"crossref","unstructured":"Li, X.H., Ortner, C., Shapeev, A., Van Koten, B.: Analysis of blended atomistic\/continuum hybrid methods. arXiv:1404.4878 (2014)","DOI":"10.1007\/s00211-015-0772-z"},{"key":"772_CR40","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1051\/m2an:1999140","volume":"33","author":"C Carstensen","year":"1999","unstructured":"Carstensen, C.: Quasi-interpolation and a posteriori error analysis in n+nnite element methods, M2AN. Math Model. Numer. Anal. 33, 1187\u20131202 (1999)","journal-title":"Math Model. Numer. Anal."},{"key":"772_CR41","doi-asserted-by":"crossref","unstructured":"Glaeser, G.: Racine carr\u00e9e d\u2019une fonction diff\u00e9rentiable. Annales de l\u2019institut Fourier 13(2), 203\u2013210 (1963)","DOI":"10.5802\/aif.146"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-015-0772-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-015-0772-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-015-0772-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T15:40:45Z","timestamp":1653579645000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-015-0772-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,19]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2016,10]]}},"alternative-id":["772"],"URL":"https:\/\/doi.org\/10.1007\/s00211-015-0772-z","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,11,19]]}}}