Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Badia, S., Parks, M., Bochev, P., Gunzburger, M., Lehoucq, R.: On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7(1), 381–406 (2008)
Bernstein, N., Kermode, J.R., Csányi, G.: Hybrid atomistic simulation methods for materials systems. Rep. Prog. Phys. 72, 026501 (2009)
Blanc, X., Le Bris, C., Legoll, F.: Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. Math. Model. Numer. Anal. 39(4), 797–826 (2005)
Blanc, X., Le Bris, C., Legoll, F., Patz, C.: Finite-temperature coarse-graining of one-dimensional models: mathematical analysis and computational approaches. J. Nonlinear Sci. 20, 241–275 (2010)
Blanc, X., Le Bris, C., Lions, P.-L.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164(4), 341–381 (2002)
Dobson, M., Luskin, M., Ortner, C.: Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58(10), 1741–1757 (2010)
Dobson, M., Luskin, M., Ortner, C.: Stability, instability, and error of the force-based quasicontinuum approximation. Arch. Ration. Mech. Anal. 197, 179–202 (2010)
Dobson, M., Luskin, M., Ortner, C.: Iterative methods for the force-based quasicontinuum approximation. Comput. Methods Appl. Mech. Eng. 200, 2697–2709 (2011)
Dupuy, L.M., Tadmor, E.B., Legoll, F., Miller, R.E., Kim, W.K.: Finite-temperature quasicontinuum. manuscript, (2012)
Fischmeister, H., Exner, H., Poech, M.-H., Kohlhoff, S., Gumbsch, P., Schmauder, S., Sigi, L.S., Spiegler, R.: Modelling fracture processes in metals and composite materials. Z. Metallkde. 80, 839–846 (1989)
Garcia-Cervera, C.J., Lu, J., E, W.: A sublinear scaling algorithm for computing the electronic structure of materials. Commun. Math. Sci. 5, 999–1026 (2007)
Gavini, V., Bhattacharya, K., Ortiz, M.: Quasi-continuum orbital-free density-functional theory: a route to multi-million atom non-periodic DFT calculation. J. Mech. Phys. Solids 55, 697–718 (2007)
Kim, W.K., Tadmor, E.B., Luskin, M., Perez, D., Voter, A.: Hyper-qc: an accelerated finite-temperature quasicontinuum method using hyperdynamics. manuscript, (2012)
Liu, W.K., Park, H., Qian, D., Karpov, E.G., Kadowaki, H., Wagner, G.J.: Bridging scale methods for nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 195, 1407–1421 (2006)
Luskin, M., Ortner, C., Van Koten, B.: Formulation and optimization of the energy-based blended quasicontinuum method. arXiv:1112.2377, (2011)
Luskin, M., Ortner, C.: Linear stationary iterative methods for the force-based quasicontinuum approximation. In: Engquist, B., Runborg, O., Tsai, R. (eds.), Numerical Analysis and Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol. 82, pp. 331–368. Springer, Berlin/Heidelberg/New York (2012)
Ortner, C.: The role of the patch test in 2D atomistic-to-continuum coupling methods, to appear. arXiv:1101.5256, Math. Model. Numer. Anal. (2011)
Ortner, C., Zhang, L.: Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem, preprint. arXiv:1110.0168, (2011)
Prudhomme, S., Ben Dhia, H., 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(41–42), 3399–3409 (2008)
Shapeev, A.V.: Consistent energy-based atomistic/continuum coupling for two-body potential: 1D and 2D case. Multiscale Model. Simul. 9, 905–932 (2011)
Shilkrot, L.E., Miller, R.E., Curtin, W.A.: Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J. Mech. Phys. Solids 52, 755–787 (2004)
Shimokawa, T., Mortensen, J.J., Schiotz, J., Jacobsen, K.W.: Matching conditions in the quasicontinuum method: removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B 69(21), 214104 (2004)
Tadmor, E.B., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)
Van Koten, B., Luskin, M.: Analysis of energy-based blended quasicontinuum approximations. SIAM J. Numer. Anal. 49, 2182–2209 (2011)
E, W., Lu, J., Yang, J.Z.: Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74(21), 214115 (2006)
E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)
Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193, 1645–1669 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Luskin, M., Ortner, C. (2015). Atomistic to Continuum Coupling. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_271
Download citation
DOI: https://doi.org/10.1007/978-3-540-70529-1_271
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70528-4
Online ISBN: 978-3-540-70529-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering