Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in Three Dimensions

Very few works exist to date on development of a consistent energy-based coupling of atomistic and continuum models of materials in more than one dimension. The difficulty in constructing such a coupling consists in defining a coupled energy whose minimizers are free from uncontrollable errors on the atomistic/continuum interface. In this paper a consistent coupling in three dimensions is proposed. The main achievement of this work is in identifying and efficiently treating a modified Cauchy--Born continuum model which can be coupled to the exact atomistic model. The convergence and stability of the method is confirmed with numerical tests.

[1]  Christoph Ortner,et al.  Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem , 2011 .

[2]  Christoph Ortner,et al.  Positive Definiteness of the Blended Force-Based Quasicontinuum Method , 2011, Multiscale Model. Simul..

[3]  Xingjie Helen Li,et al.  A Generalized Quasi-Nonlocal Atomistic-to-Continuum Coupling Method with Finite Range Interaction , 2010 .

[4]  Jianfeng Lu,et al.  Convergence of a force-based hybrid method for atomistic and continuum models in three dimension , 2011, 1102.2523.

[5]  Ronald E. Miller,et al.  The Quasicontinuum Method: Overview, applications and current directions , 2002 .

[6]  Alexander V. Shapeev,et al.  Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in One and Two Dimensions , 2010, Multiscale Model. Simul..

[7]  E Weinan,et al.  Uniform Accuracy of the Quasicontinuum Method , 2006, MRS Online Proceedings Library.

[8]  Christoph Ortner.  A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D , 2011, Math. Comput..

[9]  Brian Van Koten,et al.  A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods , 2010, 1012.6031.

[10]  Stability , 1973 .

[11]  Ellad B. Tadmor,et al.  A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods , 2009 .

[12]  M. Luskin,et al.  An Analysis of the Quasi-Nonlocal Quasicontinuum Approximation of the Embedded Atom Model , 2010, 1008.3628.

[13]  M. Born,et al.  Dynamical Theory of Crystal Lattices , 1954 .

[14]  M. Luskin,et al.  Formulation and optimization of the energy-based blended quasicontinuum method , 2011, 1112.2377.

[15]  Christoph Ortner,et al.  THE ROLE OF THE PATCH TEST IN 2D ATOMISTIC-TO-CONTINUUM COUPLING METHODS ∗ , 2011, 1101.5256.

[16]  Tomotsugu Shimokawa,et al.  Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region , 2004 .

[17]  Weinan E,et al.  Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems , 2007 .

[18]  Christoph Ortner,et al.  Stability, Instability, and Error of the Force-based Quasicontinuum Approximation , 2009, 0903.0610.

[19]  Brian Van Koten,et al.  Analysis of Energy-Based Blended Quasi-Continuum Approximations , 2011, SIAM J. Numer. Anal..

[20]  V. Gavini,et al.  A field theoretical approach to the quasi-continuum method , 2011 .

[21]  Christoph Ortner,et al.  Sharp Stability Estimates for the Force-Based Quasicontinuum Approximation of Homogeneous Tensile Deformation , 2010, Multiscale Model. Simul..

[22]  Alexander V. Shapeev,et al.  Analysis of an Energy-based Atomistic/Continuum Coupling Approximation of a Vacancy in the 2D Triangular Lattice , 2011 .