Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice

We present an a priori error analysis of a practical energy based atomistic/continuum coupling method (A. V. Shapeev, Multiscale Model. Simul., 9(3):905-932, 2011) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. We establish first-order consistency and stability of the method, from which we obtain a priori error estimates in the H1-norm and the energy in terms of the mesh size and the "smoothness'' of the atomistic solution in the continuum region. From these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh, as well as convergence rates in terms of the number of degrees of freedom. Our analytical predictions are supported by extensive numerical tests.

[1]  Michael Frazier,et al.  Studies in Advanced Mathematics , 2004 .

[2]  Christoph Ortner,et al.  On the Convergence of Adaptive Nonconforming Finite Element Methods for a Class of Convex Variational Problems , 2011, SIAM J. Numer. Anal..

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

[4]  Lars B. Wahlbin,et al.  Best approximation property in the W1∞ norm for finite element methods on graded meshes , 2011, Math. Comput..

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

[6]  Patrick A. Klein,et al.  Coupled atomistic-continuum simulations using arbitrary overlapping domains , 2006, J. Comput. Phys..

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

[8]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[9]  Leszek Demkowicz,et al.  On an h-type mesh-refinement strategy based on minimization of interpolation errors☆ , 1985 .

[10]  C. Ortner,et al.  ON THE STABILITY OF BRAVAIS LATTICES AND THEIR CAUCHY-BORN APPROXIMATIONS ∗ , 2012 .

[11]  Endre Süli,et al.  A Priori Error Analysis of Two Force-Based Atomistic/Continuum Hybrid Models of a Periodic Chain , 2011 .

[12]  Endre Süli,et al.  A priori error analysis of two force-based atomistic/continuum models of a periodic chain , 2011, Numerische Mathematik.

[13]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .

[14]  Hao Wang,et al.  A Posteriori Error Estimates for Energy-Based Quasicontinuum Approximations of a Periodic Chain , 2011, 1112.5480.

[15]  Endre Süli,et al.  ANALYSIS OF A QUASICONTINUUM METHOD IN ONE DIMENSION , 2008 .

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

[17]  Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potential: 1D and 2D Case , 2010 .

[18]  T. Belytschko,et al.  A bridging domain method for coupling continua with molecular dynamics , 2004 .

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

[20]  Mitchell Luskin,et al.  An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation , 2009, SIAM J. Numer. Anal..

[21]  Christoph Ortner,et al.  Accuracy of quasicontinuum approximations near instabilities , 2009, 0905.2914.

[22]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

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

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

[25]  F. C. Frank,et al.  One-dimensional dislocations. I. Static theory , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[26]  L. Evans.  Measure theory and fine properties of functions , 1992 .

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

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

[29]  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 .

[30]  Alexander V. Shapeev.  Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in Three Dimensions , 2012, SIAM J. Sci. Comput..

[31]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

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

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

[34]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[35]  Siegfried Schmauder,et al.  A New Method for Coupled Elastic-Atomistic Modelling , 1989 .

[36]  Mariano Giaquinta,et al.  Introduction to Regularity Theory for Nonlinear Elliptic Systems , 1993 .

[37]  Pingbing Ming,et al.  Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method , 2009, Multiscale Model. Simul..

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

[39]  M. Ortiz,et al.  An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method , 1997, cond-mat/9710027.