A Concurrent Multiscale Method Based on the Meshfree Method and Molecular Dynamics Analysis

This paper presents a concurrent simulation technique for analysing the deformation of systems that need the integration of material properties from nanoscopic to macroscopic dimensional scales. In the continuum sub-domain, a weak-form meshfree based method using the radial basis function interpolation was employed, but in the atomic sub-domain, molecular dynamics analysis was used. The transition from the atomic to continuum domains was realized by transition particles which are independent of either the nodes in the continuum sub-domain or the atoms in the atomic sub-domain. A simple penalty method was used to ensure the compatibility of displacements and their gradients in the transition. A virtual cell algorithm was developed using a local quasi-continuum approach to obtain the equivalent continuum strain energy density based on the atomic potentials and Cauchy-Born rule. Numerical examples showed that the present method is very accurate and stable, and has a promising potential to a wide class of multiscale systems.

[1]  M. Ortiz,et al.  An analysis of the quasicontinuum method , 2001, cond-mat/0103455.

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

[3]  D. C. Rapaport,et al.  The Art of Molecular Dynamics Simulation , 1997 .

[4]  Hyun Gyu Kim,et al.  A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods , 1999 .

[5]  M. Gurtin,et al.  Phase Transformation and Material Instabilities in Solids , 1984 .

[6]  M. F. Kanninen,et al.  Crack extension by bond rupture in a model of BCC iron , 1972 .

[7]  W. Cai,et al.  Minimizing boundary reflections in coupled-domain simulations. , 2000, Physical review letters.

[8]  Satya N. Atluri,et al.  Multiscale Simulation Based on The Meshless Local Petrov-Galerkin (MLPG) Method , 2004 .

[9]  Robert E. Rudd,et al.  Concurrent Coupling of Length Scales in Solid State Systems , 2000 .

[10]  G. R. Liu,et al.  1013 Mesh Free Methods : Moving beyond the Finite Element Method , 2003 .

[11]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[12]  YuanTong Gu,et al.  A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids , 2001 .

[13]  Haiyi Liang,et al.  MULTISCALE COUPLING OF MESHLESS METHOD AND MOLECULAR DYNAMICS , 2006 .

[14]  G. Zanzotto,et al.  The Cauchy-Born hypothesis, nonlinear elasticity and mechanical twinning in crystals , 1996 .

[15]  Noam Bernstein,et al.  Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture , 1998 .

[16]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[17]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[18]  Gregory J. Wagner,et al.  Coupling of atomistic and continuum simulations using a bridging scale decomposition , 2003 .

[19]  Hiroaki Tanaka,et al.  Towards a deeper understanding of wear and friction on the atomic scale—a molecular dynamics analysis , 1997 .

[20]  Ronald E. Miller,et al.  Atomistic/continuum coupling in computational materials science , 2003 .

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

[22]  吴宗敏.  HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS , 1992 .

[23]  H. Fischmeister,et al.  Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model , 1991 .

[24]  J. Q. Broughton,et al.  Concurrent coupling of length scales: Methodology and application , 1999 .

[25]  Hiroaki Tanaka,et al.  Atomic scale deformation in silicon monocrystals induced by two-body and three-body contact sliding , 1998 .

[26]  M. Cross,et al.  A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate , 1998 .

[27]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[28]  J. E. Sinclair.  The influence of the interatomic force law and of kinks on the propagation of brittle cracks , 1975 .

[29]  Harold S. Park,et al.  The bridging scale for two-dimensional atomistic/continuum coupling , 2005 .

[30]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[31]  J. Z. Zhu,et al.  The finite element method , 1977 .

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

[33]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[34]  Harold S. Park,et al.  An introduction and tutorial on multiple-scale analysis in solids , 2004 .

[35]  Hiroaki Tanaka,et al.  On the Mechanics and Physics in the Nano-Indentation of Silicon Monocrystals. , 1999 .

[36]  W. E,et al.  Matching conditions in atomistic-continuum modeling of materials. , 2001, Physical review letters.

[37]  Robert E. Rudd,et al.  COARSE-GRAINED MOLECULAR DYNAMICS AND THE ATOMIC LIMIT OF FINITE ELEMENTS , 1998 .

[38]  L. Girifalco,et al.  Energy of Cohesion, Compressibility, and the Potential Energy Functions of the Graphite System , 1956 .

[39]  E. Tadmor,et al.  Finite-temperature quasicontinuum: molecular dynamics without all the atoms. , 2005, Physical review letters.

[40]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .