Multiscale coupling in function space-weak coupling between molecular dynamics and continuum mechanics

International Journal for Numerical Methods in Engineering

Volumn 79, Issue 12, 2009, Pages 1517-1535

  Fackeldey, Konstantin ^a   Krause, Rolf ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF LUGANO   (Switzerland)

Author keywords

Molecular dynamics; Mortar; Multiscale; Partition of unity method; Weak coupling

Indexed keywords

DEGREES OF FREEDOM; DIFFERENT SCALE; FINITE ELEMENT; FUNCTION SPACES; HIGH FREQUENCY HF; LOCAL AVERAGING; LOW FREQUENCY; MICRO-SCALES; MULTI-SCALE SIMULATION; MULTISCALE; MULTISCALE COUPLING; NONLINEAR PROCESS; NUMERICAL EXPERIMENTS; PARTITION OF UNITY; PARTITION OF UNITY METHOD; STABILIZING EFFECTS; TRANSFER OPERATOR; WEAK COUPLING; WEAK COUPLINGS; WEIGHT FUNCTIONS;

CONTINUUM MECHANICS; ELASTICITY; FUNCTION EVALUATION; FUNCTIONAL ANALYSIS; MOLECULAR DYNAMICS; MORTAR; SCALES (WEIGHING INSTRUMENTS);

DYNAMICS;

EID: 69249205552     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2626     Document Type: Article

References (46)

1. Tadmor E, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine A 1996; 73:1529-1563.
2. Kevrekidis PG, Kevrekidis IG, Bishop AR, Titi ES. Continuum approach to discreteness. Physical Review E 2002; 65(4):046613.
3. E W, Huang Z. A dynamic atomistic-continuum method for the simulation of crystalline materials. Journal of Computational Physics 2002; 182(1):234-261.
4. Blanc X, LeBris C, Lions P. From molecular models to continuum mechanics. Archive for Rational Mechanics and Analysis 2002; 164(4):341-381.
5. Abraham FF, Broughton JQ, Bernstein N, Kaxiras E. Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhysics Letters 1998; 44(6):783-787.
6. Knap J, Ortiz M. An analysis of the quasicontinuum method. Journal of the Mechanics and Physics of Solids 2001; 49(9):1899-1923.
7. Xiao SP, Belytschko T. A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering 2004; 193:1645-1669.
8. Belytschko T, Xiao SP. Coupling methods for continuum model with molecular model. International Journal for Multiscale Computational Engineering 2003; 1(1):115-126.
9. Curtin WA, Miller RE. Atomistic/continuum coupling in computational materials science. Modelling and Simulation in Materials Science Engineering 2003; 11:R33-R68.
10. Shilkrot LE, Curtin WA, Miller RE. A coupled atomistic/continuum model of defects in solids. Journal of the Mechanics and Physics of Solids 2002; 50:2085-2106.
11. Wagner GJ, Liu WK. Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics 2003; 190:1261-1289.
12. Liu WK, Karpov EG, Park HS. Nano Mechanics and Materials. Wiley: New York, 2006.
13. Tang S, Hou TY, Liu WK. A mathematical framework of the bridging scale method. International Journal for Numerical Methods in Engineering 2005; 65:1688-1713.
14. Fish J, Nuggehally MA, Shepard MS, Picu CP, Badia S, Parks ML, Gunzburger M. Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Computer Methods in Applied Mechanics and Engineering 2007; 196:4548-4560.
15. Nuggehally MA, Shepard MS, Picu CP, Fish J. Adaptive model selection procedure for concurrent multiscale problems. International Journal for Multiscale Computational Engineering 2007; 5:369-386.
16. Badia S, Parks M, Bochev P, Gunzburger M, Lehoucq R. On atomistic-to-continuum coupling by blending. SIAM Journal on Multiscale Modeling and Simulation 2008; 7(1):381-406.
17. Adelman SA, Doll JD. Generalized Langevin equations for atom/solid-surface scattering: general formulation for classical scattering off harmonic solids. Journal of Chemical Physics 1976; 64(4):2375-2388.
18. Givoli D. Non-reflecting boundary conditions. Journal of Computational Physics 1991; 94:1-29.
19. Cai W, DeKoning M, Bulatov VV, Yip S. Minimizing boundary reflection in coupled-domain simulations. Physical Review Letters 2000; 85:3213-3216.
20. Berenger JP. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 1994; 114:185-200.
21. To AC, Li S. Perfectly matched multiscale simulation. Physical Review B 2005; 72:035414-1-8.
22. Wagner GJ, Liu WK. Hierarchical enrichment for bridging scales and mesh-free boundary conditions. International Journal for Numerical Methods in Engineering 2001; 50:507-524.
23. Dickopf T, Krause R. Efficient simulation of multi-body contact problems on complex geometries: a flexible decomposition approach using constrained minimization. International Journal for Numerical Methods in Engineering 2009; 77(13):1834-1862.
24. Zhou M, McDowell DL. Equivalent continuum for dynamically deforming atomistic particle systems. Philosophical Magazine A 2002; 82(13):2547-2574.
25. Webb EB, Zimmerman JA, Seel SC. Reconsideration of continuum thermomechanical quantities in atomic scale simulations. Mathematics and Mechanics of Solids 2008; 13(3-4):221-266.
26. Gullett M, Horstemeyer MF, Baskes MI, Fang H. A deformation gradient tensor and strain tensors for atomistic simulations. Modelling and Simulation in Materials Science and Engineering 2008; 16:015001 (17).
27. Ciarlet PG. Mathematical Elasticity, Volume 1: Three Dimensional Elasticity. North-Holland: Amsterdam, 1988.
28. Hughes T, Feijoo G, Mazzei J, Quincy J. The variational multiscale method-a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering 1998; 166:3-24.
29. Broughton J, Abraham F, Bernstein N, Kaxiras E. Concurrent coupling of length scales: methodology and application. Physical Review B 1999; 60:2391-2403.
30. Bernardi C, Maday Y, Patera AT. A new nonconforming approach to domain decomposition: the mortar element method. In Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, volume XI. Lectures presented at the weekly seminar on applied mathematics 299, Brezis H et al. (eds). Scientific & Technical Pitman Research Notes Mathematical Series Edition. Longman: Harlow, 1994; 13-51.
31. Ben Belgacem F. The mortar finite element method with Lagrange multipliers. Numerische Mathematik 1999; 84(2):173-197.
32. Krause R. Monotone multigrid methods for Signorini's problem with friction. Ph.D. Thesis, Freie Universität Berlin, 2001.
33. Babuska I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering 1997; 40:727-758.
34. Griebel M, Schweitzer MA. A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDE. SIAM Journal on Scientific Computing 2000; 22(3):853-890.
35. Schweitzer MA. A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 29. Springer: Berlin, 2003.
36. Shepard D. A two-dimensional function for irregularly spaced data. ACM National Conference, New York, NY, U.S.A., 1968; 517-524.
37. Lancaster P, Salkauskas K. Surfaces generated by moving least squares method. Mathematics of Computation 1981; 27:141-158.
38. Chen JS, Pan C, Wu CI. Large deformation analysis of rubber based on a reproducing kernel particle method. Computational Mechanics 1997; 19:211-227.
39. Liu WK, Jun S, Li S, Adee J, Belytschko T. Reproducing kernel particle methods for structural mechanics. International Journal for Numerical Methods in Engineering 1995; 38:1655-1679.
40. Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids 1995; 20:1081-1106.
41. Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 1995; 4:389-396.
42. Wu Z. Compactly supported positive definite radial functions. Advances in Computational Mathematics 1995; 4:283-292.
43. Quarteroni A, Valli A. Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications: Oxford, 1999.
44. Scott LR, Zhang S. Higher-dimensional nonnested multigrid methods. Mathematics of Computation 1992; 58(198):457-466.
45. Fackeldey K, Krause D, Krause R. Quadrature and implementation of the weak coupling method. Proceedings of the Fourth International Conference on Multiscale Materials Modeling, Tallahassee, FL, U.S.A., 2008; 62-65.
46. Friesecke G, Theil F. Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. Journal of Nonlinear Science 2002; 12:445-478.