Multiscale Modeling of Complex Dynamic Problems: An Overview and Recent Developments

  • Mohamed JebahiEmail author
  • Frédéric Dau
  • Jean-Luc Charles
  • Ivan Iordanoff
Original Paper


Multiscale modeling aims to solve problems at the engineering (macro) scale while considering the complexity of the microstructure with minimum cost. Generally, two scales are considered in multiscale modeling: small scale, which is designed to capture the mechanical phenomena at the atomistic, molecular or molecular cluster level, and large scale which is connected to continuous description. For each scale, well-established numerical methods have been developed over the years to handle the relevant phenomena. As a first part of this paper, the most popular numerical methods, used at different scales, as well as the coupling approaches between them are classified, according to their features and applications, so that the place of those used in multiscale modeling can be distinguished. Subsequently, the class of concurrent discrete–continuum coupling approaches, which is well adapted for dynamic studies of complex multiscale problems, is reviewed. Several techniques used in this class are also detailed. Among them, the bridging domain (BD) technique is used to develop a discrete–continuum coupling approach, adapted for dynamic simulations, between the Discrete Element Method and the Constrained Natural Element Method (CNEM). This approach is applied to study the BD coupling parameters in dynamics. Several results giving more light on the setting of these parameters in practice are obtained.


  1. 1.
    Abraham FF, Walkup R, Gao H, Duchaineau M, Diaz De La Rubia T, Seager M (2002) Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture. Proc Natl Acad Sci USA 99(9):5777–5782CrossRefGoogle Scholar
  2. 2.
    Adelman S, Doll J (1976) Generalized Langevin equation approach for atom/solid-surface scattering: general formulation for classical scattering off harmonic solids. J Chem Phys 64:2375–2388CrossRefGoogle Scholar
  3. 3.
    Akbari RA, Kerfriden P, Rabczuk T, Bordas S (2012) An adaptive multiscale method for fracture based on concurrent—hierarchical hybrid modelling. In: Proceedings of the 20th UK conference of the Association for Computational Mechanics in Engineering (2012)Google Scholar
  4. 4.
    Alder B, Wainwright T (1957) Phase transition for a hard sphere system. J Chem Phys 27:1208–1209CrossRefGoogle Scholar
  5. 5.
    Alder B, Wainwright T (1959) Studies in molecular dynamics. I. General method. J Chem Phys 31:459–466MathSciNetCrossRefGoogle Scholar
  6. 6.
    André D, Iordanoff I, Charles J, Néauport J (2012) Discrete element method to simulate continuous material by using the cohesive beam model. Comput Methods Appl Mech Eng 213–216:113–125CrossRefGoogle Scholar
  7. 7.
    André D, Jebahi M, Iordanoff I, Charles JL, Néauport J (2013) Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter. Comput Methods Appl Mech Eng 265:136–147zbMATHCrossRefGoogle Scholar
  8. 8.
    Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MPLG) approach in computational mechanics. Comput Mech 22:117–127zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Aubertin P, Réthoré J, De Borst R (2009) Energy conservation of atomistic/continuum coupling. Int J Numer Methods Eng 78:1365–1386zbMATHCrossRefGoogle Scholar
  10. 10.
    Bauman PL, Ben Dhia H, Elkhodja N, Oden JT, Prudhomme S (2008) On the application of the Arlequin method to the coupling of particle and continuum models. Comput Mech 42:511–530zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47zbMATHCrossRefGoogle Scholar
  12. 12.
    Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Belytschko T, Xiao SP (2003) Coupling methods for continuum model with molecular model. Int J Multiscale Comput Eng 1(1):115–126CrossRefGoogle Scholar
  15. 15.
    Ben Dhia H (1998) Problèmes mécanique multi-échelles: la méthode Arlequin. Comptes rendus de l ’académie des sciences - Analyse numérique, pp 899–904 (1998)Google Scholar
  16. 16.
    Ben Dhia H (2008) Further insights by theoretical investigations of the multiscale Arlequin method. Int J Multiscale Comput Eng 60(3):215–232CrossRefGoogle Scholar
  17. 17.
    Ben Dhia H, Rateau G (2001) Analyse mathématique de la méthode Arlequin mixte. Comptes rendus de l ’académie des sciences - Mécanique des solides et des stuctures, pp 649–654 (2001)Google Scholar
  18. 18.
    Ben Dhia H, Rateau G (2005) The Arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62:1442–1462zbMATHCrossRefGoogle Scholar
  19. 19.
    Benson DJ (1992) Computational methods in Lagrangian and Eulerian hydrocodes. Comput Methods Appl Mech Eng 99(2–3):235–394zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Bobet A, Fakhimi A, Johnson S, Morris K, Tonon F, Yeung M (2009) Numerical models in discontinuous media: review of advances for rock mechanics applications. J Geotech Geoenviron Eng 135(11):1547–1561CrossRefGoogle Scholar
  21. 21.
    Bratberg I, Radjai F, Hansen A (2002) Dynamic rearrangements and packing regimes in randomly deposited two-dimensional granular beds. Phys Rev E 66:031,303CrossRefGoogle Scholar
  22. 22.
    Braun J, Sambridge M (1995) A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376:655–660CrossRefGoogle Scholar
  23. 23.
    Broughton J, Abraham F, Bernstein N, Kaxiras E (1999) Concurrent coupling of length scales: methodology and application. Phys Rev B 60(4):2391–2403CrossRefGoogle Scholar
  24. 24.
    Cai W, DeKoning M, Bulatov V, Yip S (2000) Minimizing boundary reflections in coupled-domain simulations. Phys Rev Lett 85:3213–3216CrossRefGoogle Scholar
  25. 25.
    Celep Z, Bažant ZP (1983) Spurious reflection of elastic waves due to gradually changing finite element size. Int J Numer Methods Eng 19:631–646zbMATHCrossRefGoogle Scholar
  26. 26.
    Chen J, Wu C, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466zbMATHCrossRefGoogle Scholar
  27. 27.
    Cueto E, Doblaré M, Gracia L (2000) Imposing essential boundary conditions in the natural element method by means of density-scaled -shapes. Int J Numer Methods Eng 49:519–546zbMATHCrossRefGoogle Scholar
  28. 28.
    Cueto E, Sukumar N, Calvo B, Cegoñino J, Doblaré M (2003) Overview and recent advances in natural neighbour Galerkin methods. Arch Comput Methods Eng 10(4):307–384zbMATHCrossRefGoogle Scholar
  29. 29.
    Cundall PA (1971) Computer model for simulating progressive large scale movements in blocky rock systems. In: Proceedings of the symposium of the International Society of Rock Mechanics, Nancy, FranceGoogle Scholar
  30. 30.
    Cundall PA (1988) Formulation of a three-dimensional distinct element model—Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int J Rock Mech Min Sci 25:107–116CrossRefGoogle Scholar
  31. 31.
    Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29(1):47–65CrossRefGoogle Scholar
  32. 32.
    Delaunay B (1934) Sur la sphère vide. A la mémoire de Georges Voronoï. Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na 6:793–800Google Scholar
  33. 33.
    Dobson M, Luskin M (2008) Analysis of a force-based quasicontinuum approximation. ESAIM Math Model Numer Anal 42(1):113–139zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Dobson M, Luskin M, Ortner C (2010) Accuracy of quasicontinuum approximations near instabilities. J Mech Phys Solids 58(10):1741–1757zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Dolbow J, Belytschko T (1999) Numerical integration of the Galerkin weak form in meshfree methods. Comput Mech 23:219–230zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Donzé FV, Richefeu V, Magnier SA (2009) Advances in discrete element method applied to soil, rock and concrete mechanics. State of the art of geotechnical engineering. Electron. J Geotech Eng 8:1–44Google Scholar
  37. 37.
    Weinan E, Huang Z (2002) A dynamic atomistic–continuum method for the simulation of crystalline materials. J Comput Phys 182:234–261Google Scholar
  38. 38.
    Felici HM (1992) A coupled Eulerian/Lagrangian method for the solution of three-dimensinal vortical flows. PhD thesis, Massachusetts Institute of Technology (1992)Google Scholar
  39. 39.
    Feyel F, Chaboche J (2000) Multiscale approach for modelling the elastoviscoplasitic behavior of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183:309–330zbMATHCrossRefGoogle Scholar
  40. 40.
    Fish J, Nuggehally M, Shephard M, Picu C, Badia S, Parks M, Gunzburger M (2007) Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput Methods Appl Mech Eng 196:4548–4560zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Foulkes W, Mitas L, Needs R, Rajagopal G (2001) Quantum Monte Carlo simulations of solids. Rev Mod Phys 73(1):33CrossRefGoogle Scholar
  42. 42.
    Fourey G, Oger G, Le Touzé D, Alessandrini B (2010) Violent fluid-structure interaction simulations using a coupled SPH/FEM method. IOP Conf Ser: Mater Sci Eng 10:012041Google Scholar
  43. 43.
    González D, Cueto E, Martínez MA, Doblaré M (2004) Numerical integration in natural neighbour Galarkin methods. Int J Numer Methods Eng 60:2077–2114zbMATHCrossRefGoogle Scholar
  44. 44.
    Griffiths DV, Mustoe GGW (2001) Modelling of elastic continua using a grillage of structural elements based on discrete element concepts. Int J Numer Methods Eng 50(7):1759–1775zbMATHCrossRefGoogle Scholar
  45. 45.
    Guidault PA, Belytschko T (2007) On the L2 and the H1 couplings for an overlapping domain decomposition method using Lagrange multipliers. Int J Numer Methods Eng 70:322–350zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Guidault PA, Belytschko T (2009) Bridging domain methods for coupled atomistic–continuum models with L2 or H1 couplings. Int J Numer Methods Eng 77:1566–1592zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Hall WS (1994) The boundary element method (solid mechanics and its applications). Springer, BerlinGoogle Scholar
  48. 48.
    Hehre W (2003) A guide to molecular mechanics and quantum chemical calculations. Wavefunction Press, IrvineGoogle Scholar
  49. 49.
    Hentz S, Donzé FV, Daudeville L (2004) Discrete element modelling of concrete submitted to dynamic loading at high strain rates. Comput Struct 82(29–30):2509–2524CrossRefGoogle Scholar
  50. 50.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864MathSciNetCrossRefGoogle Scholar
  51. 51.
    Hrennikoff A (1941) Solution of problems of elasticity by the frame-work method. ASME J Appl Mech 8:A619–A715MathSciNetGoogle Scholar
  52. 52.
    Issa JA, Nelson RN (1992) Numerical analysis of micromechanical behaviour of granular materials. Eng Comput 9:211–223CrossRefGoogle Scholar
  53. 53.
    Iyer M, Gavini V (2011) A field theoretical approach to the quasi-continuum method. J Mech Phys Solids 59(8):1506–1535zbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Jean M (1999) The non smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3–4):235–257zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Jebahi M (2013) Discrete-continuum coupling method for simulation of laser-induced damage in silica glass. PhD thesis, Bordeaux 1 University (2013)Google Scholar
  56. 56.
    Jebahi M, André D, Dau F, Charles JL, Iordanoff I (2013) Simulation of Vickers indentation of silica glass. J Non-Cryst Solids 378:15–24CrossRefGoogle Scholar
  57. 57.
    Jebahi M, Charles J, Dau F, Illoul L, Iordanoff I (2013) 3D coupling approach between discrete and continuum models for dynamic simulations (DEM–CNEM). Comput Methods Appl Mech Eng 255:196–209zbMATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Jebahi M, Charles JL, Dau F, Illoul L, Iordanoff I (2012) On the H1 discrete–continuum coupling based on the Arlequin method (DEM–CNEM). In: Proceedings of the European Congress on Computational Methods in Applied Sciences and EngineeringGoogle Scholar
  59. 59.
    Kaljevic I, Saigal S (1997) An improved element free Galerkin formulation. Int J Numer Methods Eng 40:2953–2974zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Lee CK, Zhou CE (2003) On error estimation and adaptive refinement for element free Galerkin method: Part I: Stress recovery and a posteriori error estimation. Comput Struct 82(4–5):4293–4443Google Scholar
  61. 61.
    Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55(1):1–34CrossRefGoogle Scholar
  62. 62.
    Li S, Liu X, Agrawal A, To A (2006) Perfectly matched multiscale simulations for discrete lattice systems: extension to multiple dimensions. Phys Rev B 74:045,418CrossRefGoogle Scholar
  63. 63.
    Li X, Ming P (2014) On the effect of ghost force in the quasicontinuum method: dynamic problems in one dimension. Commun Computat Phys 15:647–676MathSciNetGoogle Scholar
  64. 64.
    Lin X, Ng TT (1997) A three-dimensional discrete element model using arrays of ellipsoids. Geotechnique 47(2):319–329CrossRefGoogle Scholar
  65. 65.
    Liszka T, Orkisz J (1980) The finite difference method at arbitrary irregular grids and its applications in applied mechanics. Comput Struct 11:83–95zbMATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 50:937–951zbMATHCrossRefGoogle Scholar
  67. 67.
    Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing, SingaporeGoogle Scholar
  68. 68.
    Liu MB, Liu GR, Lam KY (2002) Coupling meshfree particle method with molecular dynamics—a novel approach for multi-scale simulations. In: Proceedings of the 1st asian workshop on meshfree methods, advances in meshfree and X-FEM methods, pp 211–216Google Scholar
  69. 69.
    Liu MB, Liu G (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25–76MathSciNetCrossRefGoogle Scholar
  70. 70.
    Lu G, Kaxiras E (2005) Overview of multiscale simulations of materials. In: Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology. American Scientific Publishers, Los AngelesGoogle Scholar
  71. 71.
    Lucy LB (1977) Numerical approach to testing the fission hypothesis. Astron J 82:1013–1024CrossRefGoogle Scholar
  72. 72.
    Luding S, Clément E, Rajchenbach J, Duran J (1996) Simulations of pattern formation in vibrated granular media. Europhys Lett 36(4):247–252CrossRefGoogle Scholar
  73. 73.
    Mair HU (1995) Hydrocode methodologies for underwater explosion structure medium/interaction. Shock Vib 2:227–248CrossRefGoogle Scholar
  74. 74.
    Melenka JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1–4):289–314CrossRefGoogle Scholar
  75. 75.
    Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44(247):335–341zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150zbMATHCrossRefGoogle Scholar
  77. 77.
    Moreau JJ (1994) Some numerical methods in multibody dynamics: application to granular materials. Eur J Mech A Solids 13(4):93–114zbMATHMathSciNetGoogle Scholar
  78. 78.
    Moreau JJ, Panagiotopoulos PD (1988) Nonsmooth mechanics and applications. Springer, BerlinzbMATHCrossRefGoogle Scholar
  79. 79.
    Nuggehally MA, Shephard MS, Picu CR, Fish J (2007) Adaptive model selection procedure for concurrent multiscale problems. Int J Multiscale Comput Eng 5(5):369–386CrossRefGoogle Scholar
  80. 80.
    Oñate E, Idelsohn S (1998) A mesh-free finite point method for advective-diffusive transport and fluid flow problems. Comput Mech 21:283–292zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL, Sacco C (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 139:315–346zbMATHCrossRefGoogle Scholar
  82. 82.
    Oñate E, Sacco C, Idelsohn S (2000) A finite point method for incompressible flow problems. Comput Vis Sci 3:67–75zbMATHCrossRefGoogle Scholar
  83. 83.
    Ortiz M, Knap J (2001) An analysis of the quasicontinuum method. J Mech Phys Solids 49(9):1899–1923zbMATHCrossRefGoogle Scholar
  84. 84.
    Park H, Karpov E, Liu W, Klein P (2005) The bridging scale for two-dimensional atomistic/continuum coupling. Philos Mag 85(1):79–113CrossRefGoogle Scholar
  85. 85.
    Parka H, Karpovb E, Kleina P, Liub W (2005) Three-dimensional bridging scale analysis of dynamic fracture. J Comput Phys 207(2):588–609CrossRefGoogle Scholar
  86. 86.
    Payne M, Teter M, Allan D, Arias T, Joannopoulos J (1992) Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev Mod Phys 64:1045CrossRefGoogle Scholar
  87. 87.
    Ragueneau F, Gatuingt F (2003) Inelastic behavior modelling of concrete in low and high strain rate dynamics. Comput Struct 81(12):1287–1299CrossRefGoogle Scholar
  88. 88.
    Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139:375–408zbMATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Rapaport DC (1980) The event scheduling problem in molecular dynamic simulation. J Comput Phys 34:184–201CrossRefGoogle Scholar
  90. 90.
    Rodney D (2003) Mixed atomistic/continuum methods: static and dynamic quasicontinuum methods. In: Proceedings of the NATO conference: thermodynamics, microstructures and plasticity, pp 265–274 (2003)Google Scholar
  91. 91.
    Rougier E, Munjiza A, John N (2004) Numerical comparison of some explicit time integration schemes used in DEM, FEM/DEM and molecular dynamics. Int J Numer Methods Eng 62:856–879CrossRefGoogle Scholar
  92. 92.
    Schlangen E, Garboczi EJ (1996) New method for simulating fracture using an elastically uniform random geometry lattice. Int J Eng Sci 34(10):1131–1144zbMATHCrossRefGoogle Scholar
  93. 93.
    Schlangen E, van Mier JGM (1992) Experimental and numerical analysis of micromechanisms of fracture of cement-based composites. Cem Concr Compos 14(2):105–118CrossRefGoogle Scholar
  94. 94.
    Schlangen E, van Mier JGM (1992) Simple lattice model for numerical simulation of fracture of concrete materials and structures. Mater Struct 25(9):534–542CrossRefGoogle Scholar
  95. 95.
    Shenoy V (1999) Quasicontinuum models of atomic-scale mechanics. PhD thesis, Brown University (1999)Google Scholar
  96. 96.
    Shimokawa T, Mortensen J, Schiøtz J, Jacobsen K (2004) 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):214,104CrossRefGoogle Scholar
  97. 97.
    Sibson R (1981) A brief description of natural neighbour interpolation. In: Barnett V (ed) Interpreting multivariate data. Wiley, Chichester, pp 21–36Google Scholar
  98. 98.
    Smith GD (1985) Numerical solution of partial differential equations: finite difference methods, 3rd edn. Oxford University Press, OxfordzbMATHGoogle Scholar
  99. 99.
    Stein E, De Borst R, Hughes TJR (2004) Encyclopedia of computational mechanics, vol 1, chap 14. Wiley, LondonCrossRefGoogle Scholar
  100. 100.
    Svensson M, Humbel S, Froese R, Matsubara T, Sieber S, Morokuma K (1996) ONIOM: a multilayered integrated MO/MM method for geometry optimizations and single point energy predictions. J Phys Chem 100(50):19,357–19,363CrossRefGoogle Scholar
  101. 101.
    Szabo A, Ostlund N (1989) Modern quantum chemistry: introduction to advanced electronic structure theory. McGraw-Hill, New YorkGoogle Scholar
  102. 102.
    Tadmor E, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Philos Mag A 73(6):1529–1563CrossRefGoogle Scholar
  103. 103.
    Tadmor E, Phillips R, Ortiz M (1996) Mixed atomistic and continuum models of deformation in solids. Langmuir 12(19):4529–4534CrossRefGoogle Scholar
  104. 104.
    Tan Y, Yang D, Sheng Y (2009) Discrete element method (DEM) modelling of fracture and damage in the machining process of polycrystalline SiC. J Eur Ceram Soc 29(6):1029–1037CrossRefGoogle Scholar
  105. 105.
    Ting JM, Khwaja M, Meachum L, Rowell J (1993) An ellipse-based discrete element model for granular materials. Int J Anal Numer Methods Geomech 17(9):603–623zbMATHCrossRefGoogle Scholar
  106. 106.
    To A, Li S (2005) Perfectly matched multiscale simulations. Phys Rev B 72:035,414CrossRefGoogle Scholar
  107. 107.
    Traversoni L (1994) Natural neighbour finite elements. In: International conference on hydraulic engineering software hydrosoft proceedings, vol 2, pp 291–297Google Scholar
  108. 108.
    Versteeg HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite volume method, 2nd edn. Pearson Education Limited, LondonGoogle Scholar
  109. 109.
    Voronoi G (1907) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal für die Reine und Angewandte Mathematik 133:97–178MathSciNetGoogle Scholar
  110. 110.
    Wagner G, Liu W (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190(1):249–274zbMATHCrossRefGoogle Scholar
  111. 111.
    Weinan E, Lu J, Yang J (2006) Uniform accuracy of the quasicontinuum method. Phys Rev B 74(21):214,115CrossRefGoogle Scholar
  112. 112.
    Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193(17–20):1645–1669zbMATHMathSciNetCrossRefGoogle Scholar
  113. 113.
    Xie W, Liu Z, Young YL (2009) Application of a coupled Eulerian-Lagrangian method to simulate interactions between deformable composite structures and compressible multiphase flow. Int J Numer Methods Eng 80:1497–1519zbMATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    Xu M, Gracie R, Belytschko T (2009) Multiscale modeling with extended bridging domain method. In: Fish J (ed) Bridging the scales in science and engineering. Oxford Press, OxfordGoogle Scholar
  115. 115.
    Yvonnet J, Chinesta F, Lorong P, Ryckelynck D (2005) The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces. Int J Therm Sci 44:559–569Google Scholar
  116. 116.
    Yvonnet J, Ryckelynck D, Lorong P, Chinesta F (2004) A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM). Int J Numer Methods Eng 60:1451–1474zbMATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    Zang M, Chen H, Lei Z (2010) Simulation on high velocity impact process of windshield by SPH/FEM coupling method. In: International conference on information engineering, Beidaihe, China, pp 381–384 (2010)Google Scholar
  118. 118.
    Zhu T, Altruni SN (1998) A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free galerkin methods. Int J Numer Methods Eng 21:211–222zbMATHGoogle Scholar
  119. 119.
    Zhu Z, Lu X, Li J (2001) A study of domain decomposition and parallel computation. Acta Mech 150(3–4):219–235Google Scholar
  120. 120.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Elsevier, AmsterdamzbMATHGoogle Scholar
  121. 121.
    Zienkiewicz OC, Taylor RL, Nithiarasu P (2005) The finite element method for fluid dynamics. Elsevier, AmsterdamGoogle Scholar
  122. 122.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2005) Finite element method: its basis & fundamentals. Elsevier, AmsterdamGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  • Mohamed Jebahi
    • 1
    • 2
    Email author
  • Frédéric Dau
    • 3
  • Jean-Luc Charles
    • 3
  • Ivan Iordanoff
    • 3
  1. 1.Laval UniversityQuebecCanada
  2. 2.I2M, UMR 5295 CNRSArts et Metiers ParisTechTalenceFrance
  3. 3.I2M, UMR 5295 CNRSArts et Metiers ParisTechTalenceFrance

Personalised recommendations