Domain Decomposition Methods for Multiscale Modeling

  • Xin BianEmail author
  • Matej Praprotnik
Living reference work entry


Domain decomposition methods (DDM), which originate from the Schwarz alternating method to solve elliptic partial different equations, are largely extended and prove to have increasing influences on multiscale modeling of materials. We discuss some of the important extensions of the DDM in the fields of multiscale modeling for soft materials such as simple and complex fluids. To this end, we typically model the fluids in two or more levels of detail, which exploits the computational efficiency of the coarse model and physical accuracy of the fine description. For simple fluids, we take a continuum perspective to couple the molecular dynamics (MD) and Navier-Stokes equations by matching the state variables and/or fluxes across the hybrid interface. For complex fluids, we take a discrete perspective to encompass the complex structure of the molecules and couple the MD with coarse-grained MD by interpolating the forces between the two levels of descriptions.



Xin Bian acknowledges Prof. George Em Karniadakis, who led him to the research field of DDM for multiscale modeling. During his postdoctoral period, Xin benefited enormously from discussions with Prof. Karniadakis and his group. Xin Bian is also grateful for the discussions and full support from Prof. Nikolaus A. Adams, without whom the completeness of this work is impossible. Matej Praprotnik would like to thank Rafael Delgado-Buscalioni, Kurt Kremer, Luigi Delle Site, Jens H. Walther, and Petros Koumoutsakos for discussions and collaboration on this topic. He also acknowledges financial support from the Slovenian Research Agency (research core funding No. P1-0002 and the project J1-7435).


  1. Agarwal A, Delle Site L (2015) Path integral molecular dynamics within the grand canonical-like adaptive resolution technique: simulation of liquid water. J Chem Phys 143:094102Google Scholar
  2. Agarwal A, Delle Site L (2016) Grand-canonical adaptive resolution centroid molecular dynamics: implementation and application. Comput Phys Commun 206:26Google Scholar
  3. Agarwal A, Wang H, Schütte C, Delle Site L (2014) Chemical potential of liquids and mixtures via adaptive resolution simulation. J Chem Phys 141:034102Google Scholar
  4. Agarwal A, Zhu J, Hartmann C, Wang H, Delle Site L (2015) Molecular dynamics in a grand ensemble: Bergmann-Lebowitz model and adaptive resolution simulation. New J Phys 17:083042Google Scholar
  5. Alekseeva U, Winkler RG, Sutmann G (2016) Hydrodynamics in adaptive resolution particle simulations: multiparticle collision dynamics. J Comput Phys 314:14–34Google Scholar
  6. Allen MP, Tildesley DJ (1989) Computer simulation of liquids. Clarendon Press, OxfordGoogle Scholar
  7. Altenhoff AM, Walther JH, Koumoutsakos P (2007) A stochastic boundary forcing for dissipative particle dynamics. J Comp Phys 225:1125–1136Google Scholar
  8. Bevc S, Junghans C, Kremer K, Praprotnik M (2013) Adaptive resolution simulation of salt solutions. New J Phys 15:105007Google Scholar
  9. Bian X, Li Z, Deng M, Karniadakis GE (2015a) Fluctuating hydrodynamics in periodic domains and heterogeneous adjacent multidomains: thermal equilibrium. Phys Rev E 92:053302Google Scholar
  10. Bian X, Li Z, Karniadakis GE (2015b) Multi-resolution flow simulations by smoothed particle hydrodynamics via domain decomposition. J Comput Phys 297:132–155Google Scholar
  11. Bian X, Deng M, Tang YH, Karniadakis GE (2016) Analysis of hydrodynamic fluctuations in heterogeneous adjacent multidomains in shear flow. Phys Rev E 93:033312Google Scholar
  12. Bian X, Deng M, Karniadakis GE (2018) Analytical and computational studies of correlations of hydrodynamic fluctuations in shear flow. Commun Comput Phys 23:93–117Google Scholar
  13. Borg MK, Lockerby DA, Reese JM (2013) A multiscale method for micro/nano flows of high aspect ratio. J Comput Phys 233:400–413Google Scholar
  14. Borg MK, Lockerby DA, Reese JM (2014) The fade mass-stat: a technique for inserting or deleting particles in molecular dynamics simulations. J Chem Phys 140(7):074110Google Scholar
  15. Chang H, Li X, Li H, Karniadakis GE (2016) Md/dpd multiscale framework for predicting morphology and stresses of red blood cells in health and disease. PLOS Comput Bio 12:e1005173Google Scholar
  16. De Fabritiis G, Delgado-Buscalioni R, Coveney PV (2004) Energy controlled insertion of polar molecules in dense fluids. J Chem Phys 121(24):12139–12142Google Scholar
  17. De Fabritiis G, Delgado-Buscalioni R, Coveney PV (2006) Multiscale modeling of liquids with molecular specificity. Phys Rev Lett 97:134501Google Scholar
  18. Delgado-Buscalioni R (2012) Tools for multiscale simulation of liquids using open molecular dynamics. In: Engquist B, Runborg O, Tsai YHR (eds) Numerical analysis of multiscale computations, vol 82. Springer, Berlin/Heidelberg, pp 145–166Google Scholar
  19. Delgado-Buscalioni R, Coveney PV (2003a) Continuum-particle hybrid coupling for mass, momentum, and energy transfers in unsteady fluid flow. Phys Rev E 67:046704Google Scholar
  20. Delgado-Buscalioni R, Coveney PV (2003b) USHER: an algorithm for particle insertion in dense fluids. J Chem Phys 119:978–987Google Scholar
  21. Delgado-Buscalioni R, Coveney P (2004) Hybrid molecularcontinuum fluid dynamics. Phil Trans R Soc A 362(1821):1639–1654Google Scholar
  22. Delgado-Buscalioni R, De Fabritiis G (2007) Embedding molecular dynamics within fluctuating hydrodynamics in multiscale simulations of liquids. Phys Rev E 76:036709Google Scholar
  23. Delgado-Buscalioni R, Flekkøy EG, Coveney PV (2005) Fluctuations and continuity in particle-continuum hybrid simulations of unsteady flows based on flux-exchange. EPL 69(6):959Google Scholar
  24. Delgado-Buscalioni R, Kremer K, Praprotnik M (2008) Concurrent triple-scale simulation of molecular liquids. J Chem Phys 128:114110Google Scholar
  25. Delgado-Buscalioni R, Kremer K, Praprotnik M (2009) Coupling atomistic and continuum hydrodynamics through a mesoscopic model: application to liquid water. J Chem Phys 131:244107Google Scholar
  26. Delgado-Buscalioni R, Sablić J, Praprotnik M (2015) Open boundary molecular dynamics. Eur Phys J Special Top 224:2331–2349Google Scholar
  27. Delle Site L (2016) Formulation of Liouville’s theorem for grand ensemble molecular simulations. Phys Rev E 93:022130Google Scholar
  28. Delle Site L (2018) Grand canonical adaptive resolution simulation for molecules with electrons: a theoretical framework based on physical consistency. Comput Phys Commun 222:94–101Google Scholar
  29. Delle Site L, Praprotnik M (2017) Molecular systems with open boundaries: theory and simulation. Phys Rep 693:1–56Google Scholar
  30. Deng M, Li X, Liang H, Caswell B, Karniadakis GE (2012) Simulation and modeling of slip flow over surfaces grafted with polymer brushes and glycocalyx fibres. J Fluid Mech 711:192–211Google Scholar
  31. Español P (1995) Hydrodynamics from dissipative particle dynamics. Phys Rev E 52(2):1734–1742Google Scholar
  32. Español P, Revenga M (2003) Smoothed dissipative particle dynamics. Phys Rev E 67(2):026705Google Scholar
  33. Español P, Warren P (1995) Statistical mechanics of dissipative particle dynamics. Europhys Lett 30(4):191–196Google Scholar
  34. Español P, Warren PB (2017) Perspective: dissipative particle dynamics. J Chem Phys 146(15):150901Google Scholar
  35. Español P, Delgado-Buscalioni R, Everaers R, Potestio R, Donadio D, Kremer K (2015) Statistical mechanics of hamiltonian adaptive resolution simulations. J Chem Phys 142:064115Google Scholar
  36. Everaers, R (2016) Thermodynamic translational invariance in concurrent multiscale simulations of liquids. Eur Phys J Special Top 225:1483–1503Google Scholar
  37. Fedosov DA, Karniadakis GE (2009) Triple-decker: interfacing atomistic-mesoscopic-continuum flow regimes. J Comput Phys 228:1157–1171Google Scholar
  38. Fedosov DA, Caswell B, Karniadakis GE (2010) A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys J 98:2215–2225Google Scholar
  39. Fedosov DA, Pan W, Caswell B, Gompper G, Karniadakis GE (2011) Predicting human blood viscosity in silico. Proc Natl Acad Sci USA 108:11772–11777Google Scholar
  40. Flekkøy EG, Wagner G, Feder J (2000) Hybrid model for combined particle and continuum dynamics. Europhys Lett 52(3):271–276Google Scholar
  41. Flekkøy EG, Delgado-Buscalioni R, Coveney PV (2005) Flux boundary conditions in particle simulations. Phys Rev E 72:026703Google Scholar
  42. Fogarty AC, Potestio R, Kremer K (2015) Adaptive resolution simulation of a biomolecule and its hydration shell: structural and dynamical properties. J Chem Phys 142:195101Google Scholar
  43. Fogelson AL, Neeves KB (2015) Fluid mechanics of blood clot formation. Ann Rev Fluid Mech 47(1):377–403Google Scholar
  44. Garcia AL, Bell JB, Crutchfield WY, Alder BJ (1999) Adaptive mesh and algorithm refinement using direct simulation Monte Carlo. J Comput Phys 154(1):134–155Google Scholar
  45. Grinberg L (2012) Proper orthogonal decomposition of atomistic flow simulations. J Comput Phys 231(16):5542–5556Google Scholar
  46. Grinberg L, Fedosov DA, Karniadakis GE (2013) Parallel multiscale simulations of a brain aneurysm. J Comput Phys 244:131–147Google Scholar
  47. Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107(11):4423–4435Google Scholar
  48. Hadjiconstantinou NG (1999) Hybrid atomisticcontinuum formulations and the moving contact-line problem. J Comput Phys 154(2):245–265Google Scholar
  49. Hadjiconstantinou NG, Patera AT (1997) Heterogeneous atomistic-continuum representations for dense fluid systems. Int J Mod Phys C 08(04):967–976Google Scholar
  50. Halverson JD, Brandes T, Lenz O, Arnold A, Bevc S, Starchenko V, Kremer K, Stuehn T, Reith D (2013) Espresso++: a modern multiscale simulation package for soft matter systems. Comput Phys Commun 184(4):1129–1149Google Scholar
  51. Hoogerbrugge PJ, Koelman JMVA (1992) Simulating microsopic hydrodynamics phenomena with dissipative particle dynamics. Europhys Lett 19(3):155–160Google Scholar
  52. Kevrekidis IG, Samaey G (2009) Equation-free multiscale computation: algorithms and applications. Annu Rev Phys Chem 60(1):321–344Google Scholar
  53. Koumoutsakos P (2005) Multiscale flow simulations using particles. Ann Rev Fluid Mech 37:457–487Google Scholar
  54. Kreis K, Fogarty A, Kremer K, Potestio R (2015) Advantages and challenges in coupling an ideal gas to atomistic models in adaptive resolution simulations. Eur Phys J Special Top 224:2289–2304Google Scholar
  55. Landau LD, Lifshitz EM (1987) Fluid mechanics. Course of theoretical physics, vol 6, 2nd edn. Pergamon Press, OxfordGoogle Scholar
  56. Laso M, Öttinger HC (1993) Calculation of viscoelastic flow using molecular models: the CONNFFESSIT approach. J Non-Newton Fluid Mech 47:1–20Google Scholar
  57. Lei H, Fedosov DA, Karniadakis GE (2011) Time-dependent and outflow boundary conditions for dissipative particle dynamics. J Comput Phys 230:3765–3779Google Scholar
  58. Li J, Liao D, Yip S (1998) Coupling continuum to molecular-dynamics simulation: reflecting particle method and the field estimator. Phys Rev E 57:7259–7267Google Scholar
  59. Malevanets A, Kapral R (1999) Mesoscopic model for solvent dynamics. J Chem Phys 110(17):8605–8613Google Scholar
  60. Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model Simul Mater Sci Eng 17(5):053001Google Scholar
  61. Mohamed K, Mohamad A (2010) A review of the development of hybrid atomistic-continuum methods for dense fluids. Microfluid Nanofluid 8(3):283–302Google Scholar
  62. Mukherji D, Kremer K (2013) Coil-globule-coil transition of PNIPAm in aqueous methanol: coupling all-atom simulations to semi-grand canonical coarse-grained reservoir. Macromolecules 46(22):9158–9163Google Scholar
  63. Nagarajan A, Junghans C, Matysiak S (2013) Multiscale simulation of liquid water using a four-to-one mapping for coarse-graining. J Chem Theory Comput 9:5168–5175Google Scholar
  64. Neumann P, Bian X (2017) MaMiCo: transient multi-instance molecular-continuum flow simulation on super-computers. Comput Phys Commun 220:390–402Google Scholar
  65. Nie XB, Chen SY, Weinan E, Robbins MO (2004) A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. J Fluid Mech 500:55–64Google Scholar
  66. Nie X, Robbins MO, Chen S (2006) Resolving singular forces in cavity flow: multiscale modeling from atomic to millimeter scales. Phys Rev Lett 96:134501Google Scholar
  67. O’Connell ST, Thompson PA (1995) Molecular dynamics-continuum hybrid computations: a tool for studying complex fluid flows. Phys Rev E 52:R5792–R5795Google Scholar
  68. Petsev ND, Leal LG, Shell MS (2015) Hybrid molecular-continuum simulations using smoothed dissipative particle dynamics. J Chem Phys 142(4):044101Google Scholar
  69. Petsev ND, Leal LG, Shell MS (2017) Coupling discrete and continuum concentration particle models for multiscale and hybrid molecular-continuum simulations. J Chem Phys 147:234112Google Scholar
  70. Pivkin IV, Karniadakis GE (2006) Controlling density fluctuations in wall-bounded dissipative particle dynamics systems. Phys Rev Lett 96:206001Google Scholar
  71. Poma A, Delle Site L (2010) Classical to path-integral adaptive resolution in molecular simulation: towards a smooth quantum-classical coupling. Phys Rev Lett 104:250201Google Scholar
  72. Poma A, Delle Site L (2011) Adaptive resolution simulation of liquid para-hydrogen: testing the robustness of the quantum-classical adaptive coupling. Phys Chem Chem Phys 13:10510Google Scholar
  73. Potestio R, Español P, Delgado-Buscalioni R, Everaers R, Kremer K, Donadio D (2013a) Monte Carlo adaptive resolution simulation of multicomponent molecular liquids. Phys Rev Lett 111:060601Google Scholar
  74. Potestio R, Fritsch S, Español P, Delgado-Buscalioni R, Kremer K, Everaers R, Donadio D (2013b) Hamiltonian adaptive resolution simulation for molecular liquids. Phys Rev Lett 110:108301Google Scholar
  75. Praprotnik M, Delle Site L, Kremer K (2005) Adaptive resolution molecular-dynamics simulation: changing the degrees of freedom on the fly. J Chem Phys 123(22):224106Google Scholar
  76. Praprotnik M, Delle Site L, Kremer K (2008) Multiscale simulation of soft matter: from scale bridging to adaptive resolution. Annu Rev Phys Chem 59:545–571Google Scholar
  77. Praprotnik M, Poblete S, Kremer K (2011) Statistical physics problems in adaptive resolution computer simulations of complex fluids. J Stat Phys 145:946–966Google Scholar
  78. Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Oxford science publications, OxfordGoogle Scholar
  79. Ren W (2007) Analytical and numerical study of coupled atomistic-continuum methods for fluids. J Comput Phys 227(2):1353–1371Google Scholar
  80. Ren W, Weinan EW (2005) Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J Comput Phys 204(1):1–26Google Scholar
  81. Rossinelli D, Tang YH, Lykov K, Alexeev D, Bernaschi M, Hadjidoukas P, Bisson M, Joubert W, Conti C, Karniadakis G, Fatica M, Pivkin I, Koumoutsakos P (2015) The in-silico lab-on-a-chip: petascale and high-throughput simulations of microfluidics at cell resolution. In: Proceedings of the international conference for high performance computing, networking, storage and analysis, SC’15. ACM, New York, pp 2:1–2:12Google Scholar
  82. Sablić J, Praprotnik M, Delgado-Buscalioni R (2016) Open boundary molecular dynamics of sheared star-polymer melts. Soft Matter 12:2416–2439Google Scholar
  83. Sablić J, Delgado-Buscalioni R, Praprotnik M (2017a) Application of the eckart frame to soft matter: rotation of star polymers under shear flow. Soft Matter 13:6988–7000Google Scholar
  84. Sablić J, Praprotnik M, Delgado-Buscalioni R (2017b) Deciphering the dynamics of star molecules in shear flow. Soft Matter 13:4971–4987Google Scholar
  85. Scukins A, Nerukh D, Pavlov E, Karabasov S, Markesteijn A (2015) Multiscale molecular dynamics/hydrodynamics implementation of two dimensional “mercedes benz” water model. Euro Phys J Special Top 224(12):2217–2238Google Scholar
  86. Smith B, Bjørstad P, Gropp W (1996) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, New YorkGoogle Scholar
  87. Tang YH, Kudo S, Bian X, Li Z, Karniadakis GE (2015) Multiscale universal interface: a concurrent framework for coupling heterogeneous solvers. J Comput Phys 297:13–31Google Scholar
  88. Thompson PA, Robbins MO (1990) Shear flow near solids: epitaxial order and flow boundary conditions. Phys Rev A 41:6830–6837Google Scholar
  89. Toselli A, Widlund OB (2005) Domain decomposition methods–algorithms and theory. Springer, Berlin/HeidelbergGoogle Scholar
  90. Tuckerman ME (2010) Statistical mechanics: theory and molecular simulation. Oxford University Press, OxfordGoogle Scholar
  91. Vázquez-Quesada A, Ellero M, Español P (2009) Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics. J Chem Phys 130(3):034901Google Scholar
  92. Walther JH, Praprotnik M, Kotsalis EM, Koumoutsakos P (2012) Multiscale simulation of water flow past a C540 fullerene. J Comput Phys 231(7):2677–2681Google Scholar
  93. Wang H, Hartmann C, Schütte C, Delle Site L (2013) Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique. Phys Rev X 3:011018Google Scholar
  94. Warshel A, Karplus M (1972) Calculation of ground and excited state potential surfaces of conjugated molecules. I. Formulation and parametrization. J Am Chem Soc 94(16):5612–5625Google Scholar
  95. Warshel A, Levitt M (1976) Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. J Mole Bio 103(2):227–249Google Scholar
  96. Weinan EW, Engquist B, Li X, Ren W, Vanden-Eijnden E (2007) Heterogeneous multiscale method: a review. Commun Comput Phys 2(3):367–450Google Scholar
  97. Weinbaum S, Tarbell JM, Damiano ER (2007) The structure and function of the endothelial glycocalyx layer. Ann Rev Biomed Eng 9(1):121–167Google Scholar
  98. Werder T, Walther JH, Koumoutsakos P (2005) Hybrid atomistic-continuum method for the simulation of dense fluid flows. J Comput Phys 205:373–390Google Scholar
  99. Wijesinghe HS, Hadjiconstantinou NG (2004) Discussion of hybrid atomistic-continuum methods for multiscale hydrodynamics. Inter J Multi Comput Eng 2(2):189–202Google Scholar
  100. Yasuda S, Yamamoto R (2010) Multiscale modeling and simulation for polymer melt flows between parallel plates. Phys Rev E 81:036308Google Scholar
  101. Zavadlav J, Praprotnik M (2017) Adaptive resolution simulations coupling atomistic water to dissipative particle dynamics. J Chem Phys 147:114110Google Scholar
  102. Zavadlav J, Melo MN, Marrink SJ, Praprotnik M (2014) Adaptive resolution simulation of an atomistic protein in MARTINI water. J Chem Phys 140:054114Google Scholar
  103. Zavadlav J, Melo MN, Marrink SJ, Praprotnik M (2015a) Adaptive resolution simulation of polarizable supramolecular coarse-grained water models. J Chem Phys 142:244118Google Scholar
  104. Zavadlav J, Podgornik R, Praprotnik M (2015b) Adaptive resolution simulation of a DNA molecule in salt solution. J Chem Theory Comput 11:5035–5044Google Scholar
  105. Zavadlav J, Marrink SJ, Praprotnik M (2016a) Adaptive resolution simulation of supramolecular water: the concurrent making, breaking, and remaking of water bundles. J Chem Theory Comput 12:4138–4145Google Scholar
  106. Zavadlav J, Podgornik R, Melo MN, Marrink SJ, Praprotnik M (2016b) Adaptive resolution simulation of an atomistic DNA molecule in MARTINI salt solution. Eur Phys J Spec Top 225:1595–1607Google Scholar
  107. Zavadlav J, Bevc S, Praprotnik M (2017) Adaptive resolution simulations of biomolecular systems. Eur Biophys J 46:821–835Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Aerodynamics and Fluid Mechanics, Department of Mechanical EngineeringTechnical University of MunichGarching bei MünchenGermany
  2. 2.Laboratory for Molecular ModelingNational Institute of ChemistryLjubljanaSlovenia
  3. 3.Department of Physics, Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Section editors and affiliations

  • Ming Dao
    • 1
  • George E Karniadakis
    • 2
  1. 1.Department of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridgeUnited States
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations