Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v9.i1.20
pages 1-16


Elias Cueto
Group of Structural Mechanics and Materials Modelling (GEMM), Aragon Institute of Engineering Research (I3A), Betancourt Building, Maria de Luna, 5, E-50018 Zaragoza, Spain
M. Laso
Laboratory of Non-Metallic Materials, Department of Chemical Engineering, Universidad Poliecnica de Madrid, Jose Gutierrez Abascal 2, E-28006 Madrid, Spain
F. Chinesta
EADS Corporate International Chair, Ecole Centrale de Nantes, 1 rue de la Noe, 44321 Nantes Cedex 3, France


We present in this paper a numerical technique for the stochastic simulation of molecular models of viscoelastic fluids based on kinetic theory. The technique is based on the use of meshless methods and allows for an updated Lagrangian description of the conservation equations. It makes use of natural neighbor Galerkin schemes that allow for a proper geometrical description of the domain as it evolves. The presented technique is especially well suited for the numerical simulation of free-surface flows. In this way, model molecules are associated with nodal positions such that they are advected with material velocities. Problems associated with lack of molecules in certain elements, for instance, as encountered in the basic implementation of CONNFFESSIT approaches, are thus avoided. We present examples of validation and also performance tests of this technique applied to finite extension nonlinear elastic (FENE) and reptation (Doi-Edwards) models.


  1. Alfaro, I., Yvonnet, J., Cueto, E., Chinesta, F., and Doblaré, M., Meshless methods with application to metal forming. DOI: 10.1016/j.cma.2004.10.017

  2. Ammar, A., Mokdad, B., Chinesta, F., and Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. DOI: 10.1016/j.jnnfm.2006.07.007

  3. Ammar, A., Ryckelynck, D., Chinesta, F., and Keunings, R., On the reduction of kinetic theory models related to finitely extensible dumbbells. DOI: 10.1016/j.jnnfm.2006.01.007

  4. Ammar, A., Mokdad, B., Chinesta, F., and Keunings, R., A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: Transient simulation using space-time separated representation. DOI: 10.1016/j.jnnfm.2007.03.009

  5. Baaijens, F. P. T., An iterative solver for the DEVSS/DG method with application to smooth and non-smooth flows of the upper convected Maxwell fluid. DOI: 10.1016/S0377-0257(97)00086-4

  6. Babuška, I., The finite element method with Lagrange multipliers. DOI: 10.1007/BF01436561

  7. Babuška, I. and Aziz, A., On the angle condition in the finite element method. DOI: 10.1137/0713021

  8. Belytschko, T., Lu, Y. Y., and Gu, L., Element-free Galerkin methods. DOI: 10.1002/nme.1620370205

  9. Bonito, A., Picasso, M., and Laso, M., Numerical simulation of 3D viscoelastic flows with free surfaces. DOI: 10.1016/

  10. Chen, J.-S., Wu, C.-T., Yoon, S., and You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods. DOI: 10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A

  11. Chiba, K., Ammar, A., and Chinesta, F., On the fiber orientation in steady recirculating flows involving short fibers suspensions. DOI: 10.1007/s00397-004-0422-3

  12. Chinesta, F., Ammar, A., Falco, A., and Laso, M., On the reduction of stochastic kinetic theory models of complex fluids. DOI: 10.1088/0965-0393/15/6/004

  13. Cueto, E., Calvo, B., and Doblaré, M., Modeling three-dimensional piece-wise homogeneous domains using the α shape based natural element method. DOI: 10.1002/nme.452

  14. Cueto, E., Cegocino, J., Calvo, B., and Doblaré, M., On the imposition of essential boundary conditions in natural neighbour Galerkin methods. DOI: 10.1002/cnm.595

  15. Cueto, E., Doblaré, M., and Gracia, L., Imposing essential boundary conditions in the natural element method by means of densityscaled α shapes. DOI: 10.1002/1097-0207(20001010)49:4<519::AID-NME958>3.0.CO;2-0

  16. Edelsbrunner, H., Kirkpatrick, D. G., and Seidel, R., On the shape of a set of points in the plane. DOI: 10.1109/TIT.1983.1056714

  17. Edelsbrunner, H. and Mücke, E., Three dimensional alpha shapes. DOI: 10.1145/174462.156635

  18. Galavís, A., González, D., Alfaro, I., and Cueto, E., Domain tracking in meshless simulations of free-surface flows. DOI: 10.1007/s00466-008-0263-5

  19. Gigras, P. G. and Khomami, B., Adaptive configuration fields: A new multiscale simulation technique for reptation-based models with a stochastic strain measure and local variations of life span distribution. DOI: 10.1016/S0377-0257(02)00126-X

  20. González, D., Cueto, E., Martinez, M. A., and Doblaré, M., Numerical integration in natural neighbour Galerkin methods. DOI: 10.1002/nme.1038

  21. González, D., Cueto, E., Chinesta, F., and Doblaré, M., A natural element updated Lagrangian strategy for free-surface fluid dynamics. DOI: 10.1016/

  22. González, D., Cueto, E., and Doblaré, M., Volumetric locking in natural neighbour Galerkin methods. DOI: 10.1002/nme.1085

  23. González, D., Cueto, E., and Doblaré, M., Higher-order natural element methods: Towards an isogeometric meshless method. DOI: 10.1002/nme.2237

  24. Herrchen, M. and Öttinger, H. C., A detailed comparison of various FENE models. DOI: 10.1016/S0377-0257(96)01498-X

  25. Keunings, R., On the Peterlin approximation for finitely extensible dumbells. DOI: 10.1016/S0377-0257(96)01497-8

  26. Keunings, R., Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory.

  27. Laso,M. and Öttinger, H. C., Calculation of viscoelastic flow using molecular models: The connffessit approach. DOI: 10.1016/0377-0257(93)80042-A

  28. Lewis, R.W., Navti, S. E., and Taylor, C., A mixed Lagrangian-Eulerian approach to modelling fluid flow during mould filling. DOI: 10.1002/(SICI)1097-0363(19971030)25:8<931::AID-FLD594>3.0.CO;2-1

  29. Martínez, M. A., Cueto, E., Alfaro, I., Doblaré, M., and Chinesta, F., Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods. DOI: 10.1002/nme.1036

  30. Martínez, M. A., Cueto, E., Doblaré, M., and Chinesta, F., Fixed mesh and meshfree techniques in the numerical simulation of injection processes involving short fiber suspensions.

  31. Owens, R. G. and Phillips, T. N., Comput. Rheol..

  32. Öttinger, H. C., Stochastic Processes in Polymeric Fluids.

  33. Poitou, A., Chinesta, F., and Chaidron, G., On the solution of fokker-planck equations in steady recirculating flows involving short fiber suspensions. DOI: 10.1016/S0377-0257(03)00100-9

  34. Pruliere, E., Ammar, A., Kissi, N. E., and Chinesta, F., Recirculating flows involving short fiber suspensions: Numerical difficulties and efficient advanced micro-macro solvers. DOI: 10.1007/s11831-008-9027-9

  35. Ryckelynck, D., Chinesta, F., Cueto, E., and Ammar, A., On the a priori model reduction: Overview and recent developments. DOI: 1007/BF02905932

  36. Sibson, R., A vector identity for the Dirichlet tesselation. DOI: 10.1017/S0305004100056589

  37. Sibson, R., A brief description of natural neighbour interpolation.

  38. Sukumar, N., Moran, B., and Belytschko, T., The natural element method in solid mechanics. DOI: 10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R

  39. Sukumar, N., Moran, B., Semenov, A. Y., and Belikov, V. V., Natural neighbor Galerkin methods.

  40. Thiessen, A. H., Precipitation averages for large areas. DOI: 10.1175/1520-0493(1911)39<1082b:PAFLA>2.0.CO;2

  41. Yvonnet, J., Ryckelynck, D., Lorong, P., and Chinesta, F., A new extension of the natural element method for non-convex and discontinuous problems: The constrained natural element method. DOI: 10.1002/nme.1016

  42. Yvonnet, J., Villon, P., and Chinesta, F., Natural element approximations involving bubbles for treating mechanical models in incompressible media. DOI: 10.1002/nme.1586

  43. Zienkiewicz, O. C. and Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique. DOI: 10.1002/nme.1620330702