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Efficient Stabilization of Advection Terms Involved in Separated Representations of Boltzmann and Fokker-Planck Equations

Published online by Cambridge University Press:  30 April 2015

F. Chinesta ,
E. Abisset-Chavanne ,
A. Ammar  and
E. Cueto
F. Chinesta*
Affiliation:
GeM Institute, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
E. Abisset-Chavanne
Affiliation:
GeM Institute, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
A. Ammar
Affiliation:
Arts et Métiers ParisTech, 2 Boulevard du Ronceray, BP 93525, F-49035 Angers cedex 01, France
E. Cueto
Affiliation:
Aragon Institute of Engineering Research, Universidad de Zaragoza, Maria de Luna, 7, E-50018 Zaragoza, Spain
*
*Corresponding author. Email addresses: [email protected] (F. Chinesta), [email protected] (E. Abisset-Chavanne), [email protected] (A. Ammar), [email protected] (E. Cueto)
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Abstract

The fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

Keywords

35G1578M1578M2578M3435Q8435Q20Boltzmann equationFokker-Planck equationproper generalized decompositionmultidimensional modelsconvective stabilization
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 4 , April 2015 , pp. 975 - 1006
Copyright
Copyright © Global-Science Press 2015 

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References

[1] Abisset-Chavanne, E., Mezher, R., Le Corre, S., Ammar, A., Chinesta, F.. Kinetic theory microstructure modeling in concentrated suspensions. Entropy, 15, 2805–2832, 2013.Google Scholar
[2]Aghighi, S., Ammar, A., Metivier, C., Normandin, M., Chinesta, F.. Non-incremental transient solution of the Rayleigh-Bnard convection model using the PGD. Journal of Non-Newtonian Fluid Mechanics, 200, 65–78, 2013.Google Scholar
[3]Ammar, A., Chinesta, F., A particle strategy for solving Fokker-Planck equation governing the fiber orientation distribution in steady recirculating flows involving short fiber suspensions, Lecture Notes on Computational Science and Engineering. Springer, 43, 1–16, 2004.Google Scholar
[4]Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics, 139, 153–176, 2006.Google Scholar
[5]Ammar, A., Mokdad, B., Chinesta, F., 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. Journal of Non-Newtonian Fluid Mechanics, 144, 98–121, 2007.Google Scholar
[6]Ammar, A., Chinesta, F., Joyot, P.. The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions. International Journal for Multiscale Computational Engineering, 6/3, 191–213, 2008.CrossRefGoogle Scholar
[7]Ammar, A., Normandin, M., Daim, F., Gonzalez, D., Cueto, E., Chinesta, F.. Non-incremental strategies based on separated representations: Applications in computational rheology. Communications in Mathematical Sciences, 8/3, 671–695, 2010.CrossRefGoogle Scholar
[8]Ammar, A., Chinesta, F., Diez, P., Huerta, A.. An error estimator for separated representations of highly multidimensional models. Computer Methods in Applied Mechanics and Engineering, 199 1872–1880, 2010.Google Scholar
[9]Ammar, A., Cueto, E., Chinesta, F.. Reduction of the Chemical Master Equation for gene regulatory networks using Proper Generalized Decompositions. International Journal for Numerical Methods in Biomedical Engineering, 28/9, 960–973, 2012.Google Scholar
[10]Bhatnagar, P.L., Gross, E.P., Krook, M.. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94/3, 511–525, 1954.Google Scholar
[11]Chaubal, C.V., Srinivasan, A., Egecioglu, O., Leal, L.G.. Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems. J. Non-Newtonian Fluid Mech., 70, 125–154, 1997.Google Scholar
[12]Chauviere, C., Lozinski, A.. Simulation of dilute polymer solutions using a Fokker-Planck equation. Computers and Fluids., 33, 687–696. 2004.Google Scholar
[13]Chen, H.D., Chen, S.Y., Matthaeus, W.H.. Recorvery of the Navier-Stokes equations using a Lattice-Gas Boltzmann Method. Physical Review A, 45/8, R5339-R5342, 1992.Google Scholar
[14]Chen, S., Doolen, G.D. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30, 329–364,1998.CrossRefGoogle Scholar
[15]Chinesta, F., Chaidron, G., Poitou, A.. On the solution of the Fokker-Planck equations in steady recirculating flows involving short fiber suspensions. J. Non-Newtonian Fluid Mech., 113/2–3, 97–125, 2004.Google Scholar
[16]Chinesta, F., Ammar, A., Falco, A., Laso, M.. On the reduction of stochastic kinetic theory models of complex fluids. Modeling and Simulation in Materials Science and Engineering, 15, 639–652, 2007.Google Scholar
[17]Chinesta, F., Mackley, M.. Microstructure evolution during liquid-liquid laminar mixing: A kinetic theory approach. International Journal of Material Forming, 1, 47–55, 2008.CrossRefGoogle Scholar
[18]Chinesta, F., Ammar, A., Cueto, E.. Recent advances and new challenges in the use of the Proper Generalized Decomposition for solving multidimensional models. Archives of Computational Methods in Engineering, 17/4, 327–350, 2010.Google Scholar
[19]Chinesta, F., Ammar, A., Leygue, A., Keunings, R.. An overview of the Proper Generalized Decomposition with applications in computational rheology. Journal of Non Newtonian Fluid Mechanics, 166, 578–592, 2011.Google Scholar
[20]Chinesta, F., Ladeveze, P., Cueto, E.. A short review in model order reduction based on Proper Generalized Decomposition. Archives of Computational Methods in Engineering, 18, 395–404, 2011.Google Scholar
[21]Chinesta, F., Leygue, A., Bordeu, F., Aguado, J.V., Cueto, E., Gonzalez, D., Alfaro, I., Ammar, A., Huerta, A.. PGD-based computational vademecum for efficient design, optimization and control. Arch. Comput. Methods Eng., 20, 31–59, 2013Google Scholar
[22]Chinesta, F.. From single-scale to two-scales kinetic theory descriptions of rods suspensions. Archives in Computational Methods in Engineering, 20/1, 1–29, 2013.Google Scholar
[23]Chinesta, F., Keunings, R., Leygue, A.. The Proper Generalized Decomposition for advanced numerical simulations. A primer. Springerbriefs, Springer, 2014.Google Scholar
[24]Derksen, J.J., Prashant, . Simulations of complex flow of thixotropic liquids. Journal of Non Newtonian Fluid Mechanics, 160/2-3, 65–75, 2009.Google Scholar
[25]Dhumieres, D., Lallemand, P.. Lattice Gas Automata For Fluid-Mechanics. Physica A, 140/1–2, 326–335, 1986.Google Scholar
[26]Espanol, P.. Statistical Mechanics of Coarse-Graining. In Novel Methods in Soft Matter Simulations. Karttunen, M., Lukkarinen, A., and Vattulainen, I., Eds. Lecture Notes in Physics, vol 640, pages 69–115. Springer Berlin Heidelberg, 2004.Google Scholar
[27]Grmela, M., Chinesta, F., Ammar, A.. Mesoscopic tube model of fluids composed of wormlike micelles. Rheologica Acta, 49/5, 495–506, 2010.Google Scholar
[28]Grmela, M., Ammar, A., Chinesta, F.. One and two-fiber orientation kinetic theories of fiber suspensions. Journal of Non-Newtonian Fluid Mechanics, 200, 17–33, 2013.Google Scholar
[29]He, X.Y., Luo, L.S., Dembo, M.. Some progress in lattice Boltzmann method. 1. Nonuniform mesh grids. Journal of Computational Physics, 129/2, 357–363, 1996.Google Scholar
[30]He, X.Y., Luo, L.S. Theory of the lattice Boltzmann method: From the Boltzmann Equation to the lattice Boltzmann equation. Physical Review E, 56/6, 6811–6817, 1997.Google Scholar
[31]Higuera, F. J. and Jimnez, J.. Boltzmann Approach to Lattice Gas Simulations. Europhys. Lett. 9 663, 1989. doi:10.12090295-5075/9/7/009Google Scholar
[32]Hou, S.L., Zou, Q., Chen, S.Y., Doolen, G., Cogley, A.C.. Simulation of cavity flow by the Lattice Boltzmann Method. Journal of Computational Physics, 118/2, 329–347, 1995.Google Scholar
[33]Inamuro, T., Ogata, T., Tajima, S., Konishi, N.. A lattice Boltzmann method for incompressible two-phase flows with large density differences. Journal of Computational Physics, 198/2, 628–644, 2004.Google Scholar
[34]Jendrejack, R., de Pablo, J., Graham, M.. A method for multiscale simulation of flowing complex fluids. J. Non-Newtonian Fluid Mech., 108, 123–142, 2002.Google Scholar
[35]Keunings, R., Micro-macro methods for the multiscale simulation viscoelastic flow using molecular models of kinetic theory, Rheology Reviews, Binding, D.M. and Walters, K. (Edts.), British Society of Rheology, 67–98, 2004.Google Scholar
[36]Ladd, A.J.C., Verberg, R.. Lattice-Boltzmann simulations of particle-fluid suspensions. Journal of Statistical Physics, 104/5–6, 1191–1251, 2001.Google Scholar
[37]Lallemand, P., Luo, L.S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Physical Review E, 61/6, 6546–6562, 2000.Google Scholar
[38]Lamari, H., Ammar, A., Leygue, A., Chinesta, F.. On the solution of the multidimensional Langer’s equation by using the Proper Generalized Decomposition Method for modeling phase transitions. Modelling and Simulation in Materials Science and Engineering, 20, 015007, 2012.Google Scholar
[39]Lozinski, A., Chauviere, C.. A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model. J. Computational Physics, 189, 607–625, 2003.Google Scholar
[40]McNamara, G.R., Zanetti, G.. Use of the Boltzmann-Equation to Simulate Lattice-Gas Automata. Physical Review Letters, 61/20, 2332–2335, 1988.Google Scholar
[41]Maitejean, G., Ammar, A., Chinesta, F, Grmela, M.. Deterministic solution of the kinetic theory model of colloidal suspensions of structureless particles. Rheologica Acta, 51/6, 527–543, 2012.Google Scholar
[42]Maitrejean, G., Ammar, A., Chinesta, F.. Simulating microstructural evolution during pasive mixing. Internantional Journal of Material Forming, 5/1, 73–81, 2012.Google Scholar
[43]Maitrejean, G., Grmela, M., Ammar, A., Chinesta, F.. Kinetic theory of colloidal suspensions: Morphology, rheology and migration. Rheologica Acta, 52/6, 557–577, 2013.Google Scholar
[44]Martys, N.S., Chen, H.D. Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Physical Review E, 53/1, 743–750, 1996.Google Scholar
[45]Mokdad, B., Pruliere, E., Ammar, A., Chinesta, F.. On the simulation of kinetic theory models of complex fluids using the Fokker-Planck approach. Applied Rheology, 17/2, 26494, 1–14, 2007.Google Scholar
[46]Mokdad, B., Ammar, A., Normandin, M., Chinesta, F., Clermont, J.R.. A fully deterministic micro-macro simulation of complex flows involving reversible network fluid models. Mathematics and Computer in Simulation, 80, 1936–1961, 2010.Google Scholar
[47]Nourgaliev, R.R., Dinh, T.N., Theofanous, T.G., Joseph, D.. The lattice Boltzmann equation method: theoretical interpretation, numerics and implications. International Journal of Multiphase Flow, 29/1, 117–169, 2003.Google Scholar
[48]Ottinger, H. C., Laso, M.. Smart polymers in finite element calculation, Int Congr. on Rheology, Brussel, Belguim, 1992.Google Scholar
[49]Pruliere, E., Ammar, A., El Kissi, N., Chinesta, F.. Recirculating flows involving short fiber suspensions: Numerical difficulties and efficient advanced micro-macro solvers. Archives of Computational Methods in Engineering, State of the Art Reviews, 16, 1–30, 2009.Google Scholar
[50]Rohde, M., Derksen, J.J., Van den Akker, H.E.A.. Volumetric method for calculating the flow around moving objects in Lattice-Boltzmann schemes. Physical Review E, 65/5, 2002.Google Scholar
[51]Rohde, M., Kandhai, D., Derksen, J.J., van den Akker, H.E.A.. A generic, mass conservative local grid refinement technique for Lattice-Boltzmann schemes. International Journal for Numerical Methods in Fluids, 51/4, 439–468, 2006.Google Scholar
[52]Somasi, M., Khomami, B., Woo, N.J., Hur, J.S., Shaqfeh, E.S.G.. Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues. J. Non-Newtonian Fluid Mech., 108/1–3, 227–255, 2002.Google Scholar
[53]Succi, S.. The Lattice Boltzmann Equation For Fluid Dynamics and Beyond. Clarendon Press, 2001Google Scholar
[54]Venkiteswaran, G., Junk, M., A QMC approach for high dimensional Fokker-Planck equations modelling polymeric liquids. Mathematics and Computers in Simulation, 68, 43–56, 2005.Google Scholar
[55]Verberg, R., Ladd, A.J.C.. Simulation of low-Reynolds-number flow via a time-independent Lattice-Boltzmann method. Physical Review E, 60/3, 3366–3373, 1999.CrossRefGoogle Scholar
[56]Yu, D.Z., Mei, R.W., Luo, L.S., Shyy, W.. Viscous flow computations with the method of lattice Boltzmann equation. Progress in Aerospace Sciences, 39/5, 329–367, 2003.Google Scholar