American Institute of Mathematical Sciences

Advanced
September  2010, 5(3): 423-460. doi: 10.3934/nhm.2010.5.423

Beyond multiscale and multiphysics: Multimaths for model coupling

Xavier Blanc 1, , Claude Le Bris 2, , Frédéric Legoll 3, and  Tony Lelièvre 2,

1. 

CEA, DAM, DIF, F-91297, Arpajon, France
2. 

Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France, France
3. 

Université Paris-Est, Institut Navier, LAMI, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2

Received  January 2010 Revised  April 2010 Published  July 2010

The purpose of this article is to present a unified view of some multiscale models that have appeared in the past decades in computational materials science. Although very different in nature at first sight, since they are employed to simulate complex fluids on the one hand and crystalline solids on the other hand, the models presented actually share a lot of similarities, many of those being in fact also present in most multiscale strategies. The mathematical and numerical difficulties that these models generate, the way in which they are utilized (in particular as numerical strategies coupling different models in different regions of the computational domain), the computational load they imply, are all very similar in nature. In particular, a common feature of these models is that they require knowledge and techniques from different areas in Mathematics: theory of partial differential equations, of ordinary differential equations, of stochastic differential equations, and all the related numerical techniques appropriate for the simulation of these equations. We believe this is a general trend of modern computational modelling.
Keywords: Fokker-Planck equation, stochastic differential equations, polymeric fluids, Oldroyd-B model, Multiscale models, discrete to continuum limit, variational problems, Hookean and FENE dumbbells models, continuum mechanics, atomistic to continuum coupling, quasicontinuum method, adaptivity, numerical analysis, materials science, non-Newtonian flows, CONNFFESSIT..
Mathematics Subject Classification: 35Q30, 35Q35, 35Q74, 60H35, 65C05, 65C30, 65K10, 65M12, 65M60, 65N12, 65N15, 70-08, 70C20, 70G75, 74B20, 74G15, 74G65, 74G70, 76A05, 76A10, 76M10, 82C31, 82C80, 82D60, 49M2.
Citation: Xavier Blanc, Claude Le Bris, Frédéric Legoll, Tony Lelièvre. Beyond multiscale and multiphysics: Multimaths for model coupling. Networks & Heterogeneous Media, 2010, 5 (3) : 423-460. doi: 10.3934/nhm.2010.5.423
[1]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[2]

Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279
[3]

Helge Holden, Nils Henrik Risebro. The continuum limit of Follow-the-Leader models — a short proof. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 715-722. doi: 10.3934/dcds.2018031
[4]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[5]

Lijian Jiang, Yalchin Efendiev, Victor Ginting. Multiscale methods for parabolic equations with continuum spatial scales. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 833-859. doi: 10.3934/dcdsb.2007.8.833
[6]

Pranay Goel, James Sneyd. Gap junctions and excitation patterns in continuum models of islets. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1969-1990. doi: 10.3934/dcdsb.2012.17.1969
[7]

G. Idone, A. Maugeri. Variational inequalities and a transport planning for an elastic and continuum model. Journal of Industrial & Management Optimization, 2005, 1 (1) : 81-86. doi: 10.3934/jimo.2005.1.81
[8]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[9]

Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, Benedetto Piccoli. A continuum-discrete model for supply chains dynamics. Networks & Heterogeneous Media, 2007, 2 (4) : 661-694. doi: 10.3934/nhm.2007.2.661
[10]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[11]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135
[12]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[13]

Matthias Hieber. Remarks on the theory of Oldroyd-B fluids in exterior domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1307-1313. doi: 10.3934/dcdss.2013.6.1307
[14]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[15]

Paolo Podio-Guidugli. On the modeling of transport phenomena in continuum and statistical mechanics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1393-1411. doi: 10.3934/dcdss.2017074
[16]

Pierre Degond, Sophie Hecht, Nicolas Vauchelet. Incompressible limit of a continuum model of tissue growth for two cell populations. Networks & Heterogeneous Media, 2020, 15 (1) : 57-85. doi: 10.3934/nhm.2020003
[17]

Xiaoming Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 489-501. doi: 10.3934/dcdsb.2010.13.489
[18]

Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
[19]

Ezio Di Costanzo, Marta Menci, Eleonora Messina, Roberto Natalini, Antonia Vecchio. A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 443-472. doi: 10.3934/dcdsb.2019189
[20]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences