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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
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