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:
- Multiscale models,
- numerical analysis,
- materials science,
- discrete to continuum limit,
- variational problems,
- continuum
mechanics,
- atomistic to continuum coupling,
- quasicontinuum method,
- adaptivity,
- polymeric fluids,
- non-Newtonian flows,
- Hookean and FENE
dumbbells models,
- Oldroyd-B model,
- Fokker-Planck equation,
- stochastic differential equations,
- CONNFFESSIT
Citation: Xavier Blanc, Claude Le Bris, Frédéric Legoll, Tony Lelièvre. Beyond multiscale and multiphysics: Multimaths for model coupling[J]. Networks and Heterogeneous Media, 2010, 5(3): 423-460. doi: 10.3934/nhm.2010.5.423
Abstract
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.
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