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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 20
TRENDS IN ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis, B.H.V. Topping
Chapter 13
Atoms, Molecules and Flows: Recent Advances and New Challenges in their MultiScale Numerical Modeling at the Beginning of the Third Millenium F. Chinesta^{1}, A. Ammar^{2}, H. Lamari^{2} and N. Ranc^{1}
^{1}Laboratoire de Mécanique des Systèmes et des Procédés, UMR 8106 CNRSENSAM, Paris, France F. Chinesta, A. Ammar, H. Lamari, N. Ranc, "Atoms, Molecules and Flows: Recent Advances and New Challenges in their MultiScale Numerical Modeling at the Beginning of the Third Millenium", in M. Papadrakakis, B.H.V. Topping, (Editors), "Trends in Engineering Computational Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 13, pp 247269, 2008. doi:10.4203/csets.20.13
Keywords: multiscale modelling, quantum mechanics, molecular dynamics, Brownian dynamics, kinetic theory, statistical mechanics, model reduction, curse of dimensionality, separated representation, finite sums decompositions.
Summary
Description of complex materials, in particular complex fluids,
involves numerous computational challenges. Accurate descriptions
of such materials need a multiscale description and the
definition of pertinent bridges between the different scales. The
finest description starts at the atomic level (where quantum
mechanics and molecular dynamics simulations are usually
encountered). The next description scale introduces some molecular
simplifications in which the molecule is described as a
multibeadspring or multibeadrods models. At this level
Brownian dynamics simulations are usually employed. However, this
level of description requires intensive computation resources with
its significant unfavorable impact on the simulation performances
(CPU time). For this reason sometimes kinetic theory descriptions
are preferred. In that description, the molecular conformation is
described from a probability density function whose evolution is
governed by the FokkerPlanck equation. This approach, despite its
mathematical simplicity introduces a density function that is
defined in a multidimensional space, and then the associated
partial differential equations must be solved in a
multidimensional domain (sometimes involving thousands
of dimensions).
In this case classical discretization techniques fail because in such cases the number of required degrees of freedom could be higher than 101000 (being the computational availabilities at the present of order 109). To circumvent the curse of dimensionality that these highdimensional partial differential equations induces stochastic techniques (such as the MonteCarlo method) could be applied and they have been applied intensively in the last decade. However, the large number of stochastic trajectories needed to reduce the statistical noise has a direct impact on its efficiency. Some alternatives exist to alleviate this drawback (Brownian Configurations Fields) but the computing time is for practical applications (industrial processes) unviable. Recently some incipient techniques based on the separated representations and the tensor product approximations bases have allowed the solving of problems in spaces involving hundreds of dimensions [1,2]. References
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