Keywords: multi-scale modelling, quantum mechanics, molecular dynamics, Brownian dynamics, kinetic theory, statistical mechanics, model reduction, curse of dimensionality, separated representation, finite sums decompositions.

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 multi-bead-spring or multi-bead-rods 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 Fokker-Planck 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 high-dimensional partial differential equations induces stochastic techniques (such as the Monte-Carlo 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].

[1]

A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, "A fast solver for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids", Journal of Non-Newtonian Fluid Mechanics, Submitted.

[2]

A. Ammar, B. Mokdad, F. Chinesta, "A fast solver for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part 2: Transient simulation using space-time separated representations", Journal of Non-Newtonian Fluid Mechanics, Submitted.

purchase the full-text of this chapter (price £20)

go to the previous chapter
go to the next chapter
return to the table of contents
return to the book description
purchase this book (price £92 +P&P)