Non incremental strategies based on separated representations: applications in computational rheology. (English) Zbl 1281.76022

Ammar, A.; Normandin, M.; Daim, F.; González, D.; Cueto, E.; Chinesta, F.

Commun. Math. Sci. 8, No. 3, 671-695 (2010).

Summary: Description of complex materials, in particular complex fluids, involves numerous computational challenges. Today, atomistic descriptions are too expensive from the computational point of view, motivating that this kind of analysis is restricted to extremely small systems. The next scale introduces some simplificative hypotheses, leading to coarse grained descriptions. At this level, Brownian dynamics simulations are usually employed. However, this level of description requires intensive computation resources with significant unfavorable impact on the simulation performances (CPU time). For these reasons sometimes kinetic theory descriptions are preferred. In those descriptions, the molecular conformation is described from a probability distribution function (PDF) whose evolution is governed by the Fokker–Planck equation. This approach, despite its mathematical simplicity introduces some computational challenges. First, the distribution function is defined in a multidimensional space, and then the associated partial differential equations must be solved in a multidimensional domain (some times involving thousands dimensions). Secondly, the analysis of transient models needs intensive computation, in particular when the system response under small amplitude oscillations (of high and very high frequency) is concerned. In some of our former works we proposed a technique based on the separated representation of the unknown field able to circumvent the curse of dimensionality. In this paper, we are addressing the second challenging point, the one related to the transient behavior. For this purpose we propose a separated representation of transient models leading to a non-incremental strategy, allowing impressive CPU time savings.

MSC:

76A99	Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
65M75	Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
76M25	Other numerical methods (fluid mechanics) (MSC2010)

Keywords:

complex fluids; computational rheology; kinetic theory; separated representations; finite sums decomposition; reduced approximation basis