My research activity started with my PhD thesis which focused on the establishment of some thermodynamical models of flow induced crystallisation. The particular case of viscoelastic behaviour of polymers was considered in a thermodynamic formalism in order to understand heat generation mechanisms for such polymer and to describe the flow induced crystallisation combined to viscoelastic behaviour. At the same time, a finite element code was developed, accounting for advection effects, heat generation and free and moving boundaries.

My research activity continued after joining the Laboratoire de Rhéologie in Grenoble (a research center which depends on the University Joseph Fourier and the French National Foundation of Science -CNRS-) in 1999. For three years I've worked in collaboration with J-R. Clermont on numerical aspect of the stream tube method (an original technique numerical technique developed at that laboratory) for simulating flows making use of its steam lines. I've also collaborated with N. El Kissi developing an academic experimental device destined to validate microscopic behavior of complex flows. Special attention was addressed to the recirculating flows usually encountered in polymer extrusion processes.

I have started a successful collaboration with Francisco Chinesta (from ENSAM, Paris) leading to a new, original and in potential scientific and technological rupture (as I describe later) research activity which could be considered as my main research activity until now and in which I expect to put my main efforts in the next years. This new activity started considering some numerical aspects of kinetic models involving polymer solutions and melts. Several original model reduction techniques were proposed, analyzed and published. These research projects were positively received by the scientific community.

However, finer physical descriptions (such as Kramer's chain used to model DNA molecules) using high number of approximation functions are involved in our models, and they are related to the multidimensional character of the configuration space. For highly dimensional configuration spaces, standard discretizations fail to describe the solution using a deterministic approach because of the high number of degrees of freedom involved in the simulation (that could reach astronomical values, e.g. 10300 or more -remember that 1080 corresponds to the presumed number of elementary particles in the universe). The coarse of dimensionality is an old unsolved drawback and is associated to important physical models concerning the physics and the nanometric scale, ranging from the quantum mechanics to the kinetic theory descriptions of statistical mechanics.

A new strategy has been proposed that allows circumventing this drawback by using a reduced approximation basis within an adaptive procedure making use of an efficient separation of variables.

The idea is to dissociate the configuration space in some elementary spaces and to build some appropriate functions over these reduced spaces. The weak formulation related to the kinetic theory description (Fokker-Planck equation) is then discretized allowing the writing of the solution as a sum of different space functions products. In this approach, time and physical space can be associated to the configuration space as additional dimensions avoiding incremental procedures. This method was applied in our former works to describe Multi-Bead-Spring models, polymer melt, fiber suspensions, heat transfer in heterogeneous media ... but it could be applied to any model concerned by the curse of dimensionality, as encountered for example in financial mathematics or in risk decision.

Collaborators | University | Collaboration topic |

R. Keunings | Université Catholique de Louvain, Belgique | Molecular models |

K. Chiba | Université d'Osaka, Japon | Fiber suspensions and network polymer models |

E. Cueto | Université de Zaragoza, Espagne | Micro-macro approach, meshless techniques |

J. Azaiez | Université de Calgary, Canada | Stability phenomena, fibers |

D. Ryckelynck | LMSP, Ensam Paris | Model reduction |

M. Laso | Université de Madrid, Espagne | Liquid Crystal Polymers |