Habilitation à diriger des recherches
When the problems of physicists help mathematicians make progress
Antoine Gloria © Inria
Last February, Antoine Gloria received, with honors, his accreditation to advise doctoral theses. With the members of the Simpaf project team, for which he was deputy director, he is launching a new research team dealing with the mathematical problems raised by physics.
Throughout the history of science, physics and mathematics have stimulated each other, physics being a fertile source of inspiration for mathematics, which in return provides a language and conceptual framework. “With Felix Otto, the director of the Max Planck Institute in Leipzig (Germany), after slightly more than three years of research, we provided an answer to a problem that had remained unresolved for over thirty years: the quantification of random homogenization, a mathematical theory facilitating numeric simulation of materials presenting random heterogeneities at a very small scale, such as composite materials.”
Physicists use this homogenization process to assess the effective properties, such a thermal conductivity or elasticity. From a mathematical point of view, this means modeling a material whose relationship between the microscopic and macroscopic levels would tend towards zero, in other words a material carrying only the trace of its microstructures. This is very effective for materials presenting periodic heterogeneities, such as Kevlar or carbon fibers. However, the theory needed to be improved for the random case as the calculation of the effective properties was much too long. This is the case for several composite materials or industrial problems such as oil extraction (which takes place in a heterogeneous porous environment). “The homogenization methods allow for the calculation of an averaged result. Like for a distant image for which one has no need to know the color of each pixel. We have developed digital methods enabling calculation of approximations of the coefficients used for homogenized equations. We also know how to estimate the level of error.”
Together with Italian researchers1 and colleagues from Inria Rocquencourt2, Antoine Gloria has applied this method to calculate the energy density (the deformations experienced depending on the force applied) of rubber, a material made up of a network of polymer chains.
« Starting with our discrete model based on these polymer chains, we obtained a continuous elasticity model in perfect conformity with our tests. Our model makes it possible to simulate all deformation scenarios (uniaxial, biaxial, triaxial…), even those that cannot be accessed by means of mechanical testing . »
In a general manner, this allows physicists to better fathom the influence of the microstructure on the properties, as in this case for rubber.
Cempi Center of Excellence Laboratory
All the researchers of the Simpaf project team, some ten full-time staff, will work within the framework of the centre of excellence laboratory “European center for mathematics, physics and their interaction” (Cempi), a winner of the latest tender for investing for the future projects in February 2012. Led by Stephan De Bièvre, a member of Simpaf, this project brings together about one hundred people from different laboratories (Lille 1 University/CNRS): the Painlevé mathematics laboratory (to which the Simpaf team is associated) and the Laser physics, atoms and molecules laboratory (PhLAM). The research will be organized along three lines: the interface between mathematics and physics, in particular the study of complex behavior such as attenuation and interference in non-linear optical systems such as fiber optics; the interface between physics and biology; the interface between mathematics and theoretical computer science. The Inria researchers will work on the first line of research, by applying the approach developed within Simpaf and the future team replacing it in the specific domain of fiber optics.
1Roberto Alicandro et Marco Cicalese (Cassino and Naples Universities)
2Marina Vidrascu, Patrick Le Tallec et François Lequeux