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Section: New Results
Miscellaneous
Participants : Laurent Boudin, Jean-Frédéric Gerbeau, Damiano Lombardi, Sébastien Martin, Marina Vidrascu, Irène Vignon-Clementel.
In [56] , a reduced-order model algorithm, based on approximations of Lax pairs, is proposed to solve nonlinear evolution partial differential equations. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the space where the solution is searched for evolves according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive waves or front propagation. Numerical examples are shown for the KdV and FKPP (nonlinear reaction diffusion) equations, in one and two dimensions.
In [21] , we investigate the asymptotic behaviour of the solutions to the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We deal with cross sections such as hard spheres or cut-off power law potentials. We use Hilbert expansions near the common thermodynamic equilibrium granted by the H-theorem. The lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable which is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses. The next-order equality in the Hilbert expansion then allows to write the macroscopic continuity equations for each component of the mixture.
In [58] , we discuss some numerical properties of the viscous numerical scheme introduced in [23] to solve the one-dimensional pressureless gases system, and study in particular, from a computational viewpoint, its asymptotic behavior when the viscosity parameter used in the scheme becomes smaller.
In [33] we study a network-based model for rubber. Since the pioneering work by Treloar, many models based on polymer chain statistics have been proposed to describe rubber elasticity. Recently, Alicandro, Cicalese, and the first author rigorously derived a continuum theory of rubber elasticity from a discrete model by variational convergence. The aim of this paper is twofold. First we further physically motivate this model, and complete the analysis by numerical simulations. Second, in order to compare this model to the literature, we present in a common language two other representative types of models, specify their underlying assumptions, check their mathematical properties, and compare them to Treloar's experiments.
In [63] our aim is to demonstrate the effectiveness of the matched asymptotic expansion method in obtaining a simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface on the overall response of a linearly elastic body. The results of some numerical experiments corroborate the precise identification of the different steps, in particular of the outer/inner regions with their normalized coordinate systems and the scale separation, leading to the model.
In cancer modeling, to be able to capture the full in-vivo scale, tumors have to be modeled with continuum models. An important step consists in qualitatively and quantitatively comparing agent-based models (which parameters can generally be identified by experiments in vitro) and continuum models. We derived a first 1D continuum model for tumor growth from the cell based model (Drasdo and Hoehme, 2005): it results in a fluid-type model which capture tumor expansion in both diffusive and compact phenotypes. The tumor expands based on the pressure gradient generated by cell proliferation, the latter being hindered by high density or pressure. In [39] this modeled is analyzed mathematically, showing the existence of traveling waves in the different regimes (with or without internal friction and diffusion due to active movement). In particular the incompressible cells limit is very singular and relates to the Hele-Shaw equation. Numerical results confirm the analysis.