## Section: New Results

### Numerical models and simulations applied to physics

#### Inverse problems in geodesy

Participants : Amine Abdelmoula, Bernard Philippe.

This work is done in collaboration with M. Moakher, from ENIT, Tunisia.

The geoid is the level surface of the earth attraction at the sea level. We aim at finding an equivalent mass system which can generate a given geoid. The mathematical definition of the problem is expressed as a non-linear least- squares problem in the Hilbert space of harmonic functions. We pursue numerical simulations. A paper is in preparation.

#### Heat transfer in soil and prehistoric fires

Participants : Édouard Canot, Mohamad Muhieddine.

This subject takes place in the ARPHYMAT (Archaeology, Physics, Mathematics) interdisciplinary project, linked to the archeological/human sciences program: "Man and fire: towards a comprehension of the evolution of thermal energy control and its technical, cultural and paleo-environmental consequences". Both physical and numerical approach is used to understand the functioning mode and the thermal history of the studied structures. The main topic of this project concerns the simulation of forced evaporation of water in a saturated soil.

2D and 3D-axisymmetric configurations of this physical problem have been solved, using the Apparent Capacity Method. Emphasis is put on performance: the Jacobian matrix is stored in a sparse structure and the Newton iterations (inside the BDF method) are solved by the UMFPACK part of the SuiteSparse package. All these modifications are done inside MUESLI, giving an easy-to-use programming interface for the user. We have proposed a new global approach to solve the system of coupled equations [14] .

In addition, we have investigated the heat conduction in a real 3D saturated porous medium. In term of numerics, the discretization is based on the hybrid mixed finite element method in space and a semi-implicit scheme in time. To solve this problem we have modified `TRACES` (Transport of RadioActive Elements in Subsurface, 2004, P. Ackerer and H. Hoteit, IMFS, Université de Strasbourg) which is a computer program for the simulation of flow reactive transport in saturated porous media. The model has been applied to prehistoric fires [34] .

Besides, we have introduced a robust numerical strategy to estimate the temperature dependent thermal capacity, the thermal conductivity and the porosity of a saturated porous medium, based on the knowledge of heating curves at selected points in the medium. In order to solve the inverse problem, the least squares criterion (in which the sensitivity coefficients appear) has been used.

Recently, the proposed method for the forward problem has been applied on evaporation in heterogeneous porous media: 1D and 2D simulations have been obtained, where we have supposed the the soil was constituted by blocks of different permeability [30] .

#### Rheology of granular systems flowing out of silo

Participant : Édouard Canot.

This work is done in the framework of a project funded by the Region Bretagne. A PhD thesis (Merline Djouwe) began in February 2009, coadvised with Patrick Richard, who is from the Physics Institute at the University of Rennes (IPR).

The study concerns the rheology of granular media flowing out of a silo. The two objectives are (i) to understand the rheological properties of such kind of granular flows and especially the effect of the micro mechanical characteristics, and (ii) to determine the most efficient ways to decompact and to unblock these systems. We expect that the results will help the understanding of the jamming transition which is of fundamental interest for discrete matter.

A first code, based on molecular dynamics (particles interaction), has been adapted to the silo geometry; it can process up to 20 000 particles, but it is not sufficient to obtain an accurate description of the granular flow. Furthermore, we currently implement another numerical code (finite differences on staggered grids) based on a continuous physical model. This second approach avoids the limitation of the particles number and we expect that it will be both efficient and accurate.

#### Simulation of fluid flow in complex geometries

Participant : Édouard Canot.

This work is done in the context of the LIRIMA laboratory ( 8.3.3 ). A PhD thesis (Fateh Saci) began in january 2010, coadvised with Fatma-Zohra Nouri, professor at the Badji Mohktar Annaba University (Algeria), in the Mathematics Department.

This work concerns the numerical simulation of fluid flows (both linear Stokes and nonlinear Navier-Stokes) in geometries with small deformation, based on analytical variable transformations, in order to solve the equations on a simpler geometrical domain. The selected approach combines a collocation method with an asymptotic development method that is based on a small parameter.