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PhD in Mathematics (M/F)

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Français - Anglais

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General information

Reference : UMR7373-MARRIG-006
Workplace : MARSEILLE 13
Date of publication : Thursday, June 25, 2020
Scientific Responsible name : Madame Raphaèle HERBIN
Type of Contract : PhD Student contract / Thesis offer
Contract Period : 36 months
Start date of the thesis : 1 September 2020
Proportion of work : Full time
Remuneration : 2 135,00 € gross monthly

Description of the thesis topic

The thesis will be co-supervised by Raphaèle Herbin (I2M) and Jean-Claude Latché (IRSN Cadarache). The student will be enrolled in ED 184 "Mathematics and Computer Science of Marseille".

In the simulations carried out for nuclear safety studies, flows are described most of the time by balance equations, and in particular Navier-Stokes equations made up of mass and momentum balances, where convection terms are dominant, whatever the chosen turbulence modelling: one-point statistical models, called RANS models (for Reynolds Averaged Navier-Stokes), or large scale simulations [6, 2]. This is the case, for example, for the simulation of fires in confined and mechanically ventilated premises or for the modelling of turbulent deflagrations, phenomena described by software (ISIS and P2REMICS respectively) based on the CALIF3S library of components for fluid mechanics, developed at IRSN.
Furthermore, both ISIS and P2REMICS are intended for industrial applications, hence the need to deal with complex three-dimensional geometries, in particular with curved boundaries, the simplest example being a pipe with a circular cross-section.
This context makes it necessary to implement in CALIF3S both stable and precise discretizations of the convection operator, of the finite volume type, on meshes that are as general as possible. In addition, the space approximation used in CALIF3S is a discretization with staggered meshes: the scalar unknowns are associated with the cells of the mesh while the velocity is associated with the faces of the cells (or, equivalently, with a so-called dual mesh, consisting of control volumes centered on the centers of the faces). Two discrete convection operators are thus implemented, the first based on the initial (or primal) mesh and the second based on the dual mesh, the geometrical characteristics of these two meshes being significantly different.

The objective of this thesis is to develop one or more discrete convection operators for flows with dominant convection in complex domains (thus excluding schemes based only on structured meshes, such as the classical staggered mesh scheme, called MAC scheme); introduced in CALIF3S, these operators will be implemented for incompressible, low Mach number or compressible flows (including Euler equations). More precisely, all or part of the following points will be dealt with:
– Pyramidal or prismatic control volumes - These types of cells are quite commonly used in industrial meshes. Prismatic meshes are obtained when a mesh is constructed by extruding a 2D mesh of a general flat surface : Typically, the mesh of a pipe can be obtained from a 2D mesh of quadrangles and triangles of one of its sections; in this case, the discretization obtained often gives very good quality solutions, because it keeps faces parallel to the flow, and the prismatic meshes keep their precision (or, at least, lose it less quickly) when the anisotropy of the mesh becomes very pronounced (very elongated meshes in the direction of the tube axis near the walls to calculate the boundary layers). Pyramidal control volumes, because they have quadrangular and triangular faces, allow the coexistence of tetrahedrons and hexahedrons in the same mesh, which is sometimes necessary for some 3D meshes to process complex geometry. These control volumes are not currently processed in CALIF3S (only tetrahedra and hexahedra are taken into account), hence the proposed extension.

The first-order convection operator is standard on primary meshes; a more precise variant can be obtained by MUSCL techniques operating in any geometry, including, among many other works, those developed in the laboratory [5]. The dual mesh operator is more difficult to obtain, given the constraints of coherence with the primary operator necessary for its positivity; we will follow here the line used for the other discretizations of CALIF3S [1, 4]. It should be noted that, for these new discretizations, it will also be necessary to build a diffusion operator [5, 3]. Theoretical aspects can be addressed: error estimates in the diffusive case and behaviour in case of mesh stretching, stability of the diffusion operator (Poincaré inequality, Korn lemma), stability for the Stokes problem (discrete inf-sup condition) ....
- Slope limiting techniques for momentum convection - In a variant (to be developed) of the pressure correction scheme implemented in CALIF3S, where convection will be explicitly treated, the gains brought by a slope limiting technique applied to previously constructed meshes will be verified (at least numerically). The adaptation of the scheme to the simulation of large scales will be of particular interest.
- Non-conforming local refinement - Non-conforming refinement possibilities are offered in CALIF3S for the meshes currently available. However, the performance of such a scheme needs to be verified, or even adapted (i.e. stabilized), for problems with dominant convection. The extension to pyramidal and prismatic cells will then be considered.
- Course of the thesis -
After an important bibliography phase, taking into account the scientific scope of the subject, the work to be carried out will be addressed in the order mentioned above. Computer developments will be carried out in the CALIF3S library developed at IRSN.

7. [1] G. Ansanay-Alex, F. Babik, J.-C. Latché, and D. Vola. An L2-stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements. International Journal for Numerical Methods in Fluids, 66:555–580, 2010.
8. [2] L.C. Berselli, T. Iliescu, and W.J. Layton. Mathematics of Large Eddy Simulation of Turbulent Flows. Springer, 2006.
9. [3] J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin. The Gradient Discretisation Method. Springer, 2018.
10. [4] J.-C. Latché, B. Piar, and K. Saleh. A discrete kinetic energy preserving convection operator for variable density flows on locally refined staggered meshes. submitted, 2018.
11. [5] L. Piar, F. Babik, R. Herbin, and J.-C. Latché. A formally second order cell centered scheme for convection-diffusion equations on general grids. International Journal for Numerical Methods in Fluids, 71:873–890, 2013.
12. [6] P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer, 2001.

We are looking for a student who has followed a Master 2 course in numerical analysis of EDP and scientific computation and who has preferably completed his or her Master 2 thesis in numerical fluid mechanics.

The candidate must send his/her Curriculum Vitae accompanied by his/her transcripts in Master 1 and 2.

An interview will take place between the future thesis supervisors and the candidate either in Marseille or by videoconference between June 27 and July 4, 2020.

Work Context

The doctoral student will carry out his or her thesis at the Institut de Mathématique de Marseille and at the Laboratoire de l'Incendie et des Explosions (LIE/SA2I/PSN-RES), at IRSN Cadarache. The supervising authorities of the Institut de Mathématique de Marseille are the CNRS, the University of Aix-Marseille and the Ecole Centrale de Marseille, which has about one hundred and thirty teacher-researchers, about thirty CNRS researchers, about fifteen technical and administrative staff, about sixty doctoral students and about twenty post-doctoral researchers. The Institute comprises five research groups covering a broad spectrum of pure and applied mathematics (analysis, geometry, topology, logic, arithmetic, dynamics, combinatorics, probability, statistics, applied analysis, etc.), as well as a large number of application fields (scientific or industrial).

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