Contributions to discrete exterior calculus and applications in fluid dynamics

Organisation/Company

Université de Technologie de Troyes

Research Field

Mathematics
Computer science » Digital systems
Physics

Researcher Profile

Recognised Researcher (R2)
Leading Researcher (R4)
First Stage Researcher (R1)
Established Researcher (R3)

Country

France

Application Deadline

30 May 2024 – 22:00 (UTC)

Type of Contract

Temporary

Job Status

Full-time

Offer Starting Date

27 Jul 2024

Is the job funded through the EU Research Framework Programme?

Not funded by an EU programme

Is the Job related to staff position within a Research Infrastructure?

No

Offer Description

In the field of numerical simulation, it is important to follow from the start a good methodology for the construction of what is called the numerical scheme, corresponding to the discretization of the equations modeling the considered physical phenomenon. The main difficulty that arises during this discretization process is the loss of various properties, due to the transition to an approximation (linked to the discretization), which can be calculated on a computer. This is generally referred to as a loss of symmetry. We can for example cite among the properties whose conservation it is necessary to guarantee, the mass or the energy; the properties of certain mathematical operators used in the equations must also be preserved.

 Discrete exterior computation is a method to build a numerical scheme that allows some of these quantities to be retained, although the resulting solution remains a computer-calculated approximation. By ensuring this conservation, the occurrence of parasitic solutions or physical inconsistencies is limited. In addition, this allows us to stay as close as possible to the modern formalism of physics equations, which uses the mathematical framework of differential geometry and exterior calculus.

 The objective of this thesis is to contribute to this emerging and promising field of research, initiated in the early 2000s [1,2]. The development and improvement of numerical tools within this paradigm will form the core of the work to be carried out, by applying it in particular to computational fluid dynamics.

The PhD student will first understand the context of the DEC by:
• acquiring the vocabulary of differential geometry and exterior calculus necessary for the DEC,
• carrying out a bibliographic study relating to the scientific objectives of the thesis,
• appropriating himself/herself the existing code.

Then, the PhD student will carry out his/her first numerical simulations using the last point, by optimizing and improving the code used. The results of these simulations can be compared with those obtained before improvement, but also with popular codes such as FEniCS or OpenFOAM, which do not use DEC.

The continuation of the thesis work may be oriented, depending on the progress of the PhD student and his/her motivation, towards the following research perspectives:
• development of a 3D version of the DEC using the analytical discretization of the Hodge operator already developed in 2D in the thesis of R. Ayoub [3],
• expression of the Navier-Stokes equations on a manifold,
• development of DEC in a space-time formalism, similar to what is used in relativity. This formalism has already been studied for example in solid mechanics [4], but not within the framework of the DEC.

Aziz Hamdouni and Dina Razafindralandy, from the University of La Rochelle, will collaborate on this thesis work and will be required to discuss with the PhD student. In particular, they supervised the thesis of R. Ayoub [3], and are part, along with the other supervisors of the proposed thesis, of the Research Group (GdR) of the CNRS n°2043, entitled Differential Geometry and Mechanics. The PhD student will integrate into this community.

Participation in scientific seminars, symposiums and conferences will of course be an integral part of the PhD student’s training.

[1] A. N. Hirani, PhD thesis, Discrete exterior calculus. California Institute of Technology, 2003.
[2] M. Desbrun, A. Hirani, M. Leok, and J. Marsden, “Discrete exterior calculus,” arXiv :math/0508341, 2005.
[3] Ayoub, R. (2020). Développement d’une méthode de discrétisation des EDPs basée sur le calcul extérieur discret (Doctoral dissertation, Université de La Rochelle).
[4] E Rouhaud, R Kerner, I Choucair, R El Nahas, A Charles, B Panicaud. Space-Time Thermo-Mechanics for a Material Continuum – International Conference on Geometric Science of Information, 2021

Funding category: Contrat doctoral
UTT Salary
PHD title: Doctorat en Sciences pour l’Ingénieur, spécialité Optimisation et Sûreté des Systèmes
PHD Country: France

Requirements
Specific Requirements

The candidate must have solid training and experience in computer programming and applied mathematics, particularly in the areas of numerical analysis and scientific computing. In addition to that, master diploma in applied mathematics and a first experience with the world of research are necessary.

With regard to computer programming, the candidate must know how to be autonomous and motivated in learning new software and in appropriating an already existing code. The languages ​​used will be in particular Fortran, and possibly C, C++ and Python. Experience of these languages, especially the first, would be a plus.

Regarding applied mathematics, the candidate must be willing and motivated to learn new mathematical notions, and for their application in a numerical framework (scientific calculation and numerical simulation). In particular, notions of differential geometry (manifolds and differential forms) will have to be mastered in order to be able to adequately use discrete exterior calculus. It is obviously a significant plus if the candidate is already familiar with these theoretical notions.

Additional Information
Work Location(s)

Number of offers available

1

Company/Institute

Université de Technologie de Troyes

Country

France

City

Troyes

Geofield

Where to apply

Website

https://www.abg.asso.fr/fr/candidatOffres/show/id_offre/120816

Contact

Website

http://www.utt.fr

STATUS: EXPIRED