2022-04900 - PhD Position F/M Model order reduction with mesh adaptation
Le descriptif de l’offre ci-dessous est en Anglais

Type de contrat : CDD

Niveau de diplôme exigé : Bac + 5 ou équivalent

Fonction : Doctorant

A propos du centre ou de la direction fonctionnelle

The Inria Bordeaux Sud-Ouest centre is one of Inria's nine centres and has around twenty research teams. The Inria centre is a major and recognized player in the field of digital sciences. It is at the heart of a rich R&D and innovation ecosystem: highly innovative SMEs, large industrial groups, competitiveness clusters, research and higher education players, laboratories of excellence, technological research institute...

Contexte et atouts du poste

A wealth of applications in science and engineering involve the solution to computational fluid dynamics (CFD) problems for many different system configurations. For this class of problems, it is important to reduce the marginal cost associated with a given simulation over a range of parameters. Model order reduction (MOR) techniques rely on an offline/online decomposition to reduce marginal costs. During the offline phase, we rely on high-fidelity (hf) simulations to generate a reduced-order model (ROM) to estimate the solution over a range of parameters. During the online or deployment phase, given a new value of the parameter, we query the ROM  to estimate the solution field. We particularly focus on the development of automated procedures for both training and deployment stages: autonomy refers to the ability to complete the analysis with minimal user intervention.

Approximation to  advection-dominated PDEs  poses several fundamental challenges to state-of-the-art model order reduction methods. First, despite the recent advances in high-performance computing and numerical analysis, hf numerical approximation of these problems requires extensive computational resources: as a result, reduced-order approximations are built using a moderate number of solutions (snapshots). Second, it is well known that linear-approximation-based methods are completely inadequate to deal with parameter-dependent discontinuities: this motivates the development of nonlinear approximation methods. Third, MOR techniques rely on the projection of the equation onto a unique shared hf discretization and thus rely on the assumption that the underlying hf discretization is accurate for all parameters in a prescribed range: for problems with parameter-dependent shocks, this requires accurate  adaptive mesh refinement (AMR) over a broad portion of the spatial domain and is often unfeasible.

Mesh adaptation aims at improving the accuracy of numerical simulations while reducing the computational cost through automatic optimisation of the mesh resolution during the computation. More precisely, in anisotropic mesh adaptation, the mesh elements sizes and orientations are modified in order to minimize a certain numerical error model, to guarantee an optimal mesh size for a desired accuracy. A non-linear process is considered that ensures convergence of the mesh/solution pair to the optimum with respect to the error model considered. Anisotropic mesh adaptation is also able to handle unsteady problems, with complex geometries undergoing large displacements.

Mission confiée

The aim of the PhD project is to study a novel integrated model reduction mesh adaptation approach for nonlinear  advection-dominated systems of partial differential equations (PDEs), notably aerodynamic and hydraulic flows. Relevant solutions to these problems are characterized by parameter-dependent shocks and contact discontinuities.

To this aim, registration-based model reduction will be combined with parametric mesh adaptation.
Registration-based model reduction relies on a parametric mapping to identify and then track relevant features of the solution field and ultimately improve performance of linear compression methods such as proper orthogonal decomposition (POD). Parametric mesh adaptation refers to the task of determining an accurate mesh for a range of system configurations. Provided that the mapping is effective to track moving features of the solution, mesh adaptation should lead to considerably more parsimonious discretizations compared to uniform refinement, for any target accuracy.

The PhD student  is expected to develop, analyze and then implement the combined model-reduction mesh-adaptation procedure for nonlinear advection-dominated PDEs. Automated procedures will be developed for both training and deployment stages:  rigorous a priori and a posteriori analyses will inform the construction of the ROM and the mesh at training stage; a posteriori  error indicators will be developed to certify the accuracy of the online predictions. Applications to aeronautic configurations and coastal hazard (e.g. tsunami propagation) assessment will be considered.

Principales activités

Main activities :

  • Develop a novel integrated model reduction mesh adaptation approach for advection-dominated PDEs.
  • Implement the proposed method using existing software packages.
  • Perform extensive numerical assessments; write research reports and scientific articles.

 Additional activities (3 maximum) :

  • Present the work to colleagues, scientific and industrial partners.
  • Present the results of the project at national and international research conferences.
  • Participate in the activities of the research team at Inria.


The candidate is expected to hold an M2 (Engineering degree, MRes, or equivalent) in applied mathematics or computational mechanics. The candidate should have experience with numerical methods for PDEs, and should have good programming skills in Python or C/C++. The candidate should also be familiar with the domain of computational fluid mechanics. Experience in meshing methods and/or model order reduction is appreciated but not required.

Good communication skills (both scientific communication and interpersonal communication) are required.

Proficiency in English is a must.


  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of partial teleworking and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities


gross salary :

  • 1982€ / month (before taxes) during the first 2 years,
  • 2085€ / month (before taxs) during the third year.