ED Mathématiques et Informatique
A Diffeomorphic Mapping Approach for Model Order Reduction in Aerodynamics
by Jon LABATUT (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 14h00 - Salle Contensou centre ONERA châtillon 29 Av. de la Division Leclerc 92320 Châtillon
in front of the jury composed of
- Angelo IOLLO - Professeur - Université de Bordeaux - Directeur de these
- Damiano LOMBARDI - Directeur de recherche - INRIA Rocquencourt - Rapporteur
- Sanderse BENJAMIN - Directeur de recherche - Centrum voor Wiskunde en Informatica (CWI) - Rapporteur
- Astrid DECOENE - Professeure - Université de Bordeaux - Examinateur
- Emmanuel TRéLAT - Professeur - Sorbonne Université (Paris 6) - Examinateur
- Tommaso TADDEI - Professeur associé - Sapienza University of Rome - CoDirecteur de these
Parametric fluid dynamics simulations present significant computational challenges in aerodynamics, where high-fidelity computational fluid dynamics (CFD) models involve millions of degrees of freedom. Reduced-order models (ROMs) address this computa- tional burden by constructing low-dimensional approximations of the full-order system. However, standard projection-based approaches exhibit poor performance for advection- dominated flows. A proposed solution is to rely on coordinate transformation mappings to enhance ROMs compression. This thesis addresses the issue of developing a general framework for diffeomorphic mappings to align aerodynamics structures for application in coordinate transformation based ROMs. To answer this problem, we first establish the mathematical foundations of the method. For this, the mapping of interest is defined as the minimizer of an objective function. This constitutes the registration problem. The objective function combines a data misfit term, based on point set alignment of coherent structures, with a regularization term derived from differential operators. The mapping is defined from a parametrization of a velocity field. This guarantees diffeomorphic transformations in bounded CFD domains under tangent boundary conditions. The minimization problem is solved using a finite element discretization, gradient-based optimization, and an Expectation-Maximization algorithm for automatic coherent structure labeling. The methodology is validated on three representative test cases of increasing complex- ity: (i) coalescing Gaussian mixtures, to illustrate the alignment of merging structures; (ii) transonic Euler flow around a NACA0012 airfoil, demonstrating multi-structure registration; and (iii) viscous RANS flow around the ONERA M6 wing, showcasing the alignment of complex 3D lambda shocks. Across these cases, the proposed approach enhances ROM accuracy, and demonstrates robustness to aerodynamic problems.