Section: New Results

Particular applications of simulation methods

Analysis of a two-level parameterization optimization for antenna design

Participants : Benoît Chaigne [Doctoral student, 2007-2010] , Jean-Antoine Désideri.

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of Multigrid (MG) methods. By analogy with the Poisson equation, which is the typical example for linear MG methods, we have addressed a parametric shape inverse problem. The ideal cycle of a two-level algorithm can be defined and adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we have shown that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design have been realized. Finally, some connections with the direct and inverse Broyden-Fletcher-Goldfarb-Shanno preconditioning methods have been shown [29] .

Mesh qualification

Participants : Jean-Antoine Désideri, Maxime Nguyen, Jacques Peter [Research Engineer, ONERA/DSNA] .

M. Nguyen Dinh is conducting a CIFRE thesis at ONERA supported by AIRBUS France. The thesis topic is the qualification of CFD simulations by anisotropic mesh adaption. Methods for refining the 2D or 3D structured mesh by node movement have been examined closely. Secondly, it is investigated how could the local information on the functional gradient $∥dJ/dX∥$ be exploited in a multi-block mesh context. This raises particular questions related to conservation at the interfaces.

Hybrid meshes

Participants : Sébastien Bourasseau, Jean-Antoine Désideri, Jacques Peter [Research Engineer, ONERA/DSNA] , Pierre Trontin [Research Engineer, ONERA/DSNA] .

S. Bourasseau has started a CIFRE thesis at ONERA supported by SNECMA. The thesis is on mesh adaption in the context of hybrid meshes, that is, made of both structured and unstructured regions. Again, the aim is to exploit at best the function gradient provided by the adjoint-equation approach. Preliminary experiments have been conducted on geometries of stator blade yielding the sensitivities to global shape parameters.

Nash game approach to image processing

Participants : Abderrahmane Habbal, Rajae Aboulaich [Mohamed V University of Rabat] , Maher Moakher [University of Tunis] , Moez Kallel [University of Tunis] , Anis Theljani [University of Tunis] .

We have started in 2011 to study the application of game modeling to image processing problems. We propose an original game theory approach to simultaneously restore and segment noisy images [56] . We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm. Based on a similar idea, we formulated well known data completion (Cauchy) problems for Laplace equation as Nash games [55] and obtained results of existence, uniqueness and stability of a Nash equilibrium which turns out to be the Cauchy solution when the Cauchy data are compatible. With A. Theljani, we study the extension of the Nash data completion approach to nonlinear parabolic equations with application to image impainting.