BILAN du projet PEPS 1 EOLIN (Eolien LMI INSA)

Transcription

1 BILAN du projet PEPS 1 EOLIN (Eolien LMI INSA) Lab. de Math de l INSA de ROUEN FR CNRS 3335 et EA 3226 PLAN 1. Introduction 2. Bilan scientifique 3. Bilan financier 4. Conclusion 1 Introduction Le projet eolin [PEPS1 - Labex AMIES] a pour sujet principal l approximation d'un champ de vent à partir de données ponctuelles. Le point de départ de ce travail est la supposition que le champ de vecteurs dérive d un potentiel (par exemple la température pour le vent). Ce problème se produit par exemple dans l électromagnétisme, la météorologie, l imagerie médicale ou l analyse d images radar. Ici, nous ne voulons pas calculer explicitement un potentiel qui pourrait générer les données de champs de vecteurs. Nous voulons seulement obtenir une approximation globale de l ensemble du champ de vecteurs sur un domaine borné en tenant compte, dans la modélisation, du fait que ce champ dérive d un potentiel. Les données seront par exemple la valeur du champ du vent en un nombre fini de points (stations météorologiques, anémomètres...).

2 Membres du projet : P. ALEXANDRE (GDF SUEZ - LCV) C. LE GUYADER (LMI - INSA Rouen) Resp. du Projet D. APPRATO (Univ. de Pau - LMA) N. FORCADEL (LMI - INSA Rouen) C. GOUT (LMI - INSA Rouen) A. ZAKHAROVA (LMI - INSA Rouen) B. JOBARD (Univ. de Pau - LIUPPA) 2. Bilan scientifique On précise tout d abord les faits marquants du projet : Mars 2015 : Nouvel axe : traitement du signal + prise en compte de la topographie pour l'approximation du champ de vent. Travaux en cours. Février 2015 : Workshop HPC à Rouen [VOIR ANNEXE 4] Janvier 2015 : visualisation champs de vecteurs. [VOIR ANNEXE 3] Octobre 2014 : Article sur l'éolien de P. Alexandre : Des mathématiques dans l'éolien, p , Matapli 105, [VOIR ANNEXE 2] Prix du poster Curves and Surfaces 2014 : Wind velocity field approximation from sparse data: modelling and visualization on real dataset, T. Roy, D. Apprato, P. Alexandre, N. Forcadel, C. Gout, B. Jobard, C. Le Guyader, A. Zakharova [voir ANNEXE 1]

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4 ANNEXE 1 : Wind velocity field approximation from sparse data: Modelling and visualization on real dataset ^T. Roy, *P. Alexandre, D. Apprato, ^N. Forcadel, ^C. Gout, ^C. Le Guyader, B. Jobard et ^A. Zakharova ^ : INSA de Rouen Lab. de Math. de l INSA de Rouen - * : GDF SUEZ La Compagnie du Vent - : Université de Pau et des Pays de l Adour ABSTRACT: The problem of vector field approximation from sparse data emerges in a wide range of fields such as: motion control, computer vision, geometrical analysis, geometrical design, analysis of acoustic or electromagnetic waves, as well as in geophysics, medical imaging, fluid mechanics and so on... Many different approaches have been introduced to solve each specific problem occuring in the above fields of investigation. The originality of this work consists: - in considering that the vector field derives from a potential (conservative vector field): it occurs for instance in meteorology (winds derive from temperature potentials), oceanography (currents derive from pression potentials), image processing... For in land wind velocity field, we also take into account the topography effects. - of a rigorous study of existence-uniqueness of the solution of the problem phrased as an energy minimization, - in establishing a convergence result (while many approaches only give algorithms without mathematical study) and providing an approximation error estimate. - Taking into account the topography, - in using a specific visualization tool. More precisely, in this work, we do not want to compute a potential that could generate the vector field data. We only want to get a global approximation of the vector field dataset on a bounded domain, taking into account in the modeling that this approximation derives from a potential. Furthermore, contrary to interpolation methods, we prefer to fit the vector field dataset in the case of realistic data (when the number of vectors is large or when the data are corrupted by noise). To achieve this, we introduce a minimization problem defined as a regularized least-square problem formulated on a Sobolev space of potentials. Obviously, this problem has an infinite number of solutions, but we derive from it a problem expressed in terms of the gradient vectors. We prove that the associated problem in terms of vectors has a unique solution which is the corresponding approximation of the vector field dataset. Then, we give a convergence result when the number of vectors increases to infinity. We also give the discretization complemented by an approximation error estimate of the involved smoothing splines. We also propose to take into account the topography effect on the wind velocity field, and we finally give numerical examples using a specific visualization tool. We study a spline-based approximation of vector fields from a finite set of data. We propose to evaluate the wind velocity field at each location on the studied domain, taking into accont the topography. To do do that, we use a spline based approach, finite element method and local effect of the topography. We give existence of a solution of our problem, convergence of our solution and we also give numerical examples on explicit dataset and real datasets from METEO France. The originality of this work consists: - in considering that the vector field derives from a potential (conservative vector field): it occurs for instance in meteorology (winds derive from temperature potentials), oceanography (currents derive from pressure potentials), image processing... For in land wind velocity field, we also take into account the topography effects. - in a rigorous study of existence-uniqueness of the solution of the problem phrased as an energy minimization, - in establishing a convergence result (while many approaches only give algorithms without mathematical study) and providing an approximation error estimate. - in using a specific visualization tool. MODELLING

5 Existence-Uniqueness, convergence of the approximation, error estimates and discretization are studied.

6 Taking into account the topography Studied zone : Normandie area ; Left : data set Right : wind velocity field approximation Topography of the studied area and wind velocity field approximation ight: wind velocity field approximation Using our proposed method, we get the approximated wind velocity field at each location we want on the selected zone, for each time step. We then get the following movie :

7 ANNEXE 2

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10 ANNEXE 3 Film, 96 heures de vent en 3 minutes A partir de 7 stations de METEO FRANCE

11 ANNEXE 4 Journées Modélisation et Simulation Numérique/HPC (dans le cadre du Projet M2NUM du GRR LMN) Jeudi 19 Février Rouen (Campus du Madrillet) - Salle de conférence du CORIA (St Etienne du Rouvray) Chairwoman : Carole Le Guyader 14h15 : Violaine Louvet (ICJ et CNRS, Lyon) 14h45 : Vincent Moureau (CNRS et Coria, Rouen) 15h15-15h45 : pause 15h45 : Théophile Chaumont-Frelet (INRIA Bordeaux Sud Ouest et LMI, Rouen) 16h15 : Dimitri Komatitsch (CNRS et LMA, Aix Marseille) Vendredi 20 Février Rouen (Campus du Madrillet) - LMI 201 9h-12h : Approximation d'un champ de vent : modélisation et simulation Nombre de participants prévus : 60 Pause café entrée 15h15 et 15h45 le jeudi Accueil 8h le vendredi matin. Org. : C. Gout & V. Moureau This workshop is supported bym2num project (GRR LMN - Haute Normandie) and e@lin project (Labex AMIES)