A. .. , 6 5.1 Overall methodology for the numerical comparison between NRA, vol.87

, Input probabilistic model for Example #1

. #. Results-for-example,

. #. Results-for-example,

, Input probabilistic model for Example #2

, Score functions for normal (Case #1) and uniform (Case #2) prior distributions on an uncertain parameter ? j

, Overall strategy for the numerical tests of the proposed methodology, p.105

, Input probabilistic model for Example #1

. #. Results-for-example,

, Input probabilistic model for Example #2

. #. Results-for-example,

, Results for Example #2 considering the influence of the failure event rareness, p.109

.. .. Different,

, Overall strategy for the numerical tests of the proposed methodology, p.127

, Input probabilistic model for Example #1

. #. Results-for-example,

, Input probabilistic model for Example #2

. #. Results-for-example,

.. .. Input,

, Input probabilistic model under bi-level input uncertainty

. #. Results-for-step,

. #. Results-for-step,

, Results for Step #2 considering the influence of the failure event rareness, p.145

. #. Results-for-step, 117 7.2.1 Basic formulation of the Sobol indices on the indicator function

, Sobol indices on the indicator function adapted to the bi-level input uncertainty119 7.3.1 Bi-level input uncertainty: aggregated vs. disaggregated types of uncertainty, p.121

. .. , 124 7.4.3 Methodology based on subset sampling and data-driven tensorized G-KDE, Efficient estimation using subset sampling and kernel density estimation . . 123 7.4.1

, 125 7.5.1 Example #1: a polynomial function toy-case

. .. , 130 7.5.3 Synthesis about numerical results and discussion, p.133

.. .. Conclusion,

, Moreover, one can see that these nonlinearities seem to be specific to the variable X 4 , i.e., the azimuth angle perturbation at separation, By analyzing these cross-cuts, one can formulate the following remarks: ? firstly, one can notice that

?. Secondly, )) present possible multiple MPFPs

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, Résumé étendu de la thèse

, Contexte Les systèmes aérospatiaux sont généralement considérés comme étant des « systèmes complexes », principalement à cause de leur nature multi-disciplinaire (due au grand nombre de composants hétérogènes qu'ils rassemblent), leur caractère unitaire (i.e., faible nombre d'unités produites) et les conditions d'opérations extrêmes auxquelles ils peuvent être confrontés durant leur vie opérationnelle. La combinaison de ces facteurs fait que les systèmes aérospatiaux sont caractérisés

. ?-«-critiques, ce qui implique que les conséquences associées à une défaillance peuvent être désastreuses, tant d'un point vue économique

. ?-«-hautement-fiables, ce qui implique que le nombre de défaillances complètes (contrairement à des défaillances partielles, potentiellement mineures) observées de ces systèmes est

, une grande précision mais qui peuvent être très coûteux à mettre en oeuvre, et l'utilisation de modèles mathématiques et physiques, qui, pour être résolus et exploités, passent bien souvent par des étapes de modélisation, simulation et/ou résolution numériques. Ces modèles ont l'avantage de pouvoir se substituer aux essais expérimentaux trop coûteux qui peuvent devenir inaccessibles sous certaines conditions extrêmes voire jamais observées (e.g., simulation d'une collision satellite-débris ou collision météorite-Terre). Ce caractère exploratoire des modèles numériques a aussi un coût non négligeable dès lors que les modèles impliquent un grand nombre de variables d, La conception et l'analyse de tels systèmes reposent sur l'utilisation à la fois de moyens expérimentaux puissants et d'

, dès le cycle de conception, qui pourraient éventuelle-ment affecter le comportement ou les performances du système dans ses véritables conditions d'opération futures et mener à la défaillance redoutée. En effet, les incertitudes proviennent de multiples sources qui doivent être identifiées, caractérisées et traitées en vue d'assurer la fiabilité du système étudié. Pour ce faire, une méthodologie générale de quantification des incertitudes est disponible et adoptée de façon quasi unanime dans de nombreuses branches de l'ingénierie confrontées à des problématiques similaires. Cette méthodologie (qui peut se décliner de plusieurs manières suivant les domaines et le but recherché, Iooss, 2006.

?. Le-modèle-numérique, est-à-dire que l'ensemble des étapes énoncées précédemment sont considérées comme nonintrusives vis-à-vis du code numérique de simulation. Dès lors, ce code pourrait être possiblement coûteux à évaluer, non-linéaire et en assez grande dimension. Toutefois, dans cette thèse, les cas étudiés sont à la fois représentatifs de certaines des difficultés rencontrées dans les codes industriels (codes non-linéaires, zones de défaillances multiples, probabilités rares) tout en restant abordables du point de vue du coût de calcul

, Dans le contexte de systèmes hautement-fiables, cette probabilité est supposée associée à un événement rare et est donc supposée très faible. Dès lors, elle devient très coûteuse à estimer par simulations de

, Ainsi, les variables de base et leur structure de dépendance sont considérées à travers l'utilisation de variables aléatoires et d'une copule formant un vecteur aléatoire de loi jointe supposée connue, à l'exception de certains paramètres de distribution de certaines lois marginales qui sont peu connus (e.g., à cause d'un manque de données) ou dont la connaissance est uniquement liée à un choix d'expert. Dès lors, l'ensemble de la démarche de quantification des incertitudes, ? la modélisation des incertitudes en entrée est réalisée dans un cadre probabiliste

C. Dans-ce, cette thèse traite du problème de l'analyse de fiabilité et de l'analyse de sensibilité de modèles numériques boîtes-noires simulant des systèmes caractérisés par des défaillances de type événements rares. Les entrées sont des variables incertaines modélisées dans un cadre probabiliste et certains paramètres de distribution sont supposés méconnus ou incertains. Dès lors, le problème principal de la thèse concerne la prise en compte d'un double niveau d'incertitudes lors des deux analyses mentionnées précédemment, Ce double niveau est formé par : ? les incertitudes phénoménologiques

, ? les incertitudes portant sur le modèle probabiliste lui-même, qui caractérisent le manque de connaissance que l'analyste peut avoir dans le choix de certains paramètres de distribution

, ont déjà soulevé l'importance cruciale de tenir compte de ce double niveau d'incertitudes dans l'analyse de fiabilité, il apparaît que ce problème est toujours d'actualité, et ce pour plusieurs raisons. Tout d'abord, de nouveaux algorithmes d'estimation de probabilités d'événements rares ont été développés au cours des dernières décennies (e.g., les techniques de type "adaptive importance sampling" ou celles de "subset sampling"). De plus, la complexité des codes de calculs s'est accrue (e.g., chaînage de codes et approches multi-disciplinaires) parallèlement à l'émergence des techniques de métamodélisation, Ce problème est d'un intérêt majeur dans le domaine de la quantification des incertitudes dans les codes numériques. Si de nombreux travaux pionniers, tels ceux de Ditlevsen (Ditlevsen, 1979a; Ditlevsen, 1979b) ou de Der Kiureghian, 1986.

T. Dans-cette, le focus est mis sur la prise en compte du double niveau d'incertitudes à travers l'ensemble de la méthodologie de quantification des incertitudes (i.e., les étapes A-B-C-D mentionnées plus haut), dans un contexte d'estimation de probabilité d'événement rare pour Appendix E

, subset sampling) est menée et validée à travers l'utilisation de plusieurs cas-tests. In fine, l'approche présentant les meilleurs performances vis-à-vis de plusieurs critères définis dans ce chapitre (ici, l'approche ARA est la plus performante), Une comparaison numérique du couplage de ces deux approches avec plusieurs algorithmes d'estimation de probabilités d'événements rares

. Song, En effet, partant d'une stratégie d'estimation de la probabilité de dé-faillance prédictive dans le cadre de l'approche ARA, de nouveaux estimateurs de sensibilités fiabilistes locales sont proposés afin d'évaluer la robustesse de l'estimation de la probabilité vis-à-vis du double niveau d'incertitudes. Plus spécifiquement, on suppose que, de par la structure bayésienne hiérarchique du modèle probabiliste des entrées, on souhaite tester la robustesse de la mesure de fiabilité estimée par rapport aux choix des hyper-paramètres de la densité a priori charactérisant l'incertitude épistémique sur le paramètre incertain. De par la nature du problème, des indices de sensibilités locaux à base de score functions (Rubinstein, Chapitre 6 -Analyse de sensibilité fiabiliste locale en présence d'incertitudes sur les paramètres de distribution Ce chapitre vise à étendre les résultats obtenus dans le chapitre précédent au cadre de l'analyse de sensibilité fiabiliste, 1986.

. Li, Si ces contributions sont relativement récentes, l'originalité des travaux présentés dans ce chapitre résulte dans leur couplage et leur adaptation à la contrainte du double niveau d'incertitudes ce qui en accroit les difficultés intrinsèques d'utilisation (e.g., dimension, complexité des densités à estimer). Pour finir, cette méthodologie est appliquée à deux cas-tests afin de démontrer son efficacité, Chapitre 7 -Analyse de sensibilité fiabiliste globale en présence d'incertitudes sur les paramètres de distribution Ce chapitre vise à compléter l'approche locale proposée dans le chapitre précédent par une approche globale, vol.Lemaître, 2012.

, Approche imbriquée vs. approche augmentée (NRA vs. ARA)

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, Analyse de sensibilité fiabiliste locale (local ROSA)

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, Analyse de sensibilité fiabiliste globale (global ROSA) Rédaction d'un chapitre d'ouvrage scientifique en cours

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