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In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.

Η διατριβή αποσκοπεί στην ανάπτυξη κατάλληλης μεθοδολογίας και των αντίστοιχων υπολογιστικών εργαλείων με σκοπό τον προσδιορισμό της αξιοπιστίας κατασκευών από σύνθετα υλικά, όταν λαμβάνεται υπόψη όχι μόνο η στοχαστικότητα της φόρτισης και των ιδιοτήτων αντοχής του ορθότροπου υλικού, αλλά και η μεταβλητότητα των ελαστικών του ιδιοτήτων και των συντελεστών θερμικής διαστολής. Ταυτόχρονα, στόχος της εργασίας είναι η μέθοδος που θα αναπτυχθεί να είναι ευκολόχρηστη και αρκετά ακριβής, ώστε να καταστεί πρακτική για χρήση σε σχεδιασμό τέτοιων κατασκευών όπου μέχρι το τελικό αποτέλεσμα απαιτούνται πολλές επαναληπτικές δοκιμές. Στην εργασία διερευνάται η μεταβλητότητα των μηχανικών ιδιοτήτων συνθέτων υλικών, κυρίως αυτών που χρησιμοποιούνται στην κατασκευή πτερυγίων Ανεμογεννητριών, η στατιστική μοντελοποίηση της, καθώς και μέθοδοι για τον προσδιορισμό της αξιοπιστίας της κατασκευής. Επικεντρώνοντας το ενδιαφέρον στον δομικό σχεδιασμό πτερυγίων ανεμογεννητριών αναπτύχθηκε κατάλληλο υπολογιστικό εργαλείο για την εκτίμηση της αξιοπιστίας του πτερυγίου υπό στατική φόρτιση, στο επίπεδο της στρώσης του υλικού λαμβάνοντας υπόψη την στοχαστικότητα των ελαστικών-θερμικών ιδιοτήτων και των ιδιοτήτων αντοχής του υλικού, ενώ από την πλευρά της φόρτισης περιλαμβάνεται η μεταβλητότητα τόσο των ορθών τάσεων όσο και των διατμητικών τάσεων που αναπτύσσονται στο επίπεδο της στρώσης, ξεκινώντας από την στοχαστικότητα της εξωτερικής φόρτισης του πτερυγίου. Οι προτεινόμενες μέθοδοι εκτίμησης της πιθανότητας αστοχίας του πτερυγίου ανεμογεννήτριας αποδεικνύονται εφαρμόσιμες και σε άλλες κατασκευές από σύνθετα υλικά, με κύριο γνώρισμά τους την παραλαβή της φόρτισης σε μεγάλο βαθμό από το σύνθετο υλικό. / The dissertation aim was to develop a suitable methodology and the relevant numerical tools for the determination of the reliability of composite materials structures, when not only the variability of loading and the orthotropic material strength properties but also the variability of the material's elastic properties and thermal expansion coefficients. In parallel, scope of the work is to develop a method that is easily applicable and relative accurate, so as to be practical for use during the design of such structures, for which a large numbers of iterations is required up to the design final solution. In the current work the variability of the mechanical properties of composite materials, focusing on materials used for manufacturing wind turbine blades, is studied, the statistical modeling of the structure, as well as methods for the estimation of the structural reliability. Concentrating the interest in the structural design of wind turbine blades an adequate numerical tool was developed for the estimation of the wind turbine blade reliability under static loading, at the layer level, taking into account the stochastic material elastic properties, thermal expansion coefficients and strength properties, while on the loading side both the variability of normal and shear stresses which are developed at the layer level is taken into consideration, starting at the variability of the blade's external loading. The proposed methods developed for the failure probability estimation of the wind turbine blade are proved applicable to other composite material structures, as well, for cases where the structural load is undertaken in the largest extend by the composite material.

De nombreuses incertitudes interviennent lors de l’évaluation de l’aléa de départ des éboulements rocheux. Ce mémoire se propose de les étudier, en suivant la typologie de Baecher & Christian (2005) qui identifie (1) l’incertitude de décision, (2) l’incertitude de connaissance et (3) l’incertitude aléatoire. L’incertitude de décision intervient au travers de l’évaluation de l’aléa par expertise. Nous avons mis au point une expérimentation visant à étudier l’influence du niveau d’expertise de la personne en charge de l’étude et de la méthode utilisée (qualitative et quantitative) sur l’évaluation de l’aléa. Nous avons montré que l’utilisation d’une méthode qualitative permet de guider la pensée et de réduire la dispersion des niveaux d’aléas finaux, ce qui n’est pas le cas d’une méthode quantitative. Nous montrons également que dans des cas spectaculaires, l’expérience de l’ingénieur lui permet de ne pas surévaluer l’aléa, et que pour des cas plus classiques, il n’y a pas d’influence du niveau d’expertise dès lors que la méthode est suffisamment détaillée. L’incertitude aléatoire se manifeste par la variabilité temporelle des éboulements rocheux. Un travail sur des bases de données a été entrepris. Il a permis de mettre en évidence une corrélation statistique entre les éboulements rocheux et certains facteurs météorologiques (pluies et températures). Une méthode évaluant la probabilité d’occurrence des éboulements en fonction de l’intensité des pluies ou la valeur des températures a alors été développée. Elle peut être considérée comme un outil d’aide à la décision dans la gestion du risque. Un travail sur ces bases de données a également permis de montrer l’existence d’une composante de bruit de fond aléatoire, ne montrant aucune corrélation statistique, caractérisée par le fait que le temps qui sépare deux chutes de blocs suit une loi de Poisson. L’incertitude de connaissance résulte d’un manque d’informations concernant, en particulier, les facteurs préparatoires. Afin de mieux appréhender ces incertitudes, des modèles numériques (éléments distincts - UDEC) de glissement d’un bloc sur un plan incliné, dépendants de la présence et de la proportion de ponts rocheux, ont été réalisés. Ils ont permis de montrer l’existence de deux phases dans la rupture, contrôlées par les reports des contraintes le long du joint entre le bloc et le plan, à mesure que de la rupture des ponts rocheux se propage. Cependant, ces deux phases ne se retrouvent pas dans l’étude des déplacements du bloc. Nous montrons également l’influence de la position des ponts rocheux sur la vitesse de propagation de la rupture en fonction de la pente. Ces travaux fournissent un outil opérationnel et des indications sur le processus d’évaluation de l’aléa qui pourraient aider l’ingénieur en charge d’une étude d’aléa à justifier et à affiner son estimation / The present dissertation proposes to study uncertainties in rockfall hazard assessment process, on the basis of the Baecher & Christian typology (2005), which identifies (1) decision uncertainties, (2) knowledge uncertainties, and (3) natural variability. Decision uncertainties are due to the subjectivity of experts’ assessment. An experiment has been realized in order to evaluate the influence of the expertise level, and the chosen method (qualitative or quantitative) on the rockfall hazard. The rockfall hazard levels obtained by the qualitative method are quite uniform while the quantitative method produces more disparate results. We have shown that, classically, the expertise level has no influence on the assessment if the method is precisely detailed. In the case of spectacular sites, experts do not overestimate the hazard. The natural variability is associated to the temporal variability of the rockfalls. We have studied the statistical correlation between meteorological and rockfall databases. We have developed a method that takes into account the probability of occurrence of the studied triggering factor (rainfalls and temperatures). This new approach is easy to use, and also helps to determine the conditional probability of rockfall according to a given meteorological factor. This approach will help to optimize risk management in studied areas based on their meteorological conditions. A work on these databases also has allowed the unpredictability of the rockfalls to be highlighted. Indeed, the delay between two rockfalls follows a Poisson distribution. Knowledge uncertainties may concern the preparatory factors. We have studied them with numerical models of a rock block sliding along a planar joint, depending on the proportion and position of rock bridges (distinct elements - UDEC). Two stages of the failure process have been highlighted; they have been controlled by the stress redistribution induced by the failure, and the propagation of the rock bridges. However, these two phases are not identified when studying block displacements. We also have shown the influence of the position of rock bridges on the propagation of failure. This research provides an operational tool, and guidelines on assessment process that may help engineers in charge of a rockfall risk analysis to justify and refine the obtained risk estimations

Vid en ombyggnation av Henriksdals reningsverk har ett vertikalschakt utformats och anlagts. Under berguttaget tillämpades extensometermätningar för att följa upp deformationer i en bergpelare mot en befintlig tunnel. Mätningar utfördes automatiserat och regelbundet. För uppföljning av deformationer tillämpades tre larmnivåer avgränsade av två larmgränser vid 1.0 mm samt 3.0 mm deformation och var förbundna till olika motåtgärder. Målet med detta arbete harvarit att studera hur tillämpade larmgränser har presterat relativt dokumenterade och prognostiserade deformationer ur ett sannolikhetsbaserat perspektiv. Samt vid behov formulera och föreslå nya larmgränser. En litteraturstudie om modellering av bergmassor samt mekaniska egenskaper hos förekommande bergmassa av gnejs genomfördes. Därefter modellerades objektet för vidare numerisk analys genom FEM i Plaxis 3D. Med grund i uppställd modell kunde in-situ spänningsfältet skattas och modellen optimeras. En känslighetsanalys för bergmassans hållfasthetsparametrar utfördes och de fyra mest känsliga parametrarna: γ, ν, σci, Qbas, applicerades statistiskt i efterföljande studier. Föreslagna och tillämpade larmgränser studerades genom utförande av Monte Carlo-simuleringar. Använda deformationsbaserade larmgränser återfanns inte vara tillfredsställande och nya individuella töjningsbaserade gränsvärden framställdes. De ursprungliga deformationsbaserade larmgränserna resulterade för längre mätankare i utlösta larm och visades vara ouppnåeliga för kortare mätankare. En resulterande sannolikhet för överskridande av föreslagna varningsgränser vid dragtöjning skattades till storleksordningen 10−2 till 10−1 samt för stoppgränserna vid dragtöjning till 10−4. Sannolikheterna för överskridande av föreslagna larmgränser vid trycktöjning var inte möjliga att uppskatta för det studerade fallet. / During a rebuild of Henriksdal wastewater treatment plant, a vertical rock shaft was constructed. During the excavation, extensometers were used for monitoring deformations in a rock pillar against an existing tunnel. Measurements were automatically and regularly performed. Three alarm levels for follow up of the deformations were used and associated with different counter measures. The alarm levels were divided by two alarm limits at 1.0 mm and 3.0 mm deformation. The goal with this work was to study how the applied alarm limits from a probabilistic point of view have performed relative to obtained and forecasted deformations. And, if necessary, propose new alarm limits. A literature study about modeling of rock masses and mechanical properties of the occurring gneiss at site was performed. The object was thereafter modeled for further numerical analyzes using FEM in Plaxis 3D. Based on the performed model, the in-situ stress field could be estimated,and the model was further optimized. A sensitivity analysis of the rock mass strength parameters was conducted and the four most sensitive parameters: γ, ν, σci, Qbas, were applied as probabilistic in the following studies. Proposed and applied alarm limits were studied through Monte Carlo simulation. Applied deformation-based alarm limits were found not to be sufficient and two new strain-based were prescribed. The deformation-based limits were for the longer extensometer anchors resulting in triggered alarms and were only considered to be applicable for anchor lengths between 3.0 to 5.0 m. The probability of exceeding the proposed warning limits for tensile strain was estimated to be of a magnitude 10−2 to 10−1 and for the stop limits to be of a magnitude 10−4. The probabilities of exceeding the proposed alarm limits during compressive strain were not possible to estimate for the studied case.

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A resistência à ruptura de materiais frágeis como o concreto de granulometria fina é fortemente dependente de sua heterogeneidade, do tamanho da amostra e da heterogeneidade das tensões. A partir de uma abordagem probabilista, fundamentada no modelo de Weibull, este trabalho trata da influência de tais efeitos sobre a resistência à tração de painéis compósitos formados por concreto de granulometria fina e telas soldadas. Desde que as armaduras se mantenham em regime elástico, mostra-se que o modelo de Weibull de dois parâmetros pode ser empregado com sucesso na previsão da ruptura dos painéis compósitos em estudo. Os parâmetros foram então identificados com medidas de ensaios de flexão em três e em quatro pontos de barras de concreto de granulometria fina de dois tamanhos diferentes. A utilização do modelo de Weibull confirmou, por outro lado, a validade da teoria de misturas como ferramenta para estimar as resistências de rupturas locais correspondentes a diferentes níveis de formação de fissuras / The rupture strength of brittle materials such as the microconcrete is strongly dependent on the microstructural heterogeneity, the sample size and the stress heterogeneity. Using a probabilistic approach, based on the Weibull distribution, this work deals with the influence of such effects on the tensile strength of composite panels formed by microconcrete and welded wire grids. Provide the reinforcements behaves in elastic regime, it is shown that the model of Weibull of two parameters can be used with success to predict the failure of the composite panels studied. The parameters were then identified from measurements of flexural tests in three and in four points of two different sizes of microconcrete samples. The feasibility of the Weibull model validates, as well, the rule of mixture as a satisfactory tool to estimate local failure strengths of related to different levels of cracking

Les câbles d’acier sont utilisés comme renforts au sein des pneumatiques poids lourds, et servent notamment à supporter les efforts dus à la pression de gonflage et au poids du véhicule. Un câble est un ensemble de fils d’acier perlitique assemblés en hélices sur différentes couches. Il existe donc de nombreuses possibilités d’assemblage pour définir l’architecture d’un câble. Lors de leur sollicitation en service, les câbles sont soumis à des chargements cycliques à l’origine d’un endommagement en fatigue. Dans un contexte de réduction de la consommation et d’allègement des véhicules, la compréhension des mécanismes impliqués représente donc un enjeu majeur pour les manufacturiers de pneumatiques, en vue d’optimiser l’architecture des câbles vis-à-vis de la tenue en fatigue. Un essai de flexion cyclique représentatif de la sollicitation en service a été mis au point. Les éprouvettes testées sont des nappes composites constituées de câbles alignés au sein d’une matrice de gomme. Des essais interrompus à différents stades de l’endommagement suivis d’observations ex-situ (tomographie à rayon X, MEB) ont été réalisés. Un modèle de simulation par éléments finis de la nappe composite a été développé en vue d’étudier les interactions filgomme. La comparaison des observations aux simulations a permis de comprendre la cinétique de l’endommagement des renforts lors d’une sollicitation de flexion cyclique.L’étude de chacun des mécanismes susceptibles de contribuer à l’endommagement d’un câble a permis d’expliquer la meilleure tenue en fatigue des architectures pénétrées par la gomme. Un outil probabiliste de prédiction de la durée de vie des câbles basé sur la propagation des défauts en surface des fils a été développé. / Steel cables are used as reinforcements in heavy truck tires, in particular to support the forces resulting from the tire pressure and the vehicle's weight. A cable is a set of pearlitic steel wires assembled in helical form on different layers. There are therefore many assembly possibilities to define the cable architecture. The cables are subjected to cyclic loadings during service, resulting in fatigue damage. In a context of reduced fuel consumption and lighter vehicles, understanding the mechanisms involved is thus a major challenge for tire manufacturers, in order to optimize the architecture of cables with respect to fatigue resistance. A cyclic bending test representative of mechanical in-service loading has been developed. The tested specimens are composite layers made of cables aligned within an elastomer matrix. Interrupted tests at different stages of damage followed by ex-situ observations (X-ray tomography, SEM) were performed. A finite element model of the composite layer has been developed in order to understand wire-rubber interactions. The comparison of the observations with the simulations made it possible to understand the kinetics of cable damage during cyclic bending loading.The study of each of the mechanisms likely to contribute to the cable damage has made it possible to explain the better fatigue resistance of the architectures penetrated by the rubber. A stochastic cable fatigue life model based on wire surface defect propagation has been developed.

This thesis presents a new modeling framework and application methodology for the study of aircraft structures. The framework provides a ‘cradle-to-grave’ approach to structural analysis of a component, where structural integrity encompasses all phases of its lifespan. The methodology examines the holistic structural design of aircraft components by integrating fatigue and damage tolerance methodologies. It accomplishes this by marrying the load inputs from a fatigue analysis for new design, into a risk analysis for an existing design. The risk analysis incorporates the variability found from literature, including recorded defects, loadings, and material strength properties. The methodology is verified via formal conceptualization of the structures, which are demonstrated on an actual hydraulic accumulator and an engine nacelle inlet. The hydraulic accumulator is examined for structural integrity utilizing different base materials undergoing variable amplitude loading. Integrity is accomplished through a risk analysis by means of fault tree analysis. The engine nacelle inlet uses the damage tolerance philosophy for a sonic fatigue condition undergoing both constant amplitude loading and a theoretical flight design case. Residual strength changes are examined throughout crack growth, where structural integrity is accomplished through a risk analysis of component strength versus probability of failure. Both methodologies can be applied to nearly any structural application, not necessarily limited to aerospace.