Les matériaux composites stratifiés sont de plus en plus utilisés dans l'aéronautique, mais ils peuvent être sujets à large délaminage si soumis à impact. La nécessité d'effectuer des simulations numériques pour prédire l'endommagement devient essentielle pour l'ingénieur. Dans ce contexte, l'utilisation d'une modélisation fine semble préférable. En revanche, le coût de calcul associé serait prohibitif pour larges structures. Le but de ce travail consiste à réduire ce coût de calcul, en couplant le modèle fin, restreint à la zone active de délaminage, avec un modèle grossier appliqué au reste de la structure. En raison du comportement transitoire des problèmes d'impact, l'adaptabilité dynamique des modèles pour suivre les phénomènes évolutifs représente un point crucial de la stratégie de couplage. Des méthodes avancées sont utilisées pour coupler différents modèles. Par exemple, la méthode de Décomposition de Domaines, appliquée à l'adaptabilité dynamique, doit être combinée avec une stratégie de remaillage, considérée comme intrusive pour la mise en œuvre d'un logiciel pour Analyse à Eléments Finis. Dans ce travail, les bases d'une approche faiblement intrusive, la méthode de Substitution, sont présentés dans le domaine de la dynamique explicite. Il s'agît d'une formulation globale-locale, conçue pour appliquer un modèle grossier sur tout le domaine pour obtenir une réponse globale: ce pré-calcul est ensuite corrigé itérativement par l'application du modèle raffiné appliqué seulement où nécessaire. La vérification de la méthode de Substitution en comparaison avec la méthode de Décomposition de Domaines est présentée.

Der Vortrag stellt Idee und Grundlagen der Berechnung von Schockantwortspektren dar. Er zeigt, wie man exemplarisch für einen Halbsinusstoß das Schockantwortspektrum in der PTC FEM-Software Creo Elements / Pro Mechanica berechnen kann. Die Schockantworten eines Ein- und Zweimassenschwingers werden sowohl zeitaufgelöst als auch über die dynamische Stoßanalyse berechnet. Die modalen Superpositionsmethoden "Absolute Summe" und "SRSS" (Square Root of the Sum of the Squares - geometrischer Mittelwert) werden vorgestellt. Als reales Beispiel werden Schockanalysen für verschiedene Halbsinusimpulse mit einem Wärmebildgerät der Firma Carl Zeiss Optronics GmbH durchgeführt und mit einer zeitaufgelösten Analyse verglichen. Abschließend wird auf die Erzeugung von Antwortspektren für die Substrukturauslegung eingegangen. / The presentation explains idea and fundamentals of shock response spectra analysis. With help of the PTC FEM-software Creo Elements / Pro Mechanica the shock response spectra (SRS) for an exemplary half sine shock is calculated. The shock response of a one-mass and a two-mass oscillator are analyzed per dynamic time as well as per dynamic shock analysis. The modal superposition methods "absolute sum" and "SRSS" (Square Root of the Sum of the Squares) are explained. The method is applied for different half sine shocks on a realistic example: A thermal imaging system of the company Carl Zeiss Optronics GmbH. Finally, the creation of response spectra for global-local analysis is explained.

Der Vortrag stellt Idee und Grundlagen der Berechnung von Schockantwortspektren dar. Er zeigt, wie man exemplarisch für einen Halbsinusstoß das Schockantwortspektrum in der PTC FEM-Software Creo Elements / Pro Mechanica berechnen kann. Die Schockantworten eines Ein- und Zweimassenschwingers werden sowohl zeitaufgelöst als auch über die dynamische Stoßanalyse berechnet. Die modalen Superpositionsmethoden "Absolute Summe" und "SRSS" (Square Root of the Sum of the Squares - geometrischer Mittelwert) werden vorgestellt. Als reales Beispiel werden Schockanalysen für verschiedene Halbsinusimpulse mit einem Wärmebildgerät der Firma Carl Zeiss Optronics GmbH durchgeführt und mit einer zeitaufgelösten Analyse verglichen. Abschließend wird auf die Erzeugung von Antwortspektren für die Substrukturauslegung eingegangen. / The presentation explains idea and fundamentals of shock response spectra analysis. With help of the PTC FEM-software Creo Elements / Pro Mechanica the shock response spectra (SRS) for an exemplary half sine shock is calculated. The shock response of a one-mass and a two-mass oscillator are analyzed per dynamic time as well as per dynamic shock analysis. The modal superposition methods "absolute sum" and "SRSS" (Square Root of the Sum of the Squares) are explained. The method is applied for different half sine shocks on a realistic example: A thermal imaging system of the company Carl Zeiss Optronics GmbH. Finally, the creation of response spectra for global-local analysis is explained.

<p>Graphs are a natural choice for understanding the associations between variables, and assuming a probabilistic embedding for the graph structure leads to a variety of graphical models that enable us to understand these associations even further. In the realm of high-dimensional data, where the number of associations between interacting variables is far greater than the available number of data points, the goal is to infer a sparse graph. In this thesis, we make contributions in the domain of Bayesian graphical models, where our prior belief on the graph structure, encoded via uncertainty on the model parameters, enables the estimation of sparse graphs.</p> <p><br></p> <p>We begin with the Gaussian Graphical Model (GGM) in Chapter 2, one of the simplest and most famous graphical models, where the joint distribution of interacting variables is assumed to be Gaussian. In GGMs, the conditional independence among variables is encoded in the inverse of the covariance matrix, also known as the precision matrix. Under a Bayesian framework, we propose a novel prior--penalty dual called the `graphical horseshoe-like' prior and penalty, to estimate precision matrix. We also establish the posterior convergence of the precision matrix estimate and the frequentist consistency of the maximum a posteriori (MAP) estimator.</p> <p><br></p> <p>In Chapter 3, we develop a general framework based on local linear approximation for MAP estimation of the precision matrix in GGMs. This general framework holds true for any graphical prior, where the element-wise priors can be written as a Laplace scale mixture. As an application of the framework, we perform MAP estimation of the precision matrix under the graphical horseshoe penalty.</p> <p><br></p> <p>In Chapter 4, we focus on graphical models where the joint distribution of interacting variables cannot be assumed Gaussian. Motivated by the quantile graphical models, where the Gaussian likelihood assumption is relaxed, we draw inspiration from the domain of precision medicine, where personalized inference is crucial to tailor individual-specific treatment plans. With an aim to infer Directed Acyclic Graphs (DAGs), we propose a novel quantile DAG learning framework, where the DAGs depend on individual-specific covariates, making personalized inference possible. We demonstrate the potential of this framework in the regime of precision medicine by applying it to infer protein-protein interaction networks in Lung adenocarcinoma and Lung squamous cell carcinoma.</p> <p><br></p> <p>Finally, we conclude this thesis in Chapter 5, by developing a novel framework to compute the marginal likelihood in a GGM, addressing a longstanding open problem. Under this framework, we can compute the marginal likelihood for a broad class of priors on the precision matrix, where the element-wise priors on the diagonal entries can be written as gamma or scale mixtures of gamma random variables and those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This result paves new roads for model selection using Bayes factors and tuning of prior hyper-parameters.</p>

Global-local finite element analyses were used to study the damage tolerance of curvilinearly stiffened panels; fabricated using the modern additive manufacturing process, the so-called unitized structures, and that of adhesive joints. A damage tolerance study of the unitized structures requires cracks to be defined in the vicinity of the critical stress zone. With the damage tolerance study of unitized structures as the focus, responses of curvilinearly stiffened panels to the combined shear and compression loadings were studied for different stiffeners' height. It was observed that the magnitude of the minimum principal stress in the panel was larger than the magnitudes of the maximum principal and von Mises stresses. It was also observed that the critical buckling load factor increased significantly with the increase of stiffeners' height. To study the damage tolerance of curvilinearly stiffened panels, in the first step, buckling analysis of panels was performed to determine whether panels satisfied the buckling constraint. In the second step, stress distributions of the panel were analyzed to determine the location of the critical stress under the combined shear and compression loadings. Then, the fracture analysis of the curvilinearly stiffened panel with a crack of size 1.45 mm defined at the location of the critical stress, which was the common location with the maximum magnitude of the principal stresses and von Mises stress, was performed under combined shear and tensile loadings. This crack size was used because of the requirement of a sufficiently small crack, if the crack is in the vicinity of any stress raiser. A mesh sensitivity analysis was performed to validate the choice of the mesh density near the crack tip. All analyses were performed using global-local finite element method using MSC. Marc, and global finite element methods using MSC. Marc and ABAQUS. Negligible difference in results and 94% saving in the CPU time was achieved using the global-local finite element method over the global finite element method by using a mesh density of 8.4 element/mm ahead of the crack tip. To study the influence of different loads on basic modes of fracture, the shear and normal (tensile) loads were varied differently. It was observed that the case with the fixed shear load but variable normal loads and the case with the fixed normal load but variable shear loads were Mode-I. Under the maximum combined loading condition, the largest effective stress intensity factor was very smaller than the critical stress intensity factor. Therefore, considering the critical stress intensity factor of the panel with the crack of size 1.45 mm, the design of the stiffened panel was an optimum design satisfying damage tolerance constraints. To acquire the trends in stress intensity factors for different crack lengths under different loadings, fracture analyses of curvilinearly stiffened panels with different crack lengths were performed by using a global-local finite element method under three different load cases: a) a shear load, b) a normal load, and c) a combined shear and normal loads. It was observed that 85% data storage space and the same amount in CPU time requirement could be saved using global-local finite element method compared to the standard global finite element analysis. It was also observed that the fracture mode in panels with different crack lengths was essentially Mode-I under the normal load case; Mode-II under the shear load case; and again Mode-I under the combined load case. Under the combined loading condition, the largest effective stress intensity factor of the panel with a crack of recommended size, if the crack is not in the vicinity of any stress raiser, was very smaller than the critical stress intensity factor. This work also includes the performance evaluation of adhesive joints of two different materials. This research was motivated by our experience of an adhesive joint failure on a test-fixture that we used to experimentally validate the design of stiffened panels under a compression-shear load. In the test-fixture, steel tabs were adhesively bonded to an aluminum panel and this adhesive joint debonded before design loads on the test panel were fully applied. Therefore, the requirement of studying behavior of adhesive joints for assembling dissimilar materials was found to be necessary. To determine the failure load responsible for debonding of adhesive joints of two dissimilar materials, stress distributions in adhesive joints of the nonlinear finite element model of the test-fixture were studied under a gradually increasing compression-shear load. Since the design of the combined load test fixture was for transferring the in-plane shear and compression loads to the panel, in-plane loads might have been responsible for the debonding of the steel tabs, which was similar to the results obtained from the nonlinear finite element analysis of the combined load test fixture. Then, fundamental studies were performed on the three-dimensional finite element models of adhesive lap joints and the Asymmetric Double Cantilever Beam (ADCB) joints for shear and peel deformations subjected to a loading similar to the in-plane loading conditions in the test-fixtures. The analysis was performed using ABAQUS, and the cohesive zone modeling was used to study the debonding growth. It was observed that the stronger adhesive joints could be obtained using the tougher adhesive and thicker adherends. The effect of end constraints on the fracture resistance of the ADCB specimen under compression was also investigated. The numerical observations showed that the delamination for the fixed end ADCB joints was more gradual than for the free end ADCB joints. Finally, both the crack propagation and the characteristics of adhesive joints were studied using a global-local finite element method. Three cases were studied using the proposed global-local finite element method: a) adhesively bonded Double Cantilever Beam (DCB), b) an adhesive lap joint, and c) a three-point bending test specimen. Using global-local methods, in a crack propagation problem of an adhesively bonded DCB, more than 80% data storage space and more than 65% CPU time requirement could be saved. In the adhesive lap joints, around 70% data storage space and 70% CPU time requirement could be saved using the global-local method. For the three-point bending test specimen case, more than 90% for both data storage space and CPU time requirement could be saved using the global-local method. / Ph. D.

Model and variable selection have attracted considerable attention in areas of application where datasets usually contain thousands of variables. Variable selection is a critical step to reduce the dimension of high dimensional data by eliminating irrelevant variables. The general objective of variable selection is not only to obtain a set of cost-effective predictors selected but also to improve prediction and prediction variance. We have made several contributions to this issue through a range of advanced topics: providing a graphical view of Bayesian Variable Selection (BVS), recovering sparsity in multivariate nonparametric models and proposing a testing procedure for evaluating nonlinear interaction effect in a semiparametric model. To address the first topic, we propose a new Bayesian variable selection approach via the graphical model and the Ising model, which we refer to the ``Bayesian Ising Graphical Model'' (BIGM). There are several advantages of our BIGM: it is easy to (1) employ the single-site updating and cluster updating algorithm, both of which are suitable for problems with small sample sizes and a larger number of variables, (2) extend this approach to nonparametric regression models, and (3) incorporate graphical prior information. In the second topic, we propose a Nonnegative Garrote on a Kernel machine (NGK) to recover sparsity of input variables in smoothing functions. We model the smoothing function by a least squares kernel machine and construct a nonnegative garrote on the kernel model as the function of the similarity matrix. An efficient coordinate descent/backfitting algorithm is developed. The third topic involves a specific genetic pathway dataset in which the pathways interact with the environmental variables. We propose a semiparametric method to model the pathway-environment interaction. We then employ a restricted likelihood ratio test and a score test to evaluate the main pathway effect and the pathway-environment interaction. / Ph. D.