Recently, probabilistic methods have been used extensively to model uncertainty in many design optimization problems. An alternative approach for modeling uncertainties is fuzzy sets. Fuzzy sets usually require much less information than probabilistic methods and they rely on expert opinion. In principle, probability theory should work better in problems involving only random uncertainties, if sufficient information is available to model these uncertainties accurately. However, because such information is rarely available, probabilistic models rely on a number of assumptions regarding the magnitude of the uncertainties and their distributions and correlations. Moreover, modeling errors can introduce uncertainty in the predicted reliability of the system. Because of these assumptions and inaccuracies it is not clear if a design obtained from probabilistic optimization will actually be more reliable than a design obtained using fuzzy set optimization. Therefore, it is important to compare probabilistic methods and fuzzy sets and determine the conditions under which each method provides more reliable designs. This research work aims to be a first step in that direction. The first objective is to understand how each approach maximizes reliability. The second objective is to experimentally compare designs obtained using each method. A cantilevered truss structure is used as a test case. The truss is equipped with passive viscoelastic tuned dampers for vibration control. The structure is optimized by selecting locations for tuning masses added to the truss. The design requirement is that the acceleration at given points on the truss for a specified excitation be less than some upper limit. The properties of the dampers are the primary sources of uncertainty. They are described by their probability density functions in the probabilistic analysis. In the fuzzy set analysis, they are represented as fuzzy numbers. Two pairs of alternate optimal designs are obtained from the probabilistic and the fuzzy set optimizations, respectively. The optimizations are performed using genetic algorithms. The probabilistic optimization minimizes the system probability of failure. Fuzzy set optimization minimizes the system possibility of failure. Problem parameters (e.g., upper limits on the acceleration) are selected in a way that the probabilities of failure of the alternate designs differ significantly, so that the difference can be measured with a relatively small number of experiments in the lab. The main difference in the way each method maximizes safety is the following. Probabilistic optimization tries to reduce more the probabilities of failure of the modes that are easier to control. On the other hand, fuzzy set optimization tries to equalize the possibilities of failure of all failure modes. These optimum probabilistic and fuzzy set designs are then compared in the laboratory. Twenty-nine realizations of each optimum design are tested and the failure rates are measured. The results confirm that, for the selected problems, probabilistic methods can provide designs that are significantly more reliable than designs obtained using fuzzy set methods. / Ph. D.

There has been increasing interest in evaluating the performance of existing reinforced concrete (RC) bridges just after natural disasters or man-made events especially when the defects are invisible, or in quantifying the improvement after rehabilitations. In order to obtain an accurate assessment of the reliability of a RC bridge, it is critical to incorporate information about its current structural properties, which reflects the possible aging and deterioration. This dissertation proposes to develop an adaptive reliability analysis of RC bridges incorporating the damage detection information obtained from nondestructive testing (NDT). In this study, seismic fragility is used to describe the reliability of a structure withstanding future seismic demand. It is defined as the conditional probability that a seismic demand quantity attains or exceeds a specified capacity level for given values of earthquake intensity. The dissertation first develops a probabilistic capacity model for RC columns and the capacity model can be used when the flexural stiffness decays nonuniformly over a column height. Then, a general methodology to construct probabilistic seismic demand models for RC highway bridges with one single-column bent is presented. Next, a combination of global and local NDT methods is proposed to identify in-place structural properties. The global NDT uses the dynamic responses of a structure to assess its global/equivalent structural properties and detect potential damage locations. The local NDT uses local measurements to identify the local characteristics of the structure. Measurement and modeling errors are considered in the application of the NDT methods and the analysis of the NDT data. Then, the information obtained from NDT is used in the probabilistic capacity and demand models to estimate the seismic fragility of the bridge. As an illustration, the proposed probabilistic framework is applied to a reinforced concrete bridge with a one-column bent. The result of the illustration shows that the proposed framework can successfully provide the up-to-date structural properties and accurate fragility estimates.

On s’intéresse ici aux erreurs de modélisation liées à l’usage de modèles de flammelette sous-maille en combustion turbulente non prémélangée. Le but de cette thèse est de développer une stratégie d’estimation d’erreur a posteriori pour déterminer le meilleur modèle parmi une hiérarchie, à un coût numérique similaire à l’utilisation de ces mêmes modèles. Dans un premier temps, une stratégie faisant appel à un estimateur basé sur les résidus pondérés est développée et testée sur un système d’équations d’advection-diffusion-réaction. Dans un deuxième temps, on teste la méthodologie d’estimation d’erreur sur un autre système d’équations, où des effets d’extinction et de réallumage sont ajoutés. Lorsqu’il n’y a pas d’advection, une analyse asymptotique rigoureuse montre l’existence de plusieurs régimes de combustion déjà observés dans les simulations numériques. Nous obtenons une approximation des paramètres de réallumage et d’extinction avec la courbe en «S», un graphe de la température maximale de la flamme en fonction du nombre de Damköhler, composée de trois branches et d’une double courbure. En ajoutant des effets advectifs, on obtient également une courbe en «S» correspondant aux régimes de combustion déjà identifiés. Nous comparons les erreurs de modélisation liées aux approximations asymptotiques dans les deux régimes stables et établissons une nouvelle hiérarchie des modèles en fonction du régime de combustion. Ces erreurs sont comparées aux estimations données par la stratégie d’estimation d’erreur. Si un seul régime stable de combustion existe, l’estimateur d’erreur l’identifie correctement ; si plus d’un régime est possible, on obtient une fac˛on systématique de choisir un régime. Pour les régimes où plus d’un modèle est approprié, la hiérarchie prédite par l’estimateur est correcte. / We are interested here in the modeling errors of subgrid flamelet models in nonpremixed turbulent combustion. The goal of this thesis is to develop an a posteriori error estimation strategy to determine the best model within a hierarchy, with a numerical cost at most that of using the models in the first place. Firstly, we develop and test a dual-weighted residual estimator strategy on a system of advection-diffusion-reaction equations. Secondly, we test that methodology on another system of equations, where quenching and ignition effects are added. In the absence of advection, a rigorous asymptotic analysis shows the existence of many combustion regimes already observed in numerical simulations. We obtain approximations of the quenching and ignition parameters, alongside the S-shaped curve, a plot of the maximal flame temperature as a function of the Damköhler number, consisting of three branches and two bends. When advection effects are added, we still obtain a S-shaped curve corresponding to the known combustion regimes. We compare the modeling errors of the asymptotic approximations in the two stable regimes and establish new model hierarchies for each combustion regime. These errors are compared with the estimations obtained by using the error estimation strategy. When only one stable combustion regime exists, the error estimator correctly identifies that regime; when two or more regimes are possible, it gives a systematic way of choosing one regime. For regimes where more than one model is appropriate, the error estimator’s predicted hierarchy is correct.

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