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The ability to design the rational structure in short terms is obvious economical demand hence the engineer must have at his disposal the methodology of optimization of such structures. Grillage structures are widely used in engineering practice, e. g. in construction of so-called grillage-type foundations (further grillages). Nowadays the good-performing optimization algorithms for topology optimization of grillages – separately investigating each beam in the grillage – are elaborated therefore the main attention of this work is devoted to the simultaneous topology and size optimization of grillages, which is obviously insufficiently explored so far. The optimal grillage should meet twofold criteria: the number of piles should be minimal, and the connecting beams should receive minimal feasible bending moments what leads to minimal consumption of concrete for beams. Obviously two separate optimization problems are considered here: determination of minimal number of piles and determination of minimal volume of beams. Whereas the carrying capacity of a single pile is known, the first optimization problem can be rendered as minimization of the maximal reactive force in piles among all set of piles. Analogously, the second problem corresponds to the minimization of the maximal bending moments in connecting beams. The bending moments depend also on stiffness of beams hence the cross-sectional dimensions of beams must be identified simultaneously. Both problems can be incorporated... [to full text] / Sijynų optimizavimo rezultatai turi didelę reikšmę ekonominiu požiūriu, nes ypatingai svarbu gebėti greitai suprojektuoti pigią ir tuo pačiu racionalią bei patvarią konstrukciją. Todėl inžineriniu požiūriu šios problemos sprendimo rezultatai turi didelę reikšmę kuriant efektyvią sijynų optimizavimo technologiją. Sijynai – sudaryti iš polių ir jungiančiųjų sijų – yra labai efektyvios ir paplitusios polinių pamatų inžinerinės konstrukcijos. Šiame darbe dėmesys bus skiriamas iki šiol dar nepakankamai išnagrinėtam sijynų topologijos ir matmenų sinchroniniam optimizavimui. Šioje disertacijoje topologijos optimizavimas suprantamas kaip optimalios polių išdėstymo po jungiančiosiomis sijomis schemos ieškojimas esant duotam polių skaičiui, o matmenų optimizavimas – kaip jungiančiųjų sijų skerspjūvio optimalių matmenų ieškojimas, laikant, kad visų sijų skerspjūvis vienodas. Darbe bus bandoma apjungti topologijos ir matmenų optimizavimą į vieną algoritmo žingsnį, tuo padidinant tikimybę gauti geresnį optimizavimo sprendinį. Ši problema yra daugiakriterio globalaus optimizavimo uždavinys. Iki šiol tokie didelės apimties uždaviniai nėra iki galo išspręsti, nes jie yra pakankamai sudėtingi: tenka optimizuoti nuo didelio projektavimo kintamųjų skaičiaus priklausančią kompromisinę tikslo funkciją. Apytikriai galima laikyti, kad sijynai, kurie turi mažiausią įmanomą polių skaičių bei kurių jungiančiosios sijos yra mažiausio skerspjūvio, yra pigiausi. Matematiniu požiūriu tokių sijynų... [toliau žr. visą tekstą]

We propose interval arithmetic and interval constraint algorithms for global optimization. Both of these compute lower and upper bounds of a function over a box, and return a lower and an upper bound for the global minimum. In interval arithmetic methods, the bounds are computed using interval arithmetic evaluations. Interval constraint methods instead use domain reduction operators and consistency algorithms. The usual interval arithmetic algorithms for global optimization suffer from at least one of the following drawbacks: - Mixing the fathoming problem, in which we ask for the global minimum only, with the localization problem, in which we ask for the set of points at which the global minimum occurs. - Not handling the inner and outer approximations for epsilon-minimizer, which is the set of points at which the objective function is within epsilon of the global minimum. - Nothing is said about the quality for their results in actual computation. The properties of the algorithms are stated only in the limit for infinite running time, infinite memory, and infinite precision of the floating-point number system. To handle these drawbacks, we propose interval arithmetic algorithms for fathoming problems and for localization problems. For these algorithms we state properties that can be verified in actual executions of the algorithms. Moreover, the algorithms proposed return the best results that can be computed with given expressions for the objective function and the conditions, and a given hardware. Interval constraint methods combine interval arithmetic and constraint processing techniques, namely consistency algorithms, to obtain tighter bounds for the objective function over a box. The basic building block of interval constraint methods is the generic propagation algorithm. This explains our efforts to improve the generic propagation algorithm as much as possible. All our algorithms, namely dual, clustered, deterministic, and selective propagation algorithms, are developed as an attempt to improve the efficiency of the generic propagation algorithm. The relational box-consistency algorithm is another key algorithm in interval constraints. This algorithm keeps squashing the left and right bounds of the intervals of the variables until no further narrowing is possible. A drawback of this way of squashing is that as we proceed further, the process of squashing becomes slow. Another drawback is that, for some cases, the actual narrowing occurs late. To address these problems, we propose the following algorithms: - Dynamic Box-Consistency algorithm: instead of pruning the left and then the right bound of each domain, we alternate the pruning between all the domains. - Adaptive Box-Consistency algorithm: the idea behind this algorithm is to get rid of the boxes as soon as possible: start with small boxes and extend them or shrink them depending on the pruning outcome. This adaptive behavior makes this algorithm very suitable for quick squashing. Since the efficiency of interval constraint optimization methods depends heavily on the sharpness of the upper bound for the global minimum, we must make some effort to find the appropriate point or box to use for computing the upper bound, and not to randomly pick one as is commonly done. So, we introduce interval constraints with exploration. These methods use non-interval methods as an exploratory step in solving a global optimization problem. The results of the exploration are then used to guide interval constraint algorithms, and thus improve their efficiency.

Recently, the environmental impact of aircraft and rising fuel prices have become an increasing concern in the aviation industry. To address these problems, organizations such as NASA have set demanding goals for reducing aircraft emissions, fuel burn, and noise. In an effort to reach the goals, a movement toward more-electric aircraft and electric propulsion has emerged. With this movement, the number of critical electrical loads on an aircraft is increasing causing power system reliability to be a point of concern. Currently, power system reliability is maintained through the use of back-up power supplies such as batteries and ram-air-turbines (RATs). However, the increasing power requirements for critical loads will quickly outgrow the capacity of the emergency devices. Therefore, reliability needs to be addressed when designing the primary power distribution system. Power system reliability is a function of component reliability and redundancy. Component reliability is often not determined until detailed component design has occurred; however, the amount of redundancy in the system is often set during the system architecting phase. In order to meet the capacity and reliability requirements of future power distribution systems, a method for redundancy allocation during the system architecting phase is needed. This thesis presents an aircraft power system design methodology that is based upon the engineering decision process. The methodology provides a redundancy allocation strategy and quantitative trade-off environment to compare architecture and technology combinations based upon system capacity, weight, and reliability criteria. The methodology is demonstrated by architecting the power distribution system of an aircraft using turboelectric propulsion. The first step in the process is determining the design criteria which includes a 40 MW capacity requirement, a 20 MW capacity requirement for the an engine-out scenario, and a maximum catastrophic failure rate of one failure per billion flight hours. The next step is determining gaps between the performance of current power distribution systems and the requirements of the turboelectric system. A baseline architecture is analyzed by sizing the system using the turboelectric system power requirements and by calculating reliability using a stochastic flow network. To overcome the deficiencies discovered, new technologies and architectures are considered. Global optimization methods are used to find technology and architecture combinations that meet the system objectives and requirements. Lastly, a dynamic modeling environment is constructed to study the performance and stability of the candidate architectures. The combination of the optimization process and dynamic modeling facilitates the selection of a power system architecture that meets the system requirements and objectives.

Computational models and simulations are essential system design tools that allow for improved decision making and cost reductions during all phases of the design process. However, the most accurate models are often computationally expensive and can therefore only be used sporadically. Consequently, designers are often forced to choose between exploring many design alternatives with less accurate, inexpensive models and evaluating fewer alternatives with the most accurate models. To achieve both broad exploration of the alternatives and accurate determination of the best alternative with reasonable costs incurred, surrogate modeling and variable accuracy modeling are used widely. A surrogate model is a mathematically tractable approximation of a more expensive model based on a limited sampling of that model, while variable accuracy modeling involves a collection of different models of the same system with different accuracies and computational costs. As compared to using only very accurate and expensive models, designers can determine the best solutions more efficiently using surrogate and variable accuracy models because obviously poor solutions can be eliminated inexpensively using only the less expensive, less accurate models. The most accurate models are then reserved for discerning the best solution from the set of good solutions. In this thesis, a Value-Based Global Optimization (VGO) algorithm is introduced. The algorithm uses kriging-like surrogate models and a sequential sampling strategy based on Value of Information (VoI) to optimize an objective characterized by multiple analysis models with different accuracies. It builds on two primary research contributions. The first is a novel surrogate modeling method that accommodates data from any number of analysis models with different accuracies and costs. The second contribution is the use of Value of Information (VoI) as a new metric for guiding the sequential sampling process for global optimization. In this manner, the cost of further analysis is explicitly taken into account during the optimization process. Results characterizing the algorithm show that VGO outperforms Efficient Global Optimization (EGO), a similar global optimization algorithm that is considered to be the current state of the art. It is shown that when cost is taken into account in the final utility, VGO achieves a higher utility than EGO with statistical significance. In further experiments, it is shown that VGO can be successfully applied to higher dimensional problems as well as practical engineering design examples.

Dans le but de réduire la consommation d’énergie et d’augmenter la part des énergies renouvelables, la conception optimale des futurs bâtiments (bâtiments intelligents) apparaît comme un facteur important. Cette thèse vise donc à développer des modèles, des méthodes innovantes d’aide à la conception pour ces bâtiments. Notre nouvelle approche de conception est une optimisation globale et simultanée de l’enveloppe, des systèmes énergétiques et de leurs stratégies de gestion dès la phase d’esquisse, qui prend en compte plusieurs critères de coût (investissement et exploitation) et de confort (thermique, visuel et aéraulique). Le problème d’optimisation multi-objectif est donc un problème de couplage fort de grande taille avec de nombreuses variables et contraintes, qui induisent des difficultés lors de sa résolution. Après avoir fait des analyses sur des cas tests, une méthode d’optimisation d’ordre 1 est choisie, en association à des modèles analytiques dérivés formellement de manière automatique. Notre méthodologie est appliquée à la conception de maisons individuelles, et plus particulièrement des maisons à énergie positive. Les résultats obtenus par cette approche globale apportent des informations importantes aux concepteurs pour l’aider à faire des choix en phase amont du processus de conception. / In order to reduce the energy consumption and to increase the use of renewable energy, the optimal design of future buildings (smart-buildings) appears as an important factor.This thesis aims to develop models, innovative methods aiding decision-making during the design of buildings. Our approach of design is a global and simultaneous optimization of envelope, energy systems and their management strategies from the sketch phase, which takes into account multi-criterions of costs (investment et exploitation) and comforts (thermal, visual, aeraulic). The multi-objective optimization problem is so a strong coupling problem of large scale with a lot of variables and constraints, which leads to difficulties to solve.After the tests, an optimization method of order 1 is chosen in combination with analytical models formally derived automatically. Our methodology is applied to the design of individual houses, especially positive energy houses. The results of this global approach provide important information to designers to help make choices from the preliminary phase of the design process.

This dissertation thesis is concerned with developing a mesoscopic model for sin- gle crystalline shape-memory alloys including thermo-dynamically consistent thermo- mechanical coupling - here the term "mesoscopic" refers to the ability of the model to capture fine spatial oscillations of the deformation gradient by means of gradient Young measures. Existence of solutions to the devised model is proved in a "phase-field-like approach" by a scale transition from a microscopic model that features a term related to the interfacial energy; this scale transition from a physically relevant model justifies the mesoscopic relaxation. Further, existence of solutions is also proved by backward- Euler time discretization which forms a conceptual numerical algorithm. Based on this conceptual algorithm a computer implementation of the model has been developed and further optimized in the rate-independent isothermal setting; some calculations using this implementation are also presented. Finally, refinements s of the analysis in the convex case as well as a limit of the phase-field-like approach in this case are exposed, too.

Ce mémoire présente une approche basée sur des méthodes garanties pour résoudre des problèmes d’optimisation de systèmes dynamiques multi-physiques. Ces systèmes trouvent des applications directes dans des domaines variés tels que la conception en ingéniérie, la modélisation de réactions chimiques, la simulation de systèmes biologiques ou la prédiction de la performance sportive.La résolution de ces problèmes d’optimisation s’effectue en deux phases. La première consiste à mettre le problème en équations sous forme d’un modèle mathématique constitué d’un ensemble de variables, d’un ensemble de contraintes algébriques et fonctionelles ainsi que de fonctions de coût. Celles-ci sont utilisées lors de la seconde phase qui consiste à d’extraire du modèle les solutions optimales selon plusieurs critères (volume, poids, etc).Les contraintes algébriques permettent de manipuler des grandeurs statiques (quantité, taille, densité, etc). Elles sont non linéaires, non convexes et parfois discontinues.Les contraintes fonctionnelles permettent de manipuler des grandeurs dynamiques. Ces contraintes peuvent être relativement simples comme la monotonie ou la périodicité, mais aussi bien plus complexe par la prise en compte de contraintes différentielles simples ou définies par morceaux. Les équations différentielles sont utilisées pour modéliser des comportements physico-chimiques (magnétiques, thermiques, etc) et d’autres caractéristiques qui varient lors de l’évolution du système.Il existe plusieurs niveaux d’approximation pour chacune de ces deux phases. Ces approximations donnent des résultats pertinents, mais elles ne permettent pas de garantir l’optimalité ni la réalisabilité des solutions.Après avoir présenté un ensemble de méthodes garanties permettant de résoudre de manière garantie des équations différentielles ordinaires, nous formalisons un modèle particulier de systèmes hybrides sous la forme d’équations différentielles ordinaires par morceaux. A l’aide de plusieurs preuves et théorèmes nous étendons la première méthode de résolution pour résoudre de manière garantie ces équations différentielles par morceaux. Dans un second temps, nous intégrons ces deux méthodes au sein d’un module de programmation par contracteurs, que nous avons implémenté. Ce module basé sur des méthodes garantie permet de résoudre des problèmes de satisfaction de contraintes algébriques et fonctionnelles. Ce module est finalement utilisé dans un algorithme d’optimisation globale déterministe modulaire permettant de résoudre les problèmes considérés. / In this thesis a set of tools based on guaranteed methods are presented in order to solve multi-physics dynamic problems. These systems can be applied in various domains such that engineering design process, model of chemical reactions, simulation of biological systems or even to predict athletic performances.The resolution of these optimization problems is made of two stages. The first one consists in defining a mathematical model by setting up the equations for the problem. The model is made of a set of variables, a set of algebraic and functional constraints and cost functions. The latter are used in the second stage in order to extract the optimal solutions from the model depending on several criteria (volume, weight, etc).Algebraic constraints are used to describe the static properties of the system (quantity, size, density, etc). They are non-linear, non-convex and sometimes discontinuous. Functional constraints are used to manipulate dynamic quantities. These constraints can be quite simple such as monotony or periodicity or they can be more complex such as simple or piecewise differential constraints. Differential equations are used to describe physico-chemical properties (magnetic, thermal, etc) and other features evolving with the component use. Several levels of approximation exist for each of these two stages. These approximations give some relevant results but they do not guarantee the feasibility nor the optimality of the solutions.After presenting a set of guaranteed methods in order to perform the guaranteed integration of ordinary differential equations, a peculiar type of hybrid system that can be modeled with piecewise ordinary differential equation is considered. A new method that computes guaranteed integration of these piecewise ordinary differential equations is developed through an extension of the initial algorithm based on several proofs and theorems. In a second step these algorithms are gathered within a contractor programming module that have been implemented. It is used to solve algebraic and functional constraint satisfaction problems with guaranteed methods. Finally, the considered optimization problems are solved with a modular deterministic global optimization algorithm that uses the previous modules.

L’optimisation globale fiable est dédiée à la recherche d’un minimum global en présence d’erreurs d’arrondis. Les seules approches fournissant une preuve numérique d’optimalité sont des méthodes d’intervalles qui partitionnent l’espace de recherche et éliminent les sous-espaces qui ne peuvent contenir de solution optimale. Ces méthodes exhaustives, appelées branch and bound par intervalles, sont étudiées depuis les années 60 et ont récemment intégré des techniques de réfutation et de contraction, issues des communautés d’analyse par intervalles et de programmation par contraintes. Il est d’une importance cruciale de calculer i) un encadrement précis de la fonction objectif et des contraintes sur un sous-domaine ; ii) une bonne approximation (un majorant) du minimum global. Les solveurs de pointe sont généralement des méthodes intégratives : ils invoquent sur chaque sous-domaine des algorithmes d’optimisation locale afin d’obtenir une bonne approximation du minimum global. Dans ce document, nous nous intéressons à un cadre coopératif combinant des méthodes d’intervalles et des algorithmes évolutionnaires. Ces derniers sont des algorithmes stochastiques faisant évoluer une population de solutions candidates (individus) dans l’espace de recherche de manière itérative, dans l’espoir de converger vers des solutions satisfaisantes. Les algorithmes évolutionnaires, dotés de mécanismes permettant de s’échapper des minima locaux, sont particulièrement adaptés à la résolution de problèmes difficiles pour lesquels les méthodes traditionnelles peinent à converger. Au sein de notre solveur coopératif Charibde, l’algorithme évolutionnaire et l’algorithme sur intervalles exécutés en parallèle échangent bornes, solutions et espace de recherche par passage de messages. Une stratégie couplant une heuristique d’exploration géométrique et un opérateur de réduction de domaine empêche la convergence prématurée de la population vers des minima locaux et évite à l’algorithme évolutionnaire d’explorer des sous-espaces sous-optimaux ou non réalisables. Une comparaison de Charibde avec des solveurs de pointe (GlobSol, IBBA, Ibex) sur une base de problèmes difficiles montre un gain de temps d’un ordre de grandeur. De nouveaux résultats optimaux sont fournis pour cinq problèmes multimodaux pour lesquels peu de solutions, même approchées, sont connues dans la littérature. Nous proposons une application aéronautique dans laquelle la résolution de conflits est modélisée par un problème d’optimisation sous contraintes universellement quantifiées, et résolue par des techniques d’intervalles spécifiques. Enfin, nous certifions l’optimalité de la meilleure solution connue pour le cluster de Lennard-Jones à cinq atomes, un problème ouvert en dynamique moléculaire. / Reliable global optimization is dedicated to finding a global minimum in the presence of rounding errors. The only approaches for achieving a numerical proof of optimality in global optimization are interval-based methods that interleave branching of the search-space and pruning of the subdomains that cannot contain an optimal solution. The exhaustive interval branch and bound methods have been widely studied since the 1960s and have benefitted from the development of refutation methods and filtering algorithms, stemming from the interval analysis and interval constraint programming communities. It is of the utmost importance: i) to compute sharp enclosures of the objective function and the constraints on a given subdomain; ii) to find a good approximation (an upper bound) of the global minimum. State-of-the-art solvers are generally integrative methods, that is they embed local optimization algorithms to compute a good upper bound of the global minimum over each subspace. In this document, we propose a cooperative framework in which interval methods cooperate with evolutionary algorithms. The latter are stochastic algorithms in which a population of individuals (candidate solutions) iteratively evolves in the search-space to reach satisfactory solutions. Evolutionary algorithms, endowed with operators that help individuals escape from local minima, are particularly suited for difficult problems on which traditional methods struggle to converge. Within our cooperative solver Charibde, the evolutionary algorithm and the intervalbased algorithm run in parallel and exchange bounds, solutions and search-space via message passing. A strategy combining a geometric exploration heuristic and a domain reduction operator prevents premature convergence toward local minima and prevents the evolutionary algorithm from exploring suboptimal or unfeasible subspaces. A comparison of Charibde with state-of-the-art solvers based on interval analysis (GlobSol, IBBA, Ibex) on a benchmark of difficult problems shows that Charibde converges faster by an order of magnitude. New optimality results are provided for five multimodal problems, for which few solutions were available in the literature. We present an aeronautical application in which conflict solving between aircraft is modeled by an universally quantified constrained optimization problem, and solved by specific interval contractors. Finally, we certify the optimality of the putative solution to the Lennard-Jones cluster problem for five atoms, an open problem in molecular dynamics.

Cette thèse s'intéresse au problème de conception optimale des routes de départ et d'arrivée dans une zone terminale autour d'un aéroport. Cette conception prend en compte la configuration et l'environnement autour des aéroports, et les différentes contraintes sous-jacentes, notamment l'évitement des obstacles et la séparation des routes. Nous proposons une formulation mathématique conduisant à un problème d'optimisation combinatoire, ainsi que des méthodes de résolution ad hoc efficaces pour le problème. Pour la résolution du problème, nous procédons en deux étapes. Nous considérons d'abord la conception d'une route de longueur minimale évitant les obstacles, en utilisant la méthode de Branch and Bound (B&B). Ensuite, nous nous intéressons à la conception de plusieurs routes en assurant en plus la séparation des routes. Deux approches différentes sont appliquées : une méthode basée sur la méthode B&B pour construire les routes séquentiellement suivant un ordre fixé à l'avance, et une méthode de recuit simulé pour construire les routes simultanément. Les résultats sur un ensemble de problèmes tests (artificiels et réels) montrent l'efficacité de notre approche. / This thesis proposes a methodology for the optimization of departure and arrival routes in the Terminal Maneuvering Area (TMA). The design of these routes takes into account the configuration and environment around airports, and the related constraints, in particular the avoidance of obstacles and the separation between routes. We propose a mathematical formulation leading to a combinatorial optimization problem, as well as efficient ad hoc resolution methods for the problem. The problem is solved in two steps. First, we design an individual route avoiding obstacles with respect to minimum route length by using a Branch and Bound (B&B) method. Afterwards, the design of multiple routes is solved by two different approaches: a B&B-based approach (where routes are generated sequentially in a given order) and a Simulated Annealing approach (where routes are generated simultaneously). The simulation results of a set of (artificial and real) test problems show the efficiency of our approach.