Dans ce travail, on s'intéresse à l'analyse mathématique et au contrôle optimal pour des problèmes de diffusion relevant de certains domaines comme l'écologie ou l'environnement et comportant des termes de pollution inconnus en général. De plus, on souhaite agir sur le système en un seul point du domaine considéré pour des raisons de coût. La modélisation de ces problèmes se traduit généralement par un système de type parabolique avec donnée manquante (initiale ou aux limites) représentant la pollution, et où l'on introduit une fonction de contrôle de ce système. La méthode suivie pour résoudre ces problèmes est celle du contrôle à moindres regrets développée par J.-L. Lions et bien adaptée aux problèmes à données manquantes.Plus précisément, on est concerné par des problèmes de type parabolique qui décrivent la diffusion d'un fluide (eau) contaminé dans un domaine (une lagune ou un estuaire) par une pollution ayant son origine sur une partie du bord. De plus, on considère que la fonction source (le contrôle) est localisée en un point, c'est ce qu'on appelle le contrôle ponctuel. On cherche alors le (ou les) contrôle(s) qui peuvent améliorer la situation au lieu de la laisser à l'abandon (sans contrôle).Les solutions ne sont pas des fonctions régulières et ne peuvent être considérées qu'au sens faible. Deux méthodes sont utilisées: la première est la méthode de transposition de Lions-Magenes, détaillée au chapitre 3 de la thèse, et la deuxième méthode consiste à régulariser la masse de Dirac (le support du contrôle est un point) présentée au chapitre4. Pour les deux méthodes, on montre l'existence d'une solution faible et on établit un système d'optimalité singulier (SOS) du contrôle ponctuel à moindres regrets.Un dernier chapitre est consacré aux schémas numériques associés au problème de contrôle ponctuel à moindres regrets, où l'on obtient des estimations d'erreur par la méthode des éléments finis. / In this thesis, we are interested in mathematical analysis and optimal control of diffusion problems where there are pollution terms. In addition, we want to act on the system in a single point of the domain for cost reasons. The systems being studied are parabolic with missing (initial or boundary) data representing pollution, where we introduce a control function. The method of low-regret control of J.-L. Lions, used here for the first time to the pointwise control, seems to be well suited. We then look for the control which can improve the situation instead of doing nothing (no control).Solutions are not regular functions and can only be considered in the weak sense. Two methods are used here: the first one is the method of transposition of Lions-Magenes, detailed in Chapter 3 of the thesis, and the second method consists in regularizing the Dirac mass, presented in chapter 4. Each one of the two methods offers a new point of view. In particular, the functional spaces where the existence of a solution is obtained are different. For both methods, however, a singular optimality system is established for the low-regret pointwise control.A final chapter is devoted to the numerical schemes associated to the low-regret pointwise optimal control, where we obtain error estimates using finite elements method (FEM).

En ce qui concerne les problèmes de Dynamique des Fluides Numériques, l’adaptation du maillage est intéressante pour sa capacité à aborder la convergence asymptotique et à obtenir une prévision précise pour des flux complexes à moindre coût. La méthode d’adaptation de maillage anisotrope réduit le nombre de degrés de liberté nécessaires pour atteindre la précision d’une solution donnée, ce qui a un impact positif sur le temps de calcul. De plus, il réduit la dissipation du schéma numérique en tenant compte automatiquement de l'anisotropie des phénomènes physiques à l'intérieur du maillage. Deux approches principales existent dans la littérature. L'adaptation du maillage basée sur les caractéristiques géométriques, qui est principalement déduite d'une estimation de l'erreur d'interpolation utilisant la hessienne du senseur choisi, contrôle l'erreur d'interpolation du capteur sur l'ensemble du domaine de calcul. Une telle approche est facile à mettre en place et a un large éventail d’applications, mais elle ne prend pas en compte l’EDP considérée utilisée pour résoudre le problème. D'autre part, l'adaptation de maillage orientée fonctionnelle, qui se concentre sur une fonctionnelle scalaire, prend en compte à la fois la solution et l'EDP dans l'estimation d'erreur grâce à l'état adjoint. Mais, la conception de cette estimation d'erreur est beaucoup plus compliquée. Cette thèse présente les résultats obtenus avec différentes méthodes de Dynamique des Fluides Numériques: les solveurs de flux arbitrairement lagrangiens-eulériens (ALE) avec schémas explicites et implicites sont présentés et couplés au mouvement de maillage, l’adaptation de maillage feature-based instationnaire pour les géométries mobiles prend en compte les changements des connectivités de maillage durant toute la simulation, l'état adjoint est étendu aux problèmes de géométries mobiles et l'adaptation de maillage instationnaire orientée fonctionnelle pour les maillages mobiles est déduite d'une estimation d'erreur a priori. Plusieurs exemples numériques issus du secteur aéronautique et du domaine de sécurité civile sont considérés. / When dealing with CFD problems, mesh adaptation is interesting for its ability to approach the asymptotic convergence and to obtain an accurate prediction for complex flows at a lower cost. Anisotropic mesh adaptation method reduces the number of degrees of freedom required to reach a given solution accuracy, thus impact favorably the CPU time. Moreover, it reduces the numerical scheme dissipation by automatically taking into account the anisotropy of the physical phenomena inside the mesh. Two main approaches exist in the literature. Feature-based mesh adaptation which is mainly deduced from an interpolation error estimate using the Hessian of the chosen sensor controls the interpolation error of the sensor over the whole computational domain. Such approach is easy to set-up and has a wide range of application, but it does not take into account the considered PDE used to solve the problem. On the other hand, goal-oriented mesh adaptation, which focuses on a scalar output function, takes into consideration both the solution and the PDE in the error estimation thanks to the adjoint state. But, the design of such error estimate is much more complicated. This thesis presents the results obtained with different CFD methods : the Arbitrary Lagrangian Eulerian (ALE) flow solvers with explicit and implicit schemes are presented and coupled to the moving mesh process, the feature-based unsteady mesh adaptation for moving geometries takes into account the changes of connectivites during the whole simulation, the adjoint state is extended to moving geometries problems and goal-oriented unsteady mesh adaptation for moving meshes is derived from an a priori error estimate. Several numerical examples are considered in the aeronautics sector and the field of civil security.

L’assimilation variationnelle de données 4D-Var est une technique très utilisée en géophysique, notamment en météorologie et océanographie. Elle consiste à estimer des paramètres d’un modèle numérique direct, en minimisant une fonction de coût mesurant l’écart entre les sorties du modèle et les mesures observées. La minimisation, qui est basée sur une méthode de gradient, nécessite le calcul du modèle adjoint (produit de la transposée de la matrice jacobienne avec le vecteur dérivé de la fonction de coût aux points d’observation). Lors de la mise en œuvre de l’AD 4D-Var, il faut faire face à des problèmes d’implémentation informatique complexes, notamment concernant le modèle adjoint, la parallélisation du code et la gestion efficace de la mémoire. Afin d’aider au développement d’applications d’AD 4D-Var, le logiciel YAO qui a été développé au LOCEAN, propose de modéliser le modèle direct sous la forme d’un graphe de flot de calcul appelé graphe modulaire. Les modules représentent des unités de calcul et les arcs décrivent les transferts des données entre ces modules. YAO est doté de directives de description qui permettent à un utilisateur de décrire son modèle direct, ce qui lui permet de générer ensuite le graphe modulaire associé à ce modèle. Deux algorithmes, le premier de type propagation sur le graphe et le second de type rétropropagation sur le graphe permettent, respectivement, de calculer les sorties du modèle direct ainsi que celles de son modèle adjoint. YAO génère alors le code du modèle direct et de son adjoint. En plus, il permet d’implémenter divers scénarios pour la mise en œuvre de sessions d’assimilation.Au cours de cette thèse, un travail de recherche en informatique a été entrepris dans le cadre du logiciel YAO. Nous avons d’abord formalisé d’une manière plus générale les spécifications deYAO. Par la suite, des algorithmes permettant l’automatisation de certaines tâches importantes ont été proposés tels que la génération automatique d’un parcours “optimal” de l’ordre des calculs et la parallélisation automatique en mémoire partagée du code généré en utilisant des directives OpenMP. L’objectif à moyen terme, des résultats de cette thèse, est d’établir les bases permettant de faire évoluer YAO vers une plateforme générale et opérationnelle pour l’assimilation de données 4D-Var, capable de traiter des applications réelles et de grandes tailles. / Variational data assimilation 4D-Var is a well-known technique used in geophysics, and in particular in meteorology and oceanography. This technique consists in estimating the control parameters of a direct numerical model, by minimizing a cost function which measures the misfit between the forecast values and some actual observations. The minimization, which is based on a gradient method, requires the computation of the adjoint model (product of the transpose Jacobian matrix and the derivative vector of the cost function at the observation points). In order to perform the 4DVar technique, we have to cope with complex program implementations, in particular concerning the adjoint model, the parallelization of the code and an efficient memory management. To address these difficulties and to facilitate the implementation of 4D-Var applications, LOCEAN is developing the YAO framework. YAO proposes to represent a direct model with a computation flow graph called modular graph. Modules depict computation units and edges between modules represent data transfer. Description directives proper to YAO allow a user to describe its direct model and to generate the modular graph associated to this model. YAO contains two core algorithms. The first one is a forward propagation algorithm on the graph that computes the output of the numerical model; the second one is a back propagation algorithm on the graph that computes the adjoint model. The main advantage of the YAO framework, is that the direct and adjoint model programming codes are automatically generated once the modular graph has been conceived by the user. Moreover, YAO allows to cope with many scenarios for running different data assimilation sessions.This thesis introduces a computer science research on the YAO framework. In a first step, we have formalized in a more general way the existing YAO specifications. Then algorithms allowing the automatization of some tasks have been proposed such as the automatic generation of an “optimal” computational ordering and the automatic parallelization of the generated code on shared memory architectures using OpenMP directives. This thesis permits to lay the foundations which, at medium term, will make of YAO a general and operational platform for data assimilation 4D-Var, allowing to process applications of high dimensions.

The determination of the mechanisms setting the strength and structure of the large scale circulation is a fundamental and long-standing problem in physical oceanography. In this thesis, we seek to explore the mechanisms contributing to the steady state and variability of the large scale flow, with a focus on better understanding the dynamics of the Atlantic meridional overturning circulation (AMOC). In the first part of this thesis, we explore the linear sensitivity of the monthly mean subtropical AMOC to surface fluxes of buoyancy and momentum. Our approach is to use a numerical adjoint. Key insights are provided into the memory of the AMOC to historic atmospheric forcing. We find that significant memory to wind forcing is confined to timescales of less than a year. In contrast, we identify significant memory to surface buoyancy forcing spanning multi-decadal timescales and characterised by a large scale oscillation in the sign of sensitivity between the eastern and western North Atlantic basin. An important result is that to understand the origins of seasonal variability in the modelled AMOC, we must examine the response to a multidecadal history of atmospheric forcing. In the second part of this thesis, a new tool is presented that enables a clean diagnosis of the force balance controlling the circulation regime for a Boussinesq fluid. Specifically, the tool is based on the development of the "rotational momentum" equations and sets of scalar "velocity potentials" and analogous "force functions". The latter allow the projection of all forces onto the acceleration of the vertical shears and external modes of overturning to be visualised in isolation. The rotational momentum decomposition is applied to the modelled circulation in idealised Atlantic and global configurations of the MITgcm, with a focus on elucidating the dynamics of the simulated AMOC. We discuss the key role played by the rotational buoyancy forcing right on the western boundary.

Originally developed in the 1970s for the optimal control of systems governed by partial differential equations, the adjoint method has found several successful applications, e.g., in meteorology with large-scale 3D or 4D atmospheric data assimilation schemes, for carbon cycle data assimilation in biogeochemistry and climate research, or in oceanographic modelling with efficient adjoint codes of ocean general circulation models. Despite the variety of applications in these research fields, adjoint methods have only very recently drawn attention from the ocean acoustics community. In ocean acoustic tomography and geoacoustic inversion, where the inverse problem is to recover unknown acoustic properties of the water column and the seabed from acoustic transmission data, the solution approaches are typically based on travel time inversion or standard matched-field processing in combination with metaheuristics for global optimization. In order to complement the adjoint schemes already in use in meteorology and oceanography with an ocean acoustic component, this thesis is concerned with the development of the adjoint of a full-field acoustic propagation model for shallow water environments. In view of the increasing importance of global ocean observing systems such as the European Seas Observatory Network, the Arctic Ocean Observing System and Maritime Rapid Environmental Assessment (MREA) systems for defence and security applications, the adjoint of an ocean acoustic propagation model can become an integral part of a coupled oceanographic and acoustic data assimilation scheme in the future. Given the acoustic pressure field measured on a vertical hydrophone array and a modelled replica field that is calculated for a specific parametrization of the environment, the developed adjoint model backpropagates the mismatch (residual) between the measured and predicted field from the receiver array towards the source. The backpropagated error field is then converted into an estimate of the exact gradient of the objective function with respect to any of the relevant physical parameters of the environment including the sound speed structure in the water column and densities, compressional/shear sound speeds, and attenuations of the sediment layers and the sub-bottom halfspace. The resulting environmental gradients can be used in combination with gradient descent methods such as conjugate gradient, or Newton-type optimization methods tolocate the error surface minimum via a series of iterations. This is particularly attractive for monitoring slowly varying environments, where the gradient information can be used to track the environmental parameters continuously over time and space. In shallow water environments, where an accurate treatment of the acoustic interaction with the bottom is of outmost importance for a correct prediction of the sound field, and field data are often recorded on non-fully populated arrays, there is an inherent need for observation over a broad range of frequencies. For this purpose, the adjoint-based approach is generalized for a joint optimization across multiple frequencies and special attention is devoted to regularization methods that incorporate additional information about the desired solution in order to stabilize the optimization process. Starting with an analytical formulation of the multiple-frequency adjoint approach for parabolic-type approximations, the adjoint method is progressively tailored in the course of the thesis towards a realistic wide-angle parabolic equation propagation model and the treatment of fully nonlocal impedance boundary conditions. A semi-automatic adjoint generation via modular graph approach enables the direct inversion of both the geoacoustic parameters embedded in the discrete nonlocal boundary condition and the acoustic properties of the water column. Several case studies based on environmental data obtained in Mediterranean shallow waters are used in the thesis to assess the capabilities of adjoint-based acoustic inversion for different experimental configurations, particularly taking into account sparse array geometries and partial depth coverage of the water column. The numerical implementation of the approach is found to be robust, provided that the initial guesses are not too far from the desired solution, and accurate, and converges in a small number of iterations. During the multi-frequency optimization process, the evolution of the control parameters displays a parameter hierarchy which clearly relates to the relative sensitivity of the acoustic pressure field to the physical parameters. The actual validation of the adjoint-generated environmental gradients for acoustic monitoring of a shallow water environment is based on acoustic and oceanographic data from the Yellow Shark '94 and the MREA '07 sea trials, conducted in the Tyrrhenian Sea, south of the island of Elba. Starting from an initial guess of the environmental control parameters, either obtained through acoustic inversion with global search or supported by archival in-situ data, the adjoint method provides an efficient means to adjust local changes with a couple of iterations and monitor the environmental properties over a series of inversions. In this thesis the adjoint-based approach is used, e.g., to fine-tune up to eight bottom geoacoustic parameters of a shallow-water environment and to track the time-varying sound speed profile in the water column. In the same way the approach can be extended to track the spatial water column and bottom structure using a mobile network of sparse arrays. Work is currently being focused on the inclusion of the adjoint approach into hybrid optimization schemes or ensemble predictions, as an essential building block in a combined ocean acoustic data assimilation framework and the subsequent validation of the acoustic monitoring capabilities with long-term experimental data in shallow water environments.

<p>The thesis addresses time-domain inverse electromagneticscattering for determining unknown characteristics of an objectfrom observations of the scattered .eld. Applications includenon-destructive characterization of media and optimization ofmaterial properties, for example the design of radar absorbingmaterials.A nother interesting application is the parameteroptimization of subcell models to avoid detailed modeling ofcomplex geometries.</p><p>The inverse problem is formulated as an optimal controlproblem where the cost function to be minimized is thedi.erence between the estimated and observed .elds, and thecontrol parameters are the unknown object characteristics. Theproblem is solved in a deterministic gradient-basedoptimization algorithm using a parallel 2D FDTD scheme for thedirect problem.This approach is computationally intensive sincethe direct problem needs to be solved in every optimizationiteration in order to compute an estimated .eld.H ighlyaccurate analytical gradients are computed from the adjointformulation.In addition to giving better accuracy than .nitedi.erences, the analytical gradients also have the advantage ofonly requiring one direct and one adjoint problem to be solvedregardless of the number of parameters.</p><p>When absorbing boundary conditions are used to truncate thecomputational domain, the equations are non-reversible and theentire time-history of the direct solution needs to be storedfor the gradient computation.Ho wever, using an additionaldirect simulation and a restart procedure it is possible tokeep the storage at an acceptable level.</p><p>The inverse method has been successfully applied to a widerange of industrial problems within the European project,IMPACT (Inverse Methods for Wave Propagation Applications inTime-Domain).T he results presented here includecharacterization of layered dispersive media, determination ofparameters in subcell models for thin sheets and narrow slotsand optimization problems where the observed .eld is given bydesign objectives.</p>

This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.

Nessa tese, mostramos que a Hamiltoniana H e o operador helicidade de uma partícula de Dirac que se movimenta em duas dimensões na presença de um tubo de fluxo magnético infinitamente fino na origem admitem, cada um, uma família de quatro parâmetros de extensões auto-adjuntas. Para cada extensão correspondem condições de contorno a serem satisfeitas pelas auto-fuções na origem. Apesar dos operadores H e formalmente comutarem antes da especificação das condições de contorno, para garantirmos a conservação da helicidade, não é suficiente obtermos as mesmas condições de contorno para ambos os operadores, ou seja, não é suficiente a determinação de um domínio comum a ambos. Mostramos que, para certas relações entre os parâmetros das extensões satisfeitas, é possível a determinação dos domínios mais gerais onde ambos os operadores H e são auto-adjuntos e onde a helicidade é conservada, simultaneamente com a preservação da simetria de Aharonov-Bohm ( + 1), onde é o fluxo magnético em unidades naturais. Nossos resultados implicam que, nem a conservação da helicidade nem a simetria de Aharonov-Bohn, resolvem o problema da escolha da condição de contorno fisicamente correta. / We show that both the Hamiltonian H and the helicity operator of a Dirac particle moving in two dimension in the presence of an infinitely thin magnetic flux tube admit each a four- parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BC\'s) to be satisfied by the eigenfunctions at the origin. Althou- gh the actions af these two operators commute before specification of boundary conditions, to ensure helicity conservation it is not sufficient to take the same BC\'s for both operators. We show that, given certain relations between the parameters of the extensions it is possible to write down the most general domain where both operators H and are self-adjoint with heli- city conservation and also Aharonov-Bohm symmetry ( + 1) preserved, where is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results im- ply that neither helicity conservation nor Aharonov-Bohm symmetry by themselves solves the problem of choosing the \"physical \"boundary conditions for this system.

O presente trabalho é o início de um estudo da aplicação do método de otimização conhecido como adjunto em escoamentos incompressíveis, viscosos e periódicos, envolvendo um problema de bastante interesse: a análise da aplicação de splitter plates em cilindros de seção circular. Conhecido por sua simplicidade, o splitter plate, que consiste em uma placa plana alinhada ao escoamento e colocada a jusante do corpo, é um dispositivo efetivo na mudança de comportamento da esteira de vórtices de Von Kárman. A partir da revisão bibliográfica foi possível entender a dinâmica do escoamento, proporcionando uma calibração dos modelos numéricos. Complementando esta etapa, foi efetuada uma análise da qualidade das malhas computacionais. Através de uma geração de diversas malhas computacionais, o espaço de soluções foi explorado buscando encontrar o mínimo arrasto para diversos comprimentos de splitter plate e diferentes números de Reynolds (Re). Foi observada a influência da placa na formação da esteira de vórtices, obtendo uma redução dos coeficientes de força do cilindro. Com esses dados, foi possível desenvolver o método de otimização voltado para análise do gradiente de sensibilidade conhecido como método adjunto baseado nas equações de Navier Stokes utilizando o problema descrito como base para validação dos resultados. A abordagem do método adjunto caracteriza-se pela busca dos extremos de funções conhecidas como medidas de mérito. Essas funções podem ser integrais de sustentação e arrasto por exemplo. Na literatura, o método adjunto é apresentado como possuindo duas grandes vantagens: a primeira é a imposição das equações do escoamento como restrições do problema, o que sempre confinará as variações da medida de mérito dentro do universo de soluções realizáveis; já a segunda é conseqüência da primeira, pois as restrições permitem uma simplificação no cálculo do gradiente de sensibilidade, reduzindo o custo computacional. Para o cálculo do gradiente de sensibilidade, o objetivo é otimizar o arrasto do cilindro sob efeito do splitter plate variando os parâmetros de controle (comprimento e posicionamento do splitter plate). A direção de busca e o cálculo do passo da geometria são obtidos a partir da relação entre a solução numérica do escoamento e as variáveis adjuntas calculadas. Nesta dissertação, será apresentada a pesquisa bibliográfica, os resultados do método tentativa e erro, a formulação do método adjunto baseado nas equações de Navier Stokes e um exemplo de sua solução numérica, demonstrando sua existência. / The report is the beginning of a research about the application of the so called adjoint method in the optimization of incompressive, viscous and periodic flows. The study involves a problem of great interest: an analysis of the implementation of splitter plates in the flow around cylinders with circular section. Widely recognized for its simplicity, the splitter plate consists of a flat plate, which is placed in the wake of a cylinder, in the stream wise direction, and it works by changing the way the shear layers interact with one another. Based on a literature survey, it is possible to understand the physics of this class of flows. As a better result, one learns what to expect from the numerical solutions and hence, one can calibrate its parameters. Moreover, we study the best configuration of the computational mesh, thus reducing the computational cost. After the generation of meshes, the universe of solutions was explored to find the minimum drag for various lengths of splitter plate and Reynolds number (Re). The influence of the plate in the interaction of the shear layers was observed in the reduction of drag coefficient. These results form a the basis for comparison, upon one can develop the optimization by the adjoint method. The adjoint method can be used to search the extreme of objective functionals. These functionals can be the lift and drag integrals for example. The theory presents two advantages to the method: first, the imposing the equations that govern the flow as variational constraints one limits the variations to the universe of realizable solutions; second, these constrains simplify the computation of the sensitivity gradient, by reducing its computational cost. To compute the sensitivity gradient, the objective functional can be defined as the average drag coefficient of the circular cylinder with a splitter plate. The control parameters are the length of the plate and the distance between it and the body, which known as gap. The search direction and the variation of geometry can be obtained by the relationship between the solutions to the flow and the adjoint equations. This final report shows the literature survey, the results of trial and error method and the formulation and one result of adjoint equations based on the incompressible NavierStokes equations.

Problemas de otimização se fazem cada vez mais presentes nos mais diversos ramos da Engenharia. Encontrar configurações ótimas para um determinado problema significa, por exemplo, melhorar desempenho, reduzir custos entre outros ganhos. Existem hoje diversas maneiras de atacar um problema de otimização, cada qual com suas particularidades, vantagens e desvantagens. Dentre os métodos de otimização que utilizam gradientes de sensibilidade, o cálculo numérico dos mesmos consiste em uma importante etapa do projeto que, dependendo do problema, pode acarretar em custos computacionais muito elevados inviabilizando a abordagem escolhida. Este trabalho visa desenvolver e apresentar uma nova metodologia para o cálculo desses gradientes de sensibilidade, com base no Método Adjunto. O Método Adjunto é um método amplamente estudado e com diversas aplicações principalmente em Engenharia Aeronáutica. Nesse trabalho, todo o conhecimento prévio é utilizado para a derivação do método para aplicá-lo a escoamentos viscosos e incompressíveis. É desenvolvido também o cálculo do gradiente de sensibilidade com respeito a parâmetros geométricos e não geométricos. Para validar a metodologia proposta são feitas simulações numéricas das equações governantes do escoamento e adjuntas utilizando dois códigos computacionais distintos, SEMTEX e FreeFem++, o primeiro baseado no Método dos Elementos Espectrais e o segundo no Método dos Elementos Finitos, mostrando assim a independência do Método Adjunto na sua formulação contínua em relação a métodos computacionais. Para a validação são cujos gradientes possam ser calculados de outras formas permitindo comparações para calibrar e aperfeiçoar o cálculo do gradiente de sensibilidade. / Optimization problems are widely present in differents fields of Engineering. Finding optimal configurations in a problem means, for example, improving performance, reducing costs, among other achievements. There are several wellknown ways to tackle an optimization problem, each one has its own advantages and disadvantages. Considering the gradient-based optimization methods, the step of their numerical calculation is extremely important, as it may result in huge computational costs, thus making the chosen method impracticable. This work aims to develop and present a new methodology to compute these sensitivity gradients based on the Adjoint Method. The Adjoint Method is a widely studied method with several applications chiefly in A eronautical Engineering. In the present work, all the previous knowledge will be used to derive the equations of the method in order to apply them to viscous incompressible flows. The calculation of the sensitivity gradient, with respect to both geometric and non-geometric paramatersm will be developed as well. To validate the proposed methodology, numerical simulations of the governing and adjoint equations are carried out, using two computational codes called SEMTEX and FreeFem++, the former is based on the Spectral Element Method and the later, on the Finite Element Method, thus showing that the Adjoint Method, in its continuous formulation, is independent of the particular numerical method that is used. In order to validate the algorithm, simple problems are chosen, for which the gradients can be computed by other methods. This choice admits comparison between numerical values of gradients in order to calibrate and improve the methodology proposed.