A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework.

In this thesis, Initial Value Problem of Level-set Method is applied to solid propellant combustion to find the grain burnback. For the performance prediction of the rocket motor, 0-D, 1-D or 3-D flow models are used depending on the type of thre grain configuration.

Numerical simulation for Resin Transfer Molding (RTM) manufacturing process is attempted by using the eXtended Finite Element Method (XFEM) combined with the level set method. XFEM allows to obtaining a good numerical precision of the pressure near the resin flow front, where its gradient is discontinuous. The enriched shape functions of XFEM are derived by using the level set values so as to correctly describe the interpolation with the resin flow front. In addition, the level set method is used to transport the resin flow front at each time step during the mold filling. The level set values are calculated by an implicit characteristic Galerkin FEM. The multi-frontal solver of IPSAP is adopted to solve the system. This work is validated by comparing the obtained results with analytic solutions.Moreover, a localization method of XFEM and level set method is proposed to increase the computing efficiency. The computation domain is reduced to the small region near the resin flow front. Therefore, the total computing time is strongly reduced by it. The efficiency test is made with simple channel or radial flow models. Several application examples are analyzed to demonstrate ability of this method. A wind turbine blade is also treated as industrial application. Finally, a Graphic User Interface (GUI) tool is developed so as to make easy the pre/post-processing of the simulation.

Wildfire hazard and its destructive consequences have become a growing issue around the world especially in the context of global warming. An effective and efficient fire simulation model will make it possible to predict the fire spread and assist firefighters in the process of controlling the damage and containing the fire area. Simulating wildfire spread remains challenging due to the complexity of fire behaviors. The raster-based method and the vector-based method are two major approaches that allow one to perform computerized fire spread simulation. In this thesis, we present a scheme we have developed that utilizes a level set method to build a fire spread simulation model. The scheme applies the strengths and overcomes some of the shortcomings of the two major types of simulation method. We store fire data and local rules at cells. Instead of calculating which are the next ignition points cell by cell, we apply Huygens' principle and elliptical spread assumption to calculate the direction and distance of the expanding fire by the level set method. The advantage to storing data at cells is that it makes our simulation model more suitable for heterogeneous fuel and complex topographic environment. Using a level set method for our simulation model makes it possible to overcome the crossover problem. Another strength of the level set method is its continuous data processing. Applying the level set method in the simulation models, we need fewer vector points than raster cells to produce a more realistic fire shape. We demonstrate this fire simulation model through two implementations using narrow band level set method and fast marching method. The simulated results are compared to the real fire image data generated from Troy and Colina fires. The simulation data are then studied and compared. The ultimate goal is to apply this simulation model to the broader picture to better predict different types of fires such as crown fire, spotting fires, etc.

A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework.

We present a level-set algorithm for topology optimisation and demonstrate its capabilities and advantages in a variety of settings. The algorithm uses discrete element densities so that interpolation schemes are avoided and the boundary of the design is always well defined. A review of the level-set method for topology optimisation, and a description of the mathematical concepts behind the level-set algorithm are given in the introductory chapters. A compact Matlab implementation of the algorithm provides explicit implementation details for the simple example of compliance minimisation with a volume constraint. The remainder of the thesis presents original results obtained using the level-set algorithm. As a new application, we use topology optimisation to maximise fracture resistance. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension. We develop a suitable fracture resistance objective functional, derive its shape derivative and apply the level-set algorithm to simple examples. Topology optimisation methods that involve intermediate density elements are not suitable to solve this problem because the boundary of the design is not well defined. Our results indicate that the algorithm correctly optimises for fracture resistance. As the method is computationally intensive, we suggest simpler objective functionals that could be used as a proxy for fracture resistance. For example, a perimeter penalty could be added to the compliance objective functional in conjunction with a non-linear elasticity law where the Young's modulus in tension is lower than in compression. The level-set method has only recently been applied to fluid flow problems. We utilise the level-set algorithm to minimise energy dissipation in Stokes flows in both two and three dimensions. The discrete element densities allow the no-slip boundary condition to be applied directly. The Stokes equations therefore need only be solved in the fluid region of the design: this results in significant computational savings compared to conventional material distribution approaches. In order to quantify the computational savings the optimisation problems are resolved using an interpolation scheme to simulate the no-slip boundary condition. This significant advantage of the level-set method for fluid flow problems has not been noted by other authors. The algorithm produces results consistent with those obtained by other topology optimisation approaches, and solves large-scale three dimensional problems with modest computational cost. The first examples of three dimensional periodic microstructure design with the level-set method are presented in this thesis. The level-set algorithm is extended to deal with multiple constraints. This is needed so that materials can be designed with symmetry requirements imposed on their effective properties. To demonstrate the capabilities of the approach, unit cells are designed separately to maximise conductivity and bulk modulus with an isotropy requirement. The resulting materials have properties very close to the relevant Hashin-Shtrikman bounds. The algorithm is then applied to multifunctional material design: unit cells are designed to give isotropic materials that have maximum bulk modulus and maximum conductivity. Cross-property bounds indicate the near-optimality of the microstructures obtained. The design space of the problem is extensively explored with different coefficients of the conductivity and bulk modulus in the objective and different volume constraints. We hypothesise the existence of theoretically optimal single-scale microstructures with the topologies of the computationally optimised microstructures we have found. Structures derived from the Schwartz primitive (P) and diamond (D) minimal surfaces have previously been presented as good multifunctional composites. These structures are elastically anisotropic. Although they have similar conductivity, they have stiffness properties inferior to several of the isotropic optimised microstructures.

We present a level-set algorithm for topology optimisation and demonstrate its capabilities and advantages in a variety of settings. The algorithm uses discrete element densities so that interpolation schemes are avoided and the boundary of the design is always well defined. A review of the level-set method for topology optimisation, and a description of the mathematical concepts behind the level-set algorithm are given in the introductory chapters. A compact Matlab implementation of the algorithm provides explicit implementation details for the simple example of compliance minimisation with a volume constraint. The remainder of the thesis presents original results obtained using the level-set algorithm. As a new application, we use topology optimisation to maximise fracture resistance. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension. We develop a suitable fracture resistance objective functional, derive its shape derivative and apply the level-set algorithm to simple examples. Topology optimisation methods that involve intermediate density elements are not suitable to solve this problem because the boundary of the design is not well defined. Our results indicate that the algorithm correctly optimises for fracture resistance. As the method is computationally intensive, we suggest simpler objective functionals that could be used as a proxy for fracture resistance. For example, a perimeter penalty could be added to the compliance objective functional in conjunction with a non-linear elasticity law where the Young's modulus in tension is lower than in compression. The level-set method has only recently been applied to fluid flow problems. We utilise the level-set algorithm to minimise energy dissipation in Stokes flows in both two and three dimensions. The discrete element densities allow the no-slip boundary condition to be applied directly. The Stokes equations therefore need only be solved in the fluid region of the design: this results in significant computational savings compared to conventional material distribution approaches. In order to quantify the computational savings the optimisation problems are resolved using an interpolation scheme to simulate the no-slip boundary condition. This significant advantage of the level-set method for fluid flow problems has not been noted by other authors. The algorithm produces results consistent with those obtained by other topology optimisation approaches, and solves large-scale three dimensional problems with modest computational cost. The first examples of three dimensional periodic microstructure design with the level-set method are presented in this thesis. The level-set algorithm is extended to deal with multiple constraints. This is needed so that materials can be designed with symmetry requirements imposed on their effective properties. To demonstrate the capabilities of the approach, unit cells are designed separately to maximise conductivity and bulk modulus with an isotropy requirement. The resulting materials have properties very close to the relevant Hashin-Shtrikman bounds. The algorithm is then applied to multifunctional material design: unit cells are designed to give isotropic materials that have maximum bulk modulus and maximum conductivity. Cross-property bounds indicate the near-optimality of the microstructures obtained. The design space of the problem is extensively explored with different coefficients of the conductivity and bulk modulus in the objective and different volume constraints. We hypothesise the existence of theoretically optimal single-scale microstructures with the topologies of the computationally optimised microstructures we have found. Structures derived from the Schwartz primitive (P) and diamond (D) minimal surfaces have previously been presented as good multifunctional composites. These structures are elastically anisotropic. Although they have similar conductivity, they have stiffness properties inferior to several of the isotropic optimised microstructures.

Dans ce travail, nous nous sommes intéressés à la simulation numérique de suspensions denses d'objets immergés dans un fluide. En s'inspirant d'une méthode de segmentation d'image, nous avons développé une méthode efficace de capture d'interface permettant d'une part de localiser les structures immergées et d'autre part de gérer les contacts numériques entre les structures.Le domaine fluide/structure est représenté à l'aide de trois fonctions labels et deux fonctions distances qui permettent de localiser chaque structure et son plus proche voisin.Les interfaces sont capturées par une seule fonction level set, celle-ci est ensuite transportée par la vitesse du fluide ou par la vitesse de chaque structure. Un algorithme de multi-label fast marching permet de réinitialiser à chaque pas de temps les fonctions labels et distances dans un périmètre proche des interfaces.La gestion des contacts numériques est effectuée grâce à une force répulsive à courte portée prenant en compte l'interaction entre les objets les plus proches.Dans un premier temps, la méthode est appliquée à l'évolution de solides rigides immergés.Un modèle de pénalisation global couplé aux fonctions labels permet de calculer en une seule fois l'ensemble des vitesses des structures rigides. Les résultats obtenus montrent l'efficacité de la méthode à gérer un grand nombre de solides.Nous avons ensuite appliqué la méthode des suspensions de vésicules immergées. Ce type de simulation requiert le calcul des forces élastiques et de courbures exercées sur les membranes. Grâce au modèle proposé, seulement une force élastique et une force de courbure sont calculées pour l'ensemble des membranes à l'aide de la fonction level set et des fonctions labels. / In this work, we propose a method to efficiently capture an arbitrary number of fluid/solid or fluid/fluid interfaces, in a level-set framework. This technique, borrowed from image analysis, is introduced in the context of the interaction of several bodies immersed in a fluid. A configuration of the bodies in the fluid/structure domain is described by three label maps providing the first and second neighbours, and their associated distance functions. Only one level set function captures the union of all interfaces and is transported with the fluid velocity or with a global velocity field which takes into account the velocity of each structure. A multi-label fast marching method is then performed in a narrow-band around the interfaces allowing to update the label and distance functions. Within this framework, the numerical treatment of contacts between the structures is achieved by a short-range repulsive force depending on the distance between the closest bodies.The method is validated through the simulation of a dense suspension of rigid bodies immersed in an incompressible fluid. A global penalization model uses the label maps to follow the solid bodies altogether without a separate computation of each body velocity. Consequently, the method shows its efficiency when dealing with a large number of rigid bodies. We also investigate the numerical simulation of vesicle suspensions for which a computation of elastic and bending forces on membranes is required. In the present model, only one elastic and bending force is computed for the whole set of membranes according to the level set function and the label maps.

Les méthodes d'optimisation de forme s’industrialisent progressivement, elles permettent la conception automatisée de structures aux propriétés optimales. Elles constituent aussi un outil d'exploration majeur pour la conception de nouveaux matériaux.Dans une première partie nous utilisons ces méthodes afin de générer des matériaux architecturés aux propriétés thermoélastiques effectives cibles et extrêmes. En plus de proposer différentes solutions, nous répertorions les différents mécanismes œuvrant au contrôle des ces propriétés. Dans ce contexte nous proposons aussi de prendre en compte l'influence des interfaces comportant un gradient de propriétés sur les architectures obtenues.Nous étudions ensuite les procédés de fabrication pouvant être utilisés afin de réaliser ces matériaux. Les méthodes de fabrication additive, considérées comme le vecteur d'une prochaine révolution industrielle, constituent une piste que nous considérerons tout particulièrement. Nous proposons plusieurs solutions pour prendre en compte les limitations et les effets collatéraux de ces procédés de fabrication au sein de processus d'optimisation de forme. Nous traitons le problème de la prise en compte des propriétés induites par la méthode de fabrication Fiber Deposition Molding (FDM), à savoir des propriétés anisotropes orientées. Nous proposons ensuite une approche pour traiter le problème des dépôts en porte-à-faux à l'aide d'un critère mécanique.Enfin, nous abordons la prise en compte des non-linéarités géométriques au sein de calculs d'optimisation de forme et discutons de leurs apports ainsi que de leurs limitations. Nous présentons plusieurs applications pour la conception automatisée d'actuateurs non linéaires. / Shape optimization methods are promising methods and are gradually becoming industrialized. They provide the ability to automatically design structures with optimal behavior. They are outstanding tools for exploration and design of new materials.We use these methods to generate architectured multi-phased materials with prescribed thermoelastic properties. We first propose several solutions and we classify them by the mechanisms they rely on in order to control the effective properties. We also propose to evaluate the influence of an interface with a gradient of properties on the obtained architectures.Eventually we focus on the plausible manufacturing solution to produce our architectured materials. In this context, additive manufacturing methods (often considered as the support of an incoming industrial revolution) is our main option. We introduce several strategies to circumvent some limitations and side effects of these manufacturing methods during optimization process. We particularly focus on Fiber Deposition Molding, which induce an important mechanical anisotropy in processed parts. Then we consider the problem of overhangings features in design and propose a way to handle them prior to additive manufacturing using a mechanical criteria.Finally we take into account geometrical non linearities in optimization process. We highlight the pros and cons of this new modeling by presenting several applications of non linear actuators design.

Nesse trabalho, é investigada a combustão pré-misturada de metano e ar em um queimador poroso radiante com área de seção transversal variável. A estabilidade de chama e a eficiência de radiação neste tipo de queimador são analisadas através de simulação numérica. O problema considerado para análise é de um queimador unidimensional de área variável com perdas de calor por radiação em suas extremidades. A metodologia numérica usada para resolver o conjunto de equações de conservação é o método dos volumes finitos com o sistema de coordenadas cartesianas. Os resultados mostram que, o aumento da área da seção transversal proporciona um aumento na faixa de estabilidade de chama do queimador quando comparado com um de área constante. A eficiência de radiação também é influenciada positivamente pela variação de área, ou seja, um aumento na área da seção de saída do dispositivo resulta em um aumento na eficiência. Também é testada uma modelagem alternativa baseada no método de curvas de nível (level-set method). Nesse modelo as equações das espécies químicas são substituídas pela equação-G, a qual descreve a dinâmica de uma frente de chama infinitamente fina. Os resultados numéricos das simulações bem como as vantagens e as limitações do modelo de curvas de nível são discutidas. Em linhas gerais os resultados do modelo de curvas de nível não foram capazes de reproduzir adequadamente o comportamento obtido com o modelo convencional. / In this work, the premixed combustion of methane and air in a porous radiant burner with a variable cross-sectional area is investigated. The flame stability and the radiant efficiency in this kind of burner are analyzed through numerical simulation. The problem considers a variable area one-dimensional burner with radiation heat losses in its extremities. The numerical method used to solve the set of conservation equations is the finite volume method with the Cartesian coordinate system. The results show that an increase of the cross-sectional area promotes an increase of the flame stability range when compared with a constant area burner. The radiant efficiency is also positively influenced by the area variation, i.e., the increase of the outlet area results in an enhanced efficiency. An alternative modelling based on the level-set method is also tested. In this model the equations of chemical species are replaced by the G-Equation, which describes the dynamics of an infinitely thin flame front. The numerical results of simulations as well as advantages and limitations of the level-set model are discussed. In general the results of the level-set model were not able to reproduce in a suitable way the behavior obtained with the conventional model.