This thesis is devoted to the numerical investigation of mechanical behavior of Dual phase (DP) steels. Such grade of advanced high strength steels (AHSS) is favorable to the automotive industry due the unique properties such as high strength and ductility with low finished cost. Many experimental and numerical studies have been done to achieve the optimized behavior of DP steels by controlling their microstructure. Experiments are costly and time consuming so in recent years numerical tools are utilized to help the metallurgist before doing experiments. Most of the numerical studies are based on classical (local) constitutive models where no material length scale parameters are incorporated in the model. Although these models are proved to be very effective in modeling the material behavior in the large scales but they fail to address some critical phenomena which are important for our goals. First, they fail to address the size effect phenomena which materials show at microstructural scale. This means that materials show stronger behavior at small scales compared to large scales. Another issue with classical models is the mesh size dependency in modeling the softening behavior of materials. This means that in the finite element context (FEM) the results will be mesh size dependent and no converged solution exist upon mesh refinement. Thereby by applying the classical (local) models one my loose the accuracy on measuring the strength and ductility of DP steels. Among the non-classical (nonlocal) models, gradient-enhanced plasticity models which consider the effect of neighboring point on the behavior of one specific point are proved to be numerically effective and versatile tools to accomplish the two concerns mentioned above. So in this thesis a gradient-enhanced plasticity model which incorporates both the energetic and dissipative material length scales is derived based on the laws of thermodynamics. This model also has a consistent yield-like function for the interface which is an essential part of the higher-order gradient theories. The main issue with utilizing these theories is the implementation which limits the application of these theories for modeling the real problems. Here a straightforward implementation method based on the classical FEM and Meshless method will be proposed which due to its simplicity it can be applied for many problems. The application of the developed model and implementation will be shown on removing the mesh size dependency and capturing the size effect in microstructure level of dual phase steels.

L'objet de cette thèse est l'étude de deux exemples de propagation pour des équations de réaction-diffusion hétérogènes.Le but de la première partie est de déterminer quels sont les effets d'échanges non locaux entre une ligne de diffusion rapide et un environnement bidimensionnel dans lequel a lieu un phénomène de réaction-diffusion de type KPP usuel. Dans le premier chapitre nous étudions comment ce couplage non local entre la ligne et le plan accélère la propagation dans la direction de la ligne ; on détermine aussi comment différentes fonctions d'échanges maximisent ou non la vitesse d'invasion. Le deuxième chapitre est consacré à la limite singulière de termes d'échanges qui convergent vers des masses de Dirac. On montre alors que la dynamique converge avec une certaine uniformité. Dans le troisième chapitre nous étudions la limite d'échanges étalés à l'infini. Ils permettent de donner un infimum sur la vitesse de propagation pour ce type de modèle qui peut cependant être supérieure à la vitesse KPP usuelle.La seconde partie de cette thèse est consacrée à l'étude de solutions entières (ou éternelles) pour des équations bistables hétérogènes. On considère un domaine bidimensionnel infini dans une direction, borné dans l'autre, qui converge vers un cylindre quand x tend vers moins l'infini. On montre alors l'existence d'une solution entière dans un tel domaine qui est égal à l'onde bistable en t tend vers moins l'infini. Cela nous conduit à étudier un modèle unidimensionnel avec un terme de réaction hétérogène, pour lequel on obtient le même résultat. / The aim of this thesis is to study two examples of propagation phenomena in heterogeneous reaction-diffusion equations.The purpose of the first part is to understand the effect of nonlocal exchanges between a line of fast diffusion and a two dimensional environment in which reaction-diffusion of KPP type occurs. The initial model was introduced in 2013 by Berestycki, Roquejoffre, and Rossi. In the first chapter we investigate how the nonlocal coupling between the line and the plane enhances the spreading in the direction of the line; we also investigate how different exchange functions may maximize or not the spreading speed.The second chapter is concerned with the singular limit of nonlocal exchanges that tend to Dirac masses. We show the convergence of the dynamics in a rather strong sense. In the third chapter we study the limit of long range exchanges with constant mass. It gives an infimum for the asymptotic speed of spreading for these models that still could be bigger than the usual KPP spreading speed.The second part of this thesis is concerned with entire solutions for heterogeneous bistable equations.We consider a two dimensional domain infinite in one direction, bounded in the other, that converges to a cylinder as x goes to minus infinity. We prove the existence of an entire solution in such a domain which is the bistable wave for t tends to minus infinity. It also lead us to investigate a one dimensional model with a non-homogeneous reaction term,for which we prove the same property.

abstract: Fracture phenomena have been extensively studied in the last several decades. Continuum mechanics-based approaches, such as finite element methods and extended finite element methods, are widely used for fracture simulation. One well-known issue of these approaches is the stress singularity resulted from the spatial discontinuity at the crack tip/front. The requirement of guiding criteria for various cracking behaviors, such as initiation, propagation, and branching, also poses some challenges. Comparing to the continuum based formulation, the discrete approaches, such as lattice spring method, discrete element method, and peridynamics, have certain advantages when modeling various fracture problems due to their intrinsic characteristics in modeling discontinuities. A novel, alternative, and systematic framework based on a nonlocal lattice particle model is proposed in this study. The uniqueness of the proposed model is the inclusion of both pair-wise local and multi-body nonlocal potentials in the formulation. First, the basic ideas of the proposed framework for 2D isotropic solid are presented. Derivations for triangular and square lattice structure are discussed in detail. Both mechanical deformation and fracture process are simulated and model verification and validation are performed with existing analytical solutions and experimental observations. Following this, the extension to general 3D isotropic solids based on the proposed local and nonlocal potentials is given. Three cubic lattice structures are discussed in detail. Failure predictions using the 3D simulation are compared with experimental testing results and very good agreement is observed. Next, a lattice rotation scheme is proposed to account for the material orientation in modeling anisotropic solids. The consistency and difference compared to the classical material tangent stiffness transformation method are discussed in detail. The implicit and explicit solution methods for the proposed lattice particle model are also discussed. Finally, some conclusions and discussions based on the current study are drawn at the end. / Dissertation/Thesis / Doctoral Dissertation Mechanical Engineering 2015

O sistema de FitzHugh-Nagumo possui um tipo especial de solução chamadas ondas viajantes, que são da forma &micro(x,t)=&oslash(x+ct) e w(x,t)=&#1137(x+ct) e além disso sabe-se que ela é estável. Tem-se o interesse de obter uma caracterização de seu perfil (&oslash,&#1137) e sua velocidade de propagação c. Fazendo uma mudança de variáveis, transformamos tal problema em encontrar equilíbrios de uma equação não local. Esta equação não local possui uma onda viajante de velocidade zero cujo perfil é o mesmo da equação original e, com esta equação, é possível aproximar, ao mesmo tempo, o perfil e a velocidade da onda viajante. Como a intenção é usar métodos numéricos para aproximar tais soluções, o problema não local foi analisado em um intervalo limitado verificando a existência e algumas propriedades espectrais em domínios limitados. / The FitzHugh-Nagumo systems have a special kind of solution named traveling wave, which has a form &micro(x,t)=&oslash(x+ct) and w(x,t)=&#1137(x+ct) and furthermore it is a stable solution. It is our interest to obtain a characterization of its profile (&oslash,&#1137) and speed of propagation c. Changing variables, we transform the problem of finding these solutions in the problem of finding an equilibria in a nonlocal equation. This nonlocal equation has a traveling wave with zero speed whose profile is the same of the original equation, and the nonlocal equation is used to approximate the profile and speed of the traveling wave at the same time. To use numerical methods for approximating such solutions, the nonlocal problem was analyzed in a finite interval to check that the existence and some spectral properties on bounded domains.

Dans cette thèse, nous étudions différentes structures de végétation issues de l’auto-organisation spatiale. Ce phénomène est visible dans des zones (semi-)arides où le potentiel d’évaporotranspiration dépasse sensiblement la moyenne des précipitations annuelles. Ce déficit hydrique freine le développement des plantes individuelles et, au niveau communautaire, stimule des comportements de « clustering » même si la topographie est isotrope. Dans ce contexte, nous adoptons une approche basée sur l’équation F-KPP non-locale permettant de formuler ces hypothèses en termes de propriétés des plantes individuelles.Une partie importante de cette thèse concerne l’étude d’un exemple de structure de végétation localisée bien connu dans la littérature, les cercles de fées. Les cercles de fées ont été découverts dans le désert de Namibie. Cependant, ces dernières années, ils ont aussi été observés en Australie. Plusieurs hypothèses ont été proposées dans la littérature. Nous proposons la compétition non-locale forte entre plantes individuelles (en utilisant un noyau de type Lorentzien) comme ingrédient principal pour expliquer la formation des cercles de fées. Le couplage non-local fort influence l’interaction entre fronts dans le régime bistable (loin de toute forme d’instabilité briseuse de symétrie). Dans le cas d’un couplage non-local faible, par exemple dans un noyau Gaussien, l’interaction entre fronts est toujours attractive. Par conséquent, les structures localisées qui résultent de l’interaction des fronts sont instables. Le couplage non-local fort peut induire la stabilisation de structures localisées que nous interprétons comme étant des cercles de fées. Notre mécanisme permet d’expliquer les principales caractéristiques des cercles de fées, comme la relation entre leur diamètre et la disponibilité des ressources. De plus, nous avons appliqué ces résultats à d’autres modèles de végétation. Nos résultats concordent avec les observations sur le terrain.Nous avons analysé la formation de « spots » de végétation dans la région Andine en Bolivie. Nous avons étudié comment un modèle standard d’interaction-redistribution génère des « spots », de longueur d’onde d'approximativement 1.36m, via une instabilité qui brise la symétrie. En considérant des paramètres réalistes, nos résultats concordent avec les observations sur le terrain.Enfin, nous avons étudié la formation de structure en forme de spirale dans un système qui couple la végétation et les herbivores dans un modèle proie-prédateur. Nous avons trouvé que le mécanisme qui induit la formation de spirales est l’excitabilité. Nos observations sur le terrain et nos résultats numériques du modèle montrent que les spirales de végétation ont une profondeur de quelques centimètres et une longueur de quelques mètres. En ce qui concerne l'échelle de temps, nos estimations donnent une période de rotation de l’ordre de 10 ans. / Option Physique du Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Image inpainting is the process of recovering missing or deteriorated data within the digital images and videos in a plausible way. It has become an important topic in the area of image processing, which leads to the understanding of the textural and structural information within the images. Image inpainting has many different applications, such as image/video restoration, text/object removal, texture synthesis, and transmission error concealment. In recent years, many algorithms have been developed to solve the image inpainting problem, which can be roughly grouped into four categories, partial differential equation-based inpainting, exemplar-based inpainting, transform domain inpainting, and hybrid image inpainting. However, the existing algorithms do not work well when the missing region to be inpainted is large, and when there are textural and structural information needed to be recovered. To address this inpainting problem, we propose multiple algorithms, 1) perceptually aware image inpainting based on the perceptual-fidelity aware mean squared error metric, 2) image inpainting using nonlocal texture matching and nonlinear filtering, and 3) multiresolution exemplar-based image inpainting. The experimental results show that our proposed algorithms outperform other existing algorithms with respect to both qualitative analysis and observer studies when inpainting the missing regions of images.

Low-energy resonant processes in collisions of electrons, atoms, ions and molecules significantly contributed to the evolution of the early Universe. Much attention has not yet been paid to processes involving lithium atoms and ions. In this thesis, we present the theoretical description of two associa- tive detachment processes of Li with H− and H with Li− within the nonlocal resonant theory. The nonlocal resonant models were constructed from poten- tial energy curves computed by the MOLPRO package of ab initio programs and from electron-molecule scattering data obtained from R-matrix calcula- tions by the UK molecular R-matrix suite of codes. The Lippman-Schwinger equation describing the nuclear motion was solved by the Schwinger-Lanczos algorithm. We developed a new method, which is based on the singular value decomposition method and separates the coupling potential. We predict sev- eral orders of magnitude difference between the temperature-dependent rate constants of the studied collisions at temperatures below 1000 K.

In this dissertation, a novel fast modeling technique called peri-ultrasound that can model both linear and nonlinear ultrasonic behavior of materials is developed and implemented. Nonlinear ultrasonic response can detect even very small material non- linearity. Quantification of the material nonlinearity at the early stages of damage is important to avoid catastrophic failure and reduce repair costs. The developed model uses the nonlocal continuum-based peridynamic theory which was found to be a good simulation tool for handling crack propagation modeling, in particular when multiple cracks grow simultaneously. The developed peri-ultrasound modeling tool has been used to model the ultrasonic response at the interface of two materials in presence of an interface crack. Also, the stress wave propagation in a half-space (or half-plane for a 2-dimensional problem) with boundary loading is investigated using peri-ultrasound modeling. In another simulation, well-established two-dimensional Lamb's problem is investigated where the results are verified against available analytical solution. Also, the interaction between the surface wave and a surface breaking crack is studied.

Les matériaux polymères sont largement utilisés pour des applications structurelles dans le secteur automobile et leurs comportements complexes nécessitent des modèles précis pour la simulation éléments finis. Les polymères possèdent un comportement dépendant du temps et de la vitesse. La dépendance à la vitesse peut être observée par un accroissement de la rigidité et de la limite élastique en fonction de la vitesse de déformation. Le long temps nécessaire pour retrouver des contraintes nulles après sollicitation du matériau met en évidence la dépendance du temps sur le comportement. De plus, particulièrement pour les polymères chargés, le phénomène de cavitation se traduisant par la création et la croissance de micro-cavités et de microfissures conduit à un changement de volume durant la déformation. Dans ce travail, un modèle de comportement est développé pour un polymère semi-cristallin chargé de talc utilisé dans l’industrie automobile. Un modèle constitutif viscoélastique-viscoplastique non-associatif avec endommagement non-local est proposé dans le but de simuler les phénomènes observés expérimentalement. Dans le modèle développé, une surface de charge non symétrique est utilisée pour prendre en compte la pression hydrostatique. La viscoplasticité non-associative couplée avec l’endommagement conduit aux déformations viscoplastiques non-isochoriques caractérisées expérimentalement. Les paramètres du modèle proviennent d’essais expérimentaux réalisés sous différentes conditions et `a différentes vitesses de déformation. Pour ces essais, plusieurs techniques de mesure, telles que la corrélation d’images et l’extensommetrie optique sont utilisées pour les mesures de champs de déplacements. La bonne corrélation entre les données expérimentales et les simulations numériques mettent en évidence la précision du modèle développé afin de modéliser le comportement des matériaux polymères semi-cristallins. / Polymer materials are widely used for structural applications in the automotive sector and their behaviours are complex and require accurate models for finite element simulations. Polymer materials exhibit rate and time dependent behaviours. The rate dependency can be observed by an increase of the stiffness and the yield stress at increasing strain rate. The long time to recover the zero stress after solicitation of the material highlight the time dependent behaviour. Furthermore, particularly for filled polymers, the cavitation phenomenon cause the creation and growth of micro-voids and microcracks called damage and leads to volume change during the deformation. In this work, a behaviourmodel for mineral filled semi-crystalline polymer used in automotive industry is developed. A constitutive viscoelastic-viscoplastic non-associated model coupled with nonlocal damage is proposed in order to simulate the phenomena observed experimentally. In the constitutive model, a non symmetric yield surface is used to take the hydrostatic pressure into account. The non associated viscoplasticity coupled with damage leads to the non-isochoric viscoplastic deformation characterised experimentally. The material parameters arise from experimental tests carried out under various loadings and strain rates. For these experimental tests, different measurement techniques like Digital Image Correlation and optical extensometry are used for the displacements and the strain field measurements. The good agreement between the experimental data and the numerical simulations highlights the accuracy of the developed model for polymer modelling.

In this thesis, we study several nonlocal models to obtain their numerical solutions accurately and efficiently. In contrast to the classical (local) partial differential equation models, these nonlocal models are integro-differential equations that do not contain spatial derivatives. As a result, these nonlocal models allow their solutions to have discontinuities. Hence, they can be widely used for fracture problems and anisotropic problems. This thesis mainly includes two parts. The first part focuses on presenting accurate and efficient numerical methods. In this part, we first introduce three meshless methods including two global schemes, namely the radial basis functions collocation method (RBFCM) and the radial ba- sis functions-based pseudo-spectral method (RBF-PSM) and a localized scheme, namely the localized radial basis functions-based pseudo-spectral method (LRBF-PSM), which also gives the development process of the RBF methods from global to local. The comparison of these methods shows that LRBF-PSM not only avoids the Runge phenomenon but also has similar accuracy to the global scheme. Since the LRBF-PSM uses only a small subset of points, the calculation consumes less CPU time. Afterwards, we improve this scheme by adding enrichment functions so that it can be effectively applied to discontinuity problems. This thesis abbreviates this enriched method as LERBF-PSM (Localized enriched radial basis functions-based pseudo-spectral method). In the second part, we focus on applying the derived methods from the first part to nonlocal topics of current research, including nonlocal diffusion models, linear peridynamic models, parabolic/hyperbolic nonlocal phase field models, and nonlocal nonlinear Schrödinger equations arising in quantum mechanics. The first point worth noting is that in order to verify the meshless nature of LRBF-PSM, we apply this method to solve a two-dimensional steady-state continuous peridynamic model in regular, irregular (L-shaped and Y-shaped) domains with uniform and non-uniform discretizations and even extend this method to three dimensions. It is also worth noting that before solving nonlinear nonlocal Schrödinger equations, according to the property of the convolution, these partial integro-differential equations are transformed into equivalent or approximate partial differential equations (PDEs) in the whole space and then the LRBF-PSM is used for the spatial discretization in a finite domain with suitable boundary conditions. Therefore, the solutions can be quickly approximated.