We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P.

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How the patchiness of landscapes affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. In this thesis, we focus on how landscape composition and the movement behaviour of individuals at an interface between patches of different quality affect the steady state of a single species. We consider a model of reaction-diffusion equations for the temporal evolution of the density of the population in space. Individual movement is described by a diffusion process, e.g., an uncorrelated random walk. Population net growth is encapsulated in the growth function that considers birth and death of individuals, including nonlinear effects that arise from competition and/or facilitation within the species. We consider the simplest case of two adjacent one-dimensional patches, e.g., two intervals on the real line that share one boundary point. Conditions are homogeneous within a patch but differ between patches. The movement behaviour of individuals between the two patches is incorporated into matching conditions of population flux and density at the interface between patches, i.e., the boundary point that the intervals share. These matching conditions turn out to be continuous in the flux but discontinuous in the density. Several authors have studied similar models recently. Most of these studies consider monostable dynamics on both patches, i.e., logistic growth. Under logistic growth, the net population growth rate is a strictly decreasing function of population density. Logistic population dynamics are very simple: the population extinction state is unstable and a positive steady state is globally asymptotically stable. In this work, we also include bistable dynamics, i.e., an Allee effect. Biologically, an Allee effect occurs when individuals cooperate at some level so that the net population growth rate is increasing with population density for at least some low or intermediate densities. Models with Allee growth typically exhibit bistability: there are two locally stable steady states, one at low density (possibly zero) and one at high density. This bistability makes mathematical analysis more challenging, but leads to more interesting results in return. Mathematically, most existing work on related models is based on linear stability analysis of the extinction state. We focus on the nonlinear models and specifically on positive steady states. We establish the existence, uniqueness and - in some cases - global asymptotic stability of a positive steady state. We classify the shape of these states depending on movement behaviour. We clarify the role of movement in this context. In particular, we investigate the following prior observation: a randomly diffusing population at steady state in a continuously varying habitat can exceed its carrying capacity. Our results clarify when and under which conditions this effect can arise in our two-patch landscape. The analysis of the model with an Allee effect on one of the two patches yields a rich and interesting structure of steady states. Under certain parameter conditions, some of these states are amenable to explicit stability calculations. These yield insights into the possible bifurcations that can occur in our system. Finally, we indicate how the model and analysis here can be extended to systems of reaction-diffusion equations on graphs that represent natural habitats with different geometries, for example watersheds.

This dissertation investigates the development of numerical algorithms for coupling computational fluid dynamics (CFD) and computational solid dynamics (CSD) solvers, and the use of these solvers for simulating fluid-solid interaction (FSI) problems involving large deformation, shock waves, and multiphase flow. The dissertation consists of two parts. The first part investigates the use of Robin interface conditions to resolve the well-known numerical added-mass instability, which affects partitioned coupling procedures for solving problems with incompressible flow and strong added-mass effect. First, a one-parameter Robin interface condition is developed by linearly combining the conventional Dirichlet and Neumann interface conditions. Next, a numerical algorithm is developed to implement the Robin interface condition in an embedded boundary method for coupling a parallel, projection-based incompressible viscous flow solver with a nonlinear finite element solid solver. Both an analytical study and a numerical study reveal that the new algorithm can clearly outperform conventional Dirichlet-Neumann procedures in terms of both stability and accuracy, when the parameter value is carefully selected. Moreover, the studies also indicate that the optimal parameter value depends on the materials and geometry of the problem. Therefore, to efficiently solve FSI problems involving non-uniform structures, a generalized Robin interface condition is presented, in which the constant parameter is replaced by a spatially varying function that depends on the local material and geometric properties of the structure. Numerical experiments using two benchmark problems show that the spatially varying Robin interface condition can clearly improve numerical accuracy compared to the constant- parameter version with the same computational cost. The second part of this dissertation focuses on simulating complex FSI problems featuring shock waves, multiphase flow (e.g., bubbles), and shock-induced material damage and fracture. A recently developed three-dimensional computational framework is employed, which couples a multiphase, compressible CFD solver and a nonlinear finite element CSD solver using an embedded boundary method and a partitioned procedure. In particular, the CFD solver applies a level-set method to capture the evolution of the bubble surface, and the CSD solver utilizes a continuum damage mechanics model and an element erosion method to simulate the dynamic fracture of the material. Two computational studies are presented. The first one investigates the dynamic response and failure of a brittle material exposed to a prescribed shock wave. The predictive capability of the computational framework is first demonstrated by simulating a series of laboratory experiments in the context of shock wave lithotripsy. Then, a parametric study is conducted to elucidate the significant effects of the shock wave's profile on material damage. In the second study, the computational framework is applied to simulate shock-induced bubble collapse near various solid and soft materials. The reciprocal effect of the material's properties (e.g., acoustic impedance, Young's modulus) on bubble dynamics is discussed in detail. / Doctor of Philosophy / Numerical simulations that couple computational fluid dynamics (CFD) solvers and computational solid dynamics (CSD) solvers have been widely used in the solution of nonlinear fluid-solid interaction (FSI) problems underlying many engineering applications. This is primarily because they are based on partitioned solutions of fluid and solid subsystems, which facilitates the use of existing numerical methods and computational codes developed for each subsystem. The first part of this dissertation focuses on developing advanced numerical algorithms for coupling the two subsystems. The aim is to resolve a major numerical instability issue that occurs when solving problems involving incompressible, heavy fluids and thin, lightweight structures. Specifically, this work first presents a new coupling algorithm based on a one-parameter Robin interface condition. An embedded boundary method is developed to enforce the Robin interface condition, which can be advantageous in solving problems involving complex geometry and large deformation. The new coupling algorithm has been shown to significantly improve numerical stability when the constant parameter is carefully selected. Next, the constant parameter is generalized into a spatially varying function whose local value is determined by the local material and geometric properties of the structure. Numerical studies show that when solving FSI problems involving non-uniform structures, using this spatially varying Robin interface condition can outperform the constant-parameter version in both stability and accuracy under the same computational cost. In the second part of this dissertation, a recently developed three-dimensional multiphase CFD - CSD coupled solver is extended to simulate complex FSI problems featuring shock wave, bubbles, and material damage and fracture. The aim is to understand the material’s response to loading induced by a shock wave and the collapse of nearby bubbles, which is important for advancing the beneficial use of shock wave and bubble collapse for material modification. Two computational studies are presented. The first one investigates the dynamic response and failure of a brittle material exposed to a prescribed shock wave. The causal relationship between shock loading and material failure, and the effects of the shock wave’s profile on material damage are discussed. The second study investigates the shock-induced bubble collapse near various solid and soft materials. The two-way interaction between bubble dynamics and materials response, and the reciprocal effects of the material’s properties are discussed in detail.

La thèse porte sur la modélisation de la transformation de l'austénite en ferrite dans les aciers en mettant l'accent sur les conditions thermodynamiques et cinétiques aux interfaces alpha/gamma en cours de croissance de la ferrite. Dans une première partie, la thèse se concentre sur la description des équilibres thermodynamiques entre alpha et gamma à l'aide de la méthode CalPhad. Nous avons développé un nouvel algorithme hybride combinant la construction d'une enveloppe convexe avec la méthode classique de Newton-Raphson. Nous montrons ses possibilités pour des aciers ternaire Fe-C-Cr et quaternaire Fe-C-Cr-Mo dans des cas particulièrement difficiles. Dans un second chapitre, un modèle à interface épaisse a été développé. Il permet de prédire l'ensemble du spectre des conditions à l'interface alpha/gamma au cours de la croissance de la ferrite, de l'équilibre complet au paraéquilibre avec des cas intermédiaires des plus intéressants. Nous montrons que de nombreux régimes cinétiques particuliers dans les systèmes Fe-C-X peuvent être prévus avec un minimum de paramètres d'ajustement, principalement le rapport entre les diffusivités de l'élément substitutionnel dans l'interface épaisse et dans le volume d'austénite. Le troisième chapitre porte sur l'étude d'un modèle de champ de phase. Une analyse approfondie des conditions à l'interface données par le modèle est réalisée en utilisant la technique des développements asymptotiques. En utilisant les connaissances fournies par cette analyse, le rôle de la mobilité intrinsèque d'interface sur la cinétique et les régimes de croissance est étudié, à la fois dans le cas simple d'alliages binaires Fe-C et dans le cas plus complexe d'alliages Fe-C-Mn / Transformation in steels focusing on the thermodynamic and kinetics conditions at the alpha/gamma interfaces during the ferrite growth. The first chapter deals with the determination of thermodynamic equilibria between alpha and gamma with CalPhad thermodynamic description. We have developed a new hybrid algorithm combining the construction of a convex hull to the more classical Newton-Raphson method to compute two phase equilibria in multicomponent alloys with two sublattices. Its capabilities are demonstrated on ternary Fe-C-Cr and quaternary Fe-C-Cr-Mo steels. In the second chapter, we present a thick interface model aiming to predict the whole spectrum of conditions at an alpha/gamma interface during ferrite growth, from full equilibrium to paraequilibrium with intermediate cases as the most interesting feature. The model, despite its numerous simplifying assumptions to facilitate its numerical implementation, allows to predict some peculiar kinetics in Fe-C-X systems with a minimum of fitting parameters, mainly the ratio between the diffusivities of the substitutional element inside the thick interface and in bulk austenite. The third chapter deals with the phase field model of austenite to ferrite transformation in steels. A thorough analysis on the conditions at the interface has been performed using the technique of matched asymptotic expansions. Special attention is given to clarify the role of the interface mobility on the growth regimes both in simple Fe-C alloys and in more complex Fe-C-Mn alloys

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How patchy landscape affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behaviour to local habitat quality and show preferences for some habitat types over others. In this dissertation, we focus on how landscape composition and the movement behaviour at an interface between habitat patches of different quality affects the steady states of a single species and a predator-prey system. First, we consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction-diffusion equation. Several recent publications by other authors explored how individual movement behaviour affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. The movement between patches is incorporated into the interface conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We establish the existence, uniqueness and global asymptotic stability of the steady state, and we classify their qualitative shape depending on movement behaviour. We clarify the role of nonrandom movement in this context, and we apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. In particular, we apply our results to study the question of why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. Secondly, we model population dynamics with a predator-prey system in a coupled ecological reaction-diffusion equation in a heterogeneous landscape to study Turing patterns that emerge from diffusion-driven instability (DDI). We derive the DDI conditions, which consist of necessary and sufficient conditions for initiation of spatial patterns in a one-dimensional homogeneous landscape. We use a finite difference scheme method to numerically explore the general conditions using the May model, and we present numerical simulations to illustrate our results. Then we extend our studies on Turing-pattern formation by considering a predator-prey system on an infinite patchy periodic landscape. The movement between patches is incorporated into the interface conditions that link the reaction-diffusion equations between patches. We use a homogenization technique to obtain an analytically tractable approximate model and determine Turing-pattern formation conditions. We use numerical simulations to present our results from this approximation method for this model. With this tool, we then explore how differential movement and habitat preference of both species in this model (prey and predator) affect DDI.

Cette thèse vise à contribuer à l'adoption du virtual testing, pratique industrielle encore embryonnaire qui consistera à optimiser et certifier par la simulation numérique le dimensionnement de pièces industrielles critiques. Le virtual testing permettra des économies colossales dans la conception des pièces mécaniques et un plus grand respect de l'environnement grâce à des designs optimisés. Afin d'atteindre un tel objectif, de nouvelles méthodes de calcul doivent être mises en place, plus sûres, plus respectueuses des architectures matérielles, plus rapides, compatibles avec les contraintes temporelles de l'ingénierie. Nous nous intéressons à la résolution parallèle de problèmes non linéaires de grande taille par des méthodes de décomposition de domaine. Notre objectif est d'atteindre une approximation de la solution exacte en minimisant les communications entre les sous-domaines. Pour cela nous souhaitons maximiser les calculs réalisés indépendamment par sous-domaine à l'aide d'approches de relocalisation non linéaire, contrôler les critères de convergence des solveurs imbriqués de manière à éviter la sur-résolution et les divergences, améliorer la construction de conditions d'interface mixtes, et non linéariser l'étape de préconditionnement du solveur. L'objectif à terme étant de traiter des problèmes de complexité industrielle, la robustesse des méthodes sera un souci constant. De manière classique, les problèmes non linéaires sont résolus en construisant une suite de systèmes linéaires qui peuvent être résolus en parallèle à l'aide de méthodes itératives, telles que les solveurs de Krylov. Nous souhaitons remettre en question cette procédure usuelle en essayant de construire une suite de petits systèmes non linéaires indépendants à résoudre en parallèle. Une telle technique implique l'utilisation de solveurs itératifs imbriqués dont les critères de convergence doivent être syntonisés dynamiquement de manière à éviter à la fois la sur-résolution et la perte de convergence. La robustesse de la méthode pourra notamment être assurée par l'emploi de conditions d'interface mixtes bien construites et de préconditionneurs bien choisis. / This thesis is aimed to contribute to the adoption of virtual testing, an industrial practice still embryonic which consists in optimizing and certifying by numerical simulations the dimensioning of critical industrial structures. The virtual testing will allow colossal savings in the design of mechanical parts and a greater respect for the environment, thanks to optimized designs. In order to achieve this goal, new calculation methods must be implemented, satisfying more requirements concerning safety, respect for hardware architectures, fastness, and compatibility with the time constraints of engineering.We are interested in the parallel resolution of large nonlinear problems by domain decomposition methods. Our goal is to approximate the exact solution by minimizing communication between subdomains. In order to do this, we want to maximize the computations performed independently by subdomain, using nonlinear relocation approaches. We also try to control the convergence criteria of the nested solvers in order to avoid over-resolution and divergences, to improve the construction of conditions Of mixed interface, and non-linearizing the preconditioning step of the solver. The ultimate objective being to deal with problems of industrial complexity, the robustness of the methods we develop will be a constant concern.Conventionally, non-linear problems are solved by constructing a sequence of linear systems that can be solved in parallel using iterative methods, such as Krylov solvers. We wish to question this usual procedure by trying to construct a sequence of small independent nonlinear systems to be solved in parallel. Such a technique involves the use of interleaved iterative solvers, whose convergence criteria must be dynamically tuned in order to avoid both over-resolution and loss of convergence. The robustness of the method can be ensured in particular by the use of well-constructed mixed interface conditions and well-chosen preconditioners;

No description available.

Cette thèse développe des estimations d’erreur a posteriori et critères d’arrêt pour les méthodes de décomposition de domaine avec des conditions de transmission de Robin optimisées entre les interfaces. Différents problèmes sont considérés: l’équation de Darcy stationnaire puis l’équation de la chaleur, discrétisées par les éléments finis mixtes avec un schéma de Galerkin discontinu de plus bas degré en temps pour le second cas. Pour l’équation de la chaleur, une méthode de décomposition de domaine globale en temps, avec mêmes ou différents pas de temps entre les différents sous domaines, est utilisée. Ce travail est finalement étendu à un modèle diphasique en utilisant une méthode de volumes finis centrés par maille en espace. Pour chaque modèle, un problème d’interface est résolu itérativement, où chaque itération nécessite la résolution d’un problème local dans chaque sous-domaine, et les informations sont ensuite transmises aux sous-domaines voisins. Pour les modèles instationnaires, les problèmes locaux dans les sous-domaines sont instationnaires et les données sont transmises par l’interface espace-temps. L’objectif de ce travail est, pour chaque modèle, de borner l’erreur entre la solution exacte et la solution approchée à chaque itération de l’algorithme de décomposition de domaine. Différentes composantes d’erreur en jeu de la méthode sont identifiées, dont celle de l’algorithme de décomposition de domaine, de façon à définir un critère d’arrêt efficace pour cette méthode. En particulier, pour l’équation de Darcy stationnaire, on bornera l’erreur par un estimateur de décomposition de domaine ainsi qu’un estimateur de discrétisation en espace. On ajoutera à la borne de l’erreur un estimateur de discrétisation en temps pour l’équation de la chaleur et pour le modèle diphasique. L’estimation a posteriori répose sur des techniques de reconstructions de pressions et de flux conformes respectivement dans les espaces H1 et H(div) et sur la résolution de problèmes locaux de Neumann dans des bandes autour des interfaces de chaque sous-domaine pour les flux. Ainsi, des critères pour arrêter les itérations de l’algorithme itératif de décomposition de domaine sont développés. Des simulations numériques pour des problèmes académiques ainsi qu’un problème plus réaliste basé sur des données industrielles sont présentées pour illustrer l’efficacité de ces techniques. En particulier, différents pas de temps entre les sous-domaines sont considérés pour cet exemple. / This work contributes to the developpement of a posteriori error estimates and stopping criteria for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We study several problems. First, we tackle the steady diffusion equation using the mixed finite element subdomain discretization. Then the heat equation using the mixed finite element method in space and the discontinuous Galerkin scheme of lowest order in time is investigated. For the heat equation, a global-in-time domain decomposition method is used for both conforming and nonconforming time grids allowing for different time steps in different subdomains. This work is then extended to a two-phase flow model using a finite volume scheme in space. For each model, the multidomain formulation can be rewritten as an interface problem which is solved iteratively. Here at each iteration, local subdomain problems are solved, and information is then transferred to the neighboring subdomains. For unsteady problems, the subdomain problems are time-dependent and information is transferred via a space-time interface. The aim of this work is to bound the error between the exact solution and the approximate solution at each iteration of the domain decomposition algorithm. Different error components, such as the domain decomposition error, are identified in order to define efficient stopping criteria for the domain decomposition algorithm. More precisely, for the steady diffusion problem, the error of the domain decomposition method and that of the discretization in space are estimated separately. In addition, the time error for the unsteady problems is identified. Our a posteriori estimates are based on the reconstruction techniques for pressures and fluxes respectively in the spaces H1 and H(div). For the fluxes, local Neumann problems in bands arround the interfaces extracted from the subdomains are solved. Consequently, an effective criterion to stop the domain decomposition iterations is developed. Numerical experiments, both academic and more realistic with industrial data, are shown to illustrate the efficiency of these techniques. In particular, different time steps in different subdomains for the industrial example are used.

<p>Nuclear engineering hosts diverse domains including, but not limited to, power plant automation, human-machine interfacing, detection and identification of special nuclear materials, modeling of reactor kinetics and dynamics that most frequently are described by systems of differential equations (DEs), either ordinary (ODEs) or partial ones (PDEs). In this work we study multiple problems related to safety and Special Nuclear Material detection, and numerical solutions for partial differential equations using neural networks. More specifically, this work is divided in six chapters. Chapter 1 is the introduction, in Chapter</p> <p>2 we discuss the development of a gamma-ray radionuclide library for the characterization</p> <p>of gamma-spectra. In Chapter 3, we present a new approach, the ”Variance Counterbalancing”, for stochastic</p> <p>large-scale learning. In Chapter 4, we introduce a systematic approach for constructing proper trial solutions to partial differential equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. Chapter 5 is about an alternative, less imposing development of neural-form trial solutions for PDEs, inside rectangular and non-rectangular convex boundaries. Chapter 6 presents an ensemble method that avoids the multicollinearity issue and provides</p> <p>enhanced generalization performance that could be suitable for handling ”few-shots”- problems frequently appearing in nuclear engineering.</p>