La résolution numérique d'un problème d'équations aux dérivées partielles à coefficients discontinus posé dans un domaine à couche mince est difficile car elle nécessite la discrétisation à l'échelle de l'épaisseur de la couche. D'un point de vue théorique, on parle de problème de perturbation singulière. D'un pont de vue numérique, on observe que le maillage comporte alors un très grand nombre d'éléments, ce qui rend les calculs longs et parfois peu précis dans la couche. Dans une première partie, nous avons étudié ces problèmes avec des méthodes asymptotiques. Il s'agit d'en calculer la solution sous forme de développements asymptotiques par rapport aux petits paramètres. Cette partie du travail nous a permis de mettre en évidence les différentes difficultés évoquées plus haut. Dans une seconde partie, nous envisageons une autre approche avec des méthodes de décomposition de domaines sans recouvrement. Dans la construction de ces méthodes, les conditions d'interface doivent être judicieusement choisies de façon à prendre en compte non seulement l'hétérogénéité entre les sous-domaines mais aussi la dissymétrie de la décomposition.

My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

Este trabalho apresenta um estudo quantitativo e qualitativo de soluções positivas para o problema de Dirichlet para a equação de Hénon (P) { - \'DELTA\'u = \'Ix! POT. \' alpha\'\' \'IuI POT. p-2\' em B, u = sobre \\partial B, onde B é a bola unitária aberta de \'R POT. N\' centrada em zero e \'alpha\' > 0. É mostrado que para p \'> OU =\' \'2 AST\' \'IND. \'alpha\'\' = { \'SUP. 2(N + \'alpha)\' \' INF. N - 2\' ; N > 2, \"INFINITO\'; N = 1,2 \'2 AST\' = { \'SUP. 2N\' \'INF. N - 2\' ; N > 2, \'INFINITO\'; N = 1,2, o problema não tem solução não trivial. Em contrapartida, para 1 < p < \'2 AST\'.\' \'IND. \'alpha\'\' com p \'DIFERENTE DE\' 2, a existência de uma solução positiva radial é garantida. Além disso, é provado a unicidade de solução positiva no caso em que 1 < p < 2. Também são apresentados resultados sobre a existência de soluções ground state quando 2 < p < \'2 AST\'. Nesse intervalo, é mostrado que qualquer solução ground state exibe a simetria Schwarz folheada e, no caso em que \'alpha\' é suficientemente grande, é provado que qualquer solução ground state não é radialmente simétrica. Por fim, é apresentado um resultado sobre a existência de múltiplas soluções positivas / This work presents a quantitative and qualitative study of positive solutions for the Dirichlet problem for the Hénon equation (P) (P) { - \'DELTA\'u = \'Ix! POT. \' alpha\'\' \'IuI POT. p-2\' in B, u = 0 on \\partial B, where B is the unit open ball in \'R POT. N\' centered at zero and \'alpha\' \'> OR =\' 0. It is shown that for p \' > OR =\' \'2 AST\' \'IND. alpha\' = \'SUP. 2 (N + alpha)\' INF. N - 2, N > 2, \' INFINITY\'; N = 1, 2, \'2 AST\' = { \'SUP. .2N INF. N - 2 ; N > 2; 1; \' INFINITY\', N = 1, 2; the problem does not have nontrivial solution. In counterpart, for 1 < p < \' 2 AST\' \'IND. alpha\' with p \' DIFFERENT\' 2, the existence of radial positive solutions will be guaranteed. Moreover, the uniqueness of positive solution is guaranteed as long as 1 < p < 2. In addition, results on the existence of ground state solutions are presented in case 2 < p < \'2 AST\'. In this interval, it is proved that any ground state solution exhibits the Foliated Schwarz symmetry and, in case \'alpha\' is sufficiently large, it is shown that the no ground state solution is radially symmetric. This works ends with a result on the existence of multiple positive solutions

A produção de fundidos com baixo nível de defeitos e com propriedades mecânicas adequadas é facilmente alcançada com a previsão do comportamento do metal durante a solidificação. A transferência de calor entre o metal-líquido e o molde, nos primeiros instantes de contato, compromete definitivamente as propriedades mecânicas e a qualidade do produto final. O comportamento da transferência de calor entre o metal e o molde foi explorado através da análise térmica experimental e confrontado com o Modelo de Schwarz Modificado (MSM). Para comprovar experimentalmente os fenômenos previstos no MSM tais como: superesfriamento aparente, posição das interfaces líquido-superaquecido/líquido- superesfriado e da sólido/líquido durante a solidificação foram realizados experimentos com alumínio puro, gálio puro e estanho. Para a análise térmica foram utilizados dois sistemas de solidificação vertical descendente (Griffiths et al., 1993; Jinho et al., 1996; Jamgotchian et al., 1987). O sistema A e o sistema B, sendo que o sistema B também permitia a solidificação vertical ascendente. Em ambos os casos, o fluxo de calor foi direcionado através de uma barra de alumínio (na temperatura ambiente) que foi inserida verticalmente no banho (técnica do “dedo frio”). A barra extrai calor do líquido em função da diferença de temperatura entre a massa líquida e a massa sólida e simula o choque térmico do metal líquido com as paredes de um molde. Os resultados experimentais e os obtidos pela simulação foram confrontados. Apresenta-se a relação experimental entre o superesfriamento e a taxa de solidificação. Discute-se o redimensionamento da Ti (temperatura de interface) na solução de Schwarz e a comprovação experimental da solução do MSM. / The production of casting with low level of defects and adjusted mechanical properties can be obtained with the previous knowledge of the metal solidification behavior. The transference of heat between the metal-liquid and the mold, in the first instants of the contact, compromises definitively the mechanical properties and the final product quality. The behavior of the transference of heat between the metal and the mold was carried out through the experimental thermal analysis and the results were compared with the Modified Schwarz Model (MSM). To experimentally prove, the cooled phenomena in the MSM such as: apparent supercooling, liquid superheated/liquid supercooled interfaces position and of the liquid/solid during the solidification experiments had been carried out with pure aluminum, pure gallium and tins. Two systems A and B, one of descending vertical solidification and another one of descending and ascending solidification, had been used for the thermal analysis. In both systems, the heat flow was directed through a bar of aluminum (in the ambient temperature) that was inserted vertically in the bath (“cold finger” technique). The bar extracts heat of the liquid, had the difference of temperature between the liquid mass and the solid mass, and simulates the thermal shock of the metal with the mold walls. The experimental results were compared with simulated data. In this work are presented experimental relation between the supercooling and the rate of solidification. The new dimension of the Ti (temperature of interface) in the Schwarz equation and the experimental evidence of MSM solution, are considered.

abstract: The advancements in additive manufacturing have made it possible to bring life to designs that would otherwise exist only on paper. An excellent example of such designs are the Triply Periodic Minimal Surface (TPMS) structures like Schwarz D, Schwarz P, Gyroid, etc. These structures are self-sustaining, i.e. they require minimal supports or no supports at all when 3D printed. These structures exist in stable form in nature, like butterfly wings are made of Gyroids. Automotive and aerospace industry have a growing demand for strong and light structures, which can be solved using TPMS models. In this research we will try and understand some of the properties of these Triply Periodic Minimal Surface (TPMS) structures and see how they perform in comparison to the conventional models. The research was concentrated on the mechanical, thermal and fluid flow properties of the Schwarz D, Gyroid and Spherical Gyroid Triply Periodic Minimal Surface (TPMS) models in particular, other Triply Periodic Minimal Surface (TPMS) models were not considered. A detailed finite element analysis was performed on the mechanical and thermal properties using ANSYS 19.2 and the flow properties were analyzed using ANSYS Fluent under different conditions. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2019

Cette étude propose une application innovante de deux concepts très étudiés par la communauté mathématique, le fibré des k-repères et la connexion de Cartan. D'une part, l'utilisation d'une connexion de Cartan particulière sur le fibré des 2-repères nous permet de proposer une généralisation de la notion de dérivée de Schwarz en dimension arbitraire, pour les difféomorphismes projectifs et conformes. D'autre part, nous avons pu élaborer une structure de BRS permettant de reproduire infinitésimalement l'action des difféomorphismes sur des champs de jauge à un terme de courbure près. Ainsi, la notion de connexion de Cartan sur le fibré des 2-repères a permis de résoudre un problème ouvert, originellement formulé par A.M. Polyakov en 1990 qui obtient formellement l'action de difféomorphismes (symétrie de l'espace-temps) à partir d'une transformation de jauge (symétrie interne). Les symétries d'espace-temps et les symétries internes peuvent ainsi être exprimées dans un formalisme similaire.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

Scherk-Schwarz reductions are a generalization of Kaluza-Klein reductions in which the higher dimensional fields are allowed to have a dependence on the compactiifed coordinates. This is possible only if the higher dimensional theory has a global symmetry and the dependence is dictated by this symmetry. In this thesis we consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of $D=4, N=4$ super-Yang-Mills coupled to supergravity. The reduction cannot be done on the action itself, but must be done either on the field equations or on a duality invariant form of the action, such as one in the doubled formalism in which potentials are introduced for both electric and magnetic fields. The resulting theory in odd dimensions has massive form fields satisfying a self-duality condition $dA sim m*A$. We apply these methods to theories in $D=4,6,8$, and obtain new gauged supergravity theories with massive form fields, with Chern-Simons like couplings and with a scalar potential in $D=3,5,7$.

Vorliegendes Buch erzählt die Familiengeschichte der sächsischen Familien Schwarz und Popp. Der Autor hat sowohl seine Vorfahren, nämlich die Familienlinien Schwarz, Einenkel , Schulze und Dehmel, als auch die Vorfahren seiner Frau, nämlich die Familienlinien Popp, Kießling, Störl und Rüdiger bis Mitte des 13. Jahrhunderts - überwiegend anhand von Kirchenbucheinträgen - erforscht und beschrieben. Die Familien stammen alle aus dem Erzgebirge und dem Vogtland. Einzelne Personen kamen aus Böhmen, Franken, Österreich und Baden-Württemberg hinzu. Im ersten Teil sind die Herkunft der Familien, die Berufe, herausragende Persönlichkeiten, Besonderheiten aus dem Leben der Personen, sowie tragische und kuriose Geschichten aus den Kirchenbucheinträgen zusammengetragen. Außerdem sind, zugeordnet zu den Familien, tabellarische Aufstellungen von Herkunftsorten und Familiennamen enthalten. Im Anhang Teil 2 sind die Ahnentafeln der Linien Schwarz und Popp dargestellt. Teil 3 enthält nochmals eine umfassende Aufstellung aller Herkunftsorte und Familiennamen. Der Autor wurde 1947 in Limbach-Oberfrohna geboren, absolvierte Schul- und Berufsausbildung ab 1954 in Oberschwaben, ist seit 1970 verheiratet und lebt seit 1972 in Bayern. Die Anlagen 1-12 auf den Seiten 53-218 sind in dem Online-Dokument aus rechtlichen Gründen nicht enthalten.

Since most financial institutions use the Sharpe Ratio to evaluate the performance of mutual funds, the objective of most fund managers is to select the portfolio that can generate the highest Sharpe Ratio. Traditionally, they can revise the objective function of the Markowitz mean-variance portfolio model and resolve non-linear programming to obtain the maximum Sharpe Ratio portfolio. In the scenario with short sales allowed, this project will propose a closed-form solution for the optimal Sharpe Ratio portfolio by applying Cauchy-Schwarz maximization. This method without using a non-linear programming computer program is easier than traditional method to implement and can save computing time and costs. Furthermore, in the scenarios with short sales disallowed, we will use Kuhn-Tucker conditions to find the optimal Sharpe Ratio portfolio. On the other hand, an efficient frontier generated by Markowitz mean-variance portfolio model normally has higher risk higher return characteristic, which often causes dilemma for decision maker. This research applies generalized loss function to create a family of decision-aid performance measures called IRp which can well tradeoff return with risk. We compare IRp with Sharpe Ratio and utility functions to confirm that IRp measures are approapriate to evaluate portfolio performance on efficient frontier and to improve asset allocation decisions. In addition, empirical data of domestic and international investment instruments will be used to examine the feasibility and fitness of the new proposed method and IRp measures. This study applies the methods of Cauchy-Schwarz maximization in multivariate statistical analysis and loss function in quality engineering to portfolio decisions. We believe these new applications will complete portfolio model theory and will be meaningful for academic and business fields.