We introduce random fluctuations on contact and recovery rates in three basic deterministic models in mathematical epidemiology and obtain stochastic counterparts. This paper addresses qualitative and quantitative analysis of stochastic SIS model with disease deaths and demographic effects, and stochastic SIR models with/without disease deaths and demographic effects. We prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria. We also investigate numerical properties of these models and prove the convergence of the Balanced Implicit Method approximation to the analytic solution. We simulate the models with fairly realistic parameters to visualize our conclusions.

In this research, the Alternating Cell Direction Implicit method is used in temporal discretisation of the incompressible Navier-Stokes equations and compared with the well known and widely used Point Gauss Seidel scheme on structured and quadrilateral unstructured meshes. A two dimensional, laminar and incompressible Navier-Stokes solver is developed for this purpose using the artificial compressibility formulation. The developed solver is used to obtain steady-state solutions with implicit time stepping methods and a third order data reconstruction scheme (U-MUSCL) is added to obtain high order spatial accuracy. The Alternating Cell Directions Implicit method and Point Gauss Seidel scheme is compared in terms of convergence iteration number and total computation time using test cases with growing complexity, including laminar flat plate, single and multi-element airfoil calculations. Both structured and quadrilateral unstructured grids are used in single element airfoil calculations. In these test cases, it is seen that a reduction between 13% and 20% is obtained in total computation time by usage of Alternating Cell Directions Implicit method when compared with the Point Gauss Seidel method.

A Ãgua à um recurso natural essencial para todas as formas de vida e a sua distribuiÃÃo deve ser realizada com qualidade e sem desperdÃcios. Um dos mÃtodos de otimizaÃÃo dos sistemas de distribuiÃÃo de Ãgua à baseado na simulaÃÃo (modelagem) de redes hidrÃulicas reais atravÃs de modelos computacionais capazes de prever o seu comportamento nas situaÃÃes diversas ao logo da sua vida Ãtil. As principais etapas desse mÃtodo sÃo calibraÃÃo; operaÃÃo e controle; projeto e otimizaÃÃo; e traÃado de redes. A ideia deste trabalho à aprimorar uma dessas tÃcnicas, a de calibraÃÃo, processo de identificaÃÃo dos parÃmetros das tubulaÃÃes (fator de atrito, rugosidade, diÃmetros e outros) em redes existentes, na qual os mesmos sÃo considerados desconhecidos. A metodologia adotada à o MÃtodo Transiente Inverso (MTI) com otimizaÃÃo da tÃcnica de seleÃÃo da soluÃÃo atravÃs do algoritmo GenÃtico (AG). O objetivo principal à calibrar os principais parÃmetros (fator de atrito e rugosidade) a fim de analisar a conveniÃncia de duas tÃcnicas: cÃlculo do fator de atrito pelo mÃtodo pelo MTI-AG e a partir da rugosidade (com uso da fÃrmula de Swamee, 1993). O estudo à realizado para duas redes sintÃticas tiradas da literatura, mas que representam sistemas reais. As condiÃÃes impostas para a anÃlise sÃo: duas manobras de vÃlvulas (lenta e brusca) responsÃveis pelo evento transiente, o monitoramento de cargas transientes em somente 20% dos nÃs das redes e a utilizaÃÃo de dois tipo de seleÃÃo de soluÃÃo do algoritmo genÃtico (com elitismo e sem elitismo). Os experimentos sÃo realizados a partir de trÃs programas: o primeiro calcula as condiÃÃes permanentes, o segundo fornece cargas transientes por meio do MÃtodo das CaracterÃsticas (MOC) na busca das soluÃÃes das equaÃÃes de movimento no escoamento transiente e o terceiro, que trabalha de forma conjugada com os demais, seleciona os melhores resultados atravÃs de iteraÃÃes realizadas conforme o algoritmo genÃtico, tÃcnica inspirada nos mecanismos de evoluÃÃo dos seres vivos. Os resultados encontrados indicam que independente das condiÃÃes impostas inicialmente à mais eficiente identificar o fator de atrito a partir das rugosidades absolutas do que calibrar esse fator pelo MTI-AG devido a grande variabilidade dos fatores de atrito durante a ocorrÃncia do evento transiente. / Water is an essential natural resource for all life forms and their distribution should be carried out with quality and without waste. One of the methods of optimization of water distribution systems is based on the simulation (modeling) of real hydraulic networks using computational models that can predict their behavior in various situations the logo of its useful life. The main steps of this method are calibration; operation and control; design and optimization; and route networks. The idea of ​​this work is to improve one such technique, the calibration parameters of the pipes identification process (friction factor, surface roughness, diameter and the like) existing in networks in which they are considered unknown. The methodology adopted is the Transient Inverse Method (MTI) with optimization of the solution selection technique by Genetic Algorithm (GA). The main objective is to calibrate the main parameters (friction and roughness factor) to analyze the convenience of two techniques: calculation of the friction factor by the method by MTI-AG and from the roughness (using the formula Swamee, 1993 ). The study is carried out for two taken synthetic networks of literature, but represent real systems. The conditions for the analysis are: two maneuvers valves (slow and abrupt) responsible for the transient event, the monitoring of transient loads in only 20% of the nodes of the networks and the use of two types of solution selection of genetic algorithm (with elitism and without elitism). The experiments are carried out through three programs: the first calculates the permanent conditions, the second provides transient loads through the method of characteristics (MOC) in the search for solutions of the equations of motion in the transient flow and the third, who works so combined with others, selects the best results through iterations performed according to the genetic algorithm, a technique inspired by the mechanisms of evolution of living beings. The results indicate that regardless of the conditions imposed is initially more efficiently identify the friction factor from the absolute roughness than calibrate this factor by MTI-AG due to the great variability of friction factors for the occurrence of the transient event.

This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi-implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)

Lors de sa rentrée dans l’atmosphère d’une planète, un engin spatial subit un échauffement important dû aux frottements des gaz atmosphériques sur la paroi. Cette élévation de température conduit à une dégradation physico-chimique du bouclier thermique de l’objet constitué de matériaux composites. Un composite est constitué de divers matériaux qui s’ablatent différemment. Dans cette thèse, nous nous intéressons essentiellement à la fusion d’un matériau durant sa phase de rentrée atmosphérique. Nous sommes donc en présence de trois phases : solide, liquide et gaz. Pour simuler ce phénomène, des méthodes numériques robustes ont été mises au point pour calculer l’écoulement diphasique compressible autour de l’objet. Le couplage entre le solide et l’écoulement fluide a aussi été étudié. Les méthodes numériques développées durant cette thèse sont basées sur une approche volumes finis. Une stratégie de décomposition d’opérateurs est utilisée pour résoudre le modèle diphasique à cinq équations avec les termes de dissipation modélisant l’écoulement fluide. L’idée principale de cette décomposition d’opérateurs est de séparer les phénomènes acoustiques et dissipatifs des phénomènes de transport. Un traitement implicite de l’étape acoustique est réalisé tandis que l’étape de transport est résolue explicitement. Le schéma semi-implicite global est alors très robuste, conservatif et préserve les discontinuités de contact. Les conditions d’interface entre les domaines fluide et solide sont déduites des bilans de masse et d’énergie à la paroi. Le front de fusion est suivi explicitement grâce à une formulation ALE des équations. La robustesse de l’approche et l’apport de la formulation semi-implicite sont finalement démontrés grâce à des expériences numériques mono et bidimensionnelles sur maillages curvilignes mobiles. / During atmospheric re-entry phase, a spacecraft undergoes a sudden increase of the temperature due to the friction of atmospheric gases. This rise drives to a physical-chemical degradation of the thermal protective system of the object made of composite material. A composite is made of several materials with ablates differently. In this thesis, we mainly focus on the melting of an object during its re-entry phase. Therefore there are three phases: solid, liquid and gas phases. In order to simulate this phenomenon, robust numerical methods have been developed to compute a compressible multiphase flow. The coupling strategy between the solid and the fluid have also been studied. Solvers developed in the present work are based on Finite Volume Method. A splitting strategy is used to compute compressible two-phase flows using the five-equation model with viscous and heat conduction effects. The main idea of the splitting is to separate the acoustic and dissipative phenomena from the transport one. An implicit treatment of the acoustic step is performed while the transport step is solved explicitly. The overall scheme resulting from this splitting operator strategy is very robust, conservative, and preserves contact discontinuities. The boundary interface condition between the solid and the multiphase flow is enforced by mass and energy balances at the wall. The melting front is tracked explicitly using an ALE formulation of the equations. The robustness of the approach and the interest of the semi-implicit formulation are demonstrated through numerical simulations in one and two dimensions on moving curvilinear grids.

付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(工学) / 甲第25293号 / 工博第5252号 / 新制||工||1999(附属図書館) / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 松尾 哲司, 教授 引原 隆士, 教授 土居 伸二 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

The heart is one the most important organs in the human body and many other live creatures. The electrical activity in the heart controls the heart function, and many heart diseases are linked to the abnormalities in the electrical activity in the heart. Mathematical equations and computer simulation can be used to model the electrical activity in the heart. The heart models are challenging to solve because of the complexity of the models and the huge size of the problems. Several cell models have been proposed to model the electrical activity in a single heart cell. These models must be coupled with a heart model to model the electrical activity in the entire heart. The bidomain model is a popular model to simulate the propagation of electricity in myocardial tissue. It is a continuum-based model consisting of non-linear ordinary differential equations (ODEs) describing the electrical activity at the cellular scale and a system of partial differential equations (PDEs) describing propagation of electricity at the tissue scale. Because of this multi-scale, ODE/PDE structure of the model, splitting methods that treat the ODEs and PDEs in separate steps are natural candidates as numerical methods. First, we need to solve the problem at the cellular scale using ODE solvers. One of the most popular methods to solve the ODEs is known as the Rush-Larsen (RL) method. Its popularity stems from its improved stability over integrators such as the forward Euler (FE) method along with its easy implementation. The RL method partitions the ODEs into two sets: one for the gating variables, which are treated by an exponential integrator, and another for the remaining equations, which are treated by the FE method. The success of the RL method can be understood in terms of its relatively good stability when treating the gating variables. However, this feature would not be expected to be of benefit on cell models for which the stiffness is not captured by the gating equations. We demonstrate that this is indeed the case on a number of stiff cell models. We further propose a new partitioned method based on the combination of a first-order generalization of the RL method with the FE method. This new method leads to simulations of stiff cell models that are often one or two orders of magnitude faster than the original RL method. After solving the ODEs, we need to use bidomain solvers to solve the bidomain model. Two well-known, first-order time-integration methods for solving the bidomain model are the semi-implicit method and the Godunov operator-splitting method. Both methods decouple the numerical procedure at the cellular scale from that at the tissue scale but in slightly different ways. The methods are analyzed in terms of their accuracy, and their relative performance is compared on one-, two-, and three-dimensional test cases. As suggested by the analysis, the test cases show that the Godunov method is significantly faster than the semi-implicit method for the same level of accuracy, specifically, between 5 and 15 times in the cases presented. Second-order bidomain solvers can generally be expected to be more effective than first-order bidomain solvers under normal accuracy requirements. However, the simplest and the most commonly applied second-order method for the PDE step, the Crank-Nicolson (CN) method, may generate unphysical oscillations. We investigate the performance of a two-stage, L-stable singly diagonally implicit Runge-Kutta method for solving the PDEs of the bidomain model and present a stability analysis. Numerical experiments show that the enhanced stability property of this method leads to more physically realistic numerical simulations compared to both the CN and Backward Euler (BE) methods.

<p align="justify">This work deals with the development of numerical methods for compressible flow simulation with application to the interaction of fluid flows and structural bodies.</p> <p align="justify">First, we develop numerical methods based on multidimensional upwind residual distribution (RD) schemes. Theoretical results for the stability and accuracy of the methods are given. Then, the RD schemes for unsteady problems are extended for computations on moving meshes. As a second approach, cell centered and vertex centered finite volume (FV) schemes are considered. The RD schemes are compared to FV schemes by means of the 1D modified equation and by the comparison of the numerical results for scalar problems and system of Euler equations. We present a number of two and three dimensional steady and unsteady test cases, illustrating properties of the numerical methods. The results are compared with the theoretical solution and experimental data.</p> <p align="justify">In the second part, a numerical method for fluid-structure interaction problems is developed. The problem is divided into three distinct sub-problems: Computational Fluid Dynamics, Computational Solid Mechanics and the problem of fluid mesh movement. The problem of Computational Solid Mechanics is formulated as a system of partial differential equations for an anisotropic elastic continuum and solved by the finite element method. The mesh movement is determined using the pseudo-elastic continuum approach and solved again by the finite element method. The coupling of the problems is achieved by a simple sub-iterative approach. Capabilities of the methods are demonstrated on computations of 2D supersonic panel flutter and 3D transonic flutter of the AGARD 445.6 wing. In the first case, the results are compared with the theoretical solution and the numerical computations given in the references. In the second case the comparison with experimental data is presented.</p>

Least Square Kinetic Upwind Method (LSKUM) belongs to the class of mesh-less method that solves compressible Euler equations of gas dynamics. LSKUM is kinetic theory based upwind scheme that operates on any cloud of points. Euler equations are derived from Boltzmann equation (of kinetic theory of gases) after taking suitable moments. The basic update scheme is formulated at Boltzmann level and mapped to Euler level by suitable moments. Mesh-less solvers need only cloud of points to solve the governing equations. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. Additionally blanking and de-blanking strategy along with overlay point cloud can be adapted in some applications like store separation to avoid regeneration of points. Naturally, the mesh-less solvers have advantage in tackling complex geometries and moving components over solvers that need grids. Conventionally, higher order accuracy for space derivative term is achieved by two step defect correction formula which is computationally expensive. The present solver uses low dissipation single step modified CIR (MCIR) scheme which is similar to first order LSKUM formulation and provides spatial accuracy closer to second order. The maximum time step taken to march solution in time is limited by stability criteria in case of explicit time integration procedure. Because of this, explicit scheme takes a large number of iterations to achieve convergence. The popular explicit time integration schemes like four stages Runge-Kutta (RK4) are slow in convergence due to this reason. The above problem can be overcome by using the implicit time integration procedure. The implicit schemes are unconditionally stable i.e. very large time steps can be used to accelerate the convergence. Also it offers superior robustness. The implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is very attractive due to its low numerical complexity, moderate memory requirement and unconditional stability for linear wave equation. Also this scheme is more efficient than explicit counterparts and can be implemented easily on parallel computers. It is based on the factorization of the implicit operator into three parts namely lower triangular matrix, upper triangular matrix and diagonal terms. The use of LU-SGS results in a matrix free implicit framework which is very economical as against other expensive procedures which necessarily involve matrix inversion. With implementation of the implicit LU-SGS scheme larger time steps can be used which in turn will reduce the computational time substantially. LU-SGS has been used widely for many Finite Volume Method based solvers. The split flux Jacobian formulation as proposed by Jameson is most widely used to make implicit procedure diagonally dominant. But this procedure when applied to mesh-less solvers leads to block diagonal matrix which again requires expensive inversion. In the present work LU-SGS procedure is adopted for mesh-less approach to retain diagonal dominancy and implemented in 2-D and 3-D solvers in matrix free framework. In order to assess the efficacy of the implicit procedure, both explicit and implicit 2-D solvers are tested on NACA 0012 airfoil for various flow conditions in subsonic and transonic regime. To study the performance of the solvers on different point distributions two types of the cloud of points, one unstructured distribution (4074 points) and another structured distribution (9600 points) have been used. The computed 2-D results are validated against NASA experimental data and AGARD test case. The density residual and lift coefficient convergence history is presented in detail. The maximum speed up obtained by use of implicit procedure as compared to explicit one is close to 6 and 14 for unstructured and structured point distributions respectively. The transonic flow over ONERA M6 wing is a classic test case for CFD validation because of simple geometry and complex flow. It has sweep angle of 30° and 15.6° at leading edge and trailing edge respectively. The taper ratio and aspect ratio of the wing are 0.562 and 3.8 respectively. At M∞=0.84 and α=3.06° lambda shock appear on the upper surface of the wing. 3¬D explicit and implicit solvers are tested on ONERA M6 wing. The computed pressure coefficients are compared with experiments at section of 20%, 44%, 65%, 80%, 90% and 95% of span length. The computed results are found to match very well with experiments. The speed up obtained from implicit procedure is over 7 for ONERA M6 wing. The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft use some form of the delta wing. To demonstrate the effectiveness of 3-D solver in handling control surfaces and small gaps, implicit 3-D code is used to compute flow past clipped delta wing with aileron deflection of 6° at M∞ = 0.9 and α = 1° and 3°. The leading edge backward sweep is 50.4°. The aileron is hinged from 56.5% semi-span to 82.9% of semi-span and at 80% of the local chord from leading edge. The computed results are validated with NASA experiments

<p align="justify">This work deals with the development of numerical methods for compressible flow simulation with application to the interaction of fluid flows and structural bodies.</p><p><p><p align="justify">First, we develop numerical methods based on multidimensional upwind residual distribution (RD) schemes. Theoretical results for the stability and accuracy of the methods are given. Then, the RD schemes for unsteady problems are extended for computations on moving meshes. As a second approach, cell centered and vertex centered finite volume (FV) schemes are considered. The RD schemes are compared to FV schemes by means of the 1D modified equation and by the comparison of the numerical results for scalar problems and system of Euler equations. We present a number of two and three dimensional steady and unsteady test cases, illustrating properties of the numerical methods. The results are compared with the theoretical solution and experimental data.</p><p><p><p align="justify">In the second part, a numerical method for fluid-structure interaction problems is developed. The problem is divided into three distinct sub-problems: Computational Fluid Dynamics, Computational Solid Mechanics and the problem of fluid mesh movement. The problem of Computational Solid Mechanics is formulated as a system of partial differential equations for an anisotropic elastic continuum and solved by the finite element method. The mesh movement is determined using the pseudo-elastic continuum approach and solved again by the finite element method. The coupling of the problems is achieved by a simple sub-iterative approach. Capabilities of the methods are demonstrated on computations of 2D supersonic panel flutter and 3D transonic flutter of the AGARD 445.6 wing. In the first case, the results are compared with the theoretical solution and the numerical computations given in the references. In the second case the comparison with experimental data is presented.</p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished