<p>Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.</p>

The work presented here is part of an ongoing effort to develop a highly accurate and numerically efficient turbulence simulation technique. The paper consists of four main parts, viz., the general discussion of the procedure known as Additive Turbulent Decomposition, the derivation of the "synthetic velocity" subgrid-scale model of the high wavenumber turbulent fluctuations necessary for its implementation, the numerical investigation of this model and a priori tests of said models physical validity. Through these investigations we have demonstrated that this procedure, coupled with the use of the "Poor Mans Navier-Stokes" equation subgrid-scale model, has the potential to be a faster, more accurate replacement of currently popular turbulence simulation techniques since: 1. The procedure is consistent with the direct solution of the Navier-Stokes equations if the subgrid-scale model is valid, i.e, the equations to be solved are never filtered, only solutions. 2. Model parameter values are "set" by their relationships to N.S. physics found from their derivation from the N.S. equation and can be calculated "on the fly" with the use of a local high-pass filtering of grid-scale results. 3. Preliminary studies of the PMNS equation model herein have shown it to be a computationally inexpensive and a priori valid model in its ability to reproduce high wavenumber fluctuations seen in an experimental turbulent flow.

Cette thèse porte sur la conception des méthodes robustes de réduction d’ordre de modèles numériques de type Éléments Finis (EF) avec contrôle de la précision. La réduction d’ordre est en général nécessaire pour réduire drastiquement les temps de calcul et permettre ainsi une analyse paramétrique, une étude de faisabilité ou de performance de système (avion, unité de production, procédé complexe, etc). Dans cette étude, la technique de décomposition orthogonale aux valeurs propres (POD) sera utilisée pour construire des modèles réduits locaux. Informatiquement parlant, le “modèle” sera considéré comme une base de données de résultats de calcul avec capacité d’extrapolation et d’interpolation locale. Une stratégie adaptative pour stocker et accéder à la base de données est étudiée en étendant l’algorithme In situ Adaptive Tabulation (ISAT) proposé initialement par Pope. En fonction de l’usage et des exigences en précision des résultats, la base de données est enrichie en ligne (online) par des appels au modèle fin en respectant une précision spécifiée jusqu’à couvrir le domaine paramétrique entier, après quoi l’évaluation d’une solution devient très peu couteuse. L’approche couplée POD-ISAT proposée dans cette thèse fournit une méthode de réduction de modèle EF très performante. La méthodologie est évaluée sur un cas réel de conditionnement d’air en régime stationnaire de cabine d’avion dépendant de plusieurs paramètres de conception (température et vitesse d’entrée d’air, mode de ventilation personnalisée, conductivité thermique du fuselage, etc.). Pour les problèmes d’évolution en temps, nous explorons une piste de chainage de modèles et d’utilisation d’algorithme de parallélisation en temps tel que l’algorithme pararéel initialement proposé par Lions, Maday et Turinici (2001). Nous proposons ici une variante quasi-Newton de l’algorithme pararéel que nous appelons algorithme Broyden-pararéel. Il est appliqué au calcul de la diffusion d’un gaz dans la cabine d’avion. Cette thèse s’insère dans le cadre du projet CSDL (Complex System Design Lab, Fond Unique Interministériel) visant à développer une plate-forme logicielle multidisciplinaire pour la conception de systèmes complexes. / In this thesis, an efficient Reduced Order Modeling (ROM) technique with control of accuracy for parameterized Finite Element solutions is proposed. The ROM methodology is usually necessary to drastically reduce the computational time and allow for tasks like parameter analysis, system performance assessment (aircraft, complex process, etc.). In this thesis, a ROM using Proper Orthogonal Decomposition (POD) will be used to build local models. The “model” will be considered as a database of simulation results store and retrieve the database is studied by extending the algorithm In Situ Adaptive Tabulation (ISAT) originally proposed by Pope (1997). Depending on the use and the accuracy requirements, the database is enriched in situ (i.e. online) by call of the fine (reference) model and construction of a local model with an accuracy region in the parameter space. Once the trust regions cover the whole parameter domain, the computational cost of a solution becomes inexpensive. The coupled POD-ISAT, here proposed, provides a promising effective ROM approach for parametric finite element model. POD is used for the low-order representation of the spatial fields and ISAT for the local representation of the solution in the design parameter space. This method is tested on a Engineering case of stationary air flow in an aircraft cabin. This is a coupled fluid-thermal problem depending on several design parameters (inflow temperature, inflow velocity, fuselage thermal conductivity, etc.). For evolution problems, we explore the use of time-parallel strategies, namely the parareal algorithm originally proposed by Lions, Maday and Turinici (2001). A quasi-Newton variant of the algorithm called Broyden-parareal algorithm is here proposed. It is applied to the computation of the gas diffusion in an aircraft cabin. This thesis is part of the project CSDL (Complex System Design Lab) funded by FUI (Fond Unique Interministériel) aimed at providing a software platform for multidisciplinary design of complex systems.

Submitted by (ana.araujo@ufrpe.br) on 2016-07-07T12:07:15Z No. of bitstreams: 1 Jose Rodrigo Santos Silva.pdf: 4054411 bytes, checksum: c22cd915da573fd5a5ce7b45feb85a0f (MD5) / Made available in DSpace on 2016-07-07T12:07:15Z (GMT). No. of bitstreams: 1 Jose Rodrigo Santos Silva.pdf: 4054411 bytes, checksum: c22cd915da573fd5a5ce7b45feb85a0f (MD5) Previous issue date: 2011-02-18 / The velocity-area method is a standard procedure for measurement of river discharge, with wide application in hydrometric studies, standardized at the international level by the norm ISO 748:2007 of the International Standards Organization. This method requires measurement of velocity at several verticals of the river, at different depths for each vertical. In general, a relatively high number of measurements is necessary do determine the discharge. Recently a technique was proposed which results in a robust estimate of river discharge using a reduced number of measurement points, based on elementary properties of fluid dynamics, stemming from the Navier-Stokes equations, and the use of continuous interpolation between the verticals for calculating velocity across the entire river cross section. In the present work the Monte Carlo Markov Chain (MCMC) method is used to search for the optimum positions for velocity measurement, with the objective of reducing the number of measurement points without significant loss of precision, and therefore maximizing the efficiency of the estimate. A dedicated computer algorithm was developed in C programming language and applied to measurements collected on the river Exu, state of Pernambuco, Brazil, in April 2008. It is found that the discharge estimates with three or more measurement points exhibit variations well within uncertainty limits corresponding to the full 27 point estimate using the traditional velocity-area method. Simulation results indicate that the best positions for velocity measurement are close to the surface, and that significant savings in cost and labor may be accomplished by positioning the measurements at strategic points, without precision loss. / O método área-velocidade é um procedimento utilizado para medir a descarga de rios. Esta é uma técnica bastante difundida na hidrometria, e é normatizada internacionalmente pela ISO 748:2007 da International Standard Organization. Este método requer a medição da velocidade em diversas verticais do rio, e em diferentes profundidades de cada vertical. Em geral é necessário um número relativamente elevado de medições para determinar a vazão. Recentemente foi proposta uma técnica que resulta em uma estimativa robusta da descarga fluvial com reduzido número de pontos de medida, que se baseia nas propriedades básicas da dinâmica de fluidos e nas equações de Navier- Stokes, além de utilizar uma interpolação continua para o cálculo das velocidades em toda a seção vertical. No presente trabalho, o método Monte Carlo Markov Chain (MCMC) é utilizado na busca da melhor posição das medidas de velocidade a serem realizados na seção vertical do rio, tal que seja possível reduzir o número de medições e maximizar a eficiência da estimativa. O algoritmo foi desenvolvido em linguagem C e aplicado em medidas de velocidade colhidas no riacho Exu, no estado de Pernanbuco, em abril de 2008. Estimativas de vazão realizadas a partir de 3 medidas de velocidade sobre a seção vertical mostraram-se eficientes, apresentando diferenças da estimativa obtida com 27 pontos através do método área-velocidade tradicional dentro de limites de incerteza. Os resultados de simulação indicam que os melhores locais de medição da velocidade sob a seção vertical situam-se perto da superfície do rio, e que uma economia significativa no custo e no trabalho pode ser conseguida através posicionamento dos pontos de medição em locais estratégicos, sem perda da precisão da estimativa.

Yes / Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

<p>This thesis is a numerical investigation of two-dimensional steady flows past a circular obstacle. In the fluid dynamics research there are few computational results concerning the structure of the steady wake flows at Reynolds numbers larger than 100, and the state-of-the-art results go back to the work of Fornberg (1980) Fornberg (1985). The radial velocity component approaches its asymptotic value relatively slowly if the solution is ``physically reasonable''. This presents a difficulty when using the standard approach such as domain truncation. To get around this problem, in the present research we will develop a spectral technique for the solution of the steady Navier-Stokes system. We introduce the ``bootstrap" method which is motivated by the mathematical fact that solutions of the Oseen system have the same asymptotic structure at infinity as the solutions of the steady Navier-Stokes system with the same boundary conditions. Thus, in the ``bootstrap" method, the streamfunction is calculated as a perturbation to the solution to the Oseen system. Solutions are calculated for a range of Reynolds number and we also investigate the solutions behaviour when the Reynolds number goes to infinity. The thesis compares the numerical results obtained using the proposed spectral ``bootstrap" method and a finite--difference approach for unbounded domains against previous results. For Reynolds numbers lower than 100, the wake is slender and similar to the flow hypothesized by Kirchoff (1869) and Levi-Civita (1907). For large Reynolds numbers the wake becomes wider and appears more similar to the Prandtl-Batchelor flow, see Batchelor (1956).</p> / Doctor of Science (PhD)

During intense rain events a stormwater system can fill rapidly and undergo a transition from open channel flow to pressurized flow. This transition can create large discrete pockets of trapped air in the system. These pockets are pressurized in the horizontal reaches of the system and then are released through vertical vents. In extreme cases, the transition and release of air pockets can create a geyser feature. The current models are inadequate for simulating mixed flows with complicated air-water interactions, such as geysers. Additionally, the simulation of air escaping in the vertical dropshaft is greatly simplified, or completely ignored, in the existing models. In this work a two-phase numerical model solving the Navier-Stokes equations is developed to investigate the key factors that form geysers. A projection method is used to solve the Navier-Stokes Equation. An advanced two-phase flow model, Volume of Fluid (VOF), is implemented in the Navier-Stokes solver to capture and advance the interface. This model has been validated with standard two-phase flow test problems that involve significant interface topology changes, air entrainment and violent free surface motion. The results demonstrate the capability of handling complicated two-phase interactions. The numerical results are compared with experimental data and theoretical solutions. The comparisons consistently show satisfactory performance of the model. The model is applied to a real stormwater system and accurately simulates the pressurization process in a horizontal channel. The two-phase model is applied to simulate air pockets rising and release motion in a vertical riser. The numerical model demonstrates the dominant factors that contribute to geyser formation, including air pocket size, pressurization of main pipe and surcharged state in the vertical riser. It captures the key dynamics of two-phase flow in the vertical riser, consistent with experimental results, suggesting that the code has an excellent potential of extending its use to practical applications.

[pt] Nesse trabalho, mostramos como simular um fluido em duas dimensões em um domínio com fronteiras arbitrárias. Nosso trabalho é baseado no esquema stable fluids desenvolvido por Joe Stam. A implementação é feita na GPU (Graphics Processing Unit), permitindo velocidade de interação com o fluido. Fazemos uso da linguagem Cg (C for Graphics), desenvolvida pela companhia NVidia. Nossas principais contribuições são o tratamento das múltiplas fronteiras, onde aplicamos interpolação bilinear para atingir melhores resultados, armazenamento das condições de fronteira usa apenas um canal de textura, e o uso de confinamento de vorticidade. / [en] In this work we show how to simulate fluids in two dimensions in a domain with arbitrary bondaries. Our work is based on the stable fluid scheme developed by Jo Stam. The implementation is done in GPU (Graphics Processinfg Unit), thus allowing fluid interaction speed. We use the language Cg (C for Graphics) developed by the company Nvídia. Our main contributions are the treatment of domains with multiple boundaries, where we apply bilinear interpolation to obtain better results, the storage of the bondaty conditions in a unique texturre channel, and the use of vorticity confinement.

A simulação numérica de escoamentos de fluidos em uma grande variedade de aplicações requer a utilização de técnicas numéricas de alta eficência e recursos computacionais de alto desempenho. O objetivo deste trabalho é iniciar uma investigação de escoamentos de fluido durante o enchimento de compartimentos de reservatórios. Uma abordagem inicial foi tratar problemas de escoamento em um canal, rebuscando a geometria do domínio para contemplar problemas mais complexos. Este trabalho apresenta o desenvolvimento e os resultados obtidos de um método numérico para simulação de escoamento de fluido incompressível em um domínio tridimensional, onde as equações de Navier-Stokes são desenvolvidas em uma formulação euleriana e discretizadas pelo método de elementos finitos. Os termos convectivos destas equações foram tratados pelo método semi-lagrangeano e o método de Galerkin foi utilizado para discretização espacial, um método baseado em decomposição LU foi utilizado para desacoplar as componentes de velocidade e pressão, sendo esta última calculada utilizando-se uma aproximação hidrostática. O domínio tridimensional foi representado por uma malha manipulada por uma estrutura de dados topológica, formada por células que definem elementos prismáticos lineares. Foram realizados experimentos sob várias alterações na geometria do domínio e também sob diferentes condições iniciais. Os resultados mostraram uma boa aproximação do método, quando analisado comparativamente a uma solução analítica / The numerical simulation of fluid flow over many applications require the use of numerical techniques of high efficiency and demand high computational power. This work aims at initiating an investigation about fluid flows while filling reservoirs. The initial approach was to deal with fluid flows in a retangular duct, as increasing the complexity of its geometry in order to model more complex cases. This document describes the development of a numerical method for the simulation of incompressible fluid flow over a threedimentional domain, where the Navier-Stokes equations were written under an Eulerian formulation and discretized by the Finite Elements Method. The semi-Lagrangean method was used to discretize the convective terms and the components of velocity and pressure were decoupled through the use of a method based on LU decomposition, where the final pressure was determined by using a hidrostatic aproximation. The threedimentional domain was represented by a mesh, manipulated by a topologic data structure, formed with cells that define linear prismatic elements. Many experiments were performed under different geometries of the domain and also under different initial conditions. The result showed a good approximation of the described method, when compared with an analitical solution

Ce travail de thèse a pour but d'étudier des écoulements non stationnaires de fluides incompressibles Newtoniens et non isothermes. Le problème est décrit par les lois de conservation de la masse, de la quantité de mouvement et de l'énergie. Nous nous intéressons au couplage entre le système de Navier-Stokes et l’équation de la chaleur hyperbolique (le résultat de la combinaison entre la loi de conservation d'énergie et la loi de Cattaneo). Cette dernière est une modification de la loi de Fourier utilisée habituellement, elle permet de surmonter « le paradoxe de la chaleur » et d'obtenir une description plus précise de la propagation de la chaleur. Le système couplé est un problème hyperbolique-parabolique dont la viscosité dépend de la température, alors que la capacité thermique et le terme de dissipation dépendent de la vitesse. Afin d’obtenir un résultat d'existence de solutions du problème couplé, nous démontrons d'abord l'existence et l'unicité de la solution du problème hyperbolique puis nous introduisons une discrétisation en temps et nous étudions la convergence des solutions approchées vers celles du problème original. Dans un deuxième temps nous étudions l'existence et l'unicité de la solution du système de Navier-Stokes muni des conditions aux limites de type Tresca puis de type Coulomb en dimension 2 et 3. Dans le chapitre 3, nous proposons une discrétisation en temps du problème d'écoulement dans le cas de la condition au limite de type Tresca et nous établissons la convergence des solutions approchées. Le dernier chapitre de ce mémoire est consacré à l'étude du problème couplé dans le cas de conditions aux limites de type Tresca. L'existence d'une solution est obtenue par un argument théorique de point fixe en dimension 2 et également par une méthode de discrétisation en temps qui conduit à résoudre sur chaque sous intervalle de temps un problème découplé pour la vitesse et la pression d'une part et la température d'autre part / The main objective of this thesis is to study nonstationary flows of incompressible Newtonian and non isothermal fluids. The problem is described by the laws of conservation of mass, momentum and energy. We consider the coupling between the Navier-Stokes system and the hyperbolic heat equation (the result of combination between the law of conservation of energy and the Cattaneo’s law). This one is a modification of the commonly used Fourier's law, it overcomes "the heat paradox" and gives a more accurate description of heat propagation. The coupled system is an hyperbolic-parabolic problem where the viscosity depends on the temperature but the thermal capacity and the dissipative term depend on the velocity. To obtain an existence result for the coupled system, we first prove the existence and uniqueness of the solution of the hyperbolic problem then we introduce a time discretization and we study the convergence of the approximate solutions to those of the original problem. In the second chapter, we study the existence and uniqueness of the solution of Navier-Stokes system with Tresca or Coulomb boundary conditions in dimension 2 and 3. In the third chapter, we propose a time discretization of the flow problem in the case of Tresca boundary conditions and we establish the convergence of the approximate solutions. The last chapter is devoted to the study of the coupled problem in the case of Tresca free boundary conditions. The existence of a solution is obtained by a theoretical argument (fixed-point theorem) in dimension 2 and also by a method of time discretization leading, on each time subinterval, to a decoupled problem for the velocity and pressure of a hand and the temperature of the other hand