The flow between rotating coaxial disks

Numerical approximations of nonunique solutions of the Navier-Stokes equations are obtained for steady viscous incompressible axisymmetric flow between two infinite rotating coaxial disks. For example, nineteen solutions have been found for the case when the disks are rotating with the same speed but in opposite direction. Bifurcation and perturbed bifurcation phenomena are observed. An efficient method is used to compute solution branches. The stability of solutions is analyzed. The rate of convergence of Newton's method at singular points is discussed. In particular, recovery of quadratic convergence at "normal limit points" and bifurcation points is indicated. Analytical construction of some of the computed solutions using singular perturbation techniques is discussed.

Experimentos numéricos para deposição de parafinas com modelo Multisólido

Este trabalho objetiva a construção de estruturas robustas e computacionalmente eficientes para a solução do problema de deposição de parafinas do ponto de vista do equilíbrio sólido-líquido. São avaliados diversos modelos termodinâmicos para a fase líquida: equação de estado de Peng-Robinson e os modelos de coeficiente de atividade de Solução Ideal, Wilson, UNIQUAC e UNIFAC. A fase sólida é caracterizada pelo modelo Multisólido. A previsão de formação de fase sólida é inicialmente prevista por um teste de estabilidade termodinâmica. Posteriormente, o sistema de equações não lineares que caracteriza o equilíbrio termodinâmico e as equações de balanço material é resolvido por três abordagens numéricas: método de Newton multivariável, método de Broyden e método Newton-Armijo. Diversos experimentos numéricos foram conduzidos de modo a avaliar os tempos de computação e a robustez frente a diversos cenários de estimativas iniciais dos métodos numéricos para os diferentes modelos e diferentes misturas. Os resultados indicam para a possibilidade de construção de arcabouços computacionais eficientes e robustos, que podem ser empregados acoplados a simuladores de escoamento em dutos, por exemplo.

Optimization on manifolds: Methods and applications

This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however,we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton's method in some detail as well as referencing the more important of the other approaches.There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. © 2010 Springer -Verlag Berlin Heidelberg.

Fast and effective analysis methods for fiber Raman amplifiers

A series of new single-step methods and their corresponding algorithms with automatic step size adjustment for model equations of fiber Raman amplifiers are proposed and compared in this paper. On the basis of the Newton-Raphson method, multiple shooting algorithms for the two-point boundary value problems involved in solving Raman amplifier propagation equations are constructed. A verified example shows that, compared with the traditional Runge-Kutta methods, the proposed methods can increase the accuracy by more than two orders of magnitude under the same conditions. The simulations for Raman amplifier propagation equations demonstrate that our methods can increase the computing speed by more than 5 times, extend the step size significantly, and improve the stability in comparison with the Dormand-Prince method. The numerical results show that the combination of the multiple shooting algorithms and the proposed methods has the capacity to rapidly and effectively solve the model equations of multipump Raman amplifiers under various conditions such as co-, counter- and bi-directionally pumped schemes, as well as dual-order pumped schemes.

Bézier曲线间最近距离的计算方法

针对Bzier曲线间最近距离计算问题,提出一种简捷、可靠的计算方法.该方法以Bernstein多项式算术运算为工具,建立Bzier曲线间最近距离的计算模型;然后充分利用Bzier曲面的凸包性质和de Casteljau分割算法进行求解.该方法几何意义明确,能有效地避免迭代初始值的选择和非线性方程组的求解,并可进一步推广应用于计算Bzier曲线/曲面间的最近距离.实验结果表明,该方法简捷、可靠且容易实现,与Newton-Raphson方法的融合可进一步提高该方法的运行速度.

A Network Charge-Orineted MOS Transistor Model

The MOS transistor physical model as described in [3] is presented here as a network model. The goal is to obtain an accurate model, suitable for simulation, free from certain problems reported in the literature [13], and conceptually as simple as possible. To achieve this goal the original model had to be extended and modified. The paper presents the derivation of the network model from physical equations, including the corrections which are required for simulation and which compensate for simplifications introduced in the original physical model. Our intrinsic MOS model consists of three nonlinear voltage-controlled capacitors and a dependent current source. The charges of the capacitors and the current of the current source are functions of the voltages $V_{gs}$, $V_{bs}$, and $V_{ds}$. The complete model consists of the intrinsic model plus the parasitics. The apparent simplicity of the model is a result of hiding information in the characteristics of the nonlinear components. The resulted network model has been checked by simulation and analysis. It is shown that the network model is suitable for simulation: It is defined for any value of the voltages; the functions involved are continuous and satisfy Lipschitz conditions with no jumps at region boundaries; Derivatives have been computed symbolically and are available for use by the Newton-Raphson method. The model"s functions can be measured from the terminals. It is also shown that small channel effects can be included in the model. Higher frequency effects can be modeled by using a network consisting of several sections of the basic lumped model. Future plans include a detailed comparison of the network model with models such as SPICE level 3 and a comparison of the multi- section higher frequency model with experiments.

QUANTITATIVE THREE DIMENSIONAL ELASTICITY IMAGING

Neoplastic tissue is typically highly vascularized, contains abnormal concentrations of extracellular proteins (e.g. collagen, proteoglycans) and has a high interstitial fluid pres- sure compared to most normal tissues. These changes result in an overall stiffening typical of most solid tumors. Elasticity Imaging (EI) is a technique which uses imaging systems to measure relative tissue deformation and thus noninvasively infer its mechanical stiffness. Stiffness is recovered from measured deformation by using an appropriate mathematical model and solving an inverse problem. The integration of EI with existing imaging modal- ities can improve their diagnostic and research capabilities. The aim of this work is to develop and evaluate techniques to image and quantify the mechanical properties of soft tissues in three dimensions (3D). To that end, this thesis presents and validates a method by which three dimensional ultrasound images can be used to image and quantify the shear modulus distribution of tissue mimicking phantoms. This work is presented to motivate and justify the use of this elasticity imaging technique in a clinical breast cancer screening study. The imaging methodologies discussed are intended to improve the specificity of mammography practices in general. During the development of these techniques, several issues concerning the accuracy and uniqueness of the result were elucidated. Two new algorithms for 3D EI are designed and characterized in this thesis. The first provides three dimensional motion estimates from ultrasound images of the deforming ma- terial. The novel features include finite element interpolation of the displacement field, inclusion of prior information and the ability to enforce physical constraints. The roles of regularization, mesh resolution and an incompressibility constraint on the accuracy of the measured deformation is quantified. The estimated signal to noise ratio of the measured displacement fields are approximately 1800, 21 and 41 for the axial, lateral and eleva- tional components, respectively. The second algorithm recovers the shear elastic modulus distribution of the deforming material by efficiently solving the three dimensional inverse problem as an optimization problem. This method utilizes finite element interpolations, the adjoint method to evaluate the gradient and a quasi-Newton BFGS method for optimiza- tion. Its novel features include the use of the adjoint method and TVD regularization with piece-wise constant interpolation. A source of non-uniqueness in this inverse problem is identified theoretically, demonstrated computationally, explained physically and overcome practically. Both algorithms were test on ultrasound data of independently characterized tissue mimicking phantoms. The recovered elastic modulus was in all cases within 35% of the reference elastic contrast. Finally, the preliminary application of these techniques to tomosynthesis images showed the feasiblity of imaging an elastic inclusion.

Energy conserving schemes for the simulation of musical instrument contact dynamics

Collisions are an innate part of the function of many musical instruments. Due to the nonlinear nature of contact forces, special care has to be taken in the construction of numerical schemes for simulation and sound synthesis. Finite difference schemes and other time-stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a Newtonian description of the system. However because impact forces are non-analytic functions of the phase space variables, algorithm stability can rarely be established this way. This paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling. The proposed numerical formulations follow from discretising Hamilton׳s equations of motion, generally leading to an implicit system of nonlinear equations that can be solved with Newton׳s method. The approach is first outlined for point mass collisions and then extended to distributed settings, such as vibrating strings and beams colliding with rigid obstacles. Stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples. The methodology is exemplified through a case study on tanpura string vibration, with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument.

Finite element modeling of debonding failures in FRP-strengthened RC beams: A dynamic approach

This paper is concerned with the finite element simulation of debonding failures in FRP-strengthened concrete beams. A key challenge for such simulations is that common solution techniques such as the Newton-Raphson method and the arc-length method often fail to converge. This paper examines the effectiveness of using a dynamic analysis approach in such FE simulations, in which debonding failure is treated as a dynamic problem and solved using an appropriate time integration method. Numerical results are presented to show that an appropriate dynamic approach effectively overcomes the convergence problem and provides accurate predictions of test results.

Investigation of tanpura string vibrations using a two-dimensional time-domain model incorporating coupling and bridge friction

Tanpura string vibrations have been investigated previously using numerical models based on energy conserving schemes derived from a Hamiltonian description in one-dimensional form. Such time-domain models have the property that, for the lossless case, the numerical Hamiltonian (representing total energy of the system) can be proven to be constant from one time step
to the next, irrespective of any of the system parameters; in practice the Hamiltonian can be shown to be conserved within machine precision. Models of this kind can reproduce a jvari effect, which results from the bridge-string interaction. However the one-dimensional formulation has recently been shown to fail to replicate the jvaris strong dependence on the thread placement. As a first step towards simulations which accurately emulate this sensitivity to the thread placement, a twodimensional model is proposed, incorporating coupling of controllable level between the two string polarisations at the string termination opposite from the barrier. In addition, a friction force acting when the string slides across the bridge in horizontal direction is introduced, thus effecting a further damping mechanism. In this preliminary study, the string is terminated at the position of the thread. As in the one-dimensional model, an implicit scheme has to be used to solve the system, employing Newton's method to calculate the updated positions and momentums of each string segment. The two-dimensional model is proven to be energy conserving when the loss parameters are set to zero, irrespective of the coupling constant. Both frequency-dependent and independent losses are then added to the string, so that the model can be compared to analogous instruments. The influence of coupling and the bridge friction are investigated.

Desenvolvimento e estudo de modelos de FACTS para o trânsito de energia: aplicação ao estudo das condições técnicas de ligação de linhas MAT

Dissertação para obtenção do grau de Mestre em Engenharia Electrotécnica Ramo de Energia

Étude numérique des origines hémodynamiques des oscillations dans des réseaux de capillaires

En simulant l’écoulement du sang dans un réseau de capillaires (en l’absence de contrôle biologique), il est possible d’observer la présence d’oscillations de certains paramètres comme le débit volumique, la pression et l’hématocrite (volume des globules rouges par rapport au volume du sang total). Ce comportement semble être en concordance avec certaines expériences in vivo. Malgré cet accord, il faut se demander si les fluctuations observées lors des simulations de l’écoulement sont physiques, numériques ou un artefact de modèles irréalistes puisqu’il existe toujours des différences entre des modélisations et des expériences in vivo. Pour répondre à cette question de façon satisfaisante, nous étudierons et analyserons l’écoulement du sang ainsi que la nature des oscillations observées dans quelques réseaux de capillaires utilisant un modèle convectif et un modèle moyenné pour décrire les équations de conservation de masse des globules rouges. Ces modèles tiennent compte de deux effets rhéologiques importants : l’effet Fåhraeus-Lindqvist décrivant la viscosité apparente dans un vaisseau et l’effet de séparation de phase schématisant la distribution des globules rouges aux points de bifurcation. Pour décrire ce dernier effet, deux lois de séparation de phase (les lois de Pries et al. et de Fenton et al.) seront étudiées et comparées. Dans ce mémoire, nous présenterons une description du problème physiologique (rhéologie du sang). Nous montrerons les modèles mathématiques employés (moyenné et convectif) ainsi que les lois de séparation de phase (Pries et al. et Fenton et al.) accompagnés d’une analyse des schémas numériques implémentés. Pour le modèle moyenné, nous employons le schéma numérique explicite traditionnel d’Euler ainsi qu’un nouveau schéma implicite qui permet de résoudre ce problème d’une manière efficace. Ceci est fait en utilisant une méthode de Newton- Krylov avec gradient conjugué préconditionné et la méthode de GMRES pour les itérations intérieures ainsi qu’une méthode quasi-Newton (la méthode de Broyden). Cette méthode inclura le schéma implicite d’Euler et la méthode des trapèzes. Pour le schéma convectif, la méthode explicite de Kiani et al. sera implémentée ainsi qu’une nouvelle approche implicite. La stabilité des deux modèles sera également explorée. À l’aide de trois différentes topologies, nous comparerons les résultats de ces deux modèles mathématiques ainsi que les lois de séparation de phase afin de déterminer dans quelle mesure les oscillations observées peuvent être attribuables au choix des modèles mathématiques ou au choix des méthodes numériques.

Nichtüberlappende Gebietszerlegungsmethoden für lineare und quasilineare (monotone und nichtmonotone) Probleme

In dieser Arbeit werden nichtüberlappende Gebietszerlegungsmethoden einerseits hinsichtlich der zu lösenden Problemklassen verallgemeinert und andererseits in bisher nicht untersuchten Kontexten betrachtet. Dabei stehen funktionalanalytische Untersuchungen zur Wohldefiniertheit, eindeutigen Lösbarkeit und Konvergenz im Vordergrund. Im ersten Teil werden lineare elliptische Dirichlet-Randwertprobleme behandelt, wobei neben Problemen mit dominantem Hauptteil auch solche mit singulärer Störung desselben, wie konvektions- oder reaktionsdominante Probleme zugelassen sind. Der zweite Teil befasst sich mit (gleichmäßig) monotonen koerziven quasilinearen elliptischen Dirichlet-Randwertproblemen. In beiden Fällen wird das Lipschitz-Gebiet in endlich viele Lipschitz-Teilgebiete zerlegt, wobei insbesondere Kreuzungspunkte und Teilgebiete ohne Außenrand zugelassen sind. Anschließend werden Transmissionsprobleme mit frei wählbaren $L^{\infty}$-Parameterfunktionen hergeleitet, wobei die Konormalenableitungen als Funktionale auf geeigneten Funktionenräumen über den Teilrändern ($H_{00}^{1/2}(\Gamma)$) interpretiert werden. Die iterative Lösung dieser Transmissionsprobleme mit einem Ansatz von Deng führt auf eine Substrukturierungsmethode mit Robin-artigen Transmissionsbedingungen, bei der eine Auswertung der Konormalenableitungen aufgrund einer geschickten Aufdatierung der Robin-Daten nicht notwendig ist (insbesondere ist die bekannte Robin-Robin-Methode von Lions als Spezialfall enthalten). Die Konvergenz bezüglich einer partitionierten $H^1$-Norm wird für beide Problemklassen gezeigt. Dabei werden keine über $H^1$ hinausgehende Regularitätsforderungen an die Lösungen gestellt und die Gebiete müssen keine zusätzlichen Glattheitsvoraussetzungen erfüllen. Im letzten Kapitel werden nichtmonotone koerzive quasilineare Probleme untersucht, wobei das Zugrunde liegende Gebiet nur in zwei Lipschitz-Teilgebiete zerlegt sein soll. Das zugehörige nichtlineare Transmissionsproblem wird durch Kirchhoff-Transformation in lineare Teilprobleme mit nichtlinearen Kopplungsbedingungen überführt. Ein optimierungsbasierter Lösungsansatz, welcher einen geeigneten Abstand der rücktransformierten Dirichlet-Daten der linearen Teilprobleme auf den Teilrändern minimiert, führt auf ein optimales Kontrollproblem. Die dabei entstehenden regularisierten freien Minimierungsprobleme werden mit Hilfe eines Gradientenverfahrens unter minimalen Glattheitsforderungen an die Nichtlinearitäten gelöst. Unter zusätzlichen Glattheitsvoraussetzungen an die Nichtlinearitäten und weiteren technischen Voraussetzungen an die Lösung des quasilinearen Ausgangsproblems, kann zudem die quadratische Konvergenz des Newton-Verfahrens gesichert werden.

A titration model: an analysis and solution

The Newton‐Raphson method is proposed for the solution of the nonlinear equation arising from a theoretical model of an acid/base titration. It is shown that it is necessary to modify the form of the equation in order that the iteration is guaranteed to converge. A particular example is considered to illustrate the analysis and method, and a BASIC program is included that can be used to predict the pH of any weak acid/weak base titration.

A search for invariant relative satellite motion

This report presents the canonical Hamiltonian formulation of relative satellite motion. The unperturbed Hamiltonian model is shown to be equivalent to the well known Hill-Clohessy-Wilshire (HCW) linear formulation. The in°uence of perturbations of the nonlinear Gravitational potential and the oblateness of the Earth; J2 perturbations are also modelled within the Hamiltonian formulation. The modelling incorporates eccentricity of the reference orbit. The corresponding Hamiltonian vector ¯elds are computed and implemented in Simulink. A numerical method is presented aimed at locating periodic or quasi-periodic relative satellite motion. The numerical method outlined in this paper is applied to the Hamiltonian system. Although the orbits considered here are weakly unstable at best, in the case of eccentricity only, the method ¯nds exact periodic orbits. When other perturbations such as nonlinear gravitational terms are added, drift is signicantly reduced and in the case of the J2 perturbation with and without the nonlinear gravitational potential term, bounded quasi-periodic solutions are found. Advantages of using Newton's method to search for periodic or quasi-periodic relative satellite motion include simplicity of implementation, repeatability of solutions due to its non-random nature, and fast convergence. Given that the use of bounded or drifting trajectories as control references carries practical di±culties over long-term missions, Principal Component Analysis (PCA) is applied to the quasi-periodic or slowly drifting trajectories to help provide a closed reference trajectory for the implementation of closed loop control. In order to evaluate the e®ect of the quality of the model used to generate the periodic reference trajectory, a study involving closed loop control of a simulated master/follower formation was performed. 2 The results of the closed loop control study indicate that the quality of the model employed for generating the reference trajectory used for control purposes has an important in°uence on the resulting amount of fuel required to track the reference trajectory. The model used to generate LQR controller gains also has an e®ect on the e±ciency of the controller.