This work considers the synthesis of brass instrument sounds using time-domain numerical methods. The operation of such a brass instrument is as follows. The player's lips are set into motion by forcing air through them, which in turn creates a pressure disturbance in the instrument mouthpiece. These disturbances produce waves that propagate along the air column, here described using one spatial dimension, to set up a series of resonances that interact with the vibrating lips of the player. Accurate description of these resonances requires the inclusion of attenuation of the wave during propagation, due to the boundary layer effects in the tube, along with how sound radiates from the instrument. A musically interesting instrument must also be flexible in the control of the available resonances, achieved, for example, by the manipulation of valves in trumpet-like instruments. These features are incorporated into a synthesis framework that allows the user to design and play a virtual instrument. This is all achieved using the finite-difference time-domain method. Robustness of simulations is vital, so a global energy measure is employed, where possible, to ensure numerical stability of the algorithms. A new passive model of viscothermal losses is proposed using tools from electrical network theory. An embedded system is also presented that couples a one-dimensional tube to the three-dimensional wave equation to model sound radiation. Additional control of the instrument using a simple lip model as well a time varying valve model to modify the instrument resonances is presented and the range of the virtual instrument is explored. Looking towards extensions of this tool, three nonlinear propagation models are compared, and differences related to distortion and response to changing bore profiles are highlighted. A preliminary experimental investigation into the effects of partially open valve configurations is also performed.

Les travaux effectués durant cette thèse s’inscrivent dans le cadre d’une collaboration entre l’équipe CEM de l’Institut Pascal et l’équipe EDPAN du Laboratoire de Mathématiques de l’Université Blaise Pascal de Clermont-Ferrand. Nous présentons ici une étude qui, partant de l’analyse du processus de Retournement Temporel en électromagnétisme, a débouché sur le développement d’une méthode originale baptisée Linear Combination of Configuration Fields (LCCF) ou, en français, Combinaison Linéaire de Configurations de Champs. Après avoir introduit l’ensemble des outils et méthodes utilisés dans ces travaux, ce mémoire détaille le processus de Retournement Temporel de base ainsi qu’un ajout apporté à celui-ci. Par la suite, la méthode LCCF s’étant révélée applicable à plusieurs problèmes d’identification de sources en électromagnétisme, nous nous consacrons à la présentation détaillée des différentes variantes de celle-ci et nous illustrons son utilisation sur de nombreux exemples numériques. / This Ph.D thesis is the result of a collaboration between the CEM team of Pascal Institute and the EDPAN team of the Laboratory of Mathematics of the Blaise Pascal University in Clermont-Ferrand. We present here a study based on Time Reversal process in Electromagnetics. This work led to the development of a novel method called Linear Combination of Configuration Field (LCCF). This thesis first introduces the tools and the numerical methods used during this work. Then, we describe the Time Reversal process and a possible improvement to the basic technic. Afterwards, several possible applications of the LCCF method to electromagnetic source identification problems are detailed and we illustrate each of it on various numerical examples.

Comparisons between measurements of the ground-state hyperfine structure and gravitational acceleration of hydrogen and antihydrogen could provide a test of fundamental physical theories such as CPT (charge conjugation, parity, time-reversal) and gravitational symmetries. Currently, antihydrogen traps are based on Malmberg-Penning traps. The number of antiprotons in Malmberg-Penning traps with sufficiently low energy to be suitable for trappable antihydrogen production may be reduced by the electrostatic space charge of the positrons and/or collisions among antiprotons. Alternative trap designs may be needed for future antihydrogen experiments. A computational tool is developed to simulate charged particle motion in customizable magnetic fields generated by combinations of current loops and current lines. The tool is used to examine charged particle confinement in two systems consisting of dual, levitated current loops. The loops are coaxial and arranged to produce a magnetic null curve. Conditions leading to confinement in the system are quantified and confinement modes near the null curve and encircling one or both loops are identified. Furthermore, the tool is used to examine and quantify charged particle motion parallel to the null curve in the large radius limit of the dual, levitated current loops. An alternative to new trap designs is to identify the effects of the positron space in existing traps and to find modes of operation where the space charge is beneficial. Techniques are developed to apply the Boltzmann density relation along curved magnetic field lines. Equilibrium electrostatic potential profiles for a positron plasma are computed by solving Poisson's equation using a finite-difference method. Equilibria are computed in a model Penning trap with an axially varying magnetic field. Also, equilibria are computed for a positron plasma in a model of the ALPHA trap. Electric potential wells are found to form self-consistently. The technique is expanded to compute equilibria for a two-species plasma with an antiproton plasma confined by the positron space charge. The two-species equilibria are used to estimate timescales associated with three-body recombination, losses due to collisions between antiprotons, and temperature equilibration. An equilibrium where the three-body recombination rate is the smallest is identified.

In this study, the development and testing of a finite-difference model for wind-induced flow in shallow lakes, and, in particular, a new technique for improving the land--water boundary representation, are documented. The model solves nonlinear, as well as linear, versions of the two-dimensional depth-integrated shallow water equations. Finite-difference methods on rectangular grids are widely used in numerical models of environmental flows. In these models, land--water boundaries are usually approximated by a series of perpendicular line segments, which enable the impermeability condition to be easily implemented. A disadvantage of this approach is that the actual boundary is often poorly approximated, particularly in regions which have complicated coastlines, and, as a result, currents in these regions cannot be accurately predicted. A technique for improving the land--water boundary representation in finite-difference models is introduced. This technique permits the model boundary to contain diagonal line segments, in addition to the vertical and horizontal line segments used in traditional models. The new technique is based on a simple concept and can easily be included in existing finite-difference models. In order to test the new method, the linearised shallow water equations are solved numerically for oscillatory wind-driven flow in lakes with simple geometry. Predictions obtained using the new approach are compared with predictions from the traditional stepped boundary and known analytic solutions. A significant improvement in the accuracy of results is noticed when the new approach is used, particularly in currents close to shore. The increased accuracy obtained using the improved boundary representation can lead to a significant computational saving, when compared with running the rectangular grid model with smaller grid spacings. A second-order analytic solution to the nonlinear shallow water equations is developed for oscillatory wind-driven flow in a rectangular lake. Comparisons between this solution and numerical results, obtained using the traditional stepped boundary and the improved boundary, verify the finite-difference formulae used in these models, including the approximations used for the cross-advective terms close to shore. Once more, currents are predicted with greater accuracy when the new technique for representing the land--water boundary is implemented. The lake circulation model is applied to the Lower Murray Lakes, South Australia, and predicted water levels at Tauwitchere Barrage are shown to agree very well with observations. The model is then used to examine the effectiveness of two schemes that have been proposed to increase wind-induced circulation, and therefore potentially decrease salinity, in Lake Albert, demonstrating the model's use as an efficient and effective tool for analysing flow behaviour in lakes. / Thesis (Ph.D.)--Mathematical Sciences (Applied Mathematics), 2005.

Cette thèse constitue un ensemble de travaux et de réflexions sur la question de la modélisation des applications électromagnétiques en nano-optique en utilisant la méthode des différences finies dans le domaine temporel (FDTD). Dans un premier temps, des codes FDTD bidimentionnels pour le calcul de bandes interdites photoniques ont été mis en oeuvre. Ces algorithmes tiennent comptes de la dispersion des métaux nobles dans la gamme optique décrite par le modèle de Drude ou de Drude-Lorentz. Ces programmes FDTD permettent de tenir compte de la propagation soit dans le plan perpendiculaire au plan d'invariance (appelé "cas dans le plan" ou "in-plane" en anglais) pour les deux polarisations TE et TM ainsi que le cas d'une propagation quelconque hors du plan (ou off-plane). Plusieurs diagrammes de bandes sont calculés et présentés pour les structures carrées et triangulaires dans les cas diélectriques et métalliques. Ensuite, nous avons implémenté un code BOR-FDTD, basé sur la discrétisation des équations de Maxwell exprimées en coordonnées cylindriques, pour la modélisation des guides d'ondes (ou d'autres objets) à symétrie de révolution. Les conditions absorbantes PML pour décrire l'espace libre sont intégrées à la BOR-FDTD ainsi que les deux modèles de Drude et de Drude-Lorentz. Des simulations ont été effectuées pour le calcul de modes propres de guides d'ondes coaxiaux et cylindriques sub-longueurs d'ondes faits en métal parfait et en métal réel (argent par exemple). Les résultats montrent la possibilité de guider des signaux optiques sans beaucoup de pertes dans un guide coaxial fait en argent de dimensions sublongueur d'onde. Ce dernier résultat est original et constitue une très importante avancée dans le domaine de la "nanoconnectique" en optique, plus particulièrement pour l'optique intégrée. Puis, un autre code numérique FDTD-3D a été élaboré pour la modélisation des structures périodiques (type cristaux photoniques tridimensionnels) éclairées en incidence oblique. Ce code intègre aussi les couches absorbantes PML ainsi que les modèles de dispersion de Drude et de Drude-lorentz. Les résultats obtenus sont comparés à ceux issus d'autres modèles théoriques. Les applications de ce code à l'étude de radôme, à l'excitation du mode TEM de la structure métallique à ouvertures annulaires et aux calculs des spectres d'extinction Raman montrent l'efficacité de la FDTD pour la modélisation de telles structures.

<p>The heat transfer in the filling phase of injection moulding is studied, based on Gunnar Aronsson’s distance model for flow expansion ([Aronsson], 1996).</p><p>The choice of a thermoplastic materials model is motivated by general physical properties, admitting temperature and pressure dependence. Two-phase, per-phase-incompressible, power-law fluids are considered. The shear rate expression takes into account pseudo-radial flow from a point inlet.</p><p>Instead of using a finite element (FEM) solver for the momentum equations a general analytical viscosity expression is used, adjusted to current axial temperature profiles and yielding expressions for axial velocity profile, pressure distribution, frozen layer expansion and special front convection.</p><p>The nonlinear energy partial differential equation is transformed into its conservative form, expressed by the internal energy, and is solved differently in the regions of streaming and stagnant flow, respectively. A finite difference (FD) scheme is chosen using control volume discretization to keep truncation errors small in the presence of non-uniform axial node spacing. Time and pseudo-radial marching is used. A local system of nonlinear FD equations is solved. In an outer iterative procedure the position of the boundary between the “solid” and “liquid” fluid cavity parts is determined. The uniqueness of the solution is claimed. In an inner iterative procedure the axial node temperatures are found. For all physically realistic material properties the convergence is proved. In particular the assumptions needed for the Newton-Mysovskii theorem are secured. The metal mould PDE is locally solved by a series expansion. For particular material properties the same technique can be applied to the “solid” fluid.</p><p>In the circular plate application, comparisons with the commercial FEM-FD program Moldflow (Mfl) are made, on two Mfl-database materials, for which model parameters are estimated/adjusted. The resulting time evolutions of pressures and temperatures are analysed, as well as the radial and axial profiles of temperature and frozen layer. The greatest differences occur at the flow front, where Mfl neglects axial heat convection. The effects of using more and more complex material models are also investigated. Our method performance is reported.</p><p>In the polygonal star-shaped plate application a geometric cavity model is developed. Comparison runs with the commercial FEM-FD program Cadmould (Cmd) are performed, on two Cmd-database materials, in an equilateral triangular mould cavity, and materials model parameters are estimated/adjusted. The resulting average temperatures at the end of filling are compared, on rays of different angular deviation from the closest corner ray and on different concentric circles, using angular and axial (cavity-halves) symmetry. The greatest differences occur in narrow flow sectors, fatal for our 2D model for a material with non-realistic viscosity model. We present some colour plots, e.g. for the residence time.</p><p>The classical square-root increase by time of the frozen layer is used for extrapolation. It may also be part of the front model in the initial collision with the cold metal mould. An extension of the model is found which describes the radial profile of the frozen layer in the circular plate application accurately also close to the inlet.</p><p>The well-posedness of the corresponding linearized problem is studied, as well as the stability of the linearized FD-scheme.</p> / Report code: LiU-TEK-LIC-2002:66.

Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system. The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state.  Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions. We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used. Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.

Simulating wave propagation in microstrip lines with Gyrotropic magnetic substrate is considered in this thesis. Since the static internal field distribution has an important effect on the device behavior, accurate determination of the internal fields are considered as well. To avoid the losses at microwave frequencies it is assumed that the magnetic substrate is saturated in the direction of local internal field. An iterative method to obtain the magnetization distribution has been developed. It is applied to a variety of nonlinear nonuniform magnetic material configurations that one may encounter in the design stage, subject to a nonuniform applied field. One of the main characteristics of the proposed iterative method to obtain the static internal field is that the results are supported by a uniqueness theorem in magnetostatics. The series of solutions Mn,Hn, where n is the iteration number, minimize the free Gibbs energy G(M) in sequence. They also satisfy the constitutive equation M = χH at the end of each iteration better than the previous one. Therefore based on the given uniqueness theorem, the unique stable equilibrium state M is determined. To simulate wave propagation in the Gyrotropic magnetic media a new FDTD formulation is proposed. The proposed formulation considers the static part of the electromagnetic field, obtained by using the iterative approach, as parameters and updates the dynamic parts in time. It solves the Landau-Lifshitz-Gilbert equation in consistency with Maxwell’s equations in time domain. The stability of the initial static field distribution ensures that the superposition of the time varying parts due to the propagating wave will not destabilize the code. Resonances in a cavity filled with YIG are obtained. Wave propagation through a microstrip line with YIG substrate is simulated. Magnetization oscillations around local internal field are visualized. It is proved that the excitation of magnetization precession which is accompanied by the excitation of magnetostatic waves is responsible for the gap in the scattering parameter S12. Key characteristics of the wide microstrip lines are verified in a full wave FDTD simulation. These characteristics are utilized in a variety of nonreciprocal devices like edgemode isolators and phase shifters.

The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.