Dense disparity map estimation respecting image discontinuities

[EN] We present an energy based approach to estimate a dense disparity map from a set of two weakly calibrated stereoscopic images while preserving its discontinuities resulting from image boundaries. We first derive a simplified expression for the disparity that allows us to estimate it from a stereo pair of images using an energy minimization approach. We assume that the epipolar geometry is known, and we include this information in the energy model. Discontinuities are preserved by means of a regularization term based on the Nagel-Enkelmann operator. We investigate the associated Euler-Lagrange equation of the energy functional, and we approach the solution of the underlying partial differential equation (PDE) using a gradient descent method The resulting parabolic problem has a unique solution. In order to reduce the risk to be trapped within some irrelevant local minima during the iterations, we use a focusing strategy based on a linear scalespace. Experimental results on both synthetic and real images arere presented to illustrate the capabilities of this PDE and scale-space based method.

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Model predictive control in thermal management of multiprocessor systems-on-chip

MultiProcessor Systems-on-Chip (MPSoC) are the core of nowadays and next generation computing platforms. Their relevance in the global market continuously increase, occupying an important role both in everydaylife products (e.g. smartphones, tablets, laptops, cars) and in strategical market sectors as aviation, defense, robotics, medicine. Despite of the incredible performance improvements in the recent years processors manufacturers have had to deal with issues, commonly called “Walls”, that have hindered the processors development. After the famous “Power Wall”, that limited the maximum frequency of a single core and marked the birth of the modern multiprocessors system-on-chip, the “Thermal Wall” and the “Utilization Wall” are the actual key limiter for performance improvements. The former concerns the damaging effects of the high temperature on the chip caused by the large power densities dissipation, whereas the second refers to the impossibility of fully exploiting the computing power of the processor due to the limitations on power and temperature budgets. In this thesis we faced these challenges by developing efficient and reliable solutions able to maximize performance while limiting the maximum temperature below a fixed critical threshold and saving energy. This has been possible by exploiting the Model Predictive Controller (MPC) paradigm that solves an optimization problem subject to constraints in order to find the optimal control decisions for the future interval. A fully-distributedMPC-based thermal controller with a far lower complexity respect to a centralized one has been developed. The control feasibility and interesting properties for the simplification of the control design has been proved by studying a partial differential equation thermal model. Finally, the controller has been efficiently included in more complex control schemes able to minimize energy consumption and deal with mixed-criticalities tasks

Diagnose der Residuumszirkulation und ihre Bedeutung für die Klimavariabilität der Stratosphäre

Mit einem in dieser Arbeit entwickelten Diagnose-Werkzeug wird im Rahmenrneines einfachen mechanistischenModells die Residuumszirkulation in der Stratosphärernermittelt. Die Residuumszirkulation wird als eine Schlüsselgröße für diernKlimavariabilität der Stratosphäre angesehen. Für die Diagnose wird mit einemrnmechanistischem Modell die Ausbreitung und das Brechen planetarer Wellenrnbeschrieben und der daraus resultierende Wellenantrieb bestimmt. Dieser Wellenantriebrnwird verwendet, um mit der numerischen Lösung einer elliptischenrnDifferentialgleichung die Residuumszirkulation zu berechnen.rnDieses Diagnose-Werkzeug wird genutzt, um in atmosphärischen Reanalysedatenrnden Zusammenhang der Residuumszirkulation mit verschiedenen Modenrnstratosphärischer Klimavariabilität zu untersuchen. Während für unterschiedlichernPhasen der quasi-zweijährigen Schwingung und der Nordatlantischen Oszillationrndie Residuumszirkulation deutliche Unterschiede aufzeigt, kann ein Einflussrndes 11-jährigen Sonnenfleckenzyklus auf die Residuumszirkulation nichtrneindeutig nachgewiesen werden. Eine Datenstudie zeigt, dass in den WintermonatenrnDezember und Januar die Stärke der Residuumszirkulation mit derrnTemperatur der unteren polaren Stratosphäre signifikant korreliert ist.

Monodromy calculations for some differential equations

In vielen Teilgebieten der Mathematik ist es w"{u}nschenswert, die Monodromiegruppe einer homogenen linearen Differenzialgleichung zu verstehen. Es sind nur wenige analytische Methoden zur Berechnung dieser Gruppe bekannt, daher entwickeln wir im ersten Teil dieser Arbeit eine numerische Methode zur Approximation ihrer Erzeuger.rnIm zweiten Abschnitt fassen wir die Grundlagen der Theorie der Uniformisierung Riemannscher Fl"achen und die der arithmetischen Fuchsschen Gruppen zusammen. Auss erdem erkl"aren wir, wie unsere numerische Methode bei der Bestimmung von uniformisierenden Differenzialgleichungen dienlich sein kann. F"ur arithmetische Fuchssche Gruppen mit zwei Erzeugern erhalten wir lokale Daten und freie Parameter von Lam'{e} Gleichungen, welche die zugeh"origen Riemannschen Fl"achen uniformisieren. rnIm dritten Teil geben wir einen kurzen Abriss zur homologischen Spiegelsymmetrie und f"uhren die $widehat{Gamma}$-Klasse ein. Wir erkl"aren wie diese genutzt werden kann, um eine Hodge-theoretische Version der Spiegelsymmetrie f"ur torische Varit"aten zu beweisen. Daraus gewinnen wir Vermutungen "uber die Monodromiegruppe $M$ von Picard-Fuchs Gleichungen von gewissen Familien $f:mathcal{X}rightarrow bbp^1$ von $n$-dimensionalen Calabi-Yau Variet"aten. Diese besagen erstens, dass bez"uglich einer nat"urlichen Basis die Monodromiematrizen in $M$ Eintr"age aus dem K"orper $bbq(zeta(2j+1)/(2 pi i)^{2j+1},j=1,ldots,lfloor (n-1)/2 rfloor)$ haben. Und zweitens, dass sich topologische Invarianten des Spiegelpartners einer generischen Faser von $f:mathcal{X}rightarrow bbp^1$ aus einem speziellen Element von $M$ rekonstruieren lassen. Schliess lich benutzen wir die im ersten Teil entwickelten Methoden zur Verifizierung dieser Vermutungen, vornehmlich in Hinblick auf Dimension drei. Dar"uber hinaus erstellen wir eine Liste von Kandidaten topologischer Invarianten von vermutlich existierenden dreidimensionalen Calabi-Yau Variet"aten mit $h^{1,1}=1$.

Theoretical considerations to the verification of dynamic multileaf collimated fields with a SLIC-type portal image detector

The verification possibilities of dynamically collimated treatment beams with a scanning liquid ionization chamber electronic portal image device (SLIC-EPID) are investigated. The ion concentration in the liquid of a SLIC-EPID and therefore the read-out signal is determined by two parameters of a differential equation describing the creation and recombination of the ions. Due to the form of this equation, the portal image detector describes a nonlinear dynamic system with memory. In this work, the parameters of the differential equation were experimentally determined for the particular chamber in use and for an incident open 6 MV photon beam. The mathematical description of the ion concentration was then used to predict portal images of intensity-modulated photon beams produced by a dynamic delivery technique, the sliding window approach. Due to the nature of the differential equation, a mathematical condition for 'reliable leaf motion verification' in the sliding window technique can be formulated. It is shown that the time constants for both formation and decay of the equilibrium concentration in the chamber is in the order of seconds. In order to guarantee reliable leaf motion verification, these time constants impose a constraint on the rapidity of the image-read out for a given maximum leaf speed. For a leaf speed of 2 cm s(-1), a minimum image acquisition frequency of about 2 Hz is required. Current SLIC-EPID systems are usually too slow since they need about a second to acquire a portal image. However, if the condition is fulfilled, the memory property of the system can be used to reconstruct the leaf motion. It is shown that a simple edge detecting algorithm can be employed to determine the leaf positions. The method is also very robust against image noise.

Numerical Error Prediction and its applications in CFD using tau-estimation

Nowadays, Computational Fluid Dynamics (CFD) solvers are widely used within the industry to model fluid flow phenomenons. Several fluid flow model equations have been employed in the last decades to simulate and predict forces acting, for example, on different aircraft configurations. Computational time and accuracy are strongly dependent on the fluid flow model equation and the spatial dimension of the problem considered. While simple models based on perfect flows, like panel methods or potential flow models can be very fast to solve, they usually suffer from a poor accuracy in order to simulate real flows (transonic, viscous). On the other hand, more complex models such as the full Navier- Stokes equations provide high fidelity predictions but at a much higher computational cost. Thus, a good compromise between accuracy and computational time has to be fixed for engineering applications. A discretisation technique widely used within the industry is the so-called Finite Volume approach on unstructured meshes. This technique spatially discretises the flow motion equations onto a set of elements which form a mesh, a discrete representation of the continuous domain. Using this approach, for a given flow model equation, the accuracy and computational time mainly depend on the distribution of nodes forming the mesh. Therefore, a good compromise between accuracy and computational time might be obtained by carefully defining the mesh. However, defining an optimal mesh for complex flows and geometries requires a very high level expertize in fluid mechanics and numerical analysis, and in most cases a simple guess of regions of the computational domain which might affect the most the accuracy is impossible. Thus, it is desirable to have an automatized remeshing tool, which is more flexible with unstructured meshes than its structured counterpart. However, adaptive methods currently in use still have an opened question: how to efficiently drive the adaptation ? Pioneering sensors based on flow features generally suffer from a lack of reliability, so in the last decade more effort has been made in developing numerical error-based sensors, like for instance the adjoint-based adaptation sensors. While very efficient at adapting meshes for a given functional output, the latter method is very expensive as it requires to solve a dual set of equations and computes the sensor on an embedded mesh. Therefore, it would be desirable to develop a more affordable numerical error estimation method. The current work aims at estimating the truncation error, which arises when discretising a partial differential equation. These are the higher order terms neglected in the construction of the numerical scheme. The truncation error provides very useful information as it is strongly related to the flow model equation and its discretisation. On one hand, it is a very reliable measure of the quality of the mesh, therefore very useful in order to drive a mesh adaptation procedure. On the other hand, it is strongly linked to the flow model equation, so that a careful estimation actually gives information on how well a given equation is solved, which may be useful in the context of _ -extrapolation or zonal modelling. The following work is organized as follows: Chap. 1 contains a short review of mesh adaptation techniques as well as numerical error prediction. In the first section, Sec. 1.1, the basic refinement strategies are reviewed and the main contribution to structured and unstructured mesh adaptation are presented. Sec. 1.2 introduces the definitions of errors encountered when solving Computational Fluid Dynamics problems and reviews the most common approaches to predict them. Chap. 2 is devoted to the mathematical formulation of truncation error estimation in the context of finite volume methodology, as well as a complete verification procedure. Several features are studied, such as the influence of grid non-uniformities, non-linearity, boundary conditions and non-converged numerical solutions. This verification part has been submitted and accepted for publication in the Journal of Computational Physics. Chap. 3 presents a mesh adaptation algorithm based on truncation error estimates and compares the results to a feature-based and an adjoint-based sensor (in collaboration with Jorge Ponsín, INTA). Two- and three-dimensional cases relevant for validation in the aeronautical industry are considered. This part has been submitted and accepted in the AIAA Journal. An extension to Reynolds Averaged Navier- Stokes equations is also included, where _ -estimation-based mesh adaptation and _ -extrapolation are applied to viscous wing profiles. The latter has been submitted in the Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. Keywords: mesh adaptation, numerical error prediction, finite volume Hoy en día, la Dinámica de Fluidos Computacional (CFD) es ampliamente utilizada dentro de la industria para obtener información sobre fenómenos fluidos. La Dinámica de Fluidos Computacional considera distintas modelizaciones de las ecuaciones fluidas (Potencial, Euler, Navier-Stokes, etc) para simular y predecir las fuerzas que actúan, por ejemplo, sobre una configuración de aeronave. El tiempo de cálculo y la precisión en la solución depende en gran medida de los modelos utilizados, así como de la dimensión espacial del problema considerado. Mientras que modelos simples basados en flujos perfectos, como modelos de flujos potenciales, se pueden resolver rápidamente, por lo general aducen de una baja precisión a la hora de simular flujos reales (viscosos, transónicos, etc). Por otro lado, modelos más complejos tales como el conjunto de ecuaciones de Navier-Stokes proporcionan predicciones de alta fidelidad, a expensas de un coste computacional mucho más elevado. Por lo tanto, en términos de aplicaciones de ingeniería se debe fijar un buen compromiso entre precisión y tiempo de cálculo. Una técnica de discretización ampliamente utilizada en la industria es el método de los Volúmenes Finitos en mallas no estructuradas. Esta técnica discretiza espacialmente las ecuaciones del movimiento del flujo sobre un conjunto de elementos que forman una malla, una representación discreta del dominio continuo. Utilizando este enfoque, para una ecuación de flujo dado, la precisión y el tiempo computacional dependen principalmente de la distribución de los nodos que forman la malla. Por consiguiente, un buen compromiso entre precisión y tiempo de cálculo se podría obtener definiendo cuidadosamente la malla, concentrando sus elementos en aquellas zonas donde sea estrictamente necesario. Sin embargo, la definición de una malla óptima para corrientes y geometrías complejas requiere un nivel muy alto de experiencia en la mecánica de fluidos y el análisis numérico, así como un conocimiento previo de la solución. Aspecto que en la mayoría de los casos no está disponible. Por tanto, es deseable tener una herramienta que permita adaptar los elementos de malla de forma automática, acorde a la solución fluida (remallado). Esta herramienta es generalmente más flexible en mallas no estructuradas que con su homóloga estructurada. No obstante, los métodos de adaptación actualmente en uso todavía dejan una pregunta abierta: cómo conducir de manera eficiente la adaptación. Sensores pioneros basados en las características del flujo en general, adolecen de una falta de fiabilidad, por lo que en la última década se han realizado grandes esfuerzos en el desarrollo numérico de sensores basados en el error, como por ejemplo los sensores basados en el adjunto. A pesar de ser muy eficientes en la adaptación de mallas para un determinado funcional, este último método resulta muy costoso, pues requiere resolver un doble conjunto de ecuaciones: la solución y su adjunta. Por tanto, es deseable desarrollar un método numérico de estimación de error más asequible. El presente trabajo tiene como objetivo estimar el error local de truncación, que aparece cuando se discretiza una ecuación en derivadas parciales. Estos son los términos de orden superior olvidados en la construcción del esquema numérico. El error de truncación proporciona una información muy útil sobre la solución: es una medida muy fiable de la calidad de la malla, obteniendo información que permite llevar a cabo un procedimiento de adaptación de malla. Está fuertemente relacionado al modelo matemático fluido, de modo que una estimación precisa garantiza la idoneidad de dicho modelo en un campo fluido, lo que puede ser útil en el contexto de modelado zonal. Por último, permite mejorar la precisión de la solución resolviendo un nuevo sistema donde el error local actúa como término fuente (_ -extrapolación). El presenta trabajo se organiza de la siguiente manera: Cap. 1 contiene una breve reseña de las técnicas de adaptación de malla, así como de los métodos de predicción de los errores numéricos. En la primera sección, Sec. 1.1, se examinan las estrategias básicas de refinamiento y se presenta la principal contribución a la adaptación de malla estructurada y no estructurada. Sec 1.2 introduce las definiciones de los errores encontrados en la resolución de problemas de Dinámica Computacional de Fluidos y se examinan los enfoques más comunes para predecirlos. Cap. 2 está dedicado a la formulación matemática de la estimación del error de truncación en el contexto de la metodología de Volúmenes Finitos, así como a un procedimiento de verificación completo. Se estudian varias características que influyen en su estimación: la influencia de la falta de uniformidad de la malla, el efecto de las no linealidades del modelo matemático, diferentes condiciones de contorno y soluciones numéricas no convergidas. Esta parte de verificación ha sido presentada y aceptada para su publicación en el Journal of Computational Physics. Cap. 3 presenta un algoritmo de adaptación de malla basado en la estimación del error de truncación y compara los resultados con sensores de featured-based y adjointbased (en colaboración con Jorge Ponsín del INTA). Se consideran casos en dos y tres dimensiones, relevantes para la validación en la industria aeronáutica. Este trabajo ha sido presentado y aceptado en el AIAA Journal. También se incluye una extensión de estos métodos a las ecuaciones RANS (Reynolds Average Navier- Stokes), en donde adaptación de malla basada en _ y _ -extrapolación son aplicados a perfiles con viscosidad de alas. Este último trabajo se ha presentado en los Actas de la Institución de Ingenieros Mecánicos, Parte G: Journal of Aerospace Engineering. Palabras clave: adaptación de malla, predicción del error numérico, volúmenes finitos

Design and development of compact optical systems = Diseño y desarrollo de sistemas ópticos compactos

Esta tesis considera dos tipos de aplicaciones del diseño óptico: óptica formadora de imagen por un lado, y óptica anidólica (nonimaging) o no formadora de imagen, por otro. Las ópticas formadoras de imagen tienen como objetivo la obtención de imágenes de puntos del objeto en el plano de la imagen. Por su parte, la óptica anidólica, surgida del desarrollo de aplicaciones de concentración e iluminación, se centra en la transferencia de energía en forma de luz de forma eficiente. En general, son preferibles los diseños ópticos que den como resultado sistemas compactos, para ambos tipos de ópticas (formadora de imagen y anidólica). En el caso de los sistemas anidólicos, una óptica compacta permite tener costes de producción reducidos. Hay dos razones: (1) una óptica compacta presenta volúmenes reducidos, lo que significa que se necesita menos material para la producción en masa; (2) una óptica compacta es pequeña y ligera, lo que ahorra costes en el transporte. Para los sistemas ópticos de formación de imagen, además de las ventajas anteriores, una óptica compacta aumenta la portabilidad de los dispositivos, que es una gran ventaja en tecnologías de visualización portátiles, tales como cascos de realidad virtual (HMD del inglés Head Mounted Display). Esta tesis se centra por tanto en nuevos enfoques de diseño de sistemas ópticos compactos para aplicaciones tanto de formación de imagen, como anidólicas. Los colimadores son uno de los diseños clásicos dentro la óptica anidólica, y se pueden utilizar en aplicaciones fotovoltaicas y de iluminación. Hay varios enfoques a la hora de diseñar estos colimadores. Los diseños convencionales tienen una relación de aspecto mayor que 0.5. Con el fin de reducir la altura del colimador manteniendo el área de iluminación, esta tesis presenta un diseño de un colimador multicanal. En óptica formadora de imagen, las superficies asféricas y las superficies sin simetría de revolución (o freeform) son de gran utilidad de cara al control de las aberraciones de la imagen y para reducir el número y tamaño de los elementos ópticos. Debido al rápido desarrollo de sistemas de computación digital, los trazados de rayos se pueden realizar de forma rápida y sencilla para evaluar el rendimiento del sistema óptico analizado. Esto ha llevado a los diseños ópticos modernos a ser generados mediante el uso de diferentes técnicas de optimización multi-paramétricas. Estas técnicas requieren un buen diseño inicial como punto de partida para el diseño final, que será obtenido tras un proceso de optimización. Este proceso precisa un método de diseño directo para superficies asféricas y freeform que den como resultado un diseño cercano al óptimo. Un método de diseño basado en ecuaciones diferenciales se presenta en esta tesis para obtener un diseño óptico formado por una superficie freeform y dos superficies asféricas. Esta tesis consta de cinco capítulos. En Capítulo 1, se presentan los conceptos básicos de la óptica formadora de imagen y de la óptica anidólica, y se introducen las técnicas clásicas del diseño de las mismas. El Capítulo 2 describe el diseño de un colimador ultra-compacto. La relación de aspecto ultra-baja de este colimador se logra mediante el uso de una estructura multicanal. Se presentará su procedimiento de diseño, así como un prototipo fabricado y la caracterización del mismo. El Capítulo 3 describe los conceptos principales de la optimización de los sistemas ópticos: función de mérito y método de mínimos cuadrados amortiguados. La importancia de un buen punto de partida se demuestra mediante la presentación de un mismo ejemplo visto a través de diferentes enfoques de diseño. El método de las ecuaciones diferenciales se presenta como una herramienta ideal para obtener un buen punto de partida para la solución final. Además, diferentes técnicas de interpolación y representación de superficies asféricas y freeform se presentan para el procedimiento de optimización. El Capítulo 4 describe la aplicación del método de las ecuaciones diferenciales para un diseño de un sistema óptico de una sola superficie freeform. Algunos conceptos básicos de geometría diferencial son presentados para una mejor comprensión de la derivación de las ecuaciones diferenciales parciales. También se presenta un procedimiento de solución numérica. La condición inicial está elegida como un grado de libertad adicional para controlar la superficie donde se forma la imagen. Basado en este enfoque, un diseño anastigmático se puede obtener fácilmente y se utiliza como punto de partida para un ejemplo de diseño de un HMD con una única superficie reflectante. Después de la optimización, dicho diseño muestra mejor rendimiento. El Capítulo 5 describe el método de las ecuaciones diferenciales ampliado para diseños de dos superficies asféricas. Para diseños ópticos de una superficie, ni la superficie de imagen ni la correspondencia entre puntos del objeto y la imagen pueden ser prescritas. Con esta superficie adicional, la superficie de la imagen se puede prescribir. Esto conduce a un conjunto de tres ecuaciones diferenciales ordinarias implícitas. La solución numérica se puede obtener a través de cualquier software de cálculo numérico. Dicho procedimiento también se explica en este capítulo. Este método de diseño da como resultado una lente anastigmática, que se comparará con una lente aplanática. El diseño anastigmático converge mucho más rápido en la optimización y la solución final muestra un mejor rendimiento. ABSTRACT We will consider optical design from two points of view: imaging optics and nonimaging optics. Imaging optics focuses on the imaging of the points of the object. Nonimaging optics arose from the development of concentrators and illuminators, focuses on the transfer of light energy, and has wide applications in illumination and concentration photovoltaics. In general, compact optical systems are necessary for both imaging and nonimaging designs. For nonimaging optical systems, compact optics use to be important for reducing cost. The reasons are twofold: (1) compact optics is small in volume, which means less material is needed for mass-production; (2) compact optics is small in size and light in weight, which saves cost in transportation. For imaging optical systems, in addition to the above advantages, compact optics increases portability of devices as well, which contributes a lot to wearable display technologies such as Head Mounted Displays (HMD). This thesis presents novel design approaches of compact optical systems for both imaging and nonimaging applications. Collimator is a typical application of nonimaging optics in illumination, and can be used in concentration photovoltaics as well due to the reciprocity of light. There are several approaches for collimator designs. In general, all of these approaches have an aperture diameter to collimator height not greater than 2. In order to reduce the height of the collimator while maintaining the illumination area, a multichannel design is presented in this thesis. In imaging optics, aspheric and freeform surfaces are useful in controlling image aberrations and reducing the number and size of optical elements. Due to the rapid development of digital computing systems, ray tracing can be easily performed to evaluate the performance of optical system. This has led to the modern optical designs created by using different multi-parametric optimization techniques. These techniques require a good initial design to be a starting point so that the final design after optimization procedure can reach the optimum solution. This requires a direct design method for aspheric and freeform surface close to the optimum. A differential equation based design method is presented in this thesis to obtain single freeform and double aspheric surfaces. The thesis comprises of five chapters. In Chapter 1, basic concepts of imaging and nonimaging optics are presented and typical design techniques are introduced. Readers can obtain an understanding for the following chapters. Chapter 2 describes the design of ultra-compact collimator. The ultra-low aspect ratio of this collimator is achieved by using a multichannel structure. Its design procedure is presented together with a prototype and its evaluation. The ultra-compactness of the device has been approved. Chapter 3 describes the main concepts of optimizing optical systems: merit function and Damped Least-Squares method. The importance of a good starting point is demonstrated by presenting an example through different design approaches. The differential equation method is introduced as an ideal tool to obtain a good starting point for the final solution. Additionally, different interpolation and representation techniques for aspheric and freeform surface are presented for optimization procedure. Chapter 4 describes the application of differential equation method in the design of single freeform surface optical system. Basic concepts of differential geometry are presented for understanding the derivation of partial differential equations. A numerical solution procedure is also presented. The initial condition is chosen as an additional freedom to control the image surface. Based on this approach, anastigmatic designs can be readily obtained and is used as starting point for a single reflective surface HMD design example. After optimization, the evaluation shows better MTF. Chapter 5 describes the differential equation method extended to double aspheric surface designs. For single optical surface designs, neither image surface nor the mapping from object to image can be prescribed. With one more surface added, the image surface can be prescribed. This leads to a set of three implicit ordinary differential equations. Numerical solution can be obtained by MATLAB and its procedure is also explained. An anastigmatic lens is derived from this design method and compared with an aplanatic lens. The anastigmatic design converges much faster in optimization and the final solution shows better performance.

Co-simulation Methodologies for Hybrid and Electric Vehicle Dynamics

In recent decades, full electric and hybrid electric vehicles have emerged as an alternative to conventional cars due to a range of factors, including environmental and economic aspects. These vehicles are the result of considerable efforts to seek ways of reducing the use of fossil fuel for vehicle propulsion. Sophisticated technologies such as hybrid and electric powertrains require careful study and optimization. Mathematical models play a key role at this point. Currently, many advanced mathematical analysis tools, as well as computer applications have been built for vehicle simulation purposes. Given the great interest of hybrid and electric powertrains, along with the increasing importance of reliable computer-based models, the author decided to integrate both aspects in the research purpose of this work. Furthermore, this is one of the first final degree projects held at the ETSII (Higher Technical School of Industrial Engineers) that covers the study of hybrid and electric propulsion systems. The present project is based on MBS3D 2.0, a specialized software for the dynamic simulation of multibody systems developed at the UPM Institute of Automobile Research (INSIA). Automobiles are a clear example of complex multibody systems, which are present in nearly every field of engineering. The work presented here benefits from the availability of MBS3D software. This program has proven to be a very efficient tool, with a highly developed underlying mathematical formulation. On this basis, the focus of this project is the extension of MBS3D features in order to be able to perform dynamic simulations of hybrid and electric vehicle models. This requires the joint simulation of the mechanical model of the vehicle, together with the model of the hybrid or electric powertrain. These sub-models belong to completely different physical domains. In fact the powertrain consists of energy storage systems, electrical machines and power electronics, connected to purely mechanical components (wheels, suspension, transmission, clutch…). The challenge today is to create a global vehicle model that is valid for computer simulation. Therefore, the main goal of this project is to apply co-simulation methodologies to a comprehensive model of an electric vehicle, where sub-models from different areas of engineering are coupled. The created electric vehicle (EV) model consists of a separately excited DC electric motor, a Li-ion battery pack, a DC/DC chopper converter and a multibody vehicle model. Co-simulation techniques allow car designers to simulate complex vehicle architectures and behaviors, which are usually difficult to implement in a real environment due to safety and/or economic reasons. In addition, multi-domain computational models help to detect the effects of different driving patterns and parameters and improve the models in a fast and effective way. Automotive designers can greatly benefit from a multidisciplinary approach of new hybrid and electric vehicles. In this case, the global electric vehicle model includes an electrical subsystem and a mechanical subsystem. The electrical subsystem consists of three basic components: electric motor, battery pack and power converter. A modular representation is used for building the dynamic model of the vehicle drivetrain. This means that every component of the drivetrain (submodule) is modeled separately and has its own general dynamic model, with clearly defined inputs and outputs. Then, all the particular submodules are assembled according to the drivetrain configuration and, in this way, the power flow across the components is completely determined. Dynamic models of electrical components are often based on equivalent circuits, where Kirchhoff’s voltage and current laws are applied to draw the algebraic and differential equations. Here, Randles circuit is used for dynamic modeling of the battery and the electric motor is modeled through the analysis of the equivalent circuit of a separately excited DC motor, where the power converter is included. The mechanical subsystem is defined by MBS3D equations. These equations consider the position, velocity and acceleration of all the bodies comprising the vehicle multibody system. MBS3D 2.0 is entirely written in MATLAB and the structure of the program has been thoroughly studied and understood by the author. MBS3D software is adapted according to the requirements of the applied co-simulation method. Some of the core functions are modified, such as integrator and graphics, and several auxiliary functions are added in order to compute the mathematical model of the electrical components. By coupling and co-simulating both subsystems, it is possible to evaluate the dynamic interaction among all the components of the drivetrain. ‘Tight-coupling’ method is used to cosimulate the sub-models. This approach integrates all subsystems simultaneously and the results of the integration are exchanged by function-call. This means that the integration is done jointly for the mechanical and the electrical subsystem, under a single integrator and then, the speed of integration is determined by the slower subsystem. Simulations are then used to show the performance of the developed EV model. However, this project focuses more on the validation of the computational and mathematical tool for electric and hybrid vehicle simulation. For this purpose, a detailed study and comparison of different integrators within the MATLAB environment is done. Consequently, the main efforts are directed towards the implementation of co-simulation techniques in MBS3D software. In this regard, it is not intended to create an extremely precise EV model in terms of real vehicle performance, although an acceptable level of accuracy is achieved. The gap between the EV model and the real system is filled, in a way, by introducing the gas and brake pedals input, which reflects the actual driver behavior. This input is included directly in the differential equations of the model, and determines the amount of current provided to the electric motor. For a separately excited DC motor, the rotor current is proportional to the traction torque delivered to the car wheels. Therefore, as it occurs in the case of real vehicle models, the propulsion torque in the mathematical model is controlled through acceleration and brake pedal commands. The designed transmission system also includes a reduction gear that adapts the torque coming for the motor drive and transfers it. The main contribution of this project is, therefore, the implementation of a new calculation path for the wheel torques, based on performance characteristics and outputs of the electric powertrain model. Originally, the wheel traction and braking torques were input to MBS3D through a vector directly computed by the user in a MATLAB script. Now, they are calculated as a function of the motor current which, in turn, depends on the current provided by the battery pack across the DC/DC chopper converter. The motor and battery currents and voltages are the solutions of the electrical ODE (Ordinary Differential Equation) system coupled to the multibody system. Simultaneously, the outputs of MBS3D model are the position, velocity and acceleration of the vehicle at all times. The motor shaft speed is computed from the output vehicle speed considering the wheel radius, the gear reduction ratio and the transmission efficiency. This motor shaft speed, somehow available from MBS3D model, is then introduced in the differential equations corresponding to the electrical subsystem. In this way, MBS3D and the electrical powertrain model are interconnected and both subsystems exchange values resulting as expected with tight-coupling approach.When programming mathematical models of complex systems, code optimization is a key step in the process. A way to improve the overall performance of the integration, making use of C/C++ as an alternative programming language, is described and implemented. Although this entails a higher computational burden, it leads to important advantages regarding cosimulation speed and stability. In order to do this, it is necessary to integrate MATLAB with another integrated development environment (IDE), where C/C++ code can be generated and executed. In this project, C/C++ files are programmed in Microsoft Visual Studio and the interface between both IDEs is created by building C/C++ MEX file functions. These programs contain functions or subroutines that can be dynamically linked and executed from MATLAB. This process achieves reductions in simulation time up to two orders of magnitude. The tests performed with different integrators, also reveal the stiff character of the differential equations corresponding to the electrical subsystem, and allow the improvement of the cosimulation process. When varying the parameters of the integration and/or the initial conditions of the problem, the solutions of the system of equations show better dynamic response and stability, depending on the integrator used. Several integrators, with variable and non-variable step-size, and for stiff and non-stiff problems are applied to the coupled ODE system. Then, the results are analyzed, compared and discussed. From all the above, the project can be divided into four main parts: 1. Creation of the equation-based electric vehicle model; 2. Programming, simulation and adjustment of the electric vehicle model; 3. Application of co-simulation methodologies to MBS3D and the electric powertrain subsystem; and 4. Code optimization and study of different integrators. Additionally, in order to deeply understand the context of the project, the first chapters include an introduction to basic vehicle dynamics, current classification of hybrid and electric vehicles and an explanation of the involved technologies such as brake energy regeneration, electric and non-electric propulsion systems for EVs and HEVs (hybrid electric vehicles) and their control strategies. Later, the problem of dynamic modeling of hybrid and electric vehicles is discussed. The integrated development environment and the simulation tool are also briefly described. The core chapters include an explanation of the major co-simulation methodologies and how they have been programmed and applied to the electric powertrain model together with the multibody system dynamic model. Finally, the last chapters summarize the main results and conclusions of the project and propose further research topics. In conclusion, co-simulation methodologies are applicable within the integrated development environments MATLAB and Visual Studio, and the simulation tool MBS3D 2.0, where equation-based models of multidisciplinary subsystems, consisting of mechanical and electrical components, are coupled and integrated in a very efficient way.

Mathematical modelling of neural processes within the oculomotor system

It is presented a mathematical model of the oculomotor plant, based on experimental data in cats. The system that generates, from the neuronal processes at the motoneuron, the control signals to the eye muscles that moves the eye. In contrast with previous models, that base the eye movement related motoneuron behavior on a first order linear differential equation, non-linear effects are described: A dependency on the eye angular position of the model parameters.

A model-based approach for assessing in vivo combination therapy interactions

We present an approach for evaluating the efficacy of combination antitumor agent schedules that accounts for order and timing of drug administration. Our model-based approach compares in vivo tumor volume data over a time course and offers a quantitative definition for additivity of drug effects, relative to which synergism and antagonism are interpreted. We begin by fitting data from individual mice receiving at most one drug to a differential equation tumor growth/drug effect model and combine individual parameter estimates to obtain population statistics. Using two null hypotheses: (i) combination therapy is consistent with additivity or (ii) combination therapy is equivalent to treating with the more effective single agent alone, we compute predicted tumor growth trajectories and their distribution for combination treated animals. We illustrate this approach by comparing entire observed and expected tumor volume trajectories for a data set in which HER-2/neu-overexpressing MCF-7 human breast cancer xenografts are treated with a humanized, anti-HER-2 monoclonal antibody (rhuMAb HER-2), doxorubicin, or one of five proposed combination therapy schedules.

Modelagem estocástica da dispersão axial: aplicação em um reator tubular de polimerização.

Reatores tubulares de polimerização podem apresentar um perfil de velocidade bastante distorcido. Partindo desta observação, um modelo estocástico baseado no modelo de dispersão axial foi proposto para a representação matemática da fluidodinâmica de um reator tubular para produção de poliestireno. A equação diferencial foi obtida inserindo a aleatoriedade no parâmetro de dispersão, resultando na adição de um termo estocástico ao modelo capaz de simular as oscilações observadas experimentalmente. A equação diferencial estocástica foi discretizada e resolvida pelo método Euler-Maruyama de forma satisfatória. Uma função estimadora foi desenvolvida para a obtenção do parâmetro do termo estocástico e o parâmetro do termo determinístico foi calculado pelo método dos mínimos quadrados. Uma análise de convergência foi conduzida para determinar o número de elementos da discretização e o modelo foi validado através da comparação de trajetórias e de intervalos de confiança computacionais com dados experimentais. O resultado obtido foi satisfatório, o que auxilia na compreensão do comportamento fluidodinâmico complexo do reator estudado.

Equações de quarta ordem na modelagem de oscilações de pontes

Equações diferenciais de quarta ordem aparecem naturalmente na modelagem de oscilações de estruturas elásticas, como aquelas observadas em pontes pênseis. São considerados dois modelos que descrevem as oscilações no tabuleiro de uma ponte. No modelo unidimensional estudamos blow up em espaço finito de soluções de uma classe de equações diferenciais de quarta ordem. Os resultados apresentados solucionam uma conjectura apresentada em [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717752, 2013] e implicam a não existência de ondas viajantes com baixa velocidade de propagação em uma viga. No modelo bidimensional analisamos uma equação não local para uma placa longa e fina, suportada nas extremidades menores, livre nas demais e sujeita a protensão. Provamos existência e unicidade de solução fraca e estudamos o seu comportamento assintótico sob amortecimento viscoso. Estudamos ainda a estabilidade de modos simples de oscilação, os quais são classificados como longitudinais ou torcionais.

Automatic detection and treatment of oscillatory and/or stiff ordinary differential equations /

"Presented at the Differential Equation Workshop, Center for Interdisciplinary Research (Zif), University of Bielefeld, West Germany, April 21, 1980."

Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues

We consider the boundary value problems for nonlinear second-order differential equations of the form u '' + a(t)f (u) = 0, 0 < t < 1, u(0) = u (1) = 0. We give conditions on the ratio f (s)/s at infinity and zero that guarantee the existence of solutions with prescribed nodal properties. Then we establish existence and multiplicity results for nodal solutions to the problem. The proofs of our main results are based upon bifurcation techniques. (c) 2004 Elsevier Ltd. All rights reserved.