Interpretation of the pressuremeter test using numerical models based on deformation tensor equations

The pressuremeter test in boreholes has proven itself as a useful tool in geotechnical explorations, especially comparing its results with those obtained from a mathematical model ruled by a soil representative constitutive equation. The numerical model shown in this paper is aimed to be the reference framework for the interpretation of this test. The model analyses variables such as: the type of response, the initial state, the drainage regime and the constitutive equations. It is a model of finite elements able to work with a mesh without deformation or one adapted to it.

On the generative equations of fractal self-similarity in granular media and the related PSD models

The physical appearance of granular media suggests the existence of geometrical scale invariance. The paper discuss how this physico-empirical property can be mathematically encoded leading to different generative models: a smooth one encoded by a differential equation and another encoded by an equation coming from a measure theoretical property.

Hybrid analysis of nonlinear circuits: DAE models with indices zero and one

We extend in this paper some previous results concerning the differential-algebraic index of hybrid models of electrical and electronic circuits. Specifically, we present a comprehensive index characterization which holds without passivity requirements, in contrast to previous approaches, and which applies to nonlinear circuits composed of uncoupled, one-port devices. The index conditions, which are stated in terms of the forest structure of certain digraph minors, do not depend on the specific tree chosen in the formulation of the hybrid equations. Additionally, we show how to include memristors in hybrid circuit models; in this direction, we extend the index analysis to circuits including active memristors, which have been recently used in the design of nonlinear oscillators and chaotic circuits. We also discuss the extension of these results to circuits with controlled sources, making our framework of interest in the analysis of circuits with transistors, amplifiers, and other multiterminal devices.

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis.

Chaos in non lineal Alfven waves using the DNLS equations

The electro-dynamical tethers emit waves in structured denominated Alfven wings. The Derivative Nonlineal Schrödinger Equation (DNLS) possesses the capacity to describe the propagation of circularly polarized Alfven waves of finite amplitude in cold plasmas. The DNLS equation is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In this article is presented a theoretical and numerical analysis when the growth rate of the unstable wave is next to zero considering two damping models: Landau and resistive. The DNLS equation presents a chaotic dynamics when is consider only three wave truncation. The evolution to chaos possesses three routes: hard transition, period-doubling and intermittence of type I.

A dynamic programming approach to the formulation and solution of finite element equations

A method for formulating and algorithmically solving the equations of finite element problems is presented. The method starts with a parametric partition of the domain in juxtaposed strips that permits sweeping the whole region by a sequential addition (or removal) of adjacent strips. The solution of the difference equations constructed over that grid proceeds along with the addition removal of strips in a manner resembling the transfer matrix approach, except that different rules of composition that lead to numerically stable algorithms are used for the stiffness matrices of the strips. Dynamic programming and invariant imbedding ideas underlie the construction of such rules of composition. Among other features of interest, the present methodology provides to some extent the analyst's control over the type and quantity of data to be computed. In particular, the one-sweep method presented in Section 9, with no apparent counterpart in standard methods, appears to be very efficient insofar as time and storage is concerned. The paper ends with the presentation of a numerical example

Development of AMREM: a Maxwell's equations solver with Adaptive Mesh Refinement

El principal objetivo de la presente tesis es el de desarrollar y probar un código capaz de resolver las ecuaciones de Maxwell en el dominio del tiempo con Malla Refinada Adaptativa (AMR por sus siglas en inglés). AMR es una técnica de cálculo basada en dividir el dominio físico del problema en distintas mallas rectangulares paralelas a las direcciones cartesianas. Cada una de las mallas tendrá distinta resolución y aquellas con mayor resolución se sitúan allí dónde las ondas electromagnéticas se propagan o interaccionan con los materiales, es decir, dónde mayor precisión es requerida. Como las ondas van desplazándose por todo el dominio, las mayas deberán seguirlas. El principal problema al utilizar esta metodología se puede encontrar en las fronteras internas, dónde las distintas mallas se unen. Ya que el método más corrientemente utilizado para resolver las ecuaciones de Maxwell es el de las diferencias finitas en el dominio del tiempo (FDTD por sus siglas en inglés) , el trabajo comenzó tratando de adaptar AMR a FDTD. Tras descubrirse que esta interacción resultaba en problemas de inestabilidades en las fronteras internas antes citadas, se decidió cambiar a un método basado en volúmenes finitos en el dominio del tiempo (FVTD por sus siglas en inglés). Este se basa en considerar la forma en ecuaciones de conservación de las ecuaciones de Maxwell y aplicar a su resolución un esquema de Godunov. Se ha probado que es clave para el correcto funcionamiento del código la elección de un limitador de flujo que proteja los extremos de la onda de la disipación típica de los métodos de este tipo. Otro problema clásico a la hora de resolver las ecuaciones de Maxwell es el de tratar con las condiciones de frontera física cuando se simulan dominios no acotados, es decir, dónde las ondas deben salir del sistema sin producir ninguna reflexión. Normalmente la solución es la de disponer una banda absorbente en las fronteras físicas. En AMREM se ha desarrollado un nuevo método basado en los campos característicos que con menor requisito de CPU funcina suficientemente bien incluso en los casos más desfaborables. El código ha sido contrastado con soluciones analíticas de diferentes problemas y también su velocidad ha sido comparada con la de Meep, uno de los programas más conocidos del ámbito. También algunas aplicaciones han sido simuladas con el fin de demostrar el amplio espectro de campos en los que AMREM puede funcionar como una útil herramienta.

Overtopping of harbour breakwaters: a comparison of semi-empirical equations, neural networks, and physical model tests

This paper reports extensive tests of empirical equations developed by different authors for harbour breakwater overtopping. First, the existing equations are compiled and evaluated as tools for estimating the overtopping rates on sloping and vertical breakwaters. These equations are then tested using the data obtained in a number of laboratory studies performed in the Centre for Harbours and Coastal Studies of the CEDEX, Spain. It was found that the recommended application ranges of the empirical equations typically deviate from those revealed in the experimental tests. In addition, a neural network model developed within the European CLASH Project is tested. The wind effects on overtopping are also assessed using a reduced scale physical model

Optimization of the Integrated Guidance and Control for a Dual Aerodynamic Control Missile

Las futuras misiones para misiles aire-aire operando dentro de la atmósfera requieren la interceptación de blancos a mayores velocidades y más maniobrables, incluyendo los esperados vehículos aéreos de combate no tripulados. La intercepción tiene que lograrse desde cualquier ángulo de lanzamiento. Una de las principales discusiones en la tecnología de misiles en la actualidad es cómo satisfacer estos nuevos requisitos incrementando la capacidad de maniobra del misil y en paralelo, a través de mejoras en los métodos de guiado y control modernos. Esta Tesis aborda estos dos objetivos simultáneamente, al proponer un diseño integrando el guiado y el control de vuelo (autopiloto) y aplicarlo a misiles con control aerodinámico simultáneo en canard y cola. Un primer avance de los resultados obtenidos ha sido publicado recientemente en el Journal of Aerospace Engineering, en Abril de 2015, [Ibarrondo y Sanz-Aranguez, 2015]. El valor del diseño integrado obtenido es que permite al misil cumplir con los requisitos operacionales mencionados empleando únicamente control aerodinámico. El diseño propuesto se compara favorablemente con esquemas más tradicionales, consiguiendo menores distancias de paso al blanco y necesitando de menores esfuerzos de control incluso en presencia de ruidos. En esta Tesis se demostrará cómo la introducción del doble mando, donde tanto el canard como las aletas de cola son móviles, puede mejorar las actuaciones de un misil existente. Comparado con un misil con control en cola, el doble control requiere sólo introducir dos servos adicionales para accionar los canards también en guiñada y cabeceo. La sección de cola será responsable de controlar el misil en balanceo mediante deflexiones diferenciales de los controles. En el caso del doble mando, la complicación añadida es que los vórtices desprendidos de los canards se propagan corriente abajo y pueden incidir sobre las superficies de cola, alterando sus características de control. Como un primer aporte, se ha desarrollado un modelo analítico completo para la aerodinámica no lineal de un misil con doble control, incluyendo la caracterización de este efecto de acoplamiento aerodinámico. Hay dos modos de funcionamiento en picado y guiñada para un misil de doble mando: ”desviación” y ”opuesto”. En modo ”desviación”, los controles actúan en la misma dirección, generando un cambio inmediato en la sustentación y produciendo un movimiento de translación en el misil. La respuesta es rápida, pero en el modo ”desviación” los misiles con doble control pueden tener dificultades para alcanzar grandes ángulos de ataque y altas aceleraciones laterales. Cuando los controles actúan en direcciones opuestas, el misil rota y el ángulo de ataque del fuselaje se incrementa para generar mayores aceleraciones en estado estacionario, aunque el tiempo de respuesta es mayor. Con el modelo aerodinámico completo, es posible obtener una parametrización dependiente de los estados de la dinámica de corto periodo del misil. Debido al efecto de acoplamiento entre los controles, la respuesta en bucle abierto no depende linealmente de los controles. El autopiloto se optimiza para obtener la maniobra requerida por la ley de guiado sin exceder ninguno de los límites aerodinámicos o mecánicos del misil. Una segunda contribución de la tesis es el desarrollo de un autopiloto con múltiples entradas de control y que integra la aerodinámica no lineal, controlando los tres canales de picado, guiñada y cabeceo de forma simultánea. Las ganancias del autopiloto dependen de los estados del misil y se calculan a cada paso de integración mediante la resolución de una ecuación de Riccati de orden 21x21. Las ganancias obtenidas son sub-óptimas, debido a que una solución completa de la ecuación de Hamilton-Jacobi-Bellman no puede obtenerse de manera práctica, y se asumen ciertas simplificaciones. Se incorpora asimismo un mecanismo que permite acelerar la respuesta en caso necesario. Como parte del autopiloto, se define una estrategia para repartir el esfuerzo de control entre el canard y la cola. Esto se consigue mediante un controlador aumentado situado antes del bucle de optimización, que minimiza el esfuerzo total de control para maniobrar. Esta ley de alimentación directa mantiene al misil cerca de sus condiciones de equilibrio, garantizando una respuesta transitoria adecuada. El controlador no lineal elimina la respuesta de fase no-mínima característica de la cola. En esta Tesis se consideran dos diseños para el guiado y control, el control en Doble-Lazo y el control Integrado. En la aproximación de Doble-Lazo, el autopiloto se sitúa dentro de un bucle interior y se diseña independientemente del guiado, que conforma el bucle más exterior del control. Esta estructura asume que existe separación espectral entre los dos, esto es, que los tiempos de respuesta del autopiloto son mucho mayores que los tiempos característicos del guiado. En el estudio se combina el autopiloto desarrollado con una ley de guiado óptimo. Los resultados obtenidos demuestran que se consiguen aumentos muy importantes en las actuaciones frente a misiles con control canard o control en cola, y que la interceptación, cuando se lanza cerca del curso de colisión, se consigue desde cualquier ángulo alrededor del blanco. Para el misil de doble mando, la estrategia óptima resulta en utilizar el modo de control opuesto en la aproximación al blanco y utilizar el modo de desviación justo antes del impacto. Sin embargo la lógica de doble bucle no consigue el impacto cuando hay desviaciones importantes con respecto al curso de colisión. Una de las razones es que parte de la demanda de guiado se pierde, ya que el misil solo es capaz de modificar su aceleración lateral, y no tiene control sobre su aceleración axial, a no ser que incorpore un motor de empuje regulable. La hipótesis de separación mencionada, y que constituye la base del Doble-Bucle, puede no ser aplicable cuando la dinámica del misil es muy alta en las proximidades del blanco. Si se combinan el guiado y el autopiloto en un único bucle, la información de los estados del misil está disponible para el cálculo de la ley de guiado, y puede calcularse la estrategia optima de guiado considerando las capacidades y la actitud del misil. Una tercera contribución de la Tesis es la resolución de este segundo diseño, la integración no lineal del guiado y del autopiloto (IGA) para el misil de doble control. Aproximaciones anteriores en la literatura han planteado este sistema en ejes cuerpo, resultando en un sistema muy inestable debido al bajo amortiguamiento del misil en cabeceo y guiñada. Las simplificaciones que se tomaron también causan que el misil se deslice alrededor del blanco y no consiga la intercepción. En nuestra aproximación el problema se plantea en ejes inerciales y se recurre a la dinámica de los cuaterniones, eliminado estos inconvenientes. No se limita a la dinámica de corto periodo del misil, porque se construye incluyendo de modo explícito la velocidad dentro del bucle de optimización. La formulación resultante en el IGA es independiente de la maniobra del blanco, que sin embargo se ha de incluir en el cálculo del modelo en Doble-bucle. Un típico inconveniente de los sistemas integrados con controlador proporcional, es el problema de las escalas. Los errores de guiado dominan sobre los errores de posición del misil y saturan el controlador, provocando la pérdida del misil. Este problema se ha tratado aquí con un controlador aumentado previo al bucle de optimización, que define un estado de equilibrio local para el sistema integrado, que pasa a actuar como un regulador. Los criterios de actuaciones para el IGA son los mismos que para el sistema de Doble-Bucle. Sin embargo el problema matemático resultante es muy complejo. El problema óptimo para tiempo finito resulta en una ecuación diferencial de Riccati con condiciones terminales, que no puede resolverse. Mediante un cambio de variable y la introducción de una matriz de transición, este problema se transforma en una ecuación diferencial de Lyapunov que puede resolverse mediante métodos numéricos. La solución resultante solo es aplicable en un entorno cercano del blanco. Cuando la distancia entre misil y blanco es mayor, se desarrolla una solución aproximada basada en la solución de una ecuación algebraica de Riccati para cada paso de integración. Los resultados que se han obtenido demuestran, a través de análisis numéricos en distintos escenarios, que la solución integrada es mejor que el sistema de Doble-Bucle. Las trayectorias resultantes son muy distintas. El IGA preserva el guiado del misil y consigue maximizar el uso de la propulsión, consiguiendo la interceptación del blanco en menores tiempos de vuelo. El sistema es capaz de lograr el impacto donde el Doble-Bucle falla, y además requiere un orden menos de magnitud en la cantidad de cálculos necesarios. El efecto de los ruidos radar, datos discretos y errores del radomo se investigan. El IGA es más robusto, resultando menos afectado por perturbaciones que el Doble- Bucle, especialmente porque el núcleo de optimización en el IGA es independiente de la maniobra del blanco. La estimación de la maniobra del blanco es siempre imprecisa y contaminada por ruido, y degrada la precisión de la solución de Doble-Bucle. Finalmente, como una cuarta contribución, se demuestra que el misil con guiado IGA es capaz de realizar una maniobra de defensa contra un blanco que ataque por su cola, sólo con control aerodinámico. Las trayectorias estudiadas consideran una fase pre-programada de alta velocidad de giro, manteniendo siempre el misil dentro de su envuelta de vuelo. Este procedimiento no necesita recurrir a soluciones técnicamente más complejas como el control vectorial del empuje o control por chorro para ejecutar esta maniobra. En todas las demostraciones matemáticas se utiliza el producto de Kronecker como una herramienta practica para manejar las parametrizaciones dependientes de variables, que resultan en matrices de grandes dimensiones. ABSTRACT Future missions for air to air endo-atmospheric missiles require the interception of targets with higher speeds and more maneuverable, including forthcoming unmanned supersonic combat vehicles. The interception will need to be achieved from any angle and off-boresight launch conditions. One of the most significant discussions in missile technology today is how to satisfy these new operational requirements by increasing missile maneuvering capabilities and in parallel, through the development of more advanced guidance and control methods. This Thesis addresses these two objectives by proposing a novel optimal integrated guidance and autopilot design scheme, applicable to more maneuverable missiles with forward and rearward aerodynamic controls. A first insight of these results have been recently published in the Journal of Aerospace Engineering in April 2015, [Ibarrondo and Sanz-Aránguez, 2015]. The value of this integrated solution is that it allows the missile to comply with the aforementioned requirements only by applying aerodynamic control. The proposed design is compared against more traditional guidance and control approaches with positive results, achieving reduced control efforts and lower miss distances with the integrated logic even in the presence of noises. In this Thesis it will be demonstrated how the dual control missile, where canard and tail fins are both movable, can enhance the capabilities of an existing missile airframe. Compared to a tail missile, dual control only requires two additional servos to actuate the canards in pitch and yaw. The tail section will be responsible to maintain the missile stabilized in roll, like in a classic tail missile. The additional complexity is that the vortices shed from the canard propagate downstream where they interact with the tail surfaces, altering the tail expected control characteristics. These aerodynamic phenomena must be properly described, as a preliminary step, with high enough precision for advanced guidance and control studies. As a first contribution we have developed a full analytical model of the nonlinear aerodynamics of a missile with dual control, including the characterization of this cross-control coupling effect. This development has been produced from a theoretical model validated with reliable practical data obtained from wind tunnel experiments available in the scientific literature, complement with computer fluid dynamics and semi-experimental methods. There are two modes of operating a missile with forward and rear controls, ”divert” and ”opposite” modes. In divert mode, controls are deflected in the same direction, generating an increment in direct lift and missile translation. Response is fast, but in this mode, dual control missiles may have difficulties in achieving large angles of attack and high level of lateral accelerations. When controls are deflected in opposite directions (opposite mode) the missile airframe rotates and the body angle of attack is increased to generate greater accelerations in steady-state, although the response time is larger. With the aero-model, a state dependent parametrization of the dual control missile short term dynamics can be obtained. Due to the cross-coupling effect, the open loop dynamics for the dual control missile is not linearly dependent of the fin positions. The short term missile dynamics are blended with the servo system to obtain an extended autopilot model, where the response is linear with the control fins turning rates, that will be the control variables. The flight control loop is optimized to achieve the maneuver required by the guidance law without exceeding any of the missile aerodynamic or mechanical limitations. The specific aero-limitations and relevant performance indicators for the dual control are set as part of the analysis. A second contribution of this Thesis is the development of a step-tracking multi-input autopilot that integrates non-linear aerodynamics. The designed dual control missile autopilot is a full three dimensional autopilot, where roll, pitch and yaw are integrated, calculating command inputs simultaneously. The autopilot control gains are state dependent, and calculated at each integration step solving a matrix Riccati equation of order 21x21. The resulting gains are sub-optimal as a full solution for the Hamilton-Jacobi-Bellman equation cannot be resolved in practical terms and some simplifications are taken. Acceleration mechanisms with an λ-shift is incorporated in the design. As part of the autopilot, a strategy is defined for proper allocation of control effort between canard and tail channels. This is achieved with an augmented feed forward controller that minimizes the total control effort of the missile to maneuver. The feedforward law also maintains the missile near trim conditions, obtaining a well manner response of the missile. The nonlinear controller proves to eliminate the non-minimum phase effect of the tail. Two guidance and control designs have been considered in this Thesis: the Two- Loop and the Integrated approaches. In the Two-Loop approach, the autopilot is placed in an inner loop and designed separately from an outer guidance loop. This structure assumes that spectral separation holds, meaning that the autopilot response times are much higher than the guidance command updates. The developed nonlinear autopilot is linked in the study to an optimal guidance law. Simulations are carried on launching close to collision course against supersonic and highly maneuver targets. Results demonstrate a large boost in performance provided by the dual control versus more traditional canard and tail missiles, where interception with the dual control close to collision course is achieved form 365deg all around the target. It is shown that for the dual control missile the optimal flight strategy results in using opposite control in its approach to target and quick corrections with divert just before impact. However the Two-Loop logic fails to achieve target interception when there are large deviations initially from collision course. One of the reasons is that part of the guidance command is not followed, because the missile is not able to control its axial acceleration without a throttleable engine. Also the separation hypothesis may not be applicable for a high dynamic vehicle like a dual control missile approaching a maneuvering target. If the guidance and autopilot are combined into a single loop, the guidance law will have information of the missile states and could calculate the most optimal approach to the target considering the actual capabilities and attitude of the missile. A third contribution of this Thesis is the resolution of the mentioned second design, the non-linear integrated guidance and autopilot (IGA) problem for the dual control missile. Previous approaches in the literature have posed the problem in body axes, resulting in high unstable behavior due to the low damping of the missile, and have also caused the missile to slide around the target and not actually hitting it. The IGA system is posed here in inertial axes and quaternion dynamics, eliminating these inconveniences. It is not restricted to the missile short term dynamic, and we have explicitly included the missile speed as a state variable. The IGA formulation is also independent of the target maneuver model that is explicitly included in the Two-loop optimal guidance law model. A typical problem of the integrated systems with a proportional control law is the problem of scales. The guidance errors are larger than missile state errors during most of the flight and result in high gains, control saturation and loss of control. It has been addressed here with an integrated feedforward controller that defines a local equilibrium state at each flight point and the controller acts as a regulator to minimize the IGA states excursions versus the defined feedforward state. The performance criteria for the IGA are the same as in the Two-Loop case. However the resulting optimization problem is mathematically very complex. The optimal problem in a finite-time horizon results in an irresoluble state dependent differential Riccati equation with terminal conditions. With a change of variable and the introduction of a transition matrix, the equation is transformed into a time differential Lyapunov equation that can be solved with known numerical methods in real time. This solution results range limited, and applicable when the missile is in a close neighborhood of the target. For larger ranges, an approximate solution is used, obtained from solution of an algebraic matrix Riccati equation at each integration step. The results obtained show, by mean of several comparative numerical tests in diverse homing scenarios, than the integrated approach is a better solution that the Two- Loop scheme. Trajectories obtained are very different in the two cases. The IGA fully preserves the guidance command and it is able to maximize the utilization of the missile propulsion system, achieving interception with lower miss distances and in lower flight times. The IGA can achieve interception against off-boresight targets where the Two- Loop was not able to success. As an additional advantage, the IGA also requires one order of magnitude less calculations than the Two-Loop solution. The effects of radar noises, discrete radar data and radome errors are investigated. IGA solution is robust, and less affected by radar than the Two-Loop, especially because the target maneuvers are not part of the IGA core optimization loop. Estimation of target acceleration is always imprecise and noisy and degrade the performance of the two-Loop solution. The IGA trajectories are such that minimize the impact of radome errors in the guidance loop. Finally, as a fourth contribution, it is demonstrated that the missile with IGA guidance is capable of performing a defense against attacks from its rear hemisphere, as a tail attack, only with aerodynamic control. The studied trajectories have a preprogrammed high rate turn maneuver, maintaining the missile within its controllable envelope. This solution does not recur to more complex features in service today, like vector control of the missile thrust or side thrusters. In all the mathematical treatments and demonstrations, the Kronecker product has been introduced as a practical tool to handle the state dependent parametrizations that have resulted in very high order matrix equations.

Ricci flows on surfaces related to the Einstein weyl and Abelian vortex equations

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

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.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Hyers–Ulam stability of functional equations with a square-symmetric operation

The stability of the functional equation f(x ○ y) = H(f(x), f(y)) (x, y ∈ S) is investigated, where H is a homogeneous function and ○ is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.