Solución bidimensional sin malla de la ecuación no lineal de convección-difusión-reacción mediante el método de Interpolación Local Hermítica

A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional domains. The Local Hermitian Interpolation (LHI) method is employed for the spatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same results are obtained for all the non-linear solvers tested, but better convergence rates are attained with the Newton Raphson method in a double iteration scheme.Se desarrolla un esquema numérico sin malla para resolver una versión genérica de la ecuación no lineal de convección-difusión-reacción en dominios bidimensionales. El método de Interpolación Hermitiana Local (LHI) se emplea para la discretización espacial y se implementan varias estrategias para la solución del sistema de ecuaciones no lineal resultante, entre ellas la iteración Picard, el método Newton Raphson y una versión truncada del Método de Análisis de Homotopía. (JAMÓN). El método LHI es una estrategia de colocación local en la que se utilizan funciones de base radial (RBF) para construir la función de interpolación. A diferencia del método original de Kansa, el LHI se aplica localmente y los operadores diferenciales de ecuación límite y gobernante se utilizan para obtener la función de interpolación, dando una matriz de colocación simétrica y no singular. Las matrices analíticas y numéricas jacobianas se prueban para el método de Newton-Raphson y las derivadas de la ecuación de gobierno con respecto al parámetro de homotopía se obtienen analíticamente. El esquema numérico se verifica comparando los resultados obtenidos con las soluciones analíticas unidimensionales de Burgers y Richards bidimensionales. Se obtienen los mismos resultados para todos los solucionadores no lineales probados, pero se obtienen mejores tasas de convergencia con el método Newton Raphson en un esquema de doble iteración

Homotopy Continuation for Characteristic Roots of Delay Differential Equation

In this thesis we develop a homotopy continuation method to find the characteristic roots of scalar delay differential equations with multiple delays. We introduce a homotopy parameter µ € [0;1] in such a way that for µ - 0, the characteristic equation contains only one delay term and for µ - 1 the original characteristic equation is recovered. By selecting µ - 0 allows us to express all the roots of the characteristic equation in closed form in terms of Lambert W function. A numerical continuation based scheme is then developed to trace the roots as µ is varied from 0 to 1. The roots of the characteristic equation for µ - 1 correspond to the characteristic roots of the delay differential equation. We show several numerical examples to demonstrate the developed method

ELIMINACIÓN ALGEBRAICA Y EL MÉTODO NEWTON-HOMOTOPÍA: DOS MÉTODOS EFICIENTES EN LA SOLUCIÓN DE ECUACIONES COMPUESTAS POR POLINOMIOS (ALGEBRAIC ELIMINATION AND THE NEWTON-HOMOTOPY METHOD: TWO EFFICIENT METHODS FOR SOLVING POLYNOMIAL EQUATIONS)

ResumenEn este trabajo se describen de manera simple y detallada dos métodos de solución de ecuaciones compuestas por polinomios que usualmente surgen en el análisis y síntesis de mecanismos: i) la eliminación dialítica de Sylvester, ii) el método de Newton-homotopía. Ambos métodos han sido aplicados de manera exitosa en diversos problemas en el área de robótica, especialmente en el análisis cinemático directo de manipuladores espaciales. Sin embargo, estudiantes de ingeniería experimentan ciertas dificultades en su aplicación en materias fundamentales como mecanismos dada la estructura de las materias de matemáticas correspondientes. Por tal motivo, el presenta trabajo está dedicado a aquellas personas que se inician en el tema. Los métodos matemáticos aquí expuestos se aplican en un caso de estudio. Palabras Clave: Cinemática, Eliminación dialítica, Newton-Raphson, Homotopía, Polinomio. AbstractIn this paper we describe in a simple and detailed way two methods of solution of equations composed by polynomials that usually arise in the analysis and synthesis of mechanisms: i) the dialytic Sylvester method of elimination, ii) the Newton-homotopy method. Both methods have been applied successfully in diverse problems in the area of robotics, especially in the direct kinematics of spatial manipulators. However, undergraduate students undergo certain difficulties in their application in fundamental courses as mechanisms given the structure of the mathematics courses. For this motivation, the presented work is dedicated to those people who are initiated in the subject. The mathematical methods here treated are applied in a case study.Keywords: Dyalitic elimination, Homotopy, Kinematics, Newton-Raphson, Polynomial equation

Analytic continuation of Taylor series and the two-point boundary value problems of some nonlinear ordinary differential equations

We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Pad\'{e} and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Pad\'{e}-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution sought and show how this distribution controls the efficiency of the two methods. In general the method based on Pad\'{e} approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.Comment: Final versio

Optimal control with structure constraints and its application to the design of passive mechanical systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering; and, (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2002.Page 214 blank.Includes bibliographical references.Structured control (static output feedback, reduced-order control, and decentralized feedback) is one of the most important open problems in control theory and practice. In this thesis, various techniques for synthesis of structured controllers are surveyed and investigated, including H2 optimization, H[infinity] optimization, L1 control, eigenvalue and eigenstructure treatment, and multiobjective control. Unstructured control-full- state feedback and full-order control-is also discussed. Riccati-based synthesis, linear matrix inequalities (LMI), homotopy methods, gradient- and subgradientbased optimization are used. Some new algorithms and extensions are proposed, such as a subgradient-based method to maximize the minimal damping with structured feedback, a multiplier method for structured optimal H2 control with pole regional placement, and the LMI-based H2/H[infinity]/pole suboptimal synthesis with static output feedback. Recent advances in related areas are comprehensively surveyed and future research directions are suggested. In this thesis we cast the parameter optimization of passive mechanical systems as a decentralized control problem in state space, so that we can apply various decentralized control techniques to the parameter design which might be very hard traditionally. More practical constraints for mechanical system design are considered; for example, the parameters are restricted to be nonnegative, symmetric, or within some physically-achievable ranges. Marginally statable systems and hysterically damped systems are also discussed. Numerical examples and experimental results are given to illustrate the successful application of decentralized control techniques to the design of passive mechanical systems, such as multi-degree-of-freedom tuned-mass dampers, passive vehicle suspensions, and others.by Lei Zuo.S.M

Equation-oriented modeling, simulation, and optimization of integrated and intensified process and energy systems

Process intensification, defined as unconventional design and/or operation of processes that results in substantial performance improvements, represents a promising route toward reducing capital and operating expenses in the chemical/petrochemical process industry, while simultaneously achieving improved safety and environmental performance. In this dissertation, intensification is approached from three different angles: reactor design and control, process flowsheet design and optimization, and production scheduling and control. In the first part of the dissertation, three novel concepts for improving the controllability of intensified microchannel reactors are introduced. The first concept is a latent energy storage-based temperature controller, where a phase change material is confined within the walls of an autothermal reactor to improve local temperature control. The second concept is a segmented catalyst layer which modulates the rate of heat generation and consumption along the length of an autothermal reactor. Finally, the third concept is a thermally actuated valve, which uses small-scale bimetallic strips to modulate flow in a microchannel reactor in response to temperature changes. The second part of the dissertation introduces a novel framework for equation-oriented flowsheet modeling, simulation and optimization. The framework consists of a pseudo-transient reformulation of the steady-state material and energy balance equations of process unit operations as differential-algebraic equation (DAE) systems that are statically equivalent to the original model. I show that these pseudo-transient models improve the convergence properties of equation-oriented process flowsheet simulations by expanding the convergence basin in comparison to conventional steady state equation-oriented simulators. A library of pseudo-transient unit operation models is developed, and several case studies are presented. Models for more complex unit operations such as a pseudo-transient multistream heat exchanger and a dividing-wall distillation column are later introduced, and can easily be included in the flowsheet optimization framework. In the final part of the dissertation, a paradigm for calculating the optimal production schedule in a fast changing market situation is introduced. This is accomplished by including a model of the dynamics of a process and its control system into production scheduling calculations. The scheduling-relevant dynamic models are constructed to be of lower order than a detailed dynamic process model, while capturing the closed-loop behavior of a set of scheduling-relevant variables. Additionally, a method is given for carrying out these production scheduling calculations online and in "closed scheduling loop,"' i.e., recalculating scheduling decisions upon the advent of scheduling-relevant process or market events. An air separation unit operating in a demand response scenario is used as a representative case study.Chemical Engineerin

Position analysis based on multi-affine formulations

Aplicat embargament des de la data de defensa fins el 31/5/2022The position analysis problem is a fundamental issue that underlies many problems in Robotics such as the inverse kinematics of serial robots, the forward kinematics of parallel robots, the coordinated manipulation of objects, the generation of valid grasps, the constraint-based object positioning, the simultaneous localization and map building, and the analysis of complex deployable structures. It also arises in other fields, such as in computer aided design, when the location of objects in a design is given in terms of geometric constrains, or in the conformational analysis of biomolecules. The ubiquity of this problem, has motivated an intense quest for methods able of tackling it. Up to now, efficient algorithms for the general problem have remained elusive and they are only available for particular cases. Moreover, the complexity of the problem has typically led to methods difficult to be implemented. Position analysis can be decomposed into two equally important steps: obtaining a set of closure equations, and solving them. This thesis deals with both of them to obtain a general, simple, and yet efficient solution method that we call the trapezoid method. The first step is addressed relying on dual quaternions. Although it has not been properly highlighted in the past, the use of dual quaternions permits expressing the closure condition of a kinematic loop involving only lower pairs as a system of multi-affine equations. In this thesis, this property is leveraged to introduce an interval-based method specially tailored for solving multi-affine systems. The proposed method is objectively simpler (in the sense that it is easier to understand and to implement) than previous methods based on general techniques such as interval Newton methods, conversions to Bernstein basis, or linear relaxations. Moreover, it relies on two simple operations, namely, linear interpolations and projections on coordinate planes, which can be executed with a high performance. The result is a method that accurately and efficiently bounds the valid solutions of the problem at hand. To further improve the accuracy, we propose the use of redundant, multi affine equations that are derived from the minimal set of equations describing the problem. To improve the efficiency, we introduce a variable elimination methodology that preserves the multi-affinity of the system of equations. The generality and the performance of the proposed trapezoid method are extensively evaluated on different kind of mechanisms, including spherical mechanisms, generic 6R and 7R loops, over-constrained systems, and multi-loop mechanisms. The proposed method is, in all cases, significantly faster than state of the art alternatives.El problema de l'anàlisi de posició és un tema fonamental que subjau a molts problemes de la robòtica, com ara la cinemàtica inversa de robots sèrie, la cinemàtica directa de robots paral·lels, la manipulació coordinada d'objectes, la generació de prensions vàlides amb mans robòtiques, el posicionament d'objectes basat en restriccions, la localització i la creació de mapes de forma simultània, i l'anàlisi d'estructures desplegables complexes. També sorgeix en altres camps, com ara en el disseny assistit per ordinador, quan la ubicació dels objectes en un disseny es dóna en termes de restriccions geomètriques o en l'anàlisi conformacional de biomolècules. La omnipresència d'aquest problema ha motivat una intensa recerca de mètodes capaços d'afrontar-lo. Fins al moment, els algoritmes eficients per al problema general han estat esquius i només estan disponibles per a casos particulars. A més, la complexitat del problema normalment ha conduït a mètodes difícils d'implementar. L'anàlisi de posició es pot descompondre en dos passos igualment importants: l'obtenció d'un sistema d'equacions de tancament i la resolució d'aquest sistema. Aquesta tesi tracta de tots dos passos per tal d'obtenir un mètode de solució general, senzill i alhora eficient que anomenem el mètode del trapezoide. El primer pas s'aborda utilitzant quaternions duals. Tot i que no ha estat suficientment destacat en el passat, l'ús de quaternions duals permet expressar la condició de tancament d'un bucle cinemàtic que impliqui només parells inferiors com a un sistema d'equacions multi-afins. En aquesta tesi s'aprofita aquesta propietat per introduir un mètode especialment dissenyat per resoldre sistemes multi-afins. El mètode proposat és objectivament més senzill (en el sentit que és més fàcil d'entendre i d'implementar) que els mètodes anteriors que utilitzen tècniques generals com ara els mètodes de Newton basats en intervals, les conversions a la base de Bernstein o les relaxacions lineals. A més, el mètode es basa en dues operacions simples, a saber, les interpolacions lineals i les projeccions en plans de coordenades, que es poden executar de forma molt eficient. El resultat és un mètode que acota amb precisió i eficiència les solucions vàlides del problema. Per millorar encara més la precisió, proposem l'ús d'equacions multi-afins redundants derivades del conjunt mínim d'equacions que descriuen el problema. Per altra banda, per millorar l'eficiència, introduïm un metodologia d'eliminació de variables que preserva la multi-afinitat del sistema d'equacions. La generalitat i el rendiment del mètode del trapezoide s'avalua extensivament en diferents tipus de mecanismes, inclosos els mecanismes esfèrics, bucles 6R i 7R genèrics, sistemes sobre-restringits i mecanismes de múltiples bucles. El mètode proposat és, en tots els casos, significativament més ràpid que els mètodes alternatius descrits en la literatura fins al moment.Postprint (published version