The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

An efficient time advancing strategy for energy-preserving simulations

Energy-conserving numerical methods are widely employed within the broad area of convection-dominated systems. Semi-discrete conservation of energy is usually obtained by adopting the so-called skew-symmetric splitting of the non-linear convective term, defined as a suitable average of the divergence and advective forms. Although generally allowing global conservation of kinetic energy, it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone. In this paper, a general theoretical framework has been developed to derive an efficient time-advancement strategy in the context of explicit Runge–Kutta schemes. The novel technique retains the conservation properties of skew-symmetric-based discretizations at a reduced computational cost. It is found that optimal energy conservation can be achieved by properly constructed Runge–Kutta methods in which only divergence and advective forms for the convective term are used. As a consequence, a considerable improvement in computational efficiency over existing practices is achieved. The overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy. The effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of Burgers' equation.Postprint (published version

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo

Engineering Education and Research Using MATLAB

MATLAB is a software package used primarily in the field of engineering for signal processing, numerical data analysis, modeling, programming, simulation, and computer graphic visualization. In the last few years, it has become widely accepted as an efficient tool, and, therefore, its use has significantly increased in scientific communities and academic institutions. This book consists of 20 chapters presenting research works using MATLAB tools. Chapters include techniques for programming and developing Graphical User Interfaces (GUIs), dynamic systems, electric machines, signal and image processing, power electronics, mixed signal circuits, genetic programming, digital watermarking, control systems, time-series regression modeling, and artificial neural networks

Learned simulation as the engine of physical scene understanding

La cognición humana evoca las habilidades del razonamiento, la comunicación y la interacción. Esto incluye la interpretación de la física del mundo real para comprender las leyes que subyacen en ella. Algunas teorías postulan la semejanza entre esta capacidad de razonamiento con simulaciones para interpretar la física de la escena, que abarca la percepción para la comprensión del estado físico actual, y el razonamiento acerca de la evolución temporal de un sistema dado. En este contexto se propone el desarrollo de un sistema para realizar simulación aprendida. Establecido un objetivo, el algoritmo se entrena para aprender una aproximación de la dinámica real, para construir así un gemelo digital del entorno. Entonces, el sistema de simulación emulará la física subyacente con información obtenida mediante observaciones de la escena. Para ello, se empleará una cámara estéreo para adquirir datos a partir de secuencias de video. El trabajo se centra los fenómenos oscilatorios de fluidos. Los fluidos están presentes en muchas de nuestras acciones diarias y constituyen un reto físico para el sistema propuesto. Son deformables, no lineales, y presentan un carácter disipativo dominante, lo que los convierte en un sistema complejo para ser aprendido. Además, sólo se tiene acceso a mediciones parciales de su estado ya que la cámara sólo proporciona información acerca de la superficie libre. El resultado es un sistema capaz de percibir y razonar sobre la dinámica del fluido. El gemelo digital cognitivo así construido proporciona una interpretación del estado del mismo para integrar su evolución en tiempo real, aprendiendo con información observada del gemelo físico. El sistema, entrenado originalmente para un líquido concreto, se adaptará a cualquier otro a través del aprendizaje por refuerzo produciendo así resultados precisos para líquidos desconocidos. Finalmente, se emplea la realidad aumentada (RA) para ofrecer una representación visual de los resultados, así como información adicional sobre el estado del líquido que no es accesible al ojo humano. Este objetivo se alcanza mediante el uso de técnicas de aprendizaje de variedades, y aprendizaje automático, como las redes neuronales, enriquecido con información física. Empleamos sesgos inductivos basados en el conocimiento de la termodinámica para desarrollar un sistema inteligente que cumpla con estos principios para dar soluciones con sentido sobre la dinámica. El problema abordado en esta tesis constituye una dificultad de primer orden en el desarrollo de sistemas robóticos destinados a la manipulación de fluidos. En acciones como el vertido o el movimiento, la oscilación de los líquidos juega un papel importante en el desarrollo de sistemas de asistencia a personas con movilidad reducida o aplicaciones industriales. Cognition evokes human abilities for reasoning, communication, and interaction. This includes the interpretation of real-world physics so as to understand its underlying laws. Theories postulate the similarity of human reasoning about these phenomena with simulations for physical scene understanding, which gathers perception for comprehension of the current dynamical state, and reasoning for time evolution prediction of a given system. In this context, we propose the development of a system for learned simulation. Given a design objective, an algorithm is trained to learn an approximation to the real dynamics to build a digital twin of the environment. Then, the underlying physics will be emulated with information coming from observations of the scene. For this purpose, we use a commodity camera to acquire data exclusively from video recordings. We focus on the sloshing problem as a benchmark. Fluids are widely present in several daily actions and portray a physically rich challenge for the proposed systems. They are highly deformable, nonlinear, and present a dominant dissipative behavior, making them a complex entity to be emulated. In addition, we only have access to partial measurements of their dynamical state, since a commodity camera only provides information about the free surface. The result is a system capable of perceiving and reasoning about the dynamics of the fluid. This cognitive digital twin provides an interpretation of the state of the fluid to integrate its dynamical evolution in real-time, updated with information observed from the real twin. The system, trained originally for one liquid, will be able to adapt itself to any other fluid through reinforcement learning and produce accurate results for previously unseen liquids. Augmented reality is used in the design of this application to offer a visual interpretation of the solutions to the user, and include information about the dynamics that is not accessible to the human eye. This objective is to be achieved through the use of manifold learning and machine learning techniques, such as neural networks, enriched with physics information. We use inductive biases based on the knowledge of thermodynamics to develop machine intelligence systems that fulfill these principles to provide meaningful solutions to the dynamics. This problem is considered one of the main targets in fluid manipulation for the development of robotic systems. Pursuing actions such as pouring or moving, sloshing dynamics play a capital role for the correct performance of aiding systems for the elderly or industrial applications that involve liquids. <br /

Power-to-Syngas: A Parareal Optimal Control Approach

A chemical plant layout for the production of syngas from renewable power, H2O and biogas, is presented to ensure a steady productivity of syngas with a constant H2-to-CO ratio under time-dependent electricity provision. An electrolyzer supplies H2 to the reverse water-gas shift reactor. The system compensates for a drop in electricity supply by gradually operating a tri-reforming reactor, fed with pure O2 directly from the electrolyzer or from an intermediate generic buffering device. After the introduction of modeling assumptions and governing equations, suitable reactor parameters are identified. Finally, two optimal control problems are investigated, where computationally expensive model evaluations are lifted viaparareal and necessary objective derivatives are calculated via the continuous adjoint method. For the first time, modeling, simulation, and optimal control are applied to a combination of the reverse water-gas shift and tri-reforming reactor, exploring a promising pathway in the conversion of renewable power into chemicals

HPCCP/CAS Workshop Proceedings 1998

This publication is a collection of extended abstracts of presentations given at the HPCCP/CAS (High Performance Computing and Communications Program/Computational Aerosciences Project) Workshop held on August 24-26, 1998, at NASA Ames Research Center, Moffett Field, California. The objective of the Workshop was to bring together the aerospace high performance computing community, consisting of airframe and propulsion companies, independent software vendors, university researchers, and government scientists and engineers. The Workshop was sponsored by the HPCCP Office at NASA Ames Research Center. The Workshop consisted of over 40 presentations, including an overview of NASA's High Performance Computing and Communications Program and the Computational Aerosciences Project; ten sessions of papers representative of the high performance computing research conducted within the Program by the aerospace industry, academia, NASA, and other government laboratories; two panel sessions; and a special presentation by Mr. James Bailey