Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems.We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations

Advanced techniques in scientific computing: application to electromagnetics

Mención Internacional en el título de doctorDurante los últimos años, los componentes de radiofrecuencia que forman parte de un sistema de comunicaciones necesitan simulaciones cada vez más exigentes desde el punto de vista de recursos computacionales. Para ello, se han desarrollado diferentes técnicas con el método de los elementos finitos (FEM) como la conocida como adaptatividad hp, que consiste en estimar el error en el problema electromagnético para generar mallas de elementos adecuadas al problema que obtienen una aproximación de forma más efectiva que las mallas estándar; o métodos de descomposición de dominios (DDM), basado en la división del problema original en problemas más pequeños que se pueden resolver en paralelo. El principal problema de las técnicas de adaptatividad es que ofrecen buenas prestaciones para problemas bidimensionales, mientras que en tres dimensiones el tiempo de generación de las mallas adaptadas es prohibitivo. Por otra parte, DDM se ha utilizado satisfactoriamente para la simulación de problemas eléctricamente muy grandes y de gran complejidad, convirtiéndose en uno de los temas más actuales en la comunidad de electromagnetismo computacional. El principal objetivo de este trabajo es estudiar la viabilidad de algoritmos escalables (en términos de paralelización) combinando DDM no conformes y adaptatividad automática en tres dimensiones. Esto permitir ía la ejecución de algoritmos de adaptatividad independiente en cada subdominio de DDM. En este trabajo se presenta y discute un prototipo que combina técnicas de adaptatividad y DDM, que aún no se han tratado en detalle en la comunidad científica. Para ello, se implementan tres bloques fundamentales: i) funciones de base para los elementos finitos que permitan órdenes variables dentro de la misma malla; ii) DDM no conforme y sin solapamiento; y iii) algoritmos de adaptatividad en tres dimensiones. Estos tres bloques se han implementado satisfactoriamente en un código FEM mediante un método sistemático basado en el método de las soluciones manufacturadas (MMS). Además, se ha llevado a cabo una paralelización a tres niveles: a nivel de algoritmo, con DDM; a nivel de proceso, con MPI (Message Passing Interface); y a nivel de hebra, con OpenMP; todo en un código modular que facilita el mantenimiento y la introducción de nuevas características. Con respecto al primer bloque fundamental, se ha desarrollado una familia de funciones base con un enfoque sistemático que permite la expansión correcta del espacio de funciones. Por otra parte, se han introducido funciones de base jerárquicas de otros autores (con los que el grupo al que pertenece el autor de la tesis ha colaborado estrechamente en los últimos años) para facilitar la introducción de diferentes órdenes de aproximación en el mismo mallado. En lo relativo a DDM, se ha realizado un estudio cuantitativo del error generado por las disconformidades en la interfaz entre subdominios, incluidas las discontinuidades generadas por un algoritmo de adaptatividad. Este estudio es fundamental para el correcto funcionamiento de la adaptatividad, y no ha sido evaluado con detalle en la comunidad científica. Además, se ha desarrollado un algoritmo de adaptatividad con prismas triangulares, haciendo especial énfasis en las peculiaridades debidas a la elección de este elemento. Finalmente, estos tres bloques básicos se han utilizado para desarrollar, y discutir, un prototipo que une las técnicas de adaptatividad y DDM.In the last years, more and more accurate and demanding simulations of radiofrequency components in a system of communications are requested by the community. To address this need, some techniques have been introduced in finite element methods (FEM), such as hp adaptivity (which estimates the error in the problem and generates tailored meshes to achieve more accuracy with less unknowns than in the case of uniformly refined meshes) or domain decomposition methods (DDM, consisting of dividing the whole problem into more manageable subdomains which can be solved in parallel). The performance of the adaptivity techniques is good up to two dimensions, whereas for three dimensions the generation time of the adapted meshes may be prohibitive. On the other hand, large scale simulations have been reported with DDM becoming a hot topic in the computational electromagnetics community. The main objective of this dissertation is to study the viability of scalable (in terms of parallel performance) algorithms combining nonconformal DDM and automatic adaptivity in three dimensions. Specifically, the adaptivity algorithms might be run in each subdomain independently. This combination has not been detailed in the literature and a proof of concept is discussed in this work. Thus, three building blocks must be introduced: i) basis functions for the finite elements which support non-uniform approximation orders p; ii) non-conformal and non-overlapping DDM; and iii) adaptivity algorithms in 3D. In this work, these three building blocks have been successfully introduced in a FEM code with a systematic procedure based on the method of manufactured solutions (MMS). Moreover, a three-level parallelization (at the algorithm level, with DDM; at the process level, with message passing interface (MPI), and at the thread level, with OpenMP) has been developed using the paradigm of modular programming which eases the software maintenance and the introduction of new features. Regarding first building block, a family of basis functions which follows a sound mathematical approach to expand the correct space of functions is developed and particularized for triangular prisms. Also, to ease the introduction of different approximation orders in the same mesh, hierarchical basis functions from other authors are used as a black box. With respect to DDM, a thorough study of the error introduced by the non-conformal interfaces between subdomains is required for the adaptivity algorithm. Thus, a quantitative analysis is detailed including non-conformalities generated by independent refinements in neighbor subdomains. This error has not been assessed with detail in the literature and it is a key factor for the adaptivity algorithm to perform properly. An adaptivity algorithm with triangular prisms is also developed and special considerations for the implementation are explained. Finally, on top of these three building blocks, the proof of concept of adaptivity with DDM is discussed.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Daniel Segovia Vargas.- Secretario: David Pardo Zubiaur.- Vocal: Romanus Dyczij-Edlinge

Gebietszerlegungsverfahren zur Diskretisierung der vektoriellen Helmholtz-Gleichung

Die Methode der finiten Elemente (FE) ist ein weit verbreitetes Werkzeug zur Simulation elektromagnetischer Strukturen im Frequenzbereich. Der numerische Aufwand zur direkten Lösung des resultierenden Gleichungssystems steigt jedoch signifikant mit zunehmender elektrischer Größe der zugrundeliegenden Struktur, so dass der Übergang zu iterativen Lösungsstrategien unabdingbar wird. Für diese ist die Verfügbarkeit effizienter Vorkonditionierer von höchster Bedeutung. Der Fokus dieser Arbeit liegt auf der iterativen Lösung des aus der FE-Diskretisierung der vektoriellen Helmholtz-Gleichung resultierenden linearen Gleichungssystems. Zur Vorkonditionierung der Systemmatrix wird ein nicht-überlappendes Gebietszerlegungsverfahren unter Berücksichtigung von Kopplungsbedingungen höherer Ordnung aufgezeigt. Darüber hinaus wird anhand analytischer und numerischer Untersuchungen der Einfluss unterschiedlicher Kopplungsbedingungen auf das Konvergenzverhalten des iterativen Lösers untersucht. Die Simulation abstrahlender elektromagnetischer Strukturen mittels der FE-Methode erfordert darüber hinaus eine Beschränkung des Feldgebiets, welche beispielsweise durch die Einführung künstlicher, sogenannter absorbierender Randbedingungen geschieht. Aufbauend auf analytischen Untersuchungen zur Herleitung geeigneter Kopplungsbedingungen werden absorbierende Randbedingungen höherer Ordnung hergeleitet und im Rahmen einer FE-Diskretisierung umgesetzt. Um eine breitbandige Charakterisierung der betrachteten Strukturen zur ermöglichen, wird aufbauend auf dem Gebietszerlegungsverfahren ein Modellordnungsreduktionsverfahren vorgestellt. Insbesondere wird eine Reduktion des zeitlichen Aufwands zur Generierung des reduzierten Models mithilfe einer adaptiven Grobraumkorrektur erreicht.The finite-element (FE) method is a commonly used tool for simulating electromagnetic structures in the frequency domain. However, the numerical effort for the direct solution of the resulting system of equations is growing significantly with increasing electrical size of the underlying structure,so that the transition to iterative solution strategies becomes indispensable. For these, the availability of efficient preconditioners is of utmost importance. The focus of this work is on the iterative solution of the system of linear equations resulting from the FE discretization of the vector Helmholtz equation. For preconditioning the system matrix a non-overlapping domain-decomposition method, under consideration of higher-order transmission conditions, is pointed out.In addition, the effects of different kinds of transmission conditions on the convergence behavior of the iterative solver are analyzed by means of analytical and numerical investigations. Moreover, the simulation of radiating electromagnetic structures by the FE method requires the field domain to be truncated, which may be done by, e.g., introducing so-called absorbing boundary conditions. Based on analytical investigations for deriving suitable transmission conditions, absorbing boundary conditions of higher order are derived and implemented in the context of a FE discretization. To enable the broadband characterization of the considered structures, a method of model-order reduction based on the domain-decomposition approach is proposed. In particular, a reduction in the time spent for generating the reduced model is obtained, by means of an adaptive coarse-space correction

Англійська мова для студентів електромеханічних спеціальностей

Навчальний посібник розрахований на студентів напряму підготовки 6.050702 Електромеханіка. Містить уроки, що структуровані за тематичними розділами, граматичний коментар, короткі англо-український і українсько- англійський словники та додатки, які спрямовані на закріплення загальних навичок володіння англійською мовою. Акцентований на ɨсобливості термінології, що застосовується у науково-технічній галузі, зокрема, в електромеханіці та виконання запропонованих завдань, що буде сприяти формуванню навичок перекладу з англійської та української мов, сприйняттю письмової та усної англійської мови, вмінню письмового викладення англійською мовою науково-технічних та інших текстів під час професійної діяльності, спілкуванню з професійних та загальних питань тощо

Incorporation of feed-network and circuit modeling into the time-domain finite element analysis of antenna arrays and microwave circuits

In this dissertation, accurate and efficient numerical algorithms are developed to incorporate the feed-network and circuit modeling into the time-domain finite element analysis of antenna arrays and microwave circuits. First, simulation of an antenna system requires accurate modeling of interactions between the radiating elements and the associated feeding network. In this work, a feed network is represented in terms of its scattering matrix in a rational function form in the frequency domain that enables its interfacing with the time-domain finite element modeling of the antenna elements through a fast recursive time-convolution algorithm. The exchange of information between the antenna elements and the feed network occurs through the incident and reflected modal voltages/currents at properly defined port interfaces. The proposed numerical scheme allows a full utilization of the advanced antenna simulation techniques, and significantly extends the current antenna modeling capability to the system level. Second, a hybrid field-circuit solver that combines the capabilities of the time-domain finite element method and a lumped circuit analysis is developed for accurate and efficient characterization of complicated microwave circuits that include both distributive and lumped-circuit components. The distributive portion of the device is modeled by the time-domain finite element method to generate a finite element subsystem, while the lumped circuits are analyzed by a SPICE-like circuit solver to generate a circuit subsystem. A global system for both the finite-element and circuit unknowns is established by combining the two subsystems through coupling matrices to model their interactions. For simulations of even more complicated mixed-scale circuit systems that contain pre-characterized blocks of discrete circuit elements, the hybrid field-circuit analysis implemented a systematic and efficient algorithm to incorporate multiport lumped networks in terms of frequency-dependent admittance matrices. Other advanced features in the hybrid field-circuit solver include application of the tree-cotree splitting algorithm and introduction of a flexible time-stepping scheme. Various numerical examples are presented to validate the implementation and demonstrate the accuracy, efficiency, and applications of the proposed numerical algorithms

NAS technical summaries. Numerical aerodynamic simulation program, March 1992 - February 1993

NASA created the Numerical Aerodynamic Simulation (NAS) Program in 1987 to focus resources on solving critical problems in aeroscience and related disciplines by utilizing the power of the most advanced supercomputers available. The NAS Program provides scientists with the necessary computing power to solve today's most demanding computational fluid dynamics problems and serves as a pathfinder in integrating leading-edge supercomputing technologies, thus benefitting other supercomputer centers in government and industry. The 1992-93 operational year concluded with 399 high-speed processor projects and 91 parallel projects representing NASA, the Department of Defense, other government agencies, private industry, and universities. This document provides a glimpse at some of the significant scientific results for the year

Numerical Models for Induction Hardening of Gears

I modelli numerici permettono ai progettisti di evitare un approccio trial and error, lungo e costoso, nello stabilire i parametri di tempra. Questa Tesi pertanto affronta uno studio sull’influenza dei parametri di macchina sullo strato temprato di una ruota dentata temprata ad induzione tramite (FEM). Successivamente è stato proposto un confronto tra i risultati di misure sperimentali di temperatura su un processo di tempra e i risultati delle simulazioni numeriche dello stesso processo

Forced Current Excitation in Selectable Field of View Coils for 7T MRI and MRS

High field magnetic resonance imaging (MRI) provides improved signal-to-noise ratio (SNR) which can be translated to higher image resolution or reduced scan time. 7 Tesla (T) breast imaging and 7 T spine imaging are of clinical value, but they are challenging for several reasons: A bilateral breast coil requires the use of closely-spaced elements that are subject to severe mutual coupling which leads to uncontrollable current distribution and non-uniform field pattern; A spine coil at 7T requires a large field of view (FOV) in the z direction and good RF penetration into the human body. Additionally, the ability to switch FOV without the use of expensive high power RF amplifiers is desired in both applications. This capability would allow reconfigurable power distribution and avoid unnecessary heat deposition into human body. Forced-Current Excitation (FCE) is a transmission line-based method that maintains equal current distribution across an array, alleviating mutual coupling effects and allowing current/field replication across a large FOV. At the same time, the nature of this method enables selectable FOV with the inclusion of PIN diodes and a controller. In this doctoral work, the theory of FCE is explained in detail, along with its benefits and drawbacks. Electromagnetic simulation considerations of FCE-driven coils are also discussed. Two FCE-driven coils were designed and implemented: a switchable bilateral/unilateral 7T breast coil, and a segmented dipole for spine imaging at 7T with reconfigurable length. For the breast coil, shielded loop elements were used to form a volume coil, whereas for the spine coil, a segmented dipole was chosen as the final design due to improved RF penetration. Electromagnetic simulations were performed to assist the design of the two coils as well as to predict the SAR (specific absorption rate) generated in the phantom. The coils were evaluated on bench and through MRI experiments in different configurations to validate the design. The switchable breast coil provides uniform excitation in both unilateral and bilateral mode. In unilateral mode, the signal in the contralateral breast is successfully suppressed and higher power is concentrated into the breast of interest; The segmented dipole was compared to a regular dipole with the same length used for 7T spine imaging. The segmented dipole shows a large FOV in the long mode. In the short mode, the residual signal from other part of the dipole is successfully suppressed. The ability to switch FOV and reconfigure the power distribution improves the B1 generated with unit specific absorption rate towards the edge of the dipole, compared to the regular dipole

Forced Current Excitation in Selectable Field of View Coils for 7T MRI and MRS

High field magnetic resonance imaging (MRI) provides improved signal-to-noise ratio (SNR) which can be translated to higher image resolution or reduced scan time. 7 Tesla (T) breast imaging and 7 T spine imaging are of clinical value, but they are challenging for several reasons: A bilateral breast coil requires the use of closely-spaced elements that are subject to severe mutual coupling which leads to uncontrollable current distribution and non-uniform field pattern; A spine coil at 7T requires a large field of view (FOV) in the z direction and good RF penetration into the human body. Additionally, the ability to switch FOV without the use of expensive high power RF amplifiers is desired in both applications. This capability would allow reconfigurable power distribution and avoid unnecessary heat deposition into human body. Forced-Current Excitation (FCE) is a transmission line-based method that maintains equal current distribution across an array, alleviating mutual coupling effects and allowing current/field replication across a large FOV. At the same time, the nature of this method enables selectable FOV with the inclusion of PIN diodes and a controller. In this doctoral work, the theory of FCE is explained in detail, along with its benefits and drawbacks. Electromagnetic simulation considerations of FCE-driven coils are also discussed. Two FCE-driven coils were designed and implemented: a switchable bilateral/unilateral 7T breast coil, and a segmented dipole for spine imaging at 7T with reconfigurable length. For the breast coil, shielded loop elements were used to form a volume coil, whereas for the spine coil, a segmented dipole was chosen as the final design due to improved RF penetration. Electromagnetic simulations were performed to assist the design of the two coils as well as to predict the SAR (specific absorption rate) generated in the phantom. The coils were evaluated on bench and through MRI experiments in different configurations to validate the design. The switchable breast coil provides uniform excitation in both unilateral and bilateral mode. In unilateral mode, the signal in the contralateral breast is successfully suppressed and higher power is concentrated into the breast of interest; The segmented dipole was compared to a regular dipole with the same length used for 7T spine imaging. The segmented dipole shows a large FOV in the long mode. In the short mode, the residual signal from other part of the dipole is successfully suppressed. The ability to switch FOV and reconfigure the power distribution improves the B1 generated with unit specific absorption rate towards the edge of the dipole, compared to the regular dipole

Aeronautical engineering: A continuing bibliography with indexes (supplement 271)

This bibliography lists 666 reports, articles, and other documents introduced into the NASA scientific and technical information system in October, 1991. Subject coverage includes design, construction and testing of aircraft and aircraft engines; aircraft components, equipment and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics