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

3D parallel computations of turbofan noise propagation using a spectral element method

A three-dimensional code has been developed for the simulation of tone noise generated by turbofan engine inlets using computational aeroacoustics. The governing equations are the linearized Euler equations, which are further simplified to a set of equations in terms of acoustic potential, using the irrotational flow assumption, and subsequently solved in the frequency domain.Due to the special nature of acoustic wave propagation, the spatial discretization is performed using a spectral element method, where a tensor product of the nth-degree polynomials based on Chebyshev orthogonal functions is used to approximate variations within hexahedral elements. Non-reflecting boundary conditions are imposed at the far-field using a damping layer concept. This is done by augmenting the continuity equation with an additional term without modifying the governing equations as in PML methods.Solution of the linear system of equations for the acoustic problem is based on the Schur complement method, which is a nonoverlapping domain decomposition technique. The Schur matrix is first solved using a matrix-free iterative method, whose convergence is accelerated with a novel local preconditioner. The solution in the entire domain is then obtained by finding solutions in smaller subdomains.The 3D code also contains a mean flow solver based on the full potential equation in order to take into account the effects of flow variations around the nacelle on the scattering of the radiated sound field.All aspects of numerical simulations, including building and assembling the coefficient matrices, implementation of the Schur complement method, and solution of the system of equations for both the acoustic and mean flow problems are performed on multiprocessors in parallel using the resources of the CLUMEQ Supercomputer Center. A large number of test cases are presented, ranging in size from 100 000-2 000 000 unknowns for which, depending on the size of the problem, between 8-48 CPU's are used.The developed code is demonstrated to be robust and efficient in simulating acoustic propagation for a large number of problems, with an excellent parallel performance

[Research activities in applied mathematics, fluid mechanics, and computer science]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period April 1, 1995 through September 30, 1995

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Technology for large space systems: A bibliography with indexes (supplement 22)

This bibliography lists 1077 reports, articles, and other documents introduced into the NASA Scientific and Technical Information System between July 1, 1989 and December 31, 1989. Its purpose is to provide helpful information to the researcher or manager engaged in the development of technologies related to large space systems. Subject areas include mission and program definition, design techniques, structural and thermal analysis, structural dynamics and control systems, electronics, advanced materials, assembly concepts, and propulsion