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

Experimental insight into the domain decomposition method for a finite element method code

The use of Domain Decomposition Methods (DDM) for a Finite Element Method (FEM) framework was a hot topic in the past decade, leading to very powerful results in terms of scalability and widening the problems that could be full-wave simulated with the FEM, [1–3]. However, despite the promising results shown in these references, it seems not to be a widespread use of the DDM in commercial FEM softwares or publications, whereas the common research topics (adaptivity, higher-order basis functions, different element shapes) in FEM have not been explored together with DDM. In this communication, we share experimental details with different non-overlapping DDM within FEM. We explore the use of different finite element shapes with up to fourth-order basis functions. We propose a propagation problem through a rectangular waveguide as a benchmark, and we show the different implementation choices available and their impact in the performance of the code

Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach does not lead at convergence of the Schwarz method to the mono-domain DG discretization, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present an alternative discretization of the transmission conditions in the framework of a DG weak formulation, and prove that for this discretization the multidomain and mono-domain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media

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

Randomized Computations for Efficient and Robust Finite Element Domain Decomposition Methods in Electromagnetics

Numerical modeling of electromagnetic (EM) phenomenon has proved to become an effective and efficient tool in design and optimization of modern electronic devices, integrated circuits (IC) and RF systems. However the generality, efficiency and reliability/resilience of the computational EM solver is often criticised due to the fact that the underlying characteristics of the simulated problems are usually different, which makes the development of a general, \u27\u27black-box\u27\u27 EM solver to be a difficult task. In this work, we aim to propose a reliable/resilient, scalable and efficient finite elements based domain decomposition method (FE-DDM) as a general CEM solver to tackle such ultimate CEM problems to some extent. We recognize the rank deficiency property of the Dirichlet-to-Neumann (DtN) operators involved in the previously proposed FETI-2$\lambda$ DDM formulation and apply such principle to improve the computational efficiency and robustness of FETI-2$\lambda$ DDM. Specifically, the rank deficient DtN operator is computed by a randomized computation method that was originally proposed to approximate matrix singular value decomposition (SVD). Numerical results show a up to 35\% run-time and 75% memory saving of the DtN operators computation can be achieved on a realistic example. Later, such rank deficiency principle is incorporated into a new global DDM preconditioner (W-FETI) that is inspired by the matrix Woodbury identity. Numerical study of the eigenspectrum shows the validity of the proposed W-FETI global preconditioner. Several industrial-scaled examples show significant iterative convergence advantage of W-FETI that uses 35%-80% matrix-vector-products (MxVs) than state-of-the-art DDM solvers

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