217 research outputs found

    Optimized Schwarz Methods for Maxwell equations

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    Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over the last decade, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. We show here why the classical Schwarz method applied to the hyperbolic problem converges without overlap for Maxwell's equations. The reason is that the method is equivalent to a simple optimized Schwarz method for an equivalent elliptic problem. Using this link, we show how to develop more efficient Schwarz methods than the classical ones for the Maxwell's equations. We illustrate our findings with numerical results

    Optimized Schwarz methods for heterogeneous Helmholtz and Maxwell's equations

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    Both the Helmholtz equation and the time-harmonic Maxwell’s equations are difficult to solve by iterative methods in the intermediate to high frequency regime, and domain decomposition methods are among the most promising techniques for this task. We focus here on the case of dissipative and conductive media with strongly heterogeneous coefficients, and develop optimized transmission conditions for this case. We establish a link for the use of such conditions between the case of Helmholtz and Maxwell’s equations, and show that in both cases jumps aligned with the interfaces of the subdomains can improve the convergence of the subdomain iteration

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

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    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

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

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    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

    Optimized Schwarz methods for the time-harmonic Maxwell equations with damping

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    In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell's equations both in the time-harmonic and time-domain case. The optimization process has been perfomed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results

    Advanced techniques in scientific computing: application to electromagnetics

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    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

    Finite element and boundary element analysis of electromagnetic NDE phenomena

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    The endeavor to produce quality products coupled with a drive to minimize failure in major industries such as aerospace, power and transportation is the driving force behind studies of electromagnetic nondestructive evaluation (NDE) methods. Popular domain and integral methods used for the purpose of modeling electromagnetic NDE phenomena include the finite element and boundary element methods. However no single numerical modeling technique has emerged as the optimal choice for all electromagnetic NDE processes. In a computer aided design environment, where the choice of an optimum modeling technique is critical, an evaluation of the various aspects of different numerical approaches is extremely helpful;In this dissertation, a comparison is made of the relative advantages and disadvantages of the finite element (FE) and boundary element (BE) methods as applied to the DC and AC Potential drop (DCPD and ACPD) methods for characterizing fatigue cracks. The comparison covers aspects of robustness, computer resource requirements and ease of numerically implementing the methods. Two dimensional FE and BE models are used to model an infinitely thin fatigue crack using the ACPD method, and a two and three dimensional FE and BE model is used to study the compact tension and single edge notch specimen using the DCPD method. Calibration curves and field plots in the specimen are compared to experimental and analytical data. The FE and BE methods are complementary numerical techniques and are combined to exploit their individual merits in the latter part of this dissertation. A three dimensional hybrid formulation to model eddy current NDE is then developed which discretizes the interior with finite elements and the exterior with boundary elements. The three dimensional model is applied to an absolute eddy current coil over a finite block. A feasibility study to confirm the validity of the formulation is undertaken by comparing the numerical results for probe lift-off and coil impedance measurements with published data;This comparative study outlined above indicates that when the solution is required at discrete points, as in the potential drop methods, or the model needs to handle infinite boundaries, as in eddy current NDE, the boundary element model is more suitable. Since it is based 011 the Green\u27s function, the BE method is limited to linear problems. Finite element analysis gives full field solutions, making it ideal for studying energy/defect interactions. The hybrid FE/BE formulation handles non-linearity and infinite boundaries naturally, thus utilizing the best of both worlds

    HF broadband antenna design for shipboard communications: Simulation and measurements

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    The objective pursued in this work is to highlight the convenience of using electromagnetic simulation software as an alternative to the traditional scale model measurement when dealing with the design of HF antennas on real complex platforms. The experience was developed during the building process of a real vessel. A low and a medium band antennas (fan-wire type) were designed ad-hoc for this project. The HF broadband antennas’ study covered from the preliminary design stages to the final verification measurements completed onboard the ship. The experiment has demonstrated that more accurate results can be obtained when using an adequate electromagnetic simulation code, which, besides, brings important advantages in flexibility and usability. These advantages, inherent to the use of virtual models, hinge on the ability of the simulation tools to properly handle any modification of the vessel’s structure that might arise during the platform construction

    Experimental and theoretical study of an integrated silicon Mach-Zehnder interferometer for chemical sensing applications

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    This thesis involves the design, fabrication and characterization of an integrated optical waveguide sensor. Prior to fabrication, design parameters of the waveguide need to be determined and optimized. The waveguide parameters such as waveguide dimension and the refractive index of the core and cladding are obtained from the single-mode cutoff frequency calculated using either analytical or numerical methods. In this thesis, details of analytical calculations to determine the cutoff frequency in terms of the waveguide parameters will be presented. The method discussed here is Marcatili\u27s approximation. The purpose is to solve the scalar wave equation derived from Maxwell\u27s equations because it describes the mode properties inside the waveguides. The Finite Element Method is used to simulate the electric and magnetic fields inside the waveguides and to determine the propagation characteristics in optical waveguides. This method is suited for problems involving complicated geometries and variable index of refraction. Fabrication of the Integrated Mach-Zehnder Interferometer sensor involves several important standard processes such as Chemical Vapor Deposition (CVD) for thin film fabrication, photolithography for mask transfer, and etching for ridge waveguide formation. The detailed fabrication procedures of the tested Mach-Zehnder Interferometer sensors are discussed. After completion of the sensor fabrication processes, the characterizations were carried out for the thin film of Si02 and PSG, the waveguides and the Y-junction separately. The waveguides were analyzed to make sure that the sensors are working as expected. The experimental testing on the separated waveguide portions of the first batch Integrated Mach-Zehnder Interferometer (MZI) sensors are described. These testing procedures were also performed for the subsequent fabricated batches of the integrated MZI sensors until optimum performance is achieved. A new concept has been proposed for chemical sensing applications. The novelty of the approach is mainly based on utilizing the multi -wavelength or broadband source instead of single wavelength input to the integrated MZI. The shifting of output spectra resulting from the interference has shown the ability of the MZI to analyze the different concentrations of a chemical analyte. The sensitivity of the sensor is also determined from the plot of intensity versus concentration, which is around 0.013 (%ml)-1 and 0.007 (%ml)-1 for the white light source and the 1.5 ~tm broadband source, respectively, while the lowest detectable concentration of ethanol for the sensor detection is around 8% using a intensity variation method and 0.6% using a peak wavelength variation method
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