This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm.

[pt] O objetivo principal deste trabalho é o desenvolvimento de modelos computacionais para analisar a resposta eletromagnética de ferramentas de perfilagem LWD/MWD em formações geofísicas arbitrárias. Essa modelagem envolve a determinação precisa de campos eletromagnéticos em regiões tridimensionais (3D) complexas e, conseqüentemente, a solução de sistemas lineares não-hermitianos de larga escala. A modelagem numérica é realizada através da aplicação do método dos volumes finitos (FVM) no domínio da freqüência. Desenvolvem-se dois modelos computacionais, o primeiro válido em regiões isotrópicas e o segundo considerando a presença de anisotropias no meio. As equações de Maxwell são resolvidas através de duas formulações distintas: formulação por campos e formulação por potenciais vetor e escalar. A discretização por volumes finitos utiliza um esquema de grades entrelaçadas em coordenadas cilíndricas para evitar erros de aproximação de escada da geometria da ferramenta. Os modelos desenvolvidos incorporam quatro técnicas numéricas para aumentar a eficiência computacional e a precisão do método. As formulações por campos e por potenciais vetor e escalar são comparadas em termos da taxa de convergência e do tempo de processamento em cenários tridimensionais. Os modelos foram validados e testados em cenários tridimensionais complexos, tais como: (i) poços horizontais ou direcionais; (ii) formações não homogêneas com invasões de fluído de perfuração; (iii) formações anisotrópicas e (iv) poços excêntricos. Motivado pela flexibilidade dos modelos e pelos resultados numéricos obtidos em diferentes cenários tridimensionais, estende-se a metodologia para analisar a resposta de ferramentas LWD que empregam antenas inclinadas em relação ao eixo da ferramenta. Tais ferramentas podem prover dados com sensibilidade azimutal, assim como estimativas da anisotropia da formação, auxiliando o geodirecionamento de poços direcionais e horizontais. / [en] The main objective of this work is to develop computational models to analyze electromagnetic logging-while-drilling tool response in arbitrary geophysical formations. This modeling requires the determination of electromagnetic fields in three- dimensional (3-D) complex regions and consequently, the solution of large scale non-hermitian systems. The numerical modeling is done by using Finite Volume Methods (FVM) in the frequency domain. Both isotropic and anisotropic models are developed. Maxwell's equations are solved by using both the field formulation and the coupled vector-scalar potentials formulation. The proposed FVM technique utilizes an edge-based staggered-grid scheme in cylindrical coordinates to avoid staircasing errors on the tool geometry. Four numerical techniques are incorporated in the models in order to increase the computational efficiency and the accuracy of the method. The field formulation and the coupled vector-scalar potentials formulation are compared in terms of their accuracy, convergence rate, and CPU time for three-dimensional environments. The models were validated and tested in 3-D complex environments, such as:(i) horizontal and directional boreholes; (ii) multilayered geophysical formations including mud-filtrate invasions; (iii) anisotropic formations and (iv)eccentric boreholes. The methodology is extended to analyze LWD tools that are constructed with the transmitters and/or receivers tilted with respect to the axis of the drill collar. Such tools can provide improved anisotropy measurements and azimuthal sensitivity to benefit geosteering.

Diese Dissertation verallgemeinert die nichtkonformen Finite-Elemente-Methoden (FEMn) nach Morley und Crouzeix und Raviart durch neue gemischte Formulierungen für das Poisson-Problem, die Stokes-Gleichungen, die Navier-Lamé-Gleichungen der linearen Elastizität und m-Laplace-Gleichungen der Form $(-1)^m\Delta^m u=f$ für beliebiges m=1,2,3,... Diese Formulierungen beruhen auf Helmholtz-Zerlegungen. Die neuen Formulierungen gestatten die Verwendung von Ansatzräumen beliebigen Polynomgrades und ihre Diskretisierungen stimmen für den niedrigsten Polynomgrad mit den genannten nicht-konformen FEMn überein. Auch für höhere Polynomgrade ergeben sich robuste Diskretisierungen für fast-inkompressible Materialien und Approximationen für die Lösungen der Stokes-Gleichungen, die punktweise die Masse erhalten. Dieser Ansatz erlaubt außerdem eine Verallgemeinerung der nichtkonformen FEMn von der Poisson- und der biharmonischen Gleichung auf m-Laplace-Gleichungen für beliebiges m>2. Ermöglicht wird dies durch eine neue Helmholtz-Zerlegung für tensorwertige Funktionen. Die neuen Diskretisierungen lassen sich nicht nur für beliebiges m einheitlich implementieren, sondern sie erlauben auch Ansatzräume niedrigster Ordnung, z.B. stückweise affine Polynome für beliebiges m. Hat eine Lösung der betrachteten Probleme Singularitäten, so beeinträchtigt dies in der Regel die Konvergenz so stark, dass höhere Polynomgrade in den Ansatzräumen auf uniformen Gittern dieselbe Konvergenzrate zeigen wie niedrigere Polynomgrade. Deshalb sind gerade für höhere Polynomgrade in den Ansatzräumen adaptiv generierte Gitter unabdingbar. Neben der A-priori- und der A-posteriori-Analysis werden in dieser Dissertation optimale Konvergenzraten für adaptive Algorithmen für die neuen Diskretisierungen des Poisson-Problems, der Stokes-Gleichungen und der m-Laplace-Gleichung bewiesen. Diese werden auch in den numerischen Beispielen dieser Dissertation empirisch nachgewiesen. / This thesis generalizes the non-conforming finite element methods (FEMs) of Morley and Crouzeix and Raviart by novel mixed formulations for the Poisson problem, the Stokes equations, the Navier-Lamé equations of linear elasticity, and mth-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary m=1,2,3,... These formulations are based on Helmholtz decompositions. The new formulations allow for ansatz spaces of arbitrary polynomial degree and its discretizations coincide with the mentioned non-conforming FEMs for the lowest polynomial degree. Also for higher polynomial degrees, this results in robust discretizations for almost incompressible materials and approximations of the solution of the Stokes equations with pointwise mass conservation. Furthermore this approach also allows for a generalization of the non-conforming FEMs for the Poisson problem and the biharmonic equation to mth-Laplace equations for arbitrary m>2. A new Helmholtz decomposition for tensor-valued functions enables this. The new discretizations allow not only for a uniform implementation for arbitrary m, but they also allow for lowest-order ansatz spaces, e.g., piecewise affine polynomials for arbitrary m. The presence of singularities usually affects the convergence such that higher polynomial degrees in the ansatz spaces show the same convergence rate on uniform meshes as lower polynomial degrees. Therefore adaptive mesh-generation is indispensable especially for ansatz spaces of higher polynomial degree. Besides the a priori and a posteriori analysis, this thesis proves optimal convergence rates for adaptive algorithms for the new discretizations of the Poisson problem, the Stokes equations, and mth-Laplace equations. This is also demonstrated in the numerical experiments of this thesis.

<div>Accurate, fast, large-scale, and circuit-intuitive electromagnetic analysis is of critical importance to the design of integrated circuits (IC) and systems. Existing methods for the analysis of integrated circuits and systems have not satisfactorily achieved these performance goals. In this work, rapid modeling and simulation methods are developed for large-scale and circuit-intuitive electromagnetic analysis of integrated circuits and systems. The derived model is correct from zero to high frequencies where Maxwell's equations are valid. In addition, in the proposed model, we are able to analytically decompose the layout response into static and full-wave components with neither numerical computation nor approximation. This decomposed yet rigorous model greatly helps circuit diagnoses since now designers are able to analyze each component one by one, and identify which component is the root cause for the design failure. Such a decomposition also facilitates efficient layout modeling and simulation, since if an IC is dominated by RC effects, then we do not have to compute the full-wave component; and vice versa. Meanwhile, it makes parallelization straightforward. In addition, we develop fast algorithms to obtain each component of the inverse rapidly. These algorithms are also applicable for solving general partial differential equations for fast electromagnetic analysis.</div><div><br></div><div>The fast algorithms developed in this work are as follows. First, an analytical method is developed for finding the nullspace of the curl-curl operator in an arbitrary mesh for an arbitrary order of curl-conforming vector basis function. This method has been applied successfully to both a finite-difference and a finite-element based analysis of general 3-D structures. It can be used to obtain the static component of the inverse efficiently. An analytical method for finding the complementary space of the nullspace is also developed. Second, using the analytically found nullspace and its complementary space, a rigorous method is developed to overcome the low-frequency breakdown problem in the full-wave analysis of general lossy problems, where both dielectrics and conductors can be lossy and arbitrarily inhomogeneous. The method is equally valid at high frequencies without any need for changing the formulation. Third, with the static component part solved, the full-wave component is also ready to obtain. There are two ways. In the first way, the full-wave component is efficiently represented by a small number of high-frequency modes, and a fast method is created to find these modes. These modes constitute a significantly reduced order model of the complementary space of the nullspace. The second way is to utilize the relationship between the curl-curl matrix and the Laplacian matrix. An analytical method to decompose the curl-curl operator to a gradient-divergence operator and a Laplacian operator is developed. The derived Laplacian matrix is nothing but the curl-curl matrix's Laplacian counterpart. They share the same set of non-zero eigenvalues and eigenvectors. Therefore, this Laplacian matrix can be used to replace the original curl-curl matrix when operating on the full-wave component without any computational cost, and an iterative solution can converge this modified problem much faster irrespective of the matrix size. The proposed work has been applied to large-scale layout extraction and analysis. Its performance in accuracy, efficiency, and capacity has been demonstrated.</div>

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This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1