Dissertations / Theses on the topic 'Electromagnetic solver'

The research presented in this thesis discusses an electromagnetic (EM) analysis tool which is based on the partial element equivalent circuit (PEEC) method and is appropriate for combined EM and circuit simulations especially power electronics applications. EM analysis is important to ensure that a system will not affect the correct operation of other devices nor cause interference between various electrical systems. In power electronic applications, the increased switching speed can cause voltage overshoots, unbalanced current share between semiconductor modules, and unwanted resonances. Therefore, EM analysis should be carried out to perform design optimizations in order to minimize unwanted effects of high frequencies. The solver developed in this work is an appropriate solution to address the needs of EM analysis in general and power electronics in particular. The conducted research consists of performance acceleration and implementation of the solver, and verification of the simulation results by means of measurements. This work was done in two major phases.In the first phase, the solver was accelerated to optimize its performance when quasi-static (R,Lp,C)PEEC as well as full-wave (R,Lp,C,tau)PEEC simulations were carried out. The main optimizations were based on exploiting parallelism and high performance computing to solve very large problems and non-uniform mesh, which was helpful in simulating skin- and proximity effects while keeping the problem size to a minimum. The presented results and comparisons with the measurements confirmed that non-uniform mesh helped in accurately simulating large bus bar models and correctly predicting system resonances when the size of the problem was minimized. On-the-fly calculation was also developed to reduce memory usage, while increasing solution time.The second phase consists of methods to increase the performance of the solver while including some levels of approximations. In this phase sparsification techniques were used to convert a dense PEEC system into a sparse system. The sparsification was done by calculating the reluctance matrix, which can be sparsified by maintaining the accuracy at the desired level, because of the locality and the shielding effect of the reluctance matrix. Efficient algorithms were developed to perform sparse matrix-matrix multiplication and assemble the sparse coefficient matrix in a row-by-row manner to reduce the peak memory usage. The sparse system was then solved using both sparse direct and iterative solvers with proper preconditioning. The acquired results from the sparse direct solution confirmed that the memory consumption and solution time were reduced by orders of magnitude and by a factor 3 to 5. Moreover, the Schur complement was used together with the iterative approach, making it possible to solve large problems within a few iterations by preconditioning the system, and using less memory and lower computational complexity. Bus bars used in two types of power frequency converters manufactured by ABB were modelled and analysed with the developed PEEC-based solver in this research, and the simulations and measurements agreed very well. Results of simulations also led to improvement in the physical design of the bars, which reduced the inductance of the layout.With the accelerated solver, it is now possible to solve very large and complex problems on conventional computer systems, which was not possible before. This provides new possibilities to study real-world problems which are typically large in size and have complex structures.
Godkänd; 2012; 20121114 (dan); DISPUTATION Ämne: Industriell elektronik/Industrial Electronics Opponent: PhD Bruce Archambeault, IBM, Research Triangle Park, North Carolina, USA. Opponenten utför sitt uppdrag via distansöverbryggande teknik Ordförande: Docent Jonas Ekman, Institutionen för system- och rymdteknik, Luleå tekniska universitet Tid: Torsdag den 17 januari 2013, kl 13.30 Plats: A109, Luleå tekniska universitet

Thesis (M.S.) -- University of Maryland, College Park, 2004.
Thesis research directed by: Dept. of Electrical and Computer Engineering. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

This dissertation proposes a modular fast direct (MFD) analysis method for a class of problems involving a large fixed platform region and a smaller, variable design region. A modular solution algorithm is obtained by first decomposing the problem geometry into platform and design regions. The two regions are effectively detached from one another using basic equivalence concepts. Equivalence principles allow the total system model to be constructed in terms of independent interaction modules associated with the platform and design regions. These modules include interactions with the equivalent surface that bounds the design region. This dissertation discusses how to analyze (fill and factor) each of these modules separately and how to subsequently compose the solution to the original system using the separately analyzed modules. The focus of this effort is on surface integral equation formulations of electromagnetic scattering from conductors and dielectrics. In order to treat large problems, it is necessary to work with sparse representations of the underlying system matrix and other, related matrices. Fortunately, a number of such representations are available. In the following, we will primarily use the adaptive cross approximation (ACA) to fill the multilevel simply sparse method (MLSSM) representation of the system matrix. The MLSSM provides a sparse representation that is similar to the multilevel fast multipole method. Solutions to the linear systems obtained using the modular analysis strategies described above are obtained using direct methods based on the local-global solution (LOGOS) method. In particular, the LOGOS factorization provides a data sparse factorization of the MLSSM representation of the system matrix. In addition, the LOGOS solver also provides an approximate sparse factorization of the inverse of the system matrix. The availability of the inverse eases the development of the MFD method. Because the behavior of the LOGOS factorization is critical to the development of the proposed MFD method, a significant part of this dissertation is devoted to providing additional analyses, improvements, and characterizations of LOGOS-based direct solution methods. These further developments of the LOGOS factorization algorithms and their application to the development of the MFD method comprise the most significant contributions of this dissertation.

L'équation intégrale du champ électrique (EFIE) et l'équation intégrale du champ combiné (CFIE) souffrent d'un mauvais conditionnement à haute discrétisation et à bassefréquence : si la taille moyenne des arrêtes du maillage est réduite ou si la fréquence est diminuée le conditionnement du système se dégrade rapidement. Cela provoque le ralentissement ou la non convergence des solveurs itératifs. Cette dissertation présente de nouveaux paradigmes permettant l'obtention de solveurs à convergence rapide pour équations intégrales; pour prévenir la dégradation du conditionnement nous avançons l'état de l'art des techniques de préconditionnement dites de Calderon et de celles reposant sur l'utilisation des bases hiérarchiques. Pour traiter l'EFIE, nous introduisons une base hiérarchique pour maillages structurés et non-structurés dérivant des pré-ondelettes primaires et duales de Haar. De plus, nous introduisons un nouveau cadre permettant de préconditionner efficacement l'EFIE dans le cas d'objets à connexion multiples. L'applicabilité à la CFIE des préconditionneurs à bases hiérarchiques fait l'objet d'une étude aboutissant à la formalisation d'une technique de préconditionnement. Nous présentons aussi un préconditionneur multiplicatif de type Calderon (RF-CMP) qui permet l'obtention d'une matrice système Hermitienne, définie positive (HDP) et bien conditionnée, sans avoir recours, contrairement aux préconditionneurs existants, au raffinement du maillage ni à l'utilisation de fonction duales. Puisque la matrice est HPD, la méthode du gradient conjugué peut servir de solveur itératif avec une convergence garantie
The electric field integral equation (EFIE) and the combined field integral equation(CFIE) suffer from the dense-discretization and the low-frequency breakdown: if the average edgelength of the mesh is reduced, or if the frequency is decreased, then the condition number of the system matrix grows. This leads to slowly or non-converging iterative solvers. This dissertation presents new paradigms for rapidly converging integral equation solvers: to overcome the illconditioning, we advance and extend the state of the art both in hierarchical basis and in Calderón preconditioning techniques. For the EFIE, we introduce a hierarchical basis for structured and unstructured meshes based on generalized primal and dual Haar prewavelets. Furthermore, a framework is introduced which renders the hierarchical basis able to efficiently precondition the EFIE in the case that the scatterer is multiply connected. The applicability of hierarchical basis preconditioners to the CFIE is analyzed and an efficient preconditioning scheme is derived. In addition, we present a refinement-free Calderón multiplicative preconditioner (RF-CMP) that yields a system matrix which is Hermitian, positive definite (HPD), and well-conditioned. Different from existing Calderón preconditioners, no dual basis functions and thus no refinement of the mesh is required. Since the matrix is HPD—in contrast to standard discretizations of the EFIE—we can apply the conjugate gradient (CG) method as iterative solver, which guarantees convergence. Eventually, the RF-CMP is extended to the CFIE

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 67-69).
This thesis project was to write a program in C++ that solves electromagnetic scattering problems for arbitrarily shaped scatterers. This was implemented by using a surface integral formulation of Maxwell's equations, which discretizes the surface of the scatterer into thousands of triangles. The method of moments (MoM) was applied, which calculates the Green's functions between each triangle element. A matrix equation is obtained and solved using the Robin Hood (RH) method. The solution to this equation gives the scattered electromangetic field. This program is first tested on a sphere, which is compared to the analytic solution known as Mie scattering. Once these results are confirmed, the program can be used for the KATRIN experiment to ensure that no Penning traps occur in the electron spectrometer.
by David A. Chester.
S.B.

This dissertation presents and analyzes two new algorithms for sparse direct solution methods based on the use of local-global solution (LOGOS) modes. One of the new algorithms is a rigorous error control strategy for LOGOS-based matrix factorizations that utilize overlapped, localizing modes (OL-LOGOS) on a shifted grid. The use of OL-LOGOS modes is critical to obtaining asymptotically efficient factorizations from LOGOS-based methods. Unfortunately, the approach also introduces a non-orthogonal basis function structure. This can cause errors to accumulate across levels of a multilevel implementation, which has previously posed a barrier to rigorous error control for the OL-LOGOS factorization method. This limitation is overcome, and it is shown that it is possible to efficiently decouple the fundamentally non-orthogonal factorization subspaces in a manner that prevents multilevel error propagation. This renders the OL-LOGOS factorization error controllable in a relative RMS sense. The impact of the new, error-controlled OL-LOGOS factorization algorithm on computational resource utilization is discussed and several numerical examples are presented to illustrate the performance of the improved algorithm relative to previously reported results. The second algorithmic development considered is the development of efficient out-of-core (OOC) versions of the OL-LOGOS factorization algorithm that allow associated software tools to take advantage of additional resources for memory management. The proposed OOC algorithm incorporates a memory page definition that is tailored to match the flow of the OL-LOGOS factorization procedure. Efficiency of the function of the part is evaluated using a quantitative approach, because the tested massive storage device performances do not follow analytical results. The performance latency and the memory usage of the resulting OOC tools are compared with in-core performance results. Both the new error control algorithm and the OOC method have been incorporated into previously existing software tools, and the dissertation presents results for real-world simulation problems.

La résolution de problèmes d’onde par une méthode d’éléments finis de frontière (BEM) conduit à des systèmes d’équations linéaires pleins dont la taille augmente très vite pour les applications pratiques. Il est alors impératif d’employer des méthodes de résolution dites rapides. La méthode des multipôles rapides (FMM) accélère la résolution de ces systèmes par des algorithmes itératifs. La méthode des H-matrices permet d’accélérer les solveurs directs nécessaires aux cas d’application massivement multi-seconds membres. Elle a été introduite et théoriquement justifiée dans le cas de l’équation de Laplace.Néanmoins elle s’avère performante au-delà de ce qui est attendu pour des problèmes d’onde relativement haute fréquence. L’objectif de cette thèse est de comprendre pourquoi la méthode fonctionne et proposer des améliorations pour des fréquences plus élevées.Une H-matrice est une représentation hiérarchique par arbre permettant un stockage compressé des données grâce à une séparation des interactions proches (ou singulières)et lointaines (dites admissibles). Un bloc admissible a une représentation de rang faible de type UVT tandis que les interactions singulières sont représentées par des blocs pleins de petites tailles. Cette méthode permet une approximation rapide d’une matrice BEM par une H-matrice ainsi qu’une méthode de factorisation rapide de type Cholesky dont les facteurs sont également de type H-matrice.Nous montrons la nécessité d’un critère d’admissibilité dépendant de la fréquence et introduisons un critère dit de Fresnel basé sur la zone de diffraction de Fresnel. Ceci permet de contrôler la croissance du rang d’un bloc et nous proposons une estimation précise de celui-ci à haute fréquence à partir de résultats sur les fonctions d’onde sphéroïdales.Nous en déduisons une méthode de type HCA-II, robuste et fiable, d’assemblage rapide compressé à la précision voulue.Nous étudions les propriétés de cet algorithme en fonction de divers paramètres et leur influence sur le contrôle et la croissance du rang en fonction de la fréquence.Nous introduisons la notion de section efficace d’interaction entre deux clusters vérifiant le critère de Fresnel. Si celle-ci n’est pas dégénérée, le rang du bloc croît au plus linéairement avec la fréquence ; pour une interaction entre deux clusters coplanaires nous montrons une croissance comme la racine carrée de la fréquence. Ces développements sont illustrés sur des maillages représentatifs des interactions à haute fréquence
The boundary elements method (BEM) leads to dense linear systemswhose size growsrapidly in pratice ; hence the use of so-called fast methods. The fast multipole method(FMM) accelerates the resolution of BEM systems within an iterative scheme. The H-matrix method speeds up a direct resolution which is needed in massively multiple righthandsides problems. It has been provably introduced in the context of the Laplace equation.However, the use ofH-matrices for relatively high-frequency wave problems leadsto results above expectations. This thesis main goal is to provide an explanation of thesegood results and thus improve the method for higher frequencies.A H-matrix is a compressed tree-based hierarchical representation of the data associated with an admissibility criterion to separate the near (or singular) and far (or compres-sed) fields. An admissible block reads as a UVT rank deficient matrix while the singularblocks are dense with small dimensions. BEM matrices are efficiently represented byH-matrices and this method also allows for a fast Cholesky factorization whose factors arealsoH-matrices.Our work on the admissibility condition emphasizes the necessity of a frequency dependantadmissibility criterion. This new criterion is based on the Fresnel diffraction areathus labelled Fresnel admissibility condition. In that case a precise estimation of the rankof a high-frequency block is proposed thanks to the spheroidal wave functions theory.Consequently, a robust and reliable HCA-II type algorithm has been developed to ensurea compressed precision-controlled assembly. The influence of various parameters on thisnew algorithm behaviour is discussed ; in particular their influence on the control andthe growth of the rank according to the frequency.We define the interaction cross sectionfor two Fresnel-admissible clusters and show in the non-degenerate case that the rankgrowth is linear according to the frequency in the high-frequency regime ; interaction ofcoplanar clusters results in growth like the square root of the frequency. All these resultsare presented on meshes adapted to high-frequency interactions

The method of moment (MoM) [1] is a widely used method in electromagnetics to solve static and dynamic electromagnetic problems on varying geometries. However, in closely spaced geometries coupled with large aspect ratios, e.g. a large parallel plate capacitor with very small separation gap, the problem exhibits several challenges. Firstly, the close proximity of the field and source elements presents problems with convergence in numerical evaluations of the interactions between them. Secondly, the aspect ratio of the geometry gives an approximation whereby to far field points, the source contributions from locations that are far apart appear to cancel each other. This leads to high condition numbers in the system matrix. This thesis explores the potential solution to these problems as well as the application of modular fast and direct (MFD) [2] solver to expedite the solution of such problems.

Introduite récemment par Hughes et ses collaborateurs, l’analyse isogéométrique connaît un large succès pour des problèmes principalement industriels. L’idée est de faciliter la communication entre la C. A. O et la simulation numérique, sans avoir à repasser à chaque fois par des mailleurs. Ainsi, les fonctions définissants la géométrie sont utilisées pour approcher les solutions des équations à dérivées partielles. L’application aux problèmes issues de l’électromagnétisme ont été motivé par les travaux de Buffa et ses collaborateurs à Pavie. Dans cette thèse, nous avons appliqué cette méthode pour résoudre des problèmes issues de la physique des plasmas. S’il est vrai que la géométrie n’est pas définie, l’analyse isogéométrique dans sa version isoparamétrique, nous fournit un outil très puissant pour approcher les domaines de calculs. Dans un plasma, ce domaine est défini par la résolution d’un problème d’équilibre (MHD equilibrium). A partir de là, différents modèles sont utilisés pour décrire le plasma: cinétiques (gyrocinétique) ou fluides. Nous avons passé en revue les méthodes les plus classiques et plus utilisées afin de révéler l’intérêt de la méthode. Se basant sur la structure de produit tensoriel, nous avons développé des solveurs rapides pour la résolution de certains problèmes. Nous avons aussi dérivé un solveur, se basant sur les complexes de Hilbert, pour les équations de Maxwell en ”time domain”
The underlying idea behind the IsoGeometric Analysis (IGA), developed recently by Hughes et al, is to use the functions (B-splines/NURBS) that describe the geometry in order to approach the numerical solution of pdes. This reduces the communication between C. A. D system and numerical solvers, and thus is well adapted for industrial problems. The application to Maxwell’s equations was recently motivated by the works of Buffa et al. In this thesis, we apply this new method to solve problems arising from plasmas physics. Even if, in the general case, the geometry is not defined, the IsoGeometric Analysis gives a powerful tool to approach computational domains, and thus can be used in its isoparametric version. In a plasma, this domain can be defined by using an equilibrium problem (MHD equilibrium). Therefor, we can use common models to describe the plasma; this can be done using kinetic (gyrokinetic) or fluid approach. We have adapted the most classical methods using the IGA approach in order to reveal its interest. We were also able to derive some fast solver, based on the tensor product structure. We have also studied Maxwell’s equations for the time domaine problem, and gave a new fast algorithm for its resolution based on Hilbert complexes

El Método de los Momentos (MoM) ha sido ampliamente utilizado en las últimas décadas para la discretización y la solución de las formulaciones de ecuación integral que aparecen en muchos problemas electromagnéticos de antenas y dispersión. Las más utilizadas de dichas formulaciones son la Ecuación Integral de Campo Eléctrico (EFIE), la Ecuación Integral de Campo Magnético (MFIE) y la Ecuación Integral de Campo Combinada (CFIE), que no es más que una combinación lineal de las dos anteriores.
Las formulaciones MFIE y CFIE son válidas únicamente para objetos cerrados y necesitan tratar la integración de núcleos con singularidades de orden superior al de la EFIE. La falta de técnicas eficientes y precisas para el cálculo de dichas integrales singulares a llevado a imprecisiones en los resultados. Consecuentemente, su uso se ha visto restringido a propósitos puramente académicos, incluso cuando tienen una velocidad de convergencia muy superior cuando son resuelto iterativamente, debido a su excelente número de condicionamiento.
En general, la principal desventaja del MoM es el alto coste de su construcción, almacenamiento y solución teniendo en cuenta que es inevitablemente un sistema denso, que crece con el tamaño eléctrico del objeto a analizar. Por tanto, un gran número de métodos han sido desarrollados para su compresión y solución. Sin embargo, muchos de ellos son absolutamente dependientes del núcleo de la ecuación integral, necesitando de una reformulación completa para cada núcleo, en caso de que sea posible.
Esta tesis presenta nuevos enfoques o métodos para acelerar y incrementar la precisión de ecuaciones integrales discretizadas con el Método de los Momentos (MoM) en electromagnetismo computacional.
En primer lugar, un nuevo método iterativo rápido, el Multilevel Adaptive Cross Approximation (MLACA), ha sido desarrollado para acelerar la solución del sistema lineal del MoM. En la búsqueda por un esquema de propósito general, el MLACA es un método independiente del núcleo de la ecuación integral y es puramente algebraico. Mejora simultáneamente la eficiencia y la compresión con respecto a su versión mono-nivel, el ACA, ya existente. Por tanto, representa una excelente alternativa para la solución del sistema del MoM de problemas electromagnéticos de gran escala.
En segundo lugar, el Direct Evaluation Method, que ha provado ser la referencia principal en términos de eficiencia y precisión, es extendido para superar el cálculo del desafío que suponen las integrales hiper-singulares 4-D que aparecen en la formulación de Ecuación Integral de Campo Magnético (MFIE) así como en la de Ecuación Integral de Campo Combinada (CFIE). La máxima precisión asequible -precisión de máquina se obtiene en un tiempo más que razonable, sobrepasando a cualquier otra técnica existente en la bibliografía.
En tercer lugar, las integrales hiper-singulares mencionadas anteriormente se convierten en casi-singulares cuando los elementos discretizados están muy próximo pero sin llegar a tocarse. Se muestra como las reglas de integración tradicionales tampoco convergen adecuadamente en este caso y se propone una posible solución, basada en reglas de integración más sofisticadas, como la Double Exponential y la Gauss-Laguerre.
Finalmente, un esfuerzo en facilitar el uso de cualquier programa de simulación de antenas basado en el MoM ha llevado al desarrollo de un modelo matemático general de un puerto de excitación en el espacio discretizado. Con este nuevo modelo, ya no es necesaria la adaptación de los lados del mallado al puerto en cuestión.
The Method of Moments (MoM) has been widely used during the last decades for the discretization and the solution of integral equation formulations appearing in several electromagnetic antenna and scattering problems. The most utilized of these formulations are the Electric Field Integral Equation (EFIE), the Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE), which is a linear combination of the other two.
The MFIE and CFIE formulations are only valid for closed objects and need to deal with the integration of singular kernels with singularities of higher order than the EFIE. The lack of efficient and accurate techniques for the computation of these singular integrals has led to inaccuracies in the results. Consequently, their use has been mainly restricted to academic purposes, even having a much better convergence rate when solved iteratively, due to their excellent conditioning number.
In general, the main drawback of the MoM is the costly construction, storage and solution considering the unavoidable dense linear system, which grows with the electrical size of the object to analyze. Consequently, a wide range of fast methods have been developed for its compression and solution. Most of them, though, are absolutely dependent on the kernel of the integral equation, claiming for a complete re-formulation, if possible, for each new kernel.
This thesis dissertation presents new approaches to accelerate or increase the accuracy of integral equations discretized by the Method of Moments (MoM) in computational electromagnetics.
Firstly, a novel fast iterative solver, the Multilevel Adaptive Cross Approximation (MLACA), has been developed for accelerating the solution of the MoM linear system. In the quest for a general-purpose scheme, the MLACA is a method independent of the kernel of the integral equation and is purely algebraic. It improves both efficiency and compression rate with respect to the previously existing single-level version, the ACA. Therefore, it represents an excellent alternative for the solution of the MoM system of large-scale electromagnetic problems.
Secondly, the direct evaluation method, which has proved to be the main reference in terms of efficiency and accuracy, is extended to overcome the computation of the challenging 4-D hyper-singular integrals arising in the Magnetic Field Integral Equation (MFIE) and Combined Field Integral Equation (CFIE) formulations. The maximum affordable accuracy --machine precision-- is obtained in a more than reasonable computation time, surpassing any other existing technique in the literature.
Thirdly, the aforementioned hyper-singular integrals become near-singular when the discretized elements are very closely placed but not touching. It is shown how traditional integration rules fail to converge also in this case, and a possible solution based on more sophisticated integration rules, like the Double Exponential and the Gauss-Laguerre, is proposed.
Finally, an effort to facilitate the usability of any antenna simulation software based on the MoM has led to the development of a general mathematical model of an excitation port in the discretized space. With this new model, it is no longer necessary to adapt the mesh edges to the port.

Computational electromagnetic modeling involves generating system matrices by discretizing integral equations and solving the resulting system of linear equations. Many methods of solving the system of linear equations exist and one such method is the factorization of the matrix using the so called local-global solution (LOGOS) modes. Computer codes to perform the discretization of the integral equations, filling of the matrix, and the subsequent LOGOS factorization have previously been developed by others. However, these codes are limited to complex double precision arithmetic only. This thesis extends and expands the existing computer by creating a more general implementation that is able to analyze a problem not only in complex double precision but also in real double precision and both complex and real single precision. The existing code is expanded using "templates" in Fortran 90 and the resulting generic code is used test the performance of the LOGOS (both OL- and NL-LOGOS) factorization on matrices generated by discretization of the volume integral equation. As part of this effort, we demonstrate for the first time that the LOGOS factorization provides an O(N log N) complexity solution to the volume integral equation formulation of low-frequency electromagnetic problems.

Cette étude s’inscrit dans le domaine de la simulation électromagnétique des problèmes de grande taille tels que la diffraction d’ondes planes par de larges plateformes et le rayonnement d’antennes aéroportées. Elle consiste à développer une méthode combinant décomposition en sous-domaines et compression hiérarchique des équations intégrales de frontière. Pour cela, nous rappelons dans un premier temps les points importants de la méthode des équations intégrales de frontière et de leur compression hiérarchique par l’algorithme ACA (Adaptive Cross Approximation). Ensuite, nous présentons la formulation IE-DDM (Integral Equations – Domain Decomposition Method) obtenue à partir d’une représentation intégrale des sous-domaines. Les matrices résultant de la discrétisation de cette formulation sont stockées au format H-matrice (matricehiérarchique). Un solveur spécialement adapté à la résolution de la formulation IE-DDM et à sa représentation hiérarchique a été conçu. Cette étude met en évidence l’efficacité de la décomposition en sous-domaines en tant que préconditionneur des équations intégrales. De plus, la méthode développée est rapide pour la résolution des problèmes à incidences multiples ainsi que la résolution des problèmes basses fréquences

Nous présentons dans cette thèse les évolutions apportées au solveur Galerkin Discontinu Teta-CLAC, issu de la collaboration IRMA-AxesSim, au cours du projet HOROCH (2015-2018). Ce solveur permet de résoudre les équations de Maxwell en 3D, en parallèle sur un grand nombre d'accélérateurs OpenCL. L'objectif du projet HOROCH était d'effectuer des simulations de grande envergure sur un modèle numérique complet de corps humain. Ce modèle comporte 24 millions de mailles hexaédriques pour des calculs dans la bande de fréquences des objets connectés allant de 1 à 3 GHz (Bluetooth). Les applications sont nombreuses : téléphonie et accessoires, sport (maillots connectés), médecine (sondes : gélules, patchs), etc. Les évolutions ainsi apportées comprennent, entre autres : l'optimisation des kernels OpenCL à destination des CPU dans le but d'utiliser au mieux les architectures hybrides ; l'expérimentation du runtime StarPU ; le design d'un schéma d'intégration à pas de temps local ; et bon nombre d'optimisations permettant au solveur de traiter des simulations de plusieurs millions de mailles
In this thesis, we present the evolutions made to the Discontinuous Galerkin solver Teta-CLAC – resulting from the IRMA-AxesSim collaboration – during the HOROCH project (2015-2018). This solver allows to solve the Maxwell equations in 3D and in parallel on a large amount of OpenCL accelerators. The goal of the HOROCH project was to perform large-scale simulations on a complete digital human body model. This model is composed of 24 million hexahedral cells in order to perform calculations in the frequency band of connected objects going from 1 to 3 GHz (Bluetooth). The applications are numerous: telephony and accessories, sport (connected shirts), medicine (probes: capsules, patches), etc. The changes thus made include, among others: optimization of OpenCL kernels for CPUs in order to make the best use of hybrid architectures; StarPU runtime experimentation; the design of an integration scheme using local time steps; and many optimizations allowing the solver to process simulations of several millions of cells

En électromagnétisme les structures périodiques suscitent un grand intérêt. Ces structures agissent ainsi comme des filtres fréquentiels et permettent la fabrication de méta-matériaux, composites et artificiels. Elles présentent des propriétés électromagnétiques inédites pour les matériaux naturels telles que des bandes interdites. On a ainsi pu fabriquer de nouveaux dispositifs permettant de guider, de focaliser ou de stopper la propagation. C'est par exemple utile pour éviter le couplage entre différents éléments rayonnants notamment via la caractérisation des ondes de surface qui se propagent à l'interface entre l'air et la structure périodique. Ce travail de thèse s'inscrit dans ce contexte et propose une description de la méthode des éléments finis dédiée à la caractérisation des structures périodiques. La modélisation numérique aboutit à des problèmes de valeurs propres de grandes tailles. Elle implique la résolution de systèmes linéaires composés de matrices creuses. Une méthode est abordée pour résoudre ce type de problème, en optimisant et combinant différents algorithmes. Avant d'aborder les différents aspects de la méthode développée, nous établissons une liste exhaustive de l'ensemble des méthodes qui existent en énonçant leurs avantages et leurs inconvénients. Nous constatons notamment que la méthode des éléments finis permet de traiter un large éventail de structures périodiques en trois dimensions sans limitation sur leur forme géométrique. Nous présentons alors les différentes formulations de cette méthode. Ensuite les aspects algorithmiques de la méthode sont détaillés. Nous montrons notamment qu'une analyse des paramètres de résolution permet de préciser les interprétations physiques des résultats obtenus. Finalement nous présentons les performances de notre outil sur des cas d'applications issus de la littérature et nous abordons la caractérisation des ondes de surface. Pour cela, l'étude d'un réseau d'antennes patchs insérées dans des cavités métalliques est conduite. Notons pour conclure que les études conduites au cours de cette thèse ont abouti à la production d'un code utilisable dans un environnement de calcul initialement présent à l'ONERA.

Electromagnetic forward modeling and inversion methods have extensive applications in geophysical exploration, and large-scale controlled-source electromagnetic method has recently drawed lots of attention. However, to obtain a rigorous and efficient forward solver for this large-scale three-dimensional problem is difficult, since it usually requires to solve for a large number of unknowns from a system of equations describing the complicate scattering behavior of electromagnetic waves that happened within inhomogeneous media. As for the development of an efficient inversion solver, because of the nonlinear, non-unique and ill-posed properties of the problem, it is also a very challenging task.

In the first part of this dissertation, a fast three-dimensional nonlinear reconstruction method is proposed for controlled-source electromagnetic method. The borehole-to-surface and airborne electromagnetic survey methods are investigated using synthetic data. In this work, it is assumed that there is only electric contrast between the inhomogeneous object and the layered background medium, for the reason that the electric contrast is much more dominant than magnetic contrast in most cases of the earth formation. Therefore, only the EFIE is needed to solve. While the forward scattering problem is solved by the stabilized bi-conjugate gradient FFT (BCGS-FFT) method to give a rigorous and efficient modeling, the Bore iterative method along with the multiplicative regularization technique is used in the inversion through frequency hopping. In the inversion, to speed up the expensive computation of the sensitivity matrix relating every receiver station to every unknown element, a fast field evaluation (FFE) technique is proposed using the symmetry property of the layered medium Green's function combined with a database strategy. The conjugate-gradient method is then applied to minimize the cost function after each iteration. Due to the benefits of using 3D FFT acceleration, the proposed FFE technique as well as the recursive matrix method combined with an interpolation technique to evaluate the LMGF, the developed inversion solver is highly efficient, and requires very low computation time and memory. Numerical experiments for both 3D forward modeling and conductivity inversion are presented to show the accuracy and efficiency of the method.

Some recent research on artificial nanoparticles have demonstrated the improved performance in geophysical imaging using magnetodielectric materials with enhanced electric and magnetic contrasts. This gives a promising perspective to the future geophysical exploration by infusing well-designed artificial magnetodielectric materials into the subsurface objects, so that a significantly improved imaging can be achieved. As a preparation for this promising application, the second part of the dissertation presents an efficient method to solve the scattering problem of magnetodielectric materials with general anisotropy that are embedded in layered media. In this work, the volume integral equation is chosen as the target equation to solve, since it solves for fields in inhomogeneous media with less number of unknowns than the finite element method. However, for complicated materials as magnetodielectric materials with general anisotropy, it is a very challenging task, because it requires to simultaneously solve the electric field integral equation (EFIE) and magnetic field integral equation (MFIE). Besides that, the numerous evaluation of the layered medium Green's function (LMGF) for the stratified background formation adds on the difficulty and complexity of the problem. To my knowledge, there is no existing fast solver for the similar problem. In this dissertation, using the mixed order stabilized biconjugate-gradient fast Fourier transform (mixed-order BCGS-FFT) method, a fast forward modeling method is developed to solve this challenging problem. Several numerical examples are performed to validate the accuracy and efficiency of the proposed method.

Besides the above mentioned two topics, one-dimensional inversion method is presented in the third part to determine the tilted triaxial conductivity tensor in a dipping layered formation using triaxial induction measurements. The tilted triaxial conductivity tensor is described by three conductivity components and three Euler angles. Based on my knowledge, due to the highly nonlinear and ill-posed nature of the inverse problem, this study serves as the first work that investigates on the subject. There are six principal coordinate systems that can give the same conductivity tensor. Permutation is performed to eliminate the ambiguity of inversion results caused by the ambiguity of the principal coordinate system. Three new Euler angles after permutation for each layer can be found by solving a nonlinear equation. Numerical experiments are conducted on synthetic models to study the feasibility of determining triaxially anisotropic conductivity tensor from triaxial induction data. This project is accomplished during my internship in the Houston Formation Evaluation Integration Center (HFE) at Schlumberger, a world-leading oilfield service company.

Dissertation

Acoustic and electromagnetic scattering from arbitrarily shaped structures can be numerically characterized by solving various surface integral equations (SIEs). One of the most effective techniques to solve SIEs is the Nyström method. Compared to other existing methods,the Nyström method is easier to implement especially when the geometrical discretization is non-conforming and higher-order representations of the geometry and unknowns are desired. However,singularities of the Green’s function are more difficult to”manage”since they are not ”smoothened” through the use of a testing function. This dissertation describes purely numerical schemes to account for different orders of singularities that appear in acoustic and electromagnetic SIEs when they are solved by a high-order Nyström method utilizing a mesh of curved discretization elements. These schemes make use of two sets of basis functions to smoothen singular integrals: the grid robust high-order Lagrange and the high-order Silvester-Lagrange interpolation basis functions. Numerical results comparing the convergence of two schemes are presented. Moreover, an extremely scalable implementation of fast multipole method (FMM) is developed to efficiently (and iteratively) solve the linear system resulting from the discretization of the acoustic SIEs by the Nyström method. The implementation results in O(N log N) complexity for high-frequency scattering problems. This FMM-accelerated solver can handle N =2 billion on a 200,000-core Cray XC40 with 85% strong scaling efficiency. Iterative solvers are often ineffective for ill-conditioned problems. Thus, a fast direct (LU)solver,which makes use of low-rank matrix approximations,is also developed. This solver relies on tile low rank (TLR) data compression format, as implemented in the hierarchical computations on many corearchitectures (HiCMA) library. This requires to taskify the underlying SIE kernels to expose fine-grained computations. The resulting asynchronous execution permit to weaken the artifactual synchronization points,while mitigating the overhead of data motion. We compare the obtained performance results of our TLRLU factorization against the state-of-the-art dense factorizations on shared memory systems. We achieve up to a fourfold performance speedup on a 3D acoustic problem with up to 150 K unknowns in double complex precision arithmetics.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007.
Source: Dissertation Abstracts International, Volume: 68-11, Section: B, page: 7533. Adviser: Andreas C. Cangellaris. Includes bibliographical references (leaves 133-140) Available on microfilm from Pro Quest Information and Learning.

Time-domain methods are preferred over their frequency-domain counterparts for solving acoustic and electromagnetic scattering problems since they can produce wide- band data from a single simulation. Among the time-domain methods, time-domain surface integral equation solvers have recently found widespread use because they offer several benefits over differential equation solvers. This dissertation develops several second-kind surface integral equation solvers for analyzing transient acoustic scattering from rigid and penetrable objects and transient electromagnetic scattering from perfect electrically conducting and dielectric objects. For acoustically rigid, perfect electrically conducting, and dielectric scatterers, fully explicit marching-on-in-time schemes are developed for solving time domain Kirchhoff, magnetic field, and scalar potential integral equations, respectively. The unknown quantity (e.g., velocity potential, electric current, or scalar potential) on the scatterer surface is discretized using a higher-order method in space and Lagrange interpolation in time. The resulting system is cast in the form of an ordinary differen- tial equation and integrated in time using a predictor-corrector scheme to obtain the unknown expansion coefficients. The explicit scheme can use the same time step size as its implicit counterpart without sacrificing from the stability of the solution and is much faster under low-frequency excitation (i.e., for large time step size). In addition, low-frequency behavior of vector potential integral equations for perfect electrically conducting scatterers is also investigated in this dissertation. For acoustically penetrable scatterers, presence of spurious interior resonance modes in the solutions of two forms of time domain surface integral equations is investigated. Numerical results demonstrate that the solution of the form that is widely used in the literature is corrupted by the interior resonance modes. But, the amplitude of these modes in the time domain can be suppressed by increasing the accuracy of discretization especially in time. On the other hand, the proposed one in the combined form shows a resonance-free performance verified via numerical experiments. In addition to providing detailed formulations of these solvers, the dissertation presents numerical examples, which demonstrate the solvers’ accuracy, efficiency, and applicability in real-life scenarios.

With the rapid increase in available compute power and memory, and bolstered by the advent of efficient formulations and algorithms, the role of 3D full-wave computational methods for accurate modelling of complex electromagnetic (EM) structures has gained in significance. The range of problems includes Radar Cross Section (RCS) computation, analysis and design of antennas and passive microwave circuits, bio-medical non-invasive detection and therapeutics, energy harvesting etc. Further, with the rapid advances in technology trends like System-in-Package (SiP) and System-on-Chip (SoC), the fidelity of chip-to-chip communication and package-board electrical performance parameters like signal integrity (SI), power integrity (PI), electromagnetic interference (EMI) are becoming increasingly critical. Rising pin-counts to satisfy functionality requirements and decreasing layer-counts to maintain cost-effectiveness necessitates 3D full wave electromagnetic solution for accurate system modelling. Method of Moments (MoM) is one such widely used computational technique to solve a 3D electromagnetic problem with full-wave accuracy. Due to lesser number of mesh elements or discretization on the geometry, MoM has an advantage of a smaller matrix size. However, due to Green's Function interactions, the MoM matrix is dense and its solution presents a time and memory challenge. The thesis focuses on formulation and development of novel techniques that aid in fast MoM based electromagnetic solutions. With the recent paradigm shift in computer hardware architectures transitioning from single-core microprocessors to multi-core systems, it is of prime importance to parallelize the serial electromagnetic formulations in order to leverage maximum computational benefits. Therefore, the thesis explores the possibilities to expedite an electromagnetic simulation by scalable parallelization of near-linear complexity algorithms like Fast Multipole Method (FMM) on a multi-core platform. Secondly, with the best of parallelization strategies in place and near-linear complexity algorithms in use, the solution time of a complex EM problem can still be exceedingly large due to over-meshing of the geometry to achieve a desired level of accuracy. Hence, the thesis focuses on judicious placement of mesh elements on the geometry to capture the physics of the problem without compromising on accuracy- a technique called Adaptive Mesh Refinement. This facilitates a reduction in the number of solution variables or degrees of freedom in the system and hence the solution time. For multi-scale structures as encountered in chip-package-board systems, the MoM formulation breaks down for parts of the geometry having dimensions much smaller as compared to the operating wavelength. This phenomenon is popularly known as low-frequency breakdown or low-frequency instability. It results in an ill-conditioned MoM system matrix, and hence higher iteration count to converge when solved using an iterative solver framework. This consequently increases the solution time of simulation. The thesis thus proposes novel formulations to improve the spectral properties of the system matrix for real-world complex conductor and dielectric structures and hence form well-conditioned systems. This reduces the iteration count considerably for convergence and thus results in faster solution. Finally, minor changes in the geometrical design layouts can adversely affect the time-to-market of a commodity or a product. This is because the intermediate design variants, in spite of having similarities between them are treated as separate entities and therefore have to follow the conventional model-mesh-solve workflow for their analysis. This is a missed opportunity especially for design variant problems involving near-identical characteristics when the information from the previous design variant could have been used to expedite the simulation of the present design iteration. A similar problem occurs in the broadband simulation of an electromagnetic structure. The solution at a particular frequency can be expedited manifold if the matrix information from a frequency in its neighbourhood is used, provided the electrical characteristics remain nearly similar. The thesis introduces methods to re-use the subspace or Eigen-space information of a matrix from a previous design or frequency to solve the next incremental problem faster.

The indirect effects of a nearby lightning strike on an airborne vehicle with its long trailing conducting plume is not well understood. Since airborne vehicles and its payload are expensive, their loss as a result of either a direct strike or due to the induced current and voltage of a nearby lightning strike is not desirable. The electromagnetic field generated due to the induced current on the skin of the vehicle may get coupled with the internal circuitry through the apertures on the vehicle body. If the coupled electromagnetic energy is more than the damage threshold level of the sensitive devices of the control circuit, they may fail which may lead to aborting the mission or a possible degradation in the vehicle performance. It has been reported that lightning induced phenomena was the cause of malfunctioning as well as aborting of some of the lunar missions. So in the present work, the computation of induced current and voltage on the skin of the vehicle body in the presence of an ionized long trailing exhaust plume has been attempted. The lightning channel is assumed to be vertical to the ground plane and extends up to a height of 7.5 km. The radiated electric and magnetic fields from the lightning channel at different heights from 10 m to 10 km and for lateral distances varying from 20 m to 10 km from the lightning channel are computed and the field waveforms are presented. For the computation of the radiated electric and magnetic fields from the lightning channel, modified transmission line with exponential current decay (MTLE) model for representing the lightning channel and the Heidler’s expression for the lightning channel base current are used. The peak amplitude of the lightning current used is 12 kA with a maximum current derivative of 40 kA/µs. It is seen that the vertical electric field in general, is bipolar in nature and that the height at which the change in polarity reversal takes place increases with increase of lateral distance from the lightning channel. The vertical electric field just above the ground is unipolar for all lateral distances from the channel and this is because the contribution due to the image of the lightning channel dominates the vertical electric field. The horizontal electric field is always unipolar for all heights and all lateral distances from the lightning channel studied. The effect of variation in the rate of rise of lightning current (di/dt) and the velocity of lightning current on the radiated electric and magnetic fields for the above heights and distances have also been studied. It is seen that the variation in maximum current derivative does not have a significant influence on the electric field when ground is assumed as a perfect conductor but it influences significantly the horizontal electric field when ground has finite conductivity. The velocity of propagation of lightning current on the other hand has a significant influence for both perfectly as well as finitely conducting ground conditions. For the computation of the induced current and voltage on the body of the airborne vehicle due to the coupling of the above mentioned electromagnetic fields radiated from a near by lightning discharge, the vehicle and its exhaust plume have been modeled as a transmission line and Finite Difference Time Domain (FDTD) numerical technique has been used for the computation. Regardless of the vehicle size, the structure can be modeled as a nonuniform transmission line consisting of a series of sections consisting of capacitive and inductive components. These components of the vehicle and its exhaust plume are computed using the Method of Moment (MoM) technique. The interaction of the electromagnetic wave with the plume depends on the electrical conductivity as well as the gas dynamic characteristics of the plume. Hence, in this research work an attempt has also been made to study the electrical conductivity (σe) and permittivity (εe) as well as the gas dynamic properties of the exhaust plume taking into consideration its turbulent nature. In general, the airborne vehicle can be considered as perfectly conducting (conductivity 3x107 S/m) where as the plume has finite conductivity. The electrical properties of an airborne vehicle exhaust plume such as electrical conductivity and the permittivity and their distribution along axial and radial directions depend on several factors. They are (i) propellant composition, (ii) impurity content in the propellants which generate ionic charge particles in the exhaust and (iii) the characteristics of the exhaust plume intensive parameters such as temperature, pressure, velocity and the presence of shock waves. These properties of the exhaust plume are computed in the two separate regions of interest as discussed next. The first region is inside the combustion chamber and up to the nozzle throat of the vehicle and the second region is from the throat to the exterior i.e., the ambient atmosphere or the downstream of the plume. In the first region where chemical reaction kinetics have to be considered, NASA Chemical Equilibrium with Application (CEA) software package has been used to compute the intensive parameters of the fluid at the throat of the nozzle. The pressure in the combustion chamber is taken as 4410 kPa and the back pressure at the exit plane is taken as 101.325 kPa. In the second region, FLUENT software package have been used for the fluid dynamic study of the exhaust plume from the vehicle nozzle throat to the exterior domain. The data obtained from the first region using CEA provides the parameters at the nozzle throat that are used as input parameters for the second region. In the study, a conical nozzle configuration of throat radius (rt) of 0.0185 m (nozzle exit plane radius is 0.05 m), half cone angle of 18º and nozzle expansion ratio (Ae/At) of 7.011 are used. The contour plot of the intensive parameters of the exhaust plume and the mass fraction of the charged particles are presented. The vehicle exhaust flow passes through different types of expansion and compression waves. In the present work, simulation is done for a slightly under expanded nozzle i.e. nozzle exit static pressure is slightly more than the ambient static pressure. Since the exit pressure is more than the ambient pressure, the exhaust gases expand to reach the ambient pressure. As the expansion waves reach the contact discontinuity (i.e. the boundary where the outer edge of the gas flow meets the free stream air), they again reflect back inward to create compression waves. These compression waves force the flow to turn back inward and increase its pressure. If the compression waves are strong enough, they will merge into an oblique shock wave. In the present work, more than eight such barrel shocks are captured. When the shock waves are generated, Mach number reduces sharply and static temperature and static pressure increases where as the total temperature of the exhaust remains constant in the shock wave formations. The characteristics of the plume such as pressure, temperature, velocity and concentration of the charged particles (i.e., e¯, Na+ and Cl¯) and neutral species such as CO, CO2 , Cl, H, HCl, H2O, H2 , N2, Na, NaCl, O, OH and O2 along axial and radial directions in the external domain have been studied. The above parameters are used to compute the collision frequencies and plasma frequencies of the charged particles as well as the number density of the species along axial and radial directions of the exhaust plume. These parameters are used to compute the effective conductivity distribution in the axial and radial directions for an incident electromagnetic field of frequency 1 MHz. The peak value of the conductivity computed is 0.12 S/m near the exit plane and it reduces to 0.02 S/m at an axial distance of 7.5 m from the exit plane which is well within the range suggested in the published literature. It has been observed that the oscillation in the conductivity along axial direction is a reflection of the shock wave formation in the exhaust plume. The electrical conductivity and the relative permittivity of the exhaust plume have been computed for three different radii of the nozzle at the exit plane i.e., 0.025 m, 0.05 m and 0.075 m. It is seen that the distribution of the conductivity and relative permittivity along the axial direction of the exhaust are independent of the nozzle exit plane radius. To study the coupling of lightning electromagnetic field with the vehicle and its exhaust plume two cases have been considered. These are (i) when the vehicle and its exhaust plume are at certain height above the ground and (ii) when the exhaust plume is touching the ground. The dimensions of the vehicle used in the study are as follows: length of the vehicle is 20 m and the length of its exhaust plume is 75 m. The radius of the vehicle is taken as 0.5 m. The vehicle and its exhaust plume are assumed to be at a lateral distance of 250 m from the lightning channel. In case one, when the vehicle and its inhomogeneous exhaust plume tip is at a height of 10 m above the ground, both the ends are open. So the reflection coefficients of the current wave and voltage wave at the end points are -1 and +1 respectively irrespective of the characteristic impedances of the vehicle and its exhaust plume. So when the reflected current propagates it will tend to reduce along the length of the object. Hence, the induced current at the end points are zero and the currents in the end segments are less than those in the intermediate segments. The spatial distribution of the peak magnitude of the time varying induced current, |Imax|, in each segment along the length of the vehicle without and with the exhaust plume are presented. In case of vehicle without plume, the maximum value of the induced current is at the middle segment of the vehicle and its value is 4.8 A. The presence of the inhomogeneous plume enhances the maximum value of the induced current to 33 A and its position is shifted to the exhaust plume side. When the voltage wave propagates, it will enhance the induced voltage in the vehicle body. The time varying potential difference between the end points of the vehicle without plume and the vehicle with its exhaust plume which drives the induced current are computed and it is seen that the potential difference for the vehicle without plume is unipolar whereas it is bipolar for the vehicle with exhaust plume. The lightning induced current on the skin of the vehicle will generate an electromagnetic field which may couple with the internal electronic devices and circuits through the apertures. The amount of electromagnetic energy that will be transmitted through an aperture on the vehicle skin and coupled with the internal electronic equipments depends on the characteristics of the induced current on the skin of the vehicle, the electrical size, shape, orientation and location of the aperture and the location of the internal electronic devices with respect to the aperture. So the time varying induced current and its di/dt at three different locations on the vehicle body i.e., tail of the vehicle, middle of the vehicle and vehicle nose are computed. It is seen that the induced current on the vehicle and its di/dt in the absence of the plume are oscillating in nature but they are critically damped in the presence of the trailing inhomogeneous exhaust plume. It also shows that the enhancement of induced current and its di/dt at the tail are much more than at the middle or at the nose of the vehicle which is true for an electrically short vehicle i.e., lv/λmin ≈ 0.067 as cited in the literature. So the presence of an aperture on the skin of the vehicle near to tail will transmit maximum electromagnetic energy into the inside of the vehicle. Therefore during design of the electrically short airborne vehicles, any aperture should be avoided near the tail of the vehicle or internal electronic devices should be placed away from the tail of the vehicle. In case 2, when the plume is touching the ground, the transient induced current in the plume will propagate into the soil. The effective impedance for smaller currents will be quite high (the inductance and capacitance effect are not taken into consideration for calculating the impedance. So the impedance of the soil is dominated by only the resistance). However, as soon as the current exceeds a certain value, the resulting soil gradient can reach the breakdown gradient of the soil i.e., 200-500 kV/m as cited in literature resulting in soil ionization. This will effectively lower the soil impedance. These dynamic characteristics of the soil resistance with induced current are incorporated by considering the expression for the soil resistance. To study the effect of soil resistivity on the time varying induced current and the voltage, computations have been done for various resistivities of the soil i.e., 0 Ωm, 100 Ωm and 200 Ωm. For soil resistivity of 0 Ωm, the reflection coefficients at the ground and at the open ends for the current wave are +1 and -1 respectively. So at the ground end, the reflected current wave will enhance and at the open end it will diminish as it propagates along the length of the vehicle and its exhaust. As the resistivity of the soil increases, the reflection coefficient of the current at the ground end decreases from +1, so the peak magnitude of the current reduces along the length till the length is half of the total length of the plume and the vehicle. Therefore, the peak magnitude of the induced current in the ground segment is much more than the peak magnitude of the current in the segment at the open end. For a finitely conducting plume, the peak value of the potential difference between the two ends of the vehicle and its exhaust plume are 92 kV, 91 kV and 90 kV for soil resistivities of 0 Ωm 100 Ωm and 200 Ωm respectively. Therefore the influence of the soil resistivity on the induced current is found to be not much significant. The spatial distribution of the peak magnitude of the time varying induced current in each segment along the length of the vehicle with inhomogeneous exhaust plume for the above three different soil resistivities are presented at a lateral distance of 250 m from the lightning channel. It is seen that when the plume is touching the ground, the induced current on the vehicle at the tail, middle and nose sections are marginally more than when the vehicle and its exhaust are at a height of 10 m above the ground. The effects of different parameters such as peak value and maximum di/dt of lightning current, velocity of lightning current, lateral distance of the vehicle from lightning channel and the height of the tip of the exhaust plume above the ground on the induced current and voltage on the airborne vehicle have also been studied. The peak amplitude of the lightning current used are 30 kA and 100 kA in addition to 12 kA mentioned earlier for the field computation. Also maximum di/dt values of 40 kA/µs and 120 kA/µs for the lightning current have been used for the computation. It is observed that the induced current increases with increase of the peak value, maximum di/dt as well as the velocity of propagation of the lightning current where as the induced current will reduce with increase of lateral distance and height of the tip of the exhaust plume above the ground. As an offshoot of the present work, the axial and radial distribution of the parameter, σe/ωεe (loss tangent of the exhaust plume) for an incident electromagnetic wave (lightning electromagnetic field) frequency of 1 MHz have been computed to study the conducting properties of the exhaust plume. σe/ωεe of the exhaust plume at 1 MHz frequency varies from 2324 to 365. Since σe/ωεe >>1, the plume behaves as a good conductor and the displacement currents can be neglected. In addition to this, the variation of parameter σe/ωεe for frequency ranges of 0.1 MHz to 5 GHz are also studied where σe and εe are the maximum effective conductivity and permittivity of the exhaust plume at the chosen frequency of an incident EM wave. It shows that the parameter σe/ωεe is 1.8x104 at 0.1 MHz and reduces to 0.45 for 5 GHz and its value is 1 at a frequency of 2.285 GHz. Therefore at lower EM wave frequency, the exhaust plume behaves as a good conductor and that conductivity reduces with increase of the frequency. The exhaust plume in the present study behaves as a good conductor below or at the EM wave frequency of 2.285 GHz. The microwave attenuation of electromagnetic wave through the ionized plume (the angle of incidence of microwave is 90o and transmission of microwave is always transverse to the exhaust plume) has also been studied using the above electrical characteristics computed and it is seen that the attenuation follows the axial variation in the conductivity of each cross section of the plume. In the present work, a theoretical model has also been developed to compute the microwave attenuation through the vehicle exhaust plume using the electrical conductivity computed earlier for any angle of incidence of the microwave. The thesis also lists some additional topics for further studies.

碩士
國立交通大學
電信工程系所
93
When the method of moments (MOM) is applied to solve the electric field integral equation (EFIE) for electromagnetic radiation and scattering problems, the accuracy of solutions greatly depends on a proper discretization of the simulated domain. Most of time, the grid distribution needs to be manually tuned for getting an accurate solution. In this thesis, we propose a mesh refinement algorithm that adapts meshes to EFIE solutions by splitting elements (h-refinement) and relocating nodes (r-refinement). Using a divide-and-conquer Delaunay triangulation, an initial mesh is generated with equally spaced seeds on the surfaces of the simulated structure. Then the mesh is iteratively refined according the current distribution on the surface. The refinement process automatically terminates when the current distribution converges or when preset criteria, such as the smallest edge length and the maximum pass of refinement, are met. In order to expedite the iterative refinement process, the current is calculated only by the near-interaction terms of the MOM impedance matrix. The adaptive mesh refinement algorithm is further applied to solve radiation and scattering from metallic structures.

Optical whispering-gallery mode (WGM) cavities, which exhibit extraordinary spatial and temporal confinement of light, are one of the leading transducers for examining molecular recognition at low particle counts. With the advent of hybrid photonic-plasmonic and increasingly sophisticated forms of these resonators, the importance of supporting numerical methods has correspondingly become evident. In response, we adopt a full-vector finite difference approximation in order to solve for WGM's in terms of their field distributions, resonant wavelengths, and quality factors in the context of naturally discontinuous permittivity structure. A segmented Taylor series and alignment/rotation operator are utilized at such singularities in conjunction with arbitrarily spaced grid points. Simulations for microtoroids, with and without dielectric nanobeads, and plasmonic microdisks are demonstrated for short computation times and shown to be in agreement with data in the literature. Constricted surface plasmon polariton (SPP) WGM's are also featured within this document. The module of this thesis is devised as a keystone for composite WGM models that may guide experiments in the field.
Graduate

Plasmonic structures are utilized in many applications ranging from bio-medicine to solar energy generation and transfer. Numerical schemes capable of solving equations of classical electrodynamics have been the method of choice for characterizing scattering properties of such structures. However, as dimensions of these plasmonic structures reduce to nanometer scale, quantum mechanical effects start to appear. These effects cannot be accurately modeled by available classical numerical methods. One of these quantum effects is the tunneling, which is observed when two structures are located within a sub-nanometer distance of each other. At these small distances electrons “jump" from one structure to another and introduce a path for electric current to flow. Classical equations of electrodynamics and the schemes used for solving them do not account for this additional current path. This limitation can be lifted by introducing an auxiliary tunnel with material properties obtained using quantum models and applying a classical solver to the structures connected by this auxiliary tunnel. Early work on this topic focused on quantum models that are generated using a simple one-dimensional wave function to find the tunneling probability and assume a simple Drude model for the permittivity of the tunnel. These tunnel models are then used together with a classical frequency domain solver. In this thesis, a time domain surface integral equation solver for quantum corrected analysis of transient plasmonic interactions is proposed. This solver has several advantages: (i) As opposed to frequency domain solvers, it provides results at a broad band of frequencies with a single simulation. (ii) As opposed to differential equation solvers, it only discretizes surfaces (reducing number of unknowns), enforces the radiation condition implicitly (increasing the accuracy), and allows for time step selection independent of spatial discretization (increasing efficiency). The quantum model of the tunnel is obtained using density functional theory (DFT) computations, which account for the atomic structure of materials. Accuracy and applicability of this (quantum corrected) time domain surface integral equation solver will be shown by numerical examples.

Novel integral-equation methods for efficiently solving electromagnetic problems that involve more than a single length scale of interest in complex backgrounds are presented. Such multi-scale electromagnetic problems arise because of the interplay of two distinct factors: the structure under study and the background medium. Both can contain material properties (wavelengths/skin depths) and geometrical features at different length scales, which gives rise to four types of multi-scale problems: (1) twoscale, (2) multi-scale structure, (3) multi-scale background, and (4) multi-scale-squared problems, where a single-scale structure resides in a different single-scale background, a multi-scale structure resides in a single-scale background, a single-scale structure resides in a multi-scale background, and a multi-scale structure resides in a multi-scale background, respectively. Electromagnetic problems can be further categorized in terms of the relative values of the length scales that characterize the structure and the background medium as (a) high-frequency, (b) low-frequency, and (c) mixed-frequency problems, where the wavelengths/skin depths in the background medium, the structure’s geometrical features or internal wavelengths/skin depths, and a combination of these three factors dictate the field variations on/in the structure, respectively. This dissertation presents several problems arising from geophysical exploration and microwave chemistry that demonstrate the different types of multi-scale problems encountered in electromagnetic analysis and the computational challenges they pose. It also presents novel frequency-domain integral-equation methods with proper Green function kernels for solving these multi-scale problems. These methods avoid meshing the background medium and finding fields in an extended computational domain outside the structure, thereby resolving important complications encountered in type 3 and 4 multi-scale problems that limit alternative methods. Nevertheless, they have been of limited practical use because of their high computational costs and because most of the existing ‘fast integral-equation algorithms’ are not applicable to complex Green function kernels. This dissertation introduces novel FFT, multigrid, and FFT-truncated multigrid algorithms that reduce the computational costs of frequency-domain integral-equation methods for complex backgrounds and enable the solution of unprecedented type 3 and 4 multi-scale problems. The proposed algorithms are formulated in detail, their computational costs are analyzed theoretically, and their features are demonstrated by solving benchmark and challenging multi-scale problems.
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碩士
國立雲林科技大學
機械工程系
103
In order to raise motor effectiveness and heat emission efficiency, vacuum brazing and active soldering was used to join electromagnetic steel and high thermal conductivity metals. Four Al-Si-Cu based low-melting-point aluminum brazes and two active solders, Sn-Ag-Cu-Ti-Mm and Sn-Zn-Bi-Ti-Mm, were used in this study. For soldering, tradition soldering was compared with ultrasonic vibration soldering bonding. Experimental results showed that the rare earth element containing aluminum solders have high shear bonding strength. The ultrasonic vibration soldering shear strength was obviously higher than used traditional soldering. Tensile testing at elevated temperature showed that the bonding shear strength decreased.

A stochastic method is developed, implemented and investigated here for solving Laplace, Poisson's, and standard parabolic wave equations. This method is based on the properties of random walk, diffusion process, Ito formula, Dynkin formula and Monte Carlo simulations. The developed method is a local method i:e: it gives the value of the solution directly at an arbitrary point rather than extracting its value from complete field solution and thus is inherently parallel. Field computation by this method is demonstrated for electrostatic and electrodynamic propagation problems by considering simple examples and numerical results are presented to validate this method. Numerical investigations are carried out to understand efficacy and limitations of this method and to provide qualitative understanding of various parameters involved in this method.

碩士
義守大學
工業工程與管理學系碩士班
99
In the past few decades, many researchers have been working on Job-Shop Scheduling Problems (JSSPs). Some good results have been approached. The job-shop scheduling problems belong to combinatorial optimization problems. They are NP-hard problems. The traditional optimization techniques are often to use to find optimal solutions. However, they take a lot of computation time, and sometimes optimal solutions can not be reached. Therefore, meta-heuristic methods, such as Ant Colony Optimization(ACO), Tabu Search(TS), Genetic Algorithm(GA), and Particle Swarm Optimization(PSO), etc are usually used to solve the NP-hard problems. In this study, we use a heuristic method called the Electromagnetism-like Mechanism (EM).The EM method utilizes an attraction-repulsion mechanism to move the sample points (i.e., our solutions) towards the optimality. Usually, the EM method is designed for solving real-valued problems, but the JSSPs are integer-valued problems. Therefore, we must convert and encode the EM method into integer-value method to solve the job-shop scheduling problems. The objective of this study is to find the minimum makespan. In our study, the EM method shows its ability to find the near-optimal, even optimal, solutions efficiently of the JSSPs. Finally, we illustrate some benchmark problems to show that the EM algorithm is capable of solving the job-shop scheduling problems. The performance measures among EM method and other meta-heuristic methods are compared. The EM method can show its ability to compare with other meta-heuristic methods.

碩士
義守大學
工業工程與管理學系
100
The traveling salesman problems belong to the combinatorial optimization problems. These problems have been proved to be NP-complete problems that often appeared in the real world. These problems are simple but very difficult to solve. If we want to find the global optimum of these problems by traditional methods, we will either spend a lot of time and may not be cost-effective or unable to find the optimum. The Electromagnetism-like Mechanism, developed by Birbil and Fang (2003), is a meta-heuristic algorithm which utilizes an attraction-repulsion mechanism to move the sample points (i.e., our solutions) towards the optimality. In Birbil and Fang’s paper, they have shown that this mechanism of the EM algorithm can avoid the sample points from trapping into local optimum and moving toward the global optimum. Wu and Chiang (2005) applied this method to solve the traveling salesman problems. Wu and Fang (2006) improved this method to solve the traveling salesman problems and 張瓊芳(2008) used above this two methods to solving express delivery routing problem for F company. Previous literatures have shown the ability of the Electromagnetism-like Mechanism to solve the traveling salesman problems. The results are good for small dimension problems. However, the results for large dimension problems are not quite well. So it has some improvement space. Currently, the Simulated Annealing has not been initialized in the Electromagnetism-like Mechanism (SA+EM) for the initial solution for the TSP problems. In this paper, we will use this method to solve the traveling salesman problems. The performance measures will be compared with Wu and Chiang (2005) and Wu and Fang (2006).

碩士
國立屏東科技大學
工業管理系所
100
The subject on multi-floor facility layout in multi-period is firms forecast any production situations by evaluating industrial changes. The forecasting indicated the different production situations make facility re-layout, and the difficulties of implement in real world are presented. For withstanding any production situations and reducing re-layout cost, the facility layout runs with robust design. The island nations state the reasons of shortage land resources, therefore, the workshop characteristics of multi-floor factories are considered. Based on electromagnetism-like mechanism (EM) conduct the mutual attractions or repulsions between the electrons with every electric charge, and to search the global optimal solution in the effects of interactions of electrons. Through electromagnetism-like mechanism, a variety of robust designs to mutual analysis and compare each other, we finally hope to obtain a robust design for multi-floor situations in multi-period.

碩士
義守大學
工業工程與管理學系碩士班
94
In this study, the main objective is to solve the traveling salesman problems (TSP). TSP is an instance of the combinatorial optimization problems and NP-complete problems. When the problems of the dimensions are higher, it becomes difficult and hard to solve though this problem is easy to describe and understand. If we consider all feasible solutions, it may cost a lot of time and not be effective. The new meta-heuristic algorithm, called the Electromagnetism-like Mechanism (EM), was proposed by Birbil and Fang (2003). It utilizes an attraction-repulsion mechanism to force the sample point move toward better feasible region even optimality. Although EM has been used to solve TSP, the results obtained still immerge the local optimum. We introduce new mechanism and local search method into EM (called Revised electromagnetism-like mechanism) to expect to gain a better solution than original EM when we solve the problems with more dimensions. Keywords: Revised electromagnetism-like mechanism (EM), Traveling salesman problems (TSP), NP-complete problems.

碩士
義守大學
工業管理學系碩士在職專班
101
In recent years, although transportation has become more convenient, students are still faced with a daily school bus problem, especially the majority of high school students are enrolled districts; therefore, the majority of high schools are planning school bus or car to solve the route to school for each student. Convenient for the school enrollment is currently open to traffic is one important factor, and facing the impact of the declining birthrate, schools are pay more attention to and strengthen measures to recruit outstanding students enrolled browse, and student school bus or car is one of them. Type electromagnetic algorithms (Electromagnetism-like Mechanism, EM) is a heuristic algorithm proposed by Birbil and Fang in 2003, which is based on electromagnetism Coulomb's law (Coulomb's law) and boundary concepts, the use of electromagnetics E-sex repulsion, the principle of opposites attract, so particles toward Heli (optimal value) moves in the direction, this method used in solving optimization problems. School Bus Routine Problem (School Bus Routine Problem, SBRP) features as a single origin and destination and the car type; in this paper uses the EM algorithm as the research base to seek the school bus routine problem and school bus planning. School bus routine problem is NP-complete problem of type; vehicle routing problem (Vehicle Routing Problem, VRP), the traveling salesman problem (Traveling Salesman Problem, TSP), logistics and distribution of the routing problem, production scheduling ... etc., researched in many studies, have good results. The main purpose of this research, based on electromagnetism algorithm (EM), is to obtain the optimal path, and take a southern high school bus for example to study to improve its school bus routes. From the results, we can conclude that: (1) The school bus routes of academic year 2011 and 2012 have both been improved. (2) The better improvements can provide the bus company better service with shorter routes. (3) The more the stops, the better the improvements.

The aim of this research was to improve the matrix solution methods for SuperNEC MoM problems, which is an electromagnetic simulation software package used to model antennas, and develop a new preconditioner for the iterative method BICGSTAB(L). This was achieved by firstly implementing the ATLAS BLAS library optimised for a specific computer architecture. The ATLAS code primarily makes use of code generation to build and optimise applications. Comparisons show that the matrix solution times using LU decomposition optimised by ATLAS is improved by between 4.1 and 4.6 times, providing a good coding platform from which to compare other techniques. Secondly the BICGSTAB iterative solution method in SuperNEC was improved by making use of an alternative algorithm BICGSTAB(L). Systems of equations that converged slowly or not at all using BICGSTAB, converged more quickly when using BICGSTAB(L) with L set to 4, despite the high condition numbers in the coefficient matrices. Thirdly a domain decomposition method, Simply Sparse, was characterised. Investigations showed that Simply Sparse is a good compression technique for SuperNEC MoM matrices. The custom Simply Sparse solver also solves large matrix problems more quickly than LU decomposition and scales well with increased problem sizes. LU decomposition is still however quicker for problems smaller than 7000 unknowns as the overheads in compressing the coefficient matrices dominate the Simply Sparse method for small problems. Lastly a new preconditioner for BICGSTAB(L) was developed using a modified form of the Simply Sparse matrix. This was achieved by considering the Simply Sparse matrix to be equivalent to the full coefficient matrix [A] . The largest 1% to 2% of the Simply Sparse elements was selected to form the basis of the preconditioning matrix. These elements were further modified by multiplying them by a large constant i.e. 7 1×10 . The system of equations was then solved using BICGSTAB(L) with L set to 4. The new preconditioned BICGSTAB(L) algorithm is quicker than both LU decomposition and the custom Simply Sparse solution method for problems larger than 5000 unknowns.

Thin film nanostructures may be defined as assemblies, arrays, or randomly distributed nanoparticles, nanowires, or nanotubes, which together form a layer of materials supported on a substrate surface. Because such nanostructures are supported on a substrate surface, their potential applications cover a wide area in optical, magnetic, electrochemical, electromagnetic, and optoelectronic devices. The focus of the present thesis is the development of methodologies to grow certain thin film nanostructures of some transition metal oxides (TMOs), including copper oxides and LixCoO2, through CVD, sol-gel, and electromagnetic radiation-mediated approaches. The work towards this objective can be divided into three parts: first, the design, synthesis, and systematic identification of novel metalorganic precursors of copper (monometallic) and Li and Co (bimetallic); second, the growth of nanostructured oxides thin films using these precursors; and third, the application of electromagnetic radiation to control or tailor the growth of as grown nanostructures. The underlying growth mechanisms substantiated by appropriate evidence have been put forward, wherever found relevant and intriguing. It may be added that the principal objective of the work reported here has been to explore the several ideas noted above and examine possibilities, rather than to study any specific one of them in significant detail. It is hoped earnestly that this has been accomplished to a reasonable extent. Chapter 1 reviews briefly the reports available in the literature on three specific methods of growing thin films nanostructures, namely chemical vapour deposition, sol-gel processing and light-induced approach. The objective of this chapter has been to provide the background of the work done in the thesis, and is substantiated with a number of illustrative examples. Some of the fundamental concepts involved, viz., plasmons and excitons, have been defined with illustration wherever found relevant in the context of the work. Chapter 2 describes the various techniques used for synthesis and characterisation of the metalorganic complexes as well as of the thin films. This chapters covers mostly experimental details, with brief descriptions of the working principles of the analytical procedures adopted, namely, infrared spectroscopy, mass spectroscopy, elemental analysis, and thermal analysis for characterisation of the metalorganic complexes. This is followed by a similarly brief account of techniques employed to characterize the thin films prepared in this work, viz., glancing incidence X-ray diffraction (GIXRD), field-emission scanning electron microscopy (FESEM), atomic force microscopy (AFM), electrostatic force microscopy (EFM), transmission electron microscopy (TEM), glancing incidence infra-red spectroscopy (GIIR) and, UV-visible spectroscopy. The metalorganic chemical vapour deposition (MOCVD) systems built in house and used for growth of films are described in detail. The topics in the different sections of the chapter are accompanied by pertinent diagrams. Chapter 3 deals with the design, synthesis and characterisation of novel polynuclear complexes of copper and cobalt. Keeping in mind the various advantages such as low toxicity, ease of synthesis, non-pyrophoricity, and low temperature volatility, of environmentally benign complexes based on biologically compatible such as triethanolamine, diethanolamine, the objective has been to synthesize complexes containing triethanolamine and diethanolamine of transition metals such as cobalt and copper, and to investigate their applicability in MOCVD processes as a novel class of precursors. With the notion of ‘better’ and efficient design of precursors, an attempt has been made, through a semi-empirical modeling, to understand the correlation between volatility and various intrinsic molecular parameters such as lattice energy, vibrational-rotational energy, and internal symmetry. Chapter 4 discusses the growth of nanoporous Cu4O3-C composite films through the MOCVD process employing Cu4(deaH)(dea)(oAc)5.(CH3)2CO as the precursor. The various characteristic aspects of as-grown films, such as their crystallinity, morphology, and composition have been covered elaborately in various sections of this chapter. The chapter describes the efficient guiding and confining of light exploiting the photonic band gap of these nanoporous films, which indicates the potential usefulness of these and similar films as optical waveguides. A model described in the literature on absorbing photonic crystals, wherein a periodically modulated absorption entails an inevitable spatial modulation of dispersion, i.e., of the index contrast to open a photonic band gap, has been used to calculate the indices of refraction of one of these nanoporous films. The chapter also reports briefly the preliminary electrochemical investigations carried out on a typical film, examining the notion of its application as the anode in a Li-ion rechargeable battery. Chapter 5 describes the synthesis of nanocrystalline LixCoO2 films by the sol-gel method. Reports available in literature indicate that the various phases of LixCoO2 are extremely sensitive to processing temperature, making it difficult to control dimensionality of a given phase using temperature as one of process parameters. We have investigated the possibility of using incoherent light to tailor the particle size/shape of this material. The as-grown and irradiated films were characterised by X-ray diffraction, and by microscopic and spectroscopic techniques.Optical spectroscopy was carried out in order to gain insight into the physico-chemical mechanism involved in such structural and morphological transformation. Chapter 6 deals with the synthesis of self-assembled nanostructures from the pre-synthesized nanocrystals building blocks, through optical means of exciton formation and dissociation. It has been demonstrated that, upon prolonged exposure to (incoherent) ultraviolet-visible radiation, LixCoO2 nanocrystals self-assemble into acicular architectures, through intermediate excitation of excitons. Furthermore, it has been shown that such self-assembly occurs in nanocrystals, which are initially anchored to the substrate surface such as that of fused quartz. This new type of process for the self-assembly of nanocrystals, which is driven by light has been investigated by available microscopic and spectroscopic techniques. Chapter 7 describes the stabilisation of chemically reactive metallic lithium in a carbonaceous nanostructure, viz., a carbon nanotube, achieved through the MOCVD process involving a lithium-alkyl moiety. This moiety is formed in situ during deposition through partial decomposition of a metalorganic precursor synthesized in house, which contains both lithium and cobalt. It is surmised that the stabilization of metallic Li in the nanostructure in situ occurs through the partial decomposition of the metalorganic precursor. Quantitative X-ray photoelectron spectroscopy carried out on such a film reveals that as much as 33.4% metallic lithium is trapped in carbon. Lastly, Chapter 8 briefly highlights the outlook for further investigations suggested by the work undertaken for this thesis. Novel precursors derived from biologically compatible ligands can open up possibility of growing new type of micro/nano-structures, and of unusual phases in the CVD grown films. Furthermore, it is proposed that the novel method of growth and alignment of nanocrystals through irradiation with incoherent light, employed for the specific material LixCoO2, may be employed for various other metallic and semiconducting materials.