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

We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance. / Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8)

This study proposes novel optimization methodologies for the optimization of problems that reveal non-physical step discontinuities. More specifically, it is proposed to use gradient-only techniques that do not use any zeroth order information at all for step discontinuous problems. A step discontinuous problem of note is the shape optimization problem in the presence of remeshing strategies, since changes in mesh topologies may - and normally do - introduce non-physical step discontinuities. These discontinuities may in turn manifest themselves as non-physical local minima in which optimization algorithms may become trapped. Conventional optimization approaches for step discontinuous problems include evolutionary strategies, and design of experiment (DoE) techniques. These conventional approaches typically rely on the exclusive use of zeroth order information to overcome the discontinuities, but are characterized by two important shortcomings: Firstly, the computational demands of zero order methods may be very high, since many function values are in general required. Secondly, the use of zero order information only does not necessarily guarantee that the algorithms will not terminate in highly unfit local minima. In contrast, the methodologies proposed herein use only first order information, rather than only zeroth order information. The motivation for this approach is that associated gradient information in the presence of remeshing remains accurately and uniquely computable, notwithstanding the presence of discontinuities. From a computational effort point of view, a gradient-only approach is of course comparable to conventional gradient based techniques. In addition, the step discontinuities do not manifest themselves as local minima. / Thesis (PhD)--University of Pretoria, 2010. / Mechanical and Aeronautical Engineering / unrestricted

Disertační práce formuluje asymptotické odhady řešení tzv. sublineárních a superlineárních diferenciálních rovnic se zpožděním. V těchto odhadech vystupuje řešení pomocných funkcionálních rovnic a nerovností. Dále práce pojednává o kvalitativních vlastnostech diferenčních rovnic se zpožděním, které vznikly diskretizací studovaných diferenciálních rovnic. Pozornost je věnována souvislostem asympotického chování řešení rovnic ve spojitém a diskrétním tvaru, a to v obecném i v konkrétních případech. Studována je rovněž stabilita numerické diskretizace vycházející z $\theta$-metody. Práce obsahuje několik příkladů ilustrujících dosažené výsledky.

The work describes techniques for solving systems of linear and differential equations. It explains the definition of conversion from system of linear to system of differential equations. The method of the elementary transmission and the transform algorithm are presented. Both of methods are demonstrated on simply examples and properties of conversion are shown. The work compares fast and accurate solutions of methods and algorithm. For computing examples and solving experiments following programs were used: TKSL and TKSL/C. The program TKSL/C was enriched with the graphic user interface which makes the conversion of systems and computing results easier.

The aim of this MSc Thesis is to create a system for automatic generation of frequency characteristics of electrical circuits. These circuits are described by differential equations. A special simulator of RLC circuit has been created and frequence response, vector diagram can be generated. This system has been mainly suggested for application in education. The process of solving differential equations is based on the Taylor method. Systems in general is the theoretical part of this project. Different definitions of systems their divission ,basic phenomenons and mathematical devices are described there. Next chapter deals with the mathematical devices for solving differential equations which makes the basis for description of phenomenons in these systems. There are also systems TKSL and TKSL/C. In the next chapter I was investigaty the analyze of vector diagrams for simple and more difficult circuits. I have found a solution for actual circuit by this technique. The last chapter is devoted to the frequency characteristics and descriptions of simulation program for generation the frequency characteristics.

No description available.

This work entails studying unfitted finite element discretizations for convection-diffusion equations in domains that evolve in time. In particular, these partial differential equations model the evolution of the concentration of soluble surfactants in bulk-interface domains. The work in this thesis docuses on developing numerical methods which conserve the modeled physical quantities. In this work, we propose cut finite element discretizations based on the Discontinuous Galerkin framework which are both locally and globally conservative. Local conservation is achieved on so-called macro elements, and we investigate macro element partitioning of the mesh for both stationary and time-dependent domains. Additionally, we develop globally conservative methods for time-dependent problems. We analyze the proposed methods by studying the convergence of the L2-error with respect to mesh size, condition numbers of the associated linear system matrices, and the conservation error. In numerical experiments for time-dependent problems, we show that the proposed methods have optimal convergence and that the developed macro element stabilization for time-dependent problems leads to increased accuracy while retaining stable condition numbers. Moreover, the measured conservation errors verify the global conservation of the proposed methods. / Detta arbete undersöker diskretiseringar av partiella differentialekvationer i tidsberoende domäner där beräkningsnätet inte behöver anpassas till domänens rörelse. I synnerhet betraktar vi partiella differentalekvationer som modellerar koncentrationen av lösliga ytaktiva ämnen, och skurna finita elementmetoder baserade på den Diskontinuerliga Galerkinmetoden som bevarar de modellerade fysikaliska storheterna. I detta arbete föreslås diskretiseringar som är både lokalt och globalt konservativa. Lokal konservering uppnås i så kallade makroelement, och vi undersöker makroelementpartitionering för både stationära och tidsberoende domäner. Även globalt konservativa metoder utvecklas för tidsberoende problem. De föreslagna metoderna analyseras med hjälp av numeriska exempel. Vi studerar konvergensen av L2-felet med avseende på nätstorlek, konditionstalen för de linjära systemmatriserna samt konserveringsfelet. Metoderna uppvisar optimal konvergens och makroelementstabilisering som utvecklas för tidsberoende problem leder till ökad noggrannhet, samtidigt som konditionstalen förblir stabila. Dessutom veritifierar de uppmättta konserveringsfelen den globala konserveringen hos de föreslagna metoderna.

[pt] Esquemas semi-Lagrangeanos usados para a aproximação do princípio da programação dinâmica são baseados em uma discretização temporal reconstruída no espaço de estado. O uso de uma malha estruturada torna essa abordagem inviável para problemas de alta dimensão devido à maldição da dimensionalidade. Nesta tese, apresentamos uma nova abordagem para problemas de controle ótimo de horizonte infinito onde a função valor é calculada usando Funções de Base Radial (RBFs) pelo método de aproximação de mínimos quadrados móveis de Shepard em malhas irregulares. Propomos um novo método para gerar uma malha irregular guiada pela dinâmica e uma rotina de otimizada para selecionar o parâmetro responsável pelo formato nas RBFs. Esta malha ajudará a localizar o problema e aproximar o princípio da programação dinâmica em alta dimensão. As estimativas de erro para a função valor também são fornecidas. Testes numéricos para problemas de alta dimensão mostrarão a eficácia do método proposto. Além do controle ótimo de EDPs clássicas mostramos como o método também pode ser aplicado ao controle de equações não-locais. Também fornecemos um exemplo analisando a convergência numérica de uma equação não-local controlada para o modelo contínuo. / [en] Semi-Lagrangian schemes for the approximation of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for highdimensional problems due to the curse of dimensionality. In this thesis, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard s moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of nonlocal equations. We also provide an example analyzing the numerical convergence of a nonlocal controlled equation towards the continuous model.

Variable renewable energy sources typically connect to the power grid through power electronic interfaces. Increased use of such sources can cause issues for conventional distance protection. The purpose of distance protection is to detect transmission line faults and trip circuit breakers accordingly in order to maintain power system stability and power quality, and to ensure the safety of equipment and humans. Conventional distance protection solutions work satisfactorily in grids dominated by synchronous generators, but it has been recognized in industry and academia that new solutions may be needed in power grids dominated by power electronics interfaced generation due to the non-linear system impacts introduced by such sources. It was the aim of this thesis project to investigate the performance of alternative solutions used toward this end, specifically time domain distance protection solutions which are immune to the dynamics of power electronics interfaced sources. Short-circuit simulations for different fault scenarios have been performed in PSCAD, utilising models with a wind farm connected via a single overhead transmission line to a grid represented by a voltage source and impedance. The PSCAD line voltages and currents were processed in Matlab where both a more conventional phasor-based algorithm and a time domain differential equation based algorithm were implemented and simulated. A tripping logic based on detecting stable fault distance estimates within the primary protection zone was used. The implemented algorithms were compared for 96 scenarios in their ability to identify the faults and fault types, estimate the fault distances, and in their tripping speed. The results show an overall better performance for the differential equation algorithm compared to the conventional phasor-based algorithm, with more successful trips, faster tripping speed and superior distance estimate convergence on the correct fault distance. Some limitations have also been identified for the differential equation algorithm such as a fault impedance trajectory in the impedance plane which can lead to incorrect trips during external faults when using conventional tripping logic. It is suggested that more research be done regarding the differential equation algorithm with the next step being further algorithm development and testing when using fault impedance estimates and a tripping zone in the impedance plane. / Variabla förnybara energikällor använder vanligtvis kraftelektronik som ett gränssnitt för kopplingen till kraftnätet och ökningen av sådana källor kan skapa problem för konventionella distansskydd. Distansskydds syfte är att detektera jord- och fasfel på kraftledningar samt att skicka utlösningssignaler till strömbrytare för att isolera felet, vilket behövs för att upprätthålla kraftnätets stabilitet, effektkvalitet samt för att säkerställa säkerhet för komponenter och människor. Konventionella distansskydd fungerar bra i kraftnät dominerade av synkrongeneratorer men det har uppmärksammats av industrin samt den akademiska världen att nya lösningar kan behövas i kraftnät som är dominerade av energikällor med kraftelektronik-gränssnitt på grund av dess olinjära dynamik. Syftet med detta examensarbete är att undersöka prestandan av alternativa algoritmer för distansskydd som kan användas mot detta ändamål, specifikt tidsdomän algoritmer som är immuna mot dynamiken introducerad av kraftelektroniken. Kortslutningssimuleringar för olika scenarion har simulerats i PSCAD modeller med en vindkraftpark kopplad i änden på kraftledningen, där spänningarna och strömmarna sparas för vidare analys. Resultaten från PSCAD simuleringar bearbetades i Matlab där både en konventionell algoritm baserad på fasvektorer samt en tidsdomän algoritm baserad på differentialekvationer har implementerats och simulerats. Den implementerade utlösningslogiken baserades på att detektera stabila estimeringar för avståendet mellan skyddsreläet och felet på kraftledningen, där utlösningssignaler skickas för stabila estimeringar detekterade inom den primära skyddszonen för reläet. De implementerade algoritmerna jämfördes i 96 olika scenarion för deras förmåga att: identifiera fel och feltypen, estimera avståndet mellan skyddsreläet och felet samt för deras utlösningstider. Resultaten visar en i överlag bättre prestanda för differentialekvations-algoritmen jämfört med den konventionella fasvektor-baserade algoritmen med fler lyckade strömbrytarutlösningar, snabbare utlösningstider samt bättre konvergens och mer korrekt lokalisering för avståndsestimeringen mellan skyddsreläet och felet. Vissa limitationer har även identifierats för differentialekvations-algoritmen, exempelvis en felimpedans-kurva i impedansplanet vilket kan leda till inkorrekta utlösningar för externa fel om ett konventionellt impedansområden används för utlösningslogiken. Det föreslås att mer forskning bör utföras kring differentialekvations-algoritmen med nästa steg att vidare utveckla algoritmen samt utföra tester vid användande av felimpedansestimeringar och en utlösningslogik baserad på impedansområden.