11 results on '"Tischendorf, Caren"'
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2. LEAST-SQUARES COLLOCATION FOR HIGHER-INDEX LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS: ESTIMATING THE INSTABILITY THRESHOLD.
- Author
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HANKE, MICHAEL, MÄRZ, ROSWITHA, and TISCHENDORF, CAREN
- Subjects
DIFFERENTIAL-algebraic equations ,LINEAR equations ,ORDINARY differential equations ,COLLOCATION methods ,TIKHONOV regularization ,BOUNDARY value problems - Abstract
Differential-algebraic equations with higher-index give rise to essentially ill-posed problems. The overdetermined least-squares collocation for differential-algebraic equations which has been proposed recently is not much more computationally expensive than standard collocation methods for ordinary differential equations. This approach has displayed impressive convergence properties in numerical experiments, however, theoretically, till now convergence could be established merely for regular linear differential-algebraic equations with constant coefficients. We present now an estimate of the instability threshold which serves as the basic key for proving convergence for general regular linear differential-algebraic equations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
3. Topological index calculation of DAEs in circuit simulation
- Author
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Tischendorf, Caren
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index ,MNA ,modified nodal analysis ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,circuit simulation ,Hardware_INTEGRATEDCIRCUITS ,integrated circuit ,510 Mathematik ,Hardware_PERFORMANCEANDRELIABILITY ,ddc:510 ,DAE ,Hardware_LOGICDESIGN ,differential-algebraic equation - Abstract
Electric circuits are present in a number of applications, e.g. in home computers, television, credit cards, electric power networks, etc. The development of integrated circuit requires numerical simulation. Modern modeling techniques like the Modified Nodal Analysis (MNA) lead to differential algebraic equations (DAEs). Properties like the stability of solutions of such systems depend strongly on the DAE index. The paper deals with lumped circuits containing voltage sources, current sources as well as general nonlinear but time-invariant capacitances, inductances and resistances. We present network-topological criteria for the index of the DAEs obtained by the classical and the charge oriented MNA. Furthermore, the index is shown to be limited to 2 for our model-class.
- Published
- 2005
4. Structural analysis for electric circuits and consequences for MNA
- Author
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Schwarz, Diana Estévez and Tischendorf, Caren
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index ,structural properties ,510 Mathematik ,Hardware_PERFORMANCEANDRELIABILITY ,differential-algebraic equation ,modelling ,Computer Science::Hardware Architecture ,Computer Science::Emerging Technologies ,Circuit simulation ,MNA ,modified nodal analysis ,Hardware_INTEGRATEDCIRCUITS ,ddc:510 ,DAE ,Hardware_LOGICDESIGN - Abstract
The development of integrated circuits requires powerful numerical simulation programs. Of course, there is no method that treats all the different kinds of circuits successfully. The numerical simulation tools provide reliable results only if the circuit model meets the assumptions that guarantee the successful application of the integration software. Because of the large dimension of many circuits (about $10^7$ circuit elements) it is often difficult to find the circuit configurations that lead to numerical difficulties. In this paper, we analyze electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling technique, the modified nodal analysis (MNA), and discuss the index of the differential algebraic equations (DAEs) obtained by this kind of modelling.
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- 2005
5. Finding Beneficial DAE Structures in Circuit Simulation
- Author
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Schwarz, Diana Estévez, Feldmann, Uwe, März, Roswitha, Sturtzel, Sandra, and Tischendorf, Caren
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index ,structural properties ,27 Mathematik ,breakpoints ,510 Mathematik ,differential-algebraic equation ,Computer Science::Hardware Architecture ,Computer Science::Emerging Technologies ,consistent initial values ,MNA ,modified nodal analysis ,circuit simulation ,ddc:510 ,DAE - Abstract
Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. But further developments in the production engineering lead to new classes of circuits which may cause difficulties for the numerical integration. The dimension of circuit models can be quite large (105 equations). The complexity of the models demands a higher abstraction level. In this paper, we analyze electric circuits with respect to their structural properties. We discuss the relevant subspaces of the resulting differential algebraic equations (DAEs) and present algorithms for calculating the index as well as consistent initial values.
- Published
- 2005
6. Topological analysis of qualitative features in electrical circuit theory
- Author
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Riaza, Ricardo and Tischendorf, Caren
- Subjects
circuit model ,asymptotic stability ,hyperbolicity ,modified nodal analysis ,matrix pencil ,graph topology ,510 Mathematik ,ddc:510 ,differential-algebraic equation - Abstract
Several qualitative properties of equilibria in electrical circuits are analyzed in this paper. Specifically, non-singularity, hyperbolicity, and asymptotic stability are addressed in terms of the circuit topology, which is captured through the use of Modified Nodal Analysis (MNA) models. The differential-algebraic or semistate nature of these models drives the analysis of the spectrum to a matrix pencil setting, and puts the results beyond the ones already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The analysis combines results coming from graph theory, matrix analysis, matrix pencil theory, and Lyapunov theory for DAEs. The study is restricted to problems with independent sources; qualitative properties of circuits including controlled sources are the focus of future research.
- Published
- 2005
7. Structural characterization of classical and memristive circuits with purely imaginary eigenvalues.
- Author
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Riaza, Ricardo and Tischendorf, Caren
- Abstract
SUMMARY The hyperbolicity problem in circuit theory concerns the existence of purely imaginary eigenvalues in the linearization of the time-domain description of the circuit dynamics. In this paper, we characterize the circuit configurations that, in a strictly passive setting, yield purely imaginary eigenvalues for all values of the capacitances and inductances. Our framework is based on branch-oriented, semistate (differential-algebraic) circuit models that capture explicitly the circuit topology, and use several notions and results from digraph theory. So-called P-structures arising in the analysis turn out to be the key element supporting our results. The analysis is shown to hold not only for classical (RLC) circuits but also for nonlinear circuits including memristors and other mem-devices. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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8. Convergence analysis of a partial differential algebraic system from coupling a semiconductor model to a circuit model
- Author
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Matthes, Michael and Tischendorf, Caren
- Subjects
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STOCHASTIC convergence , *PARTIAL differential equations , *SEMICONDUCTORS , *COMPUTER simulation , *ELECTRIC circuit analysis , *FINITE element method , *COUPLED mode theory (Wave-motion) - Abstract
Abstract: We are interested in circuit simulation including distributed semiconductor models. The circuit itself is modeled by the modified nodal analysis. The stationary drift diffusion equations are used to describe the semiconductors. The complete system is then a partial differential-algebraic system. We discretize it first in space with finite elements and the Scharfetter–Gummel discretization. The resulting semi-discrete system can be analyzed as a differential-algebraic equation with properly stated leading term. We present topological index one criteria. They coincide with previous results for the non-discretized partial differential-algebraic equation. For the time discretization we use standard BDF methods (implicit Gear formulas). Finally we derive a convergence estimate for the whole partial differential-algebraic system close to equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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9. Least-squares collocation for linear higher-index differential–algebraic equations.
- Author
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Hanke, Michael, März, Roswitha, Tischendorf, Caren, Weinmüller, Ewa, and Wurm, Stefan
- Subjects
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ALGEBRAIC equations , *LEAST squares , *MATHEMATICAL statistics , *PROBABILITY theory , *TRIANGULATION - Abstract
Differential–algebraic equations with higher index give rise to essentially ill-posed problems. Therefore, their numerical approximation requires special care. In the present paper, we state the notion of ill-posedness for linear differential–algebraic equations more precisely. Based on this property, we construct a regularization procedure using a least-squares collocation approach by discretizing the pre-image space. Numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index-1 differential–algebraic equations. Convergence is shown for a limited class of linear higher-index differential–algebraic equations. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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10. Analysis and waveform relaxation for a differential-algebraic electrical circuit model
- Author
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Pade, Jonas, Tischendorf, Caren, Schöps, Sebastian, and Riaza, Ricardo
- Subjects
nichtlineare DAE vom Index 2 ,electrical circuits ,Netzwerk-Topologie ,nonlinear index 2 DAE ,cosimulation ,network topology ,modifizierte Knotenanalyse ,500 Naturwissenschaften und Mathematik ,SK 920 ,convergence criteria ,ddc:510 ,Waveform Relaxation ,coupled circuits ,ddc:518 ,Parallele Algorithmen ,field/circuit ,parallel algorithms ,510 Mathematik ,518 Numerische Analysis ,differential-algebraic equation ,MNA ,modified nodal analysis ,Konvergenzkriterien ,differential-algebraische Gleichung ,gekoppelte Schaltkreise ,ZN 5310 ,ddc:500 ,elektrische Schaltkreise - Abstract
Die Hauptthemen dieser Arbeit sind einerseits eine tiefgehende Analyse von nichtlinearen differential-algebraischen Gleichungen (DAEs) vom Index 2, die aus der modifizierten Knotenanalyse (MNA) von elektrischen Schaltkreisen hervorgehen, und andererseits die Entwicklung von Konvergenzkriterien für Waveform Relaxationsmethoden zum Lösen gekoppelter Probleme. Ein Schwerpunkt in beiden genannten Themen ist die Beziehung zwischen der Topologie eines Schaltkreises und mathematischen Eigenschaften der zugehörigen DAE. Der Analyse-Teil umfasst eine detaillierte Beschreibung einer Normalform für Schaltkreis DAEs vom Index 2 und Abschätzungen, die für die Sensitivität des Schaltkreises bezüglich seiner Input-Quellen folgen. Es wird gezeigt, wie diese Abschätzungen wesentlich von der topologischen Position der Input-Quellen im Schaltkreis abhängen. Die zunehmend komplexen Schaltkreise in technologischen Geräten erfordern oftmals eine Modellierung als gekoppeltes System. Waveform relaxation (WR) empfiehlt sich zur Lösung solch gekoppelter Probleme, da sie auf die Subprobleme angepasste Lösungsmethoden und Schrittweiten ermöglicht. Es ist bekannt, dass WR zwar bei Anwendung auf gewöhnliche Differentialgleichungen konvergiert, falls diese eine Lipschitz-Bedingung erfüllen, selbiges jedoch bei DAEs nicht ohne Hinzunahme eines Kontraktivitätskriteriums sichergestellt werden kann. Wir beschreiben allgemeine Konvergenzkriterien für WR auf DAEs vom Index 2. Für den Fall von Schaltkreisen, die entweder mit anderen Schaltkreisen oder mit elektromagnetischen Feldern verkoppelt sind, leiten wir außerdem hinreichende topologische Konvergenzkriterien her, die anhand von Beispielen veranschaulicht werden. Weiterhin werden die Konvergenzraten des Jacobi WR Verfahrens und des Gauss-Seidel WR Verfahrens verglichen. Simulationen von einfachen Beispielsystemen zeigen drastische Unterschiede des WR-Konvergenzverhaltens, abhängig davon, ob die Konvergenzbedingungen erfüllt sind oder nicht. The main topics of this thesis are firstly a thorough analysis of nonlinear differential-algebraic equations (DAEs) of index 2 which arise from the modified nodal analysis (MNA) for electrical circuits and secondly the derivation of convergence criteria for waveform relaxation (WR) methods on coupled problems. In both topics, a particular focus is put on the relations between a circuit's topology and the mathematical properties of the corresponding DAE. The analysis encompasses a detailed description of a normal form for circuit DAEs of index 2 and consequences for the sensitivity of the circuit with respect to its input source terms. More precisely, we provide bounds which describe how strongly changes in the input sources of the circuit affect its behaviour. Crucial constants in these bounds are determined in terms of the topological position of the input sources in the circuit. The increasingly complex electrical circuits in technological devices often call for coupled systems modelling. Allowing for each subsystem to be solved by dedicated numerical solvers and time scales, WR is an adequate method in this setting. It is well-known that while WR converges on ordinary differential equations if a Lipschitz condition is satisfied, an additional convergence criterion is required to guarantee convergence on DAEs. We present general convergence criteria for WR on higher index DAEs. Furthermore, based on our results of the analysis part, we derive topological convergence criteria for coupled circuit/circuit problems and field/circuit problems. Examples illustrate how to practically check if the criteria are satisfied. If a sufficient convergence criterion holds, we specify at which rate of convergence the Jacobi and Gauss-Seidel WR methods converge. Simulations of simple benchmark systems illustrate the drastically different convergence behaviour of WR depending on whether or not the circuit topological convergence conditions are satisfied.
- Published
- 2021
11. Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks
- Author
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Huck, Christoph, Tischendorf, Caren, Mehrmann, Volker, and Campbell, Stephen L.
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partial differential-algebraic equation ,numerical analysis ,flow networks ,Gasnetzwerke ,numerische Analysis ,510 Mathematik ,partielle Differential-algebraische Gleichung ,differential-algebraic equation ,SK 920 ,numerische Diskretisierung ,PDAE ,perturbation analysis ,numerical discretisation ,ddc:510 ,DAE ,gas networks ,Störungsanalyse ,Differential-algebraische Gleichung ,Flussnetzwerke - Abstract
Diese Arbeit beschäftigt sich mit verschiedenen mathematischen Fragestellungen hinsichtlich der Modellierung, Analysis und numerischen Simulation von Gasnetzen. Hierbei liegt der Fokus auf der mathematischen Handhabung von partiellen differential-algebraischen Gleichungen, die mit algebraischen Gleichungen gekoppelt sind. Diese bieten einen einfachen Zugang hinsichtlich der Modellierung von dynamischen Strukturen auf Netzen Somit sind sie insbesondere für Gasnetze geeignet, denen im Zuge der steigenden Bedeutung von erneuerbaren Energien ein gestiegenes Interesse seitens der Öffentlichkeit, Politik und Wissenschaft entgegen gebracht wird. Wir führen zunächst die gängigsten Elemente, die in Gasnetzen benötigt werden ein und formulieren zwei PDAE-Klassen für solche Netze: Eine für reine Rohrnetze, und eine, die zusätzliche Elemente wie Verdichter und Widerstände beinhaltet. Des Weiteren untersuchen wir die Sensitivität der Lösung der Rohrnetz-PDAE hinsichtlich Störungen. Dabei berücksichtigen wir Störungen, die nicht nur den dynamischen Teil der PDAE beeinflussen, sondern auch Störungen in den algebraischen Gleichungen und weisen Stabilitätseigenschaften für die Lösung der PDAE nach. Darüber hinaus beschäftigen wir uns mit einer neu entwickelten, an die Netztopologie angepassten Ortsdiskretisierung, welche die Stabilitätseigenschaften der PDAE auf DAE Systeme überträgt. Des Weiteren zeigen wir, wie sich die Gasnetz-DAE zu einer gewöhnlichen Differentialgleichung, welche die inhärente Dynamik der DAE widerspiegelt entkoppeln lässt. Dieses entkoppelte System kann darüber hinaus direkt aus den Topologie- und Elementinformationen des Netzes aufgestellt werden. Abschließend demonstrieren wir die Ergebnisse an Benchmark-Gasnetzen. Dabei vergleichen wir sowohl die entkoppelte Differentialgleichung mit dem ursprünglichen DAE System, zeigen aber auch, welche Vorteile die an die Netztopologie angepasste Ortsdiskretisierung gegenüber existierenden Verfahren besitzt. This thesis addresses several aspects regarding modelling, analysis and numerical simulation of gas networks. Hereby, our focus lies on (partial) differential-algebraic equations, thus systems of partial and ordinary differential equations which are coupled by algebraic equations. These coupled systems allow an easy approach towards the modelling of dynamic structures on networks. Therefore, they are well suited for gas networks, which have gained a rise of attention in society, politics and science due to the focus towards renewable energies. We give an introduction towards gas network modelling that includes the most common elements that also appear in real gas networks and present two PDAE systems: One for pipe networks and one that includes additional elements like resistors and compressors. Furthermore, we investigate the impact of perturbations onto the pipe network PDAE, where we explicitly allow perturbations to affect the system in the differential as well as in the algebraic components. We conclude that the solution of the PDAE possesses stability properties. In addition, this thesis introduces a new spatial discretisation that is adapted to the net- work topology. This topology-adapted semi-discretisation results in a DAE which possesses the same perturbation behaviour as the space continuous PDAE. Furthermore, we present a topology based decoupling procedure that allows to reformulate the DAE as an ordinary differential equation (ODE), which represents the inherent dynamics of the DAE system. This ODE, together with a decoupled set of algebraic equations, can be derived from the topology and element information directly. We conclude by demonstrating the established results for several benchmark networks. This includes a comparison of numerical solutions for the decoupled ODE and the DAE system. In addition we present the advantages of the topology-adapted spatial discretisation over existing well established methods.
- Published
- 2018
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