Consistent initialization for index-2 differential algebraic equations and its application to circuit simulation

Zur numerischen L\"osung von Algebro-Differentialgleichungen (ADGln) m\"ussen konsistente Anfangswerte berechnet werden. Diese Arbeit befasst sich mit einem Ansatz zur Behandlung dieses Problems f\"ur Index-2 DAEs unter Verwendung von Projektoren auf die zur DAE zugeh\"origen Unterr\"aume. Die Arbeit hat zwei Schwerpunkte.\\ Zum einen werden neue Struktureigenschaften aus schwachen Voraussetzungen hergeleitet. Anschlie{\ss}end wird eine Vorgehensweise zur Auswahl von geeigneten Gleichungen einer Index-2 ADGln vorgeschlagen, deren Differentiation zu einer Indexreduktion f\"uhrt. Diese Indexreduktion liefert neue Existenz- und Eindeutigkeitsergebnisse f\"ur L\"osungen von Index-2 ADGln. Die Ergebnisse umfassen eine allgemeinere Aufgabenklasse als die bisherigen Resultate. Beruhend auf dieser Vorgehensweise wird ein stufenweiser Ansatz zur Berechnung konsistenter Anfangswerte hergeleitet. Auf diese Weise werden neue Einsichten hinsichtlich der Ausnutzung von Struktureigenschaften von Index-2 ADGln gewonnen. Insbesondere stellt sich heraus, dass im Vergleich zu Index-1 ADGln der zus\"atzliche Schritt oft in der L\"osung eines linearen Systems besteht. Die sich hieraus ergebenden numerischen Folgen werden f\"ur zwei in der Schaltungssimulation h\"aufig verwendete Verfahren, das implizite Eulerverfahren und die Trapezregel, erl\"autert. \\ Zum anderen wird die Anwendung der erhaltenen Ergebnisse auf die Gleichungen, die bei der Schaltungssimulation mittels modifizierter Knotenanalyse entstehen, ausgearbeitet. Abschlie{\ss}end wird eine kurze \"Ubersicht der durchgef\"uhrten Umsetzung gegeben.\\For solving DAEs numerically, consistent initial values have to be calculated. This thesis deals with an approach for handling this problem for index-2 DAEs by considering projectors onto the spaces related to the DAE. There are two major aspects in this work.\\ On the one hand, new structural properties are deduced from weak assumptions. Subsequently, a method is proposed to choose suitable equations of an index-2 DAE, whose differentiation leads to an index reduction. This index reduction yields new theoretical results for the existence and uniqueness of solutions of index-2 DAEs which apply to a wider class of applications than previous results. Based on this method, a step-by-step approach to compute consistent initial values is developed. In this way, we gain new insights about how to deal with structural properties of index-2 DAEs. In particular, it turns out that, in comparison to index-1 DAEs, the additional step that has to be undertaken in practice often consists in solving a linear system. The numerical consequences of this fact are exemplified for two methods commonly used in circuit simulation, the implicit Euler method and the trapezoidal rule.\\ On the other hand, the application of the obtained results to the equations arising in circuit simulation by means of the modified nodal analysis (MNA) is worked out. Finally, a short overview of the specifics of their realization is given

Consistent initial values for DAE systems in circuit simulation

One of the difficulties of the numerical integration methods for differential-algebraic equations (DAEs) is computing consistent initial values before starting the integration, i.e. calculating values that satisfy the given algebraic constraints as well as the hidden constraints if higher index problems are considered. This paper presents an approach to calculate consistent initial values for index-2 DAEs starting up from possibly inconsistent ones. Firstly, the idea is exposed for linear DAEs and then it is shown how the results can be applied to those systems arising from modified nodal analysis (MNA) in circuit simulation

Structural analysis for electric circuits and consequences for MNA

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

Projected explicit and implicit Taylor series methods for DAEs

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.Peer Reviewe

Topological analysis for consistent initialization in circuit simulation

One of the difficulties of the numerical integration methods for differential-algebraic equations (DAEs) is the computation of consistent initial values before starting the integration, i.e., to calculate values that satisfy the given algebraic constraints as well as the hidden constraints if higher index problems are considered. This paper presents an algorithm that permits the consistent initialization of index 1 or 2 DAE-systems resulting from electric circuit simulation by means of modified nodal analysis (MNA). The presented approach arises from the topological properties of the network and holds for circuits that may contain some specific controlled sources

The Computation of Consistent Initial Values for Nonlinear Index-2 Differential-Algebraic Equations

The computation of consistent initial values for differential-algebraic equations (DAEs) is essential for staring a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples

Finding Beneficial DAE Structures in Circuit Simulation

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