C. Rößler:

"Regularisierungsmethoden für Differential-Algebraische Gleichungssysteme";

Supervisor: F. Breitenecker; Institut für Analysis und Scientific Computing, 2014; final examination: 2014-10-14.

The use of object{oriented simulation tools for modelling of physical or mechanical systems leads to systems of di erential{algebraic equations with a high di erential index.

The di erential index indicates the minimal number of di erentiations of the system, which are necessary to extract a system of ordinary di erential equations from the differentiated system. In general the numerical solution of di erential{algebraic equation systems with high index by conventional solution methods for ordinary di erential equations is very complex or may even be impossible. Therefore methods for solving this problem are necessary, which leads to the so{called index reduction. The aim of the index reduction is to convert the system of di erential{algebraic equations into a system

of di erential{algebraic equations of lower index or a system of ordinary di erential equations.

This work aims to provide an overview of common regularisation methods. Additionally,a classi cation of these di erent approaches is done. This classi cation divides the di erent approaches into three areas: index reduction with the use of di erentiation,

stabilization of the numerical solution by projection and (local) transformation of the state space. According to the classi cation each approach is presented and explained in detail. Then advantages and disadvantages of the di erent methods are discussed.

Three di erent methods of index reduction with the use of dierentiation are considered:

di erentiation and replacement of the constraint, the Baumgarte{Method and the Pantelides{Algorithm. There are two di erent methods using projections, called the orthogonal projection method and the symmetric projection method. The idea of the method using transformation of the state space is to obtain a system of ordinary di erential equations on a manifold.

In order to compare the di erent methods, the approaches described above are demonstrated by examples. The two examples are mechanical systems. On the one hand the equations of the motion of a pendulum on a circular path in Cartesian coordinates are considered. On the other hand, the equations of motion of the double pendulum in Cartesian coordinates, which shows chaotic behaviour, are used. For the comparison of the di erent methods, the obtained numerical solutions and the deviation from the constraint equations are considered. Therefore the numerical solutions of the distinct approaches can be compared. Finally it is possible to decide which method is suitable

for solving the given system of di erential{algebraic equations.

http://publik.tuwien.ac.at/files/PubDat_234892.pdf

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