Willi Braun

Symbolically Derived Jacobians Using Automatic Differentiation - Enhancement of the OpenModelica Compiler

Jacobian matrices are used in a wide range of applications - from solving the original DAEs to sensitivity analysis. Using Automatic Dierentiation the necessary partial derivatives can be provided eciently within a Modelica-Tool. This paper describes the corresponding implementation work within the OpenModelica Compiler (OMC) to create a symbolic derivative module. This new OMC-feature generates symbolically partial derivatives in order to calculate Jacobian matrices with respect to dierent variables. Applications

Practical realization and adaptation of Cellier's tearing method

This paper is concerned with the tearing method according to François Cellier. Tearing is used to reduce the dimension of algebraic loops, which inevitably arise in the modelling of scientific systems using differential-algebraic equations, as far as possible to achieve an efficient simulation. However, the original tearing method according to Cellier is not suitable for the application in practice, since restrictions on the solvability of equations for variables, and other features, which appear in reality, are not considered. In this work, different changes to the method are introduced and tested, which make it possible to use Cellier Tearing in practice. In addition, the modeller can influence the selection of tearing variables. Modifications of the integrated heuristic are presented, whose efficiency is statistically evaluated at the end of this work. With these changes and the new heuristics, Celliers method becomes a very suitable way to optimize the efficiency of simulation in practice.

On the importance of scaling in equation-based modelling

Equation-based modelling languages adopt a declarative modelling approach, focused on writing the model equations in a clear way and leaving the task of deriving efficient simulation code to the tool. One aspect of declarative modelling is that the use of dimensionally consistent SI units for the physical variables is preferrable; however, in many application areas this can lead to implicit nonlinear systems of equations which are badly scaled from a numerical point of view. This paper shows the negative impact of not dealing with this aspect on a benchmark test case, and then shows how the same performance of manually scaled models can be recovered by suitably exploiting information about the scaling of variables that can be declared by the modeller.