Methods for Tearing Systems of Equations in Object-oriented Modeling

Modeling of continuous systems gives a set of differential and algebraic equations. In order to utilize explicit integration routines, the highest order derivatives must be solved for. In certain cases there exist algebraic loops, i.e., subsets of the equations must be solved simultaneously. The dependency structures of such subsets are often sparse. In such cases, the solution may be found more efficiently by a technique called tearing (Kron 1963) which reduces the dimensions of subsystems. This paper gives an overview of the principles of tearing. Algorithms to determine how a set of equations should be torn are, in general, inefficient. However, physical insight often suggests how this should be done. Methods to specify tearing in the object-oriented modeling program Dymola (Elmqvist 1978, 1994) are discussed. In particular it is explained, how tearing can be defined in model libraries. This allows Dymola to perform tearing automatically and efficiently without user interaction. Examples from electrical and mechanical modeling are given, including a tearing strategy for general multibody systems with kinematic loops which allow the equations of motion to be solved by standard explicit integration algorithms.