Sparse Causalisation of Differential Algebraic Equations for Efficient Event Detection

Models written in multidomain simulation languages like Modelica describe hybrid systems of differential and algebraic equations (hybrid DAEs). Algebraic events are an integral part of those models. Their detection in general requires numerical root-finding. This method can be optimized. We show that state-of-the art simulators force unnecessary evaluations during root-finding: The calculation of a residual function is always preceded by the solution of the complete DAE for the active time step. Models are usually composed of hierarchical elements. Therefore, it is quite common, that not the whole DAE needs to be computed for a single event function, although some dependencies exist. By the application of the well known method of causalisation those dependencies can be detected. We show how a subset of the solution algorithm, called a sparse causalisation can be derived from a usual causalisation for a given event variable. When solving the sparse causalisation, the event variable is computed correctly, i.e. it holds the same result as after a complete solution. Depending on the internal structure of the model, the application of sparse causalisation can significantly reduce the amount of work necessary to detect an event.