On the passivity of a Galerkin finite element model for transient flow in hydraulic pipelines

The Galerkin finite method for the simulation of transient, laminar flow in hydraulic pipelines with an interlacing grid has gained considerable popularity because of the relatively good accuracy achieved with a moderate model order. However, the implementation used in a number of software packages has recently been shown not to guarantee model passivity. This paper reviews the concept of using an interlacing grid. Two modifications of the previously published interlacing grid method are proposed and the resulting model is shown to fulfil the passivity requirement in an example where the previously published method fails. The new model also improves the approximation error in terms of the deviation of the frequency response from the theoretical frequency response given by the transcendental transfer functions of the well-known 'frequency-dependent friction' model. This improvement is bought at the expense of a slight increase in model order due to additional nodal values for the friction model at the boundary nodes. Furthermore, the new model is given in the form of a descriptor system in order to maintain the sparse structure of the system matrices.