## Preprint 2012-13

## Felix Dietzsch, Luis Hervella-Nieto, Steffen Marburg, Rodolfo Rodríguez, Hannah Weisbecker:

### Physical and spurious modes in mixed finite element formulation for the Galbrun equation

### Abstract:

Sound propagation in moving media can be described by Galbrun equation for the oscillating component of the fluid displacement. A displacement based finite element formulation using standard Lagrangian elements produces spurious modes. Investigations in literature (e.g., IJNME 63, 974-987, 2005) have shown that Mini elements and Taylor-Hood elements suppress the effect of spurious modes. Herein, the quadratic eigenvalue problem for the mixed formulation in 2D using Mini and Taylor-Hood elements is solved. Solution confirms former results such that both element types are suitable for low Mach numbers and under certain conditions. Although the formulation is not free from spurious results, physical and spurious modes are well separated for low Mach numbers in non-dissipative systems. As reported, mini elements produce spurious modes for Mach numbers > 0.5 whereas Taylor-Hood elements perform more stable even for large Mach numbers in non-dissipative systems. If absorbing walls are considered, separation of physical and spurious modes becomes less clear. Then, eigenvalues of both types of modes are located closer to each other in the complex plane. Examples encompass the 1d duct problem, for which the spurious modes are discussed for the energy conserving problem, and an annular duct for which the dissipative case is investigated.

This preprint gave rise to the following definitive publication(s):

**Felix DIETZSCH, Luis HERVELLA-NIETO, Steffen MARBURG, Rodolfo RODRíGUEZ, Hannah WEISBECKER: ***Physical and spurious modes in mixed finite element formulation for the Galbrun equation*. Acta Acustica united with Acustica, vol. 100, 3, pp. 493-512, (2014).