Stabilized finite element formulation for the mixed convected wave equation in domains with driven flexible boundaries

A stabilized finite element (FEM) formulation for the wave equation in mixed form with convection is presented, which permits using the same interpolation fields for the acoustic pressure and the acoustic particle velocity. The formulation is based on a variational multiscale approach, in which the problem unknowns are split into a large scale component that can be captured by the computational mesh, and a small, subgrid scale component, whose influence into the large scales has to be modelled. A suitable option is that of taking the subgrid scales, or subscales, as being related to the finite element residual by means of a matrix of stabilization parameters. The design of the later turns to be the key for the good performance of the method. In addition, the mixed convected wave equation has been set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference to account for domains with moving boundaries. The movement of the boundaries in the present work consists of two components, an external prescribed motion and a motion related to the boundary elastic back reaction to the acoustic pressure, in the normal direction. A mass-damper-stiffness auxiliary equation is solved for each boundary node to include this effect. As a first benchmark example, we have considered the case of 2D simple duct acoustics with mean flow. More complex 3D examples are also presented consisting of vowel and diphthong generation, following a numerical approach to voice production. The numerical simulation of voice not only allows one to see how waves propagate inside the vocal tract, but also to collect the acoustic pressure at a node close to the mouth exit, convert it to an audio file and listen to it.