Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

Abstract

In the present contribution, we develop a mixed finite element method capable of the coupled multi-field simulation of a viscous fluid actuated by a piezoelectric resonator. Several challenges are met with in this setting, among which are the necessity of correct interface coupling, near incompressibility of the fluid, adverse geometric dimensions of flat piezoelectric transducers and different length scales of shear and pressure wave. Assuming small deformations and velocities, we present a mixed variational formulation with consistent interface coupling conditions in (mechanic) frequency domain. Both fluid and piezoelectric solid domain are discretized using Tangential-Displacement Normal-Normal-Stress elements. These elements model not only the deformation, but add an independent tensor-valued stress approximation. The method has been rigorously proven to be free from shear locking for flat prismatic or hexahedral elements. Thus, modeling of the flat geometry of piezoelectric resonators as well as resolution of the fastly decaying shear wave are facilitated. To circumvent the problem of volume locking due to the near incompressibility of the fluid, an additional independent pressure field is introduced. We present computational results indicating the capability of the method.