Forward coupling approach to model time periodic piezoelectric problems

Micro electro‐mechanical systems (MEMS), such as piezoelectric loudspeakers are widely used for ultrasound applications. Designing advanced MEMS can be challenging, since nonlinear physical effects may have a significant impact. Piezoelectricity couples the mechanical and electrostatic field directly via the material law, which is often nonlinear and hysteretic. In many applications, only the steady‐state solution is of interest. The nonlinear electrostatic response to periodic forcing can be found directly in the frequency domain with the help of harmonic balancing and an alternating time frequency scheme. For certain problems and boundary conditions it might be sufficient to only assume forward coupling, i.e. one first computes the nonlinear hysteretic electrostatic field and then, use the resulting electric field and polarization to define piezoelectrically induced strains that excite the mechanical system. There are several ways to incorporate the piezoelectrically induced strains into the mechanical system. One way is to use a constant piezoelectric coupling tensor, which is used in Voigt's linear piezoelectric model. We use a phenomenological model for the piezoelectric coupling tensor, linearly scaling the coupling tensor based on the magnitude of the polarization. In this model, we neglect mechanical depolarization under the assumption that the prevailing mechanical stresses are small and a strong applied electric field enforces a certain polarization. The errors made by the forward coupling are estimated and compared with Voigt's linear piezoelectric model, for which direct coupling is used. The result of the forward coupling approach agrees well with the directly coupled results, and it needs less computational effort. Due to the phenomenological model for the piezoelectric coupling tensor, the response of the mechanical system changes and reflects the influence of nonlinear electrostatics.