Finite elements for piezoelectric vibrations with open electric boundaries

The problem of a piezoelectric body vibrating in the free space of infinite extent has been posed as a problem with open electric boundaries. A three-dimensional finite element analysis of piezoelectric vibrations has been complemented by the modeling of the external electrostatic field. The infinite exterior region that adheres to Laplace's equation is represented as a single "superelement" obtained by the ballooning of the outer boundary. The spectral transformation Lanczos method is used to find approximations to some solutions of the generalized eigenvalue problem arising from the finite element discretization of interior and exterior regions. The proposed approach was tested on simple vibrators with different mesh discretization and element order. The method produces numerically stable results, and was used to estimate the influence of the exterior leakage field on electromechanical properties of free piezoelectric vibrations. Numerical tests show that for some materials, vibrator geometries and modes, favorable to the coupling with the exterior field, the resonant frequency and electrical parameter shifts due to open electric boundaries may be as great as several percent.