RE García, SA Langer, WC Carter “Finite element implementation of a thermodynamic description of piezoelectric microstructures” Journal of the American Ceramic Society. 88(3):742-749, 2005.
A model and numerical framework is developed for piezoelectric materials. The model treats the piezoelectric and electrostrictive effects by incorporating orientation-dependent, single-crystal properties. The method is implemented in Object Oriented Finite Element program, a public domain finite element code, so it can be applied to arbitrary two-dimensional microstructures with crystallographic anisotropy. The model is validated against analytic solutions. Consistency of the method for known cases permits application of the technique to more complicated two-dimensional systems. The piezoelectric and electrostrictive response is determined for a few simple device geometries and provides insight for design and convergence criteria.