Shape Optimization of Electrodes for Piezoelectric Actuators

Summary

This paper describes a method of shape optimization for electrodes used in piezoelectric actuators. This work is motivated by recent attempts to design piezoceramic bimorph actuators for the optimal control of two-dimensional flow fields. It has been observed experimentally that uniformly-electroded bimorphs induce undesirable three-dimensional effects into the flow. These effects complicate the optimal flow control problem considerably. Therefore, the objective is to design an electrode that is shaped in such a manner as to minimize three-dimensional effects while providing maximum actuator deflection. This paper considers electrode shape optimization for a typical two-layer composite transducer (unimorph) structure, which takes the form of a small, rigidly-clamped flap. The strong and weak forms of the governing equations of motion are derived for this system, which can be approximated as a thin plate. The finite element method is used to determine the dynamic flap response for a given electrode shape and voltage input. The optimization problem entails determining the electrode shape that, for a prescribed voltage input, minimizes the three-dimensional distortion at the trailing edge of the flap. Constraints are enforced that guarantee that the electrode shape is relatively smooth and that the electrode coverage is large enough to provide sufficient flap deflection. Theoretical results are presented that prove the existence of a solution when the shape optimization problem is formulated in this manner. Numerical results are given that validate the theory and illustrate the qualitative behavior of the shape optimization method. For the static case, the results are compared with those obtained in previous work using an ad hoc optimization algorithm that is not rigorously consistent with the shape optimization theory. Results are also given for the dynamic case in which the time-dependent response to a prescribed voltage input is considered.