Design sensitivity analysis and topology optimization of eigenvalue problems forpiezoelectric resonators

Abstract

Design sensitivity analysis (DSA) and topology design optimization methods forpiezoelectric resonators are developed. The governing equations for piezoelectric vibrationproblems are the Cauchy’s equation of motion and the charge equation of electrostatics. AMindlin plate theory using higher-order interpolations along the thickness direction that iseffective for the thickness-shear vibration analysis of piezoelectric resonators is used. Anoptimal design problem is formulated to maximize the electromechanical coupling strengthat the mode of interest. Necessary design gradients for eigenvalue problems are computedvery efficiently and accurately using an analytical DSA method with an eigenvectorexpansion concept. A robust mode-tracking method based on thickness-shear strain energyis proposed to precisely identify the thickness-shear mode at high frequencies.Through demonstrative numerical examples, the DSA method developed is verifiedto show excellent efficiency and agreement compared with the finite differencemethod. It is observed that the frequencies are almost invariant even though the sizeand shape of the crystal varies significantly. However, it is observed that theoptimal topology is sensitive to the aspect ratio even if the thicknesses are thesame.