Petrov-Galerkin zonal free element method for piezoelectric structures

This paper presents a novel Petrov-Galerkin free element method (PGPZ-FREM) based on a combination of the strong form free element method (FREM), sub-domain mapping technique, and Petrov-Galerkin method for analyzing piezoelectric structures. This is a brand new numerical method that combines the ideas of isogeometric method and meshless method. Similar to the isogeometric method, the computational domain is divided into a lot of patchs or subdomains firstly. In each subdomain, local collocation Lagrangian elements are generated according to the location of the nodes. Additionally, the Heaviside step function is selected as the weight function to simplify the calculations. By constructing equations point by point, a set of linear algebraic equations is established to solve the piezoelectric problem. Finally, the accuracy and stability of the piezoelectric zonal Petrov-Galerkin free element method are verified by numerical examples, including a symmetric piezoelectric block, a piezoelectric tuning fork, a dual-material MFC sensor, and the wing skin pressure sensing system. • A novel Petrov-Galerkin zonal free element method is proposed. • The transformation and relationship of differentials in different configurations are derived. • Complex piezoelectric examples are analyzed and verified.