Virtual Element Method for Piezoelasticity

Abstract

This paper presents a Virtual Element Method (VEM) for the simulation of 2D and 3D piezoelectric problems. Piezoelectric materials exhibit strong multiphysics coupling behavior and have the ability to convert mechanical energy into electrical energy. At the same time, the external load on the piezoelectric structure is often applied through contact with other structures. Accurate numerical simulation becomes particularly challenging, as it requires treatment of mechanical, electrical, and contact constraints within a unified framework. The VEM is a new numerical method that can handle general polygonal and polyhedral meshes, making it suitable for simulating piezoelectric problems and with contact. The core idea begins by decomposing the total energy density from the constitutive equations, thereby deriving a bilinear formulation for the coupled mechanical-electric problem. The stabilization term for the coupled problems is also discussed. Contact is considered in the VEM framework using a penalty method for the 2D case. Finally, several numerical examples in both 2D and 3D configurations are presented to demonstrate the performance of the proposed method. Matlab code of VEM for piezoelectric problems is also given to promote reproducibility and further research, see https://github.com/Qinxiaoye/VEMpiezo.