Solving electrostatic and electroelastic problems with the node's residual descent method

Piezoelectric materials are extensively used in engineering for the fabrication of sensors, transducers, and actuators. Due to the coupling characteristics, anisotropy, and arbitrariness of polarization directions, the tasks of mesh generation, numerical integration, and global equation formulation involved in numerical computations are complex and nontrivial. To solve electrostatic and electroelastic problems, an easily implementable node's residual descent method (NRDM) is established in this study. The capability and accuracy of NRDM are validated through the solution of three-dimensional electrostatic, linear elastic, and electroelastic problems, achieving relative errors of 0.17%–1.60% for stress and 0.02%–0.15% for other variables. Fundamental and higher-order variables exhibit second-order accuracy, with convergence rates ranging from 1.98 to 2.27 using a first-order generalized finite difference scheme. This comprehensively validates the double derivation technique. The electroelastic coupling problems can be addressed by explicitly calculating electric displacement and stress based on the stress–charge form of piezoelectric constitutive equations during the iteration process. Additionally, the local basis coordinate technique is seamlessly integrated, enabling the solution of anisotropic piezoelectric problems with arbitrary polarization directions through explicit tensor operations.