A meshfree Galerkin formulation for nonlinear piezoelectric semiconductors in consideration of flexoelectricity

Piezoelectric semiconductors (PSCs) are widely used in a microelectromechanical system, but there is still lack of numerical methods for analyzing the nonlinear multi-field coupling behavior of PSCs in consideration of flexoelectricity. In this study, a meshfree Galerkin formulation is presented for such a nonlinear fourth-order boundary value problem. To satisfy the C¹ continuity requirement, the moving least square approximation is employed with a quadratic base. To considering the nonlinear drift-diffusion effect in semiconductors, a nonlinear algorithm with tangent stiffness is proposed, which shows much better convergence than available direct iteration methods with secant stiffness. Further, to simulate electrically isolated PSCs, a Lagrange multiplier method is firstly employed to introduce the electroneutrality condition into the Galerkin weak form. In addition, the consistent integration with nodal smoothed derivatives is used to improve the computational efficiency. Several numerical results validate the proposed formulation and demonstrate the effects of the flexoelectric coefficient, intrinsic length scale parameter and initial carrier concentration.