A C0 interior penalty finite element method for flexoelectricity

TL;DR

This paper introduces a novel C0 interior penalty finite element method for modeling flexoelectricity, enabling stable and convergent simulations using standard finite element spaces with weakly enforced higher continuity.

Contribution

The paper presents a new C0 interior penalty approach that simplifies the finite element modeling of flexoelectricity by avoiding the need for higher-order elements.

Findings

Method is stable for sufficiently large penalty parameters.

Convergence demonstrated through 2D and 3D numerical examples.

Applicable to strain gradient elasticity as a special case.

Abstract

We propose a $\mathcal{C}^0$ Interior Penalty Method (C0-IPM) for the computational modelling of flexoelectricity, with application also to strain gradient elasticity, as a simplified case. Standard high-order $\mathcal{C}^0$ finite element approximations, with nodal basis, are considered. The proposed C0-IPM formulation involves second derivatives in the interior of the elements, plus integrals on the mesh faces (sides in 2D), that impose $\mathcal{C}^1$ continuity of the displacement in weak form. The formulation is stable for large enough interior penalty parameter, which can be estimated solving an eigenvalue problem. The applicability and convergence of the method is demonstrated with 2D and 3D numerical examples.