Continuous interior penalty finite element method for a 6th-order bending gradient elastic (nano)beam equation

Abstract

This work revolves around the design of a continuous interior penalty finite element method (CIPFEM) for a sixth-order bending gradient elastic beam equation, through the use of non-conforming finite elements consisting of piecewise C0-continuous approximation functions. The CIPFEM advantageously combines the continuous Galerkin method, the discontinuous Galerkin method and the stabilized method. Only the nodal displacements are included as degrees of freedom and the linear system’s stiffness matrix is symmetric and positive definite. The displacement is continuous and the continuity of first and higher-order derivatives is weakly enforced by means of stabilizing terms (being of importance for the method’s convergence) on both the interior and exterior boundaries. The research objective focuses on establishing the consistency and the stability property and performing an a priori error analysis of the method. Based on the conclusions reached, the method is consistent, stable and convergent.