Comparing scalar and vector forms of the finite element method by example of an elliptic cylinder

Abstract

An algorithm for constructing the stiffness matrix of a quadrangular curvilinear finite element in the form of a fragment of the middle surface of an elliptic cylinder with 18 degrees of freedom in the node is given. Implementation of the finite-difference procedure includes two variants of approximation of the sought variables, i.e., a scalar one and a vector one. Numerical examples are given and prove that the vector approximation has principal advantages over the scalar approximation for arbitrary shells with significant gradients of curvature of the lines of the middle surface or allowing for displacement as a rigid body.