An Isogeometric Element Formulation for Linear Two-Dimensional Elasticity Based on the Airy Equation

Abstract

The aim of this work is to derive a formulation for linear two-dimensional elasticity using just one degree of freedom. With the Airy stress function, a measure without further physical meaning is chosen to this single degree of freedom. The corresponding Airy equation requires higher order basis functions for the discretization of the formulation [1]. Isogeometric structural analysis (IGA) is based on shape functions of the system in Computer-Aided design (CAD) software [2]. These shape functions can fulfill the requirement of high continuity and therefore the formulation is obtained through IGA methods. Non-Uniform Rational B-splines (NURBS) are used to discretize the domain and to solve the occurring differential equations within the Galerkin method [3]. The received one-degree of freedom formulation allows to compute stresses as direct solution of the underlying system of equations. Numerical examples demonstrate the accuracy for a quadratic plate under standard, but also under complex loading. For constant or linear loading functions only one element is sufficient to receive the exact solution – a general advantage of using higher order basis functions. The correct convergence behaviour of the proposed formulation is proved by the -error norm for a complex load situation. Here, only a few refinement steps yield a good approximation with a very small error of the stresses.