On the numerical properties of high‐order spectral (Euler‐Bernoulli) beam elements
Sascha Eisenträger, Santosh Kapuria, Mayank Jain, Junqi Zhang
Abstract
Abstract In this paper, the numerical properties of a recently developed high‐order S pectral Euler‐Bernoulli B eam E lement (SBE) featuring a C 1 ‐continuous approximation of the displacement field are assessed. The C 1 ‐continuous shape functions are based on two main ingredients, which are an Hermitian interpolation scheme and the use of Gauß‐Lobatto‐Legendre (GLL) points. Employing GLL‐points does not only avoid Runge oscillations, but also yields a diagonal mass matrix when exploiting the nodal quadrature technique as a mass lumping scheme. Especially in high‐frequency transient analyses, where often explicit time integration schemes are utilized, having a diagonal mass matrix is an attractive property of the proposed element formulation. This is, however, achieved at the cost of an under‐integration of the mass matrix. Therefore, a special focus of this paper is placed on the evaluation of the numerical properties, such as the conditioning of the element matrices and the attainable rates of convergence (ROCs). To this end, the numerical behavior of the SBEs is comprehensively analyzed by means of selected benchmark examples. In a nutshell, the obtained results demonstrate that the element yields good accuracy in combination with an increased efficiency for structural dynamics exploiting the diagonal structure of the mass matrix.