Optimally blended spectral elements in structural dynamics: Selective integration and mesh distortion

Abstract

In the field of structural dynamics, spectral finite elements are well known for their appealing approximation properties. Based on a special combination of shape functions and quadrature points, a diagonal mass matrix is obtained. More recently, the so-called optimally blended spectral element method was introduced, which further improves the accuracy but comes at the cost of a non-diagonal mass matrix. In this work, we study and compare the approximation properties of the different spectral and finite element methods. For each method, an h-version (fine meshes and low-order shape functions) as well as a p-version (coarse meshes and high-order shape functions) are considered. Special attention is paid to the influence of the quadrature rule used to compute the stiffness matrix and the element distortion on the convergence behavior. The investigations reveal the importance of a correct (full) integration of the stiffness matrix in order to achieve the theoretically predicted convergence rates. However, looking at the full spectrum, novel variants of the method that apply only a single (reduced) quadrature rule for mass and stiffness matrix show a higher accuracy.