Element Order

Abstract: In this paper, we examine the infl uence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss-Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments.

Abstract: This paper presents some research results obtained recently using the p- version of the finite element method (FEM) for shape optimal design. The use of Bezier and B-spline curves to define design elements has proven to be an excellent way to model the geometry of the design problem. The p- version 2D elastic element was extended to employ part of a Bezier or B-spline curve as its element side for this purpose. This new element has been tested successfully with the patch test. Moreover, it is compatible, has no preferred direction, and contains all the required rigid-body modes (three zero eigenvalues are found in the element stiffness matrix). There are several advantages in using the p- version FEM for shape optimal design. The analysis and design models are often identical. In the p- version FEM, the stresses along an element side are as accurate as those inside the element. This feature is important because usually the critical stress constraints are found along the design element boundary. The final design can very easily be reanalyzed with a different element order to check the accuracy of the result. Some classical shape optimal design problems have been solved using the Conlin optimizer. The results indicate that similar optimal shapes can be obtained with fewer degrees of freedom than when compared with the h- version FEM. As with the h- version , less than ten structural analyses are sufficient for convergence in most of the problems.

Abstract: The discretization inherent in the finite-element method results in numerical dispersion. This dispersion is investigated for a time-harmonic plane wave propagating through an infinite, two-dimensional, finite-element mesh composed of uniform quadrilateral and triangular elements. The effects on the dispersion due to the propagation direction of the wave, the order of the elements, the node density, and the mesh geometry are studied. Results are given which can serve as a guide in selecting the appropriate element order, node density, and mesh geometry when applying the finite-element method.