The p-version of the finite element method for three-dimensional thin walled structures with physically non-linear behaviour

In this paper we present an implementation of a three-dimensional p-version for structural problems of solids with almost arbitrarily curved surfaces. Applying the blending function method, complex structures can often be modelled by a few p-elements, being the basis for a higher order approximation. Numerical examples will demonstrate, that the p-version with anisotropic Ansatz spaces allows to predict the structural behaviour of three-dimensional plates and shells with approximately the same amount of degrees of freedom as in the two-dimensional case, yet significantly more accurate due to the three-dimensional model. Furthermore, it is advantageous to compute complex structures exclusively with three-dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three-dimensional solid elements. Using the p-version with anisotropic Ansatz spaces the whole structure can be efficiently discretized with solid elements, even if the aspect ratio of the elements becomes very large. Copyright © 2001 John Wiley Sons, Ltd.

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The p‐version of the finite element method for three‐dimensional curved thin walled structures

In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity.We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implemen- tation of the p-version.

The p‐Version of the Finite Element Method

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

Locking free isogeometric formulations of curved thick beams

This paper addresses the question of accuracy of p-version finite element formulations for Reissner–Mindlin plate problems. Three model problems, a circular arc, a rhombic plate and a geometrically complex structure are investigated. Whereas displacements and bending moments turn out to be very accurate without any post-processing even for very coarse meshes, the quality of shear forces computed from constitutive equations is poor. It is shown that significantly improved results can be obtained, if shear forces are computed from equilibrium equations instead. A consistent computation of second derivatives of the shape functions is derived. © 1998 John Wiley & Sons, Ltd.

Karin Preusch

Papers

On the accuracy of p-version elements for the Reissner-Mindlin plate problem