On the incompressible constraint of the 4‐node quadrilateral element

authored by: Ulrich Hueck, Howard L. Schreyer, Peter Wriggers
Abstract: The standard plane 4‐node element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the ‘locking’‐effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher‐order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence‐free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first‐order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty‐constraint. Analytical expressions for that constant‐dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner. For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the elements are compared.
External Organisation(s): Technische Universität Darmstadt
University of New Mexico
Type: Article
Journal: International Journal for Numerical Methods in Engineering
Volume: 38
Pages: 3039-3053
No. of pages: 15
ISSN: 0029-5981
Publication date: 30.09.1995
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Engineering(all), Applied Mathematics
Electronic version(s): https://doi.org/10.1002/nme.1620381803 (Access: Unknown)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{461b5443c3e646de9bcaae4f05fc4991,
title = "On the incompressible constraint of the 4‐node quadrilateral element",
abstract = "The standard plane 4‐node element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the {\textquoteleft}locking{\textquoteright}‐effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher‐order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence‐free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first‐order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty‐constraint. Analytical expressions for that constant‐dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner. For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the elements are compared.",
keywords = "incompressibility, locking‐effect, quadrilateral element, underintegration",
author = "Ulrich Hueck and Schreyer, {Howard L.} and Peter Wriggers",
year = "1995",
month = sep,
day = "30",
doi = "10.1002/nme.1620381803",
language = "English",
volume = "38",
pages = "3039--3053",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "18",
}