TY - JOUR

T1 - On the incompressible constraint of the 4‐node quadrilateral element

AU - Hueck, Ulrich

AU - Schreyer, Howard L.

AU - Wriggers, Peter

PY - 1995/9/30

Y1 - 1995/9/30

N2 - 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.

AB - 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.

KW - incompressibility

KW - locking‐effect

KW - quadrilateral element

KW - underintegration

UR - http://www.scopus.com/inward/record.url?scp=0040625896&partnerID=8YFLogxK

U2 - 10.1002/nme.1620381803

DO - 10.1002/nme.1620381803

M3 - Article

AN - SCOPUS:0040625896

VL - 38

SP - 3039

EP - 3053

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 18

ER -