Institute of Fundamental Technological Research

The paper concerns a well-known two-dimensional nine-node quadrilateral element MITC9, which is based on two-level approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test.

As the first improvement, we propose a modification of the element's transformations, partly resolving the problem with the patch test. The source of the problem is the use of covariant components in a (local) natural co-basis, different at each sampling point.

As the second improvement, we use the corrected shape functions of Celia MA, Gray WG. An improved isoparametric transformation for finite element analysis. International Journal for Numerical Methods in Engineering 1984; 20:1447–1459, extending their applicability to the nine-node element for plane elasticity and the 3 × 3 integration. Originally, they are tested for an eight-node element for the heat conduction equation and the 4 × 4 integration.

The improved element, designated as MITC9i, is based on the Green strain and derived from the potential energy for the plane stress condition. It is subjected to a range of tests, to confirm that it passes the patch test for several types of mesh distortions, to prove its coarse mesh accuracy and the absence of locking as well as to establish its sensitivity to mesh distortions.

The improved element MITC9i performs substantially better than the MITC9 element, QUAD9** element, and our previous 9-AS element.

nine-node element, two-level approximation of strains, assumed strain method, two-dimensional MITC9, patch test, sensitivity to shape distortions, coarse mesh accuracy

The paper concerns two-dimensional nine-node quadrilateral elements based on the Green strain and the two-level approximations of strains. These approximations reduce locking well for regular meshes but cannot prevent the drop of accuracy when the side and central nodes are shifted from the middle positions.

To reduce the deterioration of accuracy when nodes are shifted, we assess the corrected shape functions of Celia and Gray (1984 [10]) as a replacement for the standard isoparametric ones. In Celia and Gray (1984 [10]), the corrected shape functions were tested for an eight-node element, the heat conduction equation and the 4×4 integration. Here, we test their applicability to nine-node elements for plane elasticity and the 3×3-point integration.

We modify and examine four elements: QUAD9⁎⁎ (Huang and Hinton, 1986 [15]), MITC9 [1] and ours 9-AS (Panasz and Wisniewski, 2008 [21]) and MITC9i (Wisniewski and Panasz, 2012 [26]). The elements are subjected to a range of tests involving several types of mesh distortions, to confirm passing of various forms of the patch test, to prove the absence of locking as well as to establish their coarse mesh accuracy and sensitivity to mesh distortions. We show that all the tested elements benefit from using the corrected shape functions, but still remain significant differences in their performance.

Two-dimensional nine-node elements, Corrected shape functions, Two-level approximations of strains, Patch tests, Shape distortions

The paper concerns 9-node quadrilateral shell elements derived for Reissner's kinematics. They are based on the Green strain and potential energy, and are applicable to large (unrestricted) rotations. The characteristic features of the developed elements are as follows:

1. Drilling rotation is included via the drill rotation constraint (RC) imposed by the penalty method. Hence, the elements have 6 dofs per node, i.e. three displacements and three rotational parameters, including drilling rotation.

2. Transverse shear and membrane locking as well as the in-plane shear over-stiffening are avoided using the two-level approximation applied to the strain (assumed strain method). This method does not affect the drilling RC.

3. A modification of the two-level approximation method is proposed, consisting in treating the sampling and the numerical integration together, which results in six sampling points being replaced by two sampling lines. The two-level approximation is applied to components in the ortho-normal basis at the element center, which differs our element from the MITC family of elements, which uses the covariant strain components.

4. Selective reduced integration (SRI) approach is revised. The total functional is split into several parts, and a suitable integration rule is found for each part, yielding an efficient element which shows very good mesh convergence.

Two 9-node shell elements are developed and subjected to a range of benchmark tests, to establish the sensitivity to mesh distortion, the coarse mesh accuracy, and to confirm the lack of locking. Our results are compared with results obtained by the MITC9 element of ADINA and the S9R5 element of ABAQUS.

Nine-node shell elements, 6 dofs/node, Drilling rotation, Two-level approximation, Assumed strain, Selective reduced integration