Reduction of mesh distortion effects for nine-node elements using corrected shape functions

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.

► The corrected shape functions of [4] are assessed as a replacement for the standard ones.
► Four elements, QUAD9⁎⁎[10], MITC9 [1] and ours 9-AS [16] and MITC9i [21], are modified and examined.
► We show that all the tested elements benefit from using the corrected shape functions.
► We show that our improved element MITC9i performs better than the other tested elements.