Keywords: nonconforming thin plate bending finite element, convergence, generalized patch test, nonconvex quadrilateral.

In this paper the sufficient conditions for convergence of the refined nonconforming quadrilateral thin plate bending finite element RPQ4 are analytically derived. This element was introduced recently by Wanji et al. [1] for solving thin clamped plate bending problems on a convex domain. The analytical proof of convergence, presented here,is based on Stummel's generalized patch test [2], the approximability condition and the degrees of freedom unisolvence requirement. The methodology of Shi [3] and Brenner and Scott [4] was followed to determine the error estimates. The nonconforming finite element passes the Irons' patch test, which is simple to employ. However, according to Stummel, Irons' patch test as the convergence condition is in general neither sufficient nor necessary for convergence of a nonconforming finite element. The analytical proofs of convergence for nonconforming finite elements employing Stummel's generalised patch test have seldom been performed. The presented proof could thus contribute to this area.

The interpolation matrix of the finite element may experience singularity or rank deficiency as a result of the strict use of Cartesian base functions, resulting in finite element degrees of freedom nonunisolvence simply denoted as finite element nonunisolvence. This may endanger the applicability of the element for randomly designed and, or very dense element meshes. This essential drawback is here resolved by the introduction of three different interpolation base shape functions, which results into a family of three similar nonconforming finite elements, from which at least one is proven to be an inherently unisolvent. It is easy to determine the appropriate finite element, usually one with the greatest value of the determinant of the interpolation matrix.

The derivations show that the RPQ4 finite element is convergent even if its quadrilateral domain is nonconvex, being theoretically advantageous over standard isoparametric elements. However, the numerical results obtained by dividing schemes using only convex quadrilaterals are somewhat more accurate compared to nonconvex quadrilaterals. The theoretical findings presented, in particular the linear rate of convergence in the energy norm for both convex and nonconvex meshes, were completely confirmed by various numerical examples. From the numerical results obtained a significant influence of unisolvence on the increase of the condition number of the structure stiffness matrix is apparent.

1

C. Wanji, Y.K. Cheung, "Refined nonconforming quadrilateral thin plate bending element", Int. J. Numer. Meth. Engng., 40, 3919-3935, 1997.

2

F. Stummel, "The generalised patch test", SIAM J. Numer. Anal., 16(3), 449-471, 1979.

3

Z.C. Shi, "A convergence condition for the quadrilateral Wilson element", Numer. Math., 44, 349-361, 1984. doi:10.1007/BF01405567

4

S.C. Brenner, L.R. Scott, "The mathematical theory of finite element methods", 2nd ed., Springer, New York, 2002.

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