On Convergence of Affine Thin Plate Bending Element

Authors: Rado Flajs, Miran Saje

Abstract:

In the present paper the displacement-based nonconforming quadrilateral affine thin plate bending finite element ARPQ4 is presented, derived directly from non-conforming quadrilateral thin plate bending finite element RPQ4 proposed by Wanji and Cheung [19]. It is found, however, that element RPQ4 is only conditionally unisolvent. The new element is shown to be inherently unisolvent. This convenient property results in the element ARPQ4 being more robust and thus better suited for computations than its predecessor. The convergence is proved and the rate of convergence estimated. The mathematically rigorous proof of convergence presented in the paper is based on Stummel-s generalized patch test and the consideration of the element approximability condition, which are both necessary and sufficient for convergence.

Keywords: Quadrilateral thin plate bending element, convergence, generalized patch test.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1081802

References:

[1] Adams, R. A.; Sobolev Spaces, Academic Press, New York, 1975.
[2] Arnold, D. N., Boffi, D., Falk, R. S.; Approximation by quadrilateral finite elements, Math. Comp., 71, 909-922, 2002.
[3] Bazeley, G. P., Cheung, Y. Q., Irons, B. M., Zienkiewicz, O. C.; Triangular elements in bending: Conforming and non-conforming solutions, in Proceedings Conference on Matrix Methods in Structural Mechanics, Wright Patterson A. F. B., Dayton, OH, 547-576, 1965.
[4] Brenner, S. C., Scott, L. R.; The Mathematical Theory of Finite Element Methods, 2nd ed., Springer, New York, 2002.
[5] Ciarlet, P. G., Lions, J. L.; Handbook of Numerical Analysis, Volume II, Finite element methods (Part 1), North-Holland, Amsterdam, 1991.
[6] Ciarlet, P. G., Raviart, P. A.; Interpolation theory over curved elements, with applications to finite element methods, Comput. Meth. Appl. Mech. Engrg., 1, 271-249, 1972.
[7] Ciarlet, P. G.; Finite Element Methods for Elliptic Problems, SIAM, Philadelphia, 2002.
[8] Demmel, J. W.; Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
[9] Flajs, R., Cen, S., Saje, M.; On convergence of nonconforming convex quadrilateral finite elements AGQ6, Comp. Meth. Appl. Mech. Engrg., 199, 1816-1827, 2010.
[10] Park, C., Sheen, D.; P1-non-conforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal., 41 (2), 624-640, 2003.
[11] Shi, Z. C.; A convergence condition for the quadrilateral Wilson element, Numer. Math., 44, 349-361, 1984.
[12] Shi, Z. C.; The F-E-M-test for convergence of non-conforming finite elements, Math. Comput., 49 (40), 391-405, 1987.
[13] Shi, Z. C.; On Stummel-s examples to the patch test, Comp. Mech., 5, 81-87, 1989.
[14] Stummel, F.; The generalised patch test, SIAM J. Numer. Anal., 16 (3), 449-471, 1979.
[15] Stummel, F.; The limitations of the patch test, Int. J. Numer. Meth. Engng., 15, 177-188, 1980.
[16] Taylor, R. L., Simo, J. C., Zienkiewicz, O. C., Chan, A. C. H.; The patch test-a condition for assessing FEM convergence, Int. J. Numer. Meth. Engng., 22, 39-62, 1986.
[17] Verf¨urth, R.; A note on polynomial approximation in Sobolev spaces, Math. Modelling Num. Anal., 33 (4), 715-719, 1999.
[18] Wang, M.; On the necessity and sufficiency of the patch test for convergence of non-conforming finite elements, SIAM J. Numer. Anal., 39 (2), 363-384, 2001.
[19] Wanji, C., Cheung, Y. K.; Refined non-conforming quadrilateral thin plate bending element, Int. J. Numer. Meth. Engng., 40, 3919-3935, 1997.
[20] Wanji, C.; Variational principles for non-conforming finite element methods, Int. J. Numer. Meth. Engng., 53, 603-619, 2002.
[21] Wanji, C.; Enhanced patch test of finite element methods, Science in China: Series G Physics, Mechanics & Astronomy, 49 (2), 213-227, 2006.
[22] Zienkiewicz, O. C., Taylor, R. L.; The finite element patch test revisited. A computer test for convergence, validation and error estimates, Comp. Meth. Appl. Mech. Engrg., 149, 223-254, 1997.