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On the incompressible constraint of the 4-node quadrilateral element. (English) Zbl 0854.73066
An analytical expression for the incompressible constraint of a 4-node quadrilateral finite element of plane stress/strain is derived. It is proved that the volume constraint arises naturally from an element matrix related to the first order terms of the Taylor series expansion of the element shape functions. The same matrix is obtained from a 1-point Gauss integration. The incompressibility condition was 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. It is shown that incompressibility could be enforced without the use of a mixed variational principle and without introduction of the pressure as an independent variable. The analytical result can be used to verify the formulation of elements in the incompressible limit by means of a numerical eigensystem analysis.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
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References:
[1] Nagtegaal, Comput. Methods Appl. Mech. Eng. 4 pp 153– (1974)
[2] Herrmann, AIAA J. 3 pp 1896– (1965)
[3] Hughes, J. Appl. Mech. ASME 44 pp 181– (1977) · doi:10.1115/1.3423994
[4] Malkus, Int. J. Solids Struct. 12 pp 731– (1976)
[5] Malkus, Comput. Methods Appl. Mech. Eng. 15 pp 63– (1978)
[6] and , Concepts and Applications of Finite Element Analysis, 3rd edn., Wiley, New York, 1989. · Zbl 0696.73039
[7] The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1987.
[8] and , The Finite Element Method, Vol. 1, 4th edn., McGraw-Hill, London, 1989.
[9] Irons, AIAA J. 4 pp 2035– (1966)
[10] Kosloff, Int. J. Numer. Anal. Meth. Geomech. 2 pp 57– (1978)
[11] Flanagan, Int. j. numer. methods eng. 17 pp 679– (1981)
[12] Belytschko, Comput. Methods Appl. Mech. Eng. 54 pp 279– (1986)
[13] Belytschko, Comput. Methods Appl. Mech. Eng. 88 pp 311– (1991)
[14] Bachrach, Comput. Methods Appl. Mech. Eng. 55 pp 43– (1986)
[15] Liu, Comput. Methods Appl. Mech. Eng. 53 pp 13– (1985)
[16] Hueck, Int. j. numer. methods eng. 38 pp 3007– (1995)
[17] Hughes, Int. j. numer. methods eng. 15 pp 1413– (1980)
[18] Simo, Comput. Methods Appl. Mech. Eng. 51 pp 177– (1985)
[19] Simo, J. Appl. Mech. ASME 53 pp 51– (1986)
[20] Simo, Int. j. numer. methods eng. 29 pp 1595– (1990)
[21] Taylor, Int. j. numer. methods eng. 10 pp 1211– (1976)
[22] Tong, Int. J. Solids Struct. 5 pp 455– (1969)
[23] Pian, Int. j. numer. methods eng. 20 pp 1685– (1984)
[24] Cheung, Comput. Struct. 42 pp 683– (1992)
[25] Yuan, Int. j. numer. methods eng. 36 pp 1747– (1993)
[26] Yuan, Int. j. numer. methods eng. 37 pp 457– (1994)
[27] Hueck, Int. j. numer. methods eng. 35 pp 1633– (1992)
[28] ’Eigensystem analysis of an irregular four-node quadrilateral’, Master’s Thesis, Department of Mechanical Engineering, The University of New Mexico, 1987.
[29] Jacquotte, Comput. Methods Appl. Mech. Eng. 44 pp 339– (1984)
[30] Linear Algebra and its Applications, 3rd edn., Harcourt Brace Jovanovich, San Diego, 1988.
[31] and , ’Stress analysis of axisymmetric solids utilizing higher-order quadrilateral finite elements’, SESM Report No. 69-3, Dept. of Civil Eng., University of California, Berkeley, 1969.
[32] Liu, Int. j. numer. methods eng. 20 pp 931– (1984)
[33] Hacker, Int. j. numer. methods eng. 28 pp 687– (1989)
[34] and , ’On the stabilization of the rectangular 4-node quadrilateral element’, Cerecam Report No. 195, University of Cape Town, South Africa, March 1993;
[35] Commun. Numer. Methods Eng. 10 pp 555– (1994)
[36] and , ’Some problems in the discrete element representation of aircraft structures’, in (ed.), Matrix Methods of Structural Analysis, Pergamon Press, Oxford, 1964, pp. 267-315.
[37] and , ’The method of orthogonal projections for handling constraints with application to incompressible four-node quadrilateral elements’, in et al. (eds.), Numerical Methods in Engineering ’92, Proc. 1st European Conf. on Numerical Methods in Engineering, 7-11 September 1992’, Brussels, Belgium, Elsevier, Amsterdam, 1992, pp. 795-802.
[38] private communication, October 1993.
[39] Weissman, Comput. Methods Appl. Mech. Eng. 98 pp 127– (1992)
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