Abstract
In this investigation, with the aid of computer algebra, a diagram showing equivalences between finite element formulations or techniques, among which including two newly formulated hybrid mixed field principles, in a 4-node plane stress/strain element was obtained. In the investigation, some known equivalence relations were confirmed; some new equivalence relations, mainly between the hybrid mixed field formulations and the others, were detected. The investigation shows that with computer algebra it is much easier to conduct parameter study in simple elements and it might provide a powerful tool for exploring more complex equivalence relations in higher order elements. It is observed from the investigation that if a finite element technique is hard to fit in a conventional variational principle, it might have a counterpart in a properly modified variational principle. The obtained equivalence diagram might provide information for establishing a more generally valid mathematical principle or theorem.
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References
M. Pröier; L. Nilsson and A. Samuelsson. The rectangular plane stress element by Turner, Pian and Wilson. Int. J. Numer. Meth. Engng., 8, 433–437, 1974.
T. H. H. Pian and T. Pin. Relations between incompatible displacement model and hybrid stress model. Int. J. Numer. Meth. Engng., 22, 173–181, 1986.
G. A. Mohr and P. L. Cook, On near equivalence of assumed stress and reduced integration formulations of the bilinear plane stress finite element, Computers & Structures, 21, 475–478, 1985.
H. Shimodaira, Equivalence between mixed models and displacement models using reduced integration, Int. J. Numer. Meth. Engng., 21, 89–104, 1985.
J. C. Simo and M. S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, Int. J. Numer. Meth. Engng., 29, 1595–1638, 1990.
H. Stolarski and T. Belytschko, On the equivalence of mode decomposition and mixed finite elements based on the Hellinger-Reissner principle. PART I: Theory, Comput. Methods Appl. Mech. Engrg., 58, 249–263, 1986.
H. Stolarski and T. Belytschko, On the equivalence of mode decomposition and mixed finite elements based on the Hellinger-Reissner principle. PART II: Application, Comput. Methods Appl Mech. Engrg., 58, 265–284, 1986.
H. Stolarski and T. Belytschko, Limitation principles for mixed finite elements based on the Hu-Washizu variational formulation, Comput. Methods Appl. Mech. Engrg., 60, 195–216, 1987.
S. T. Yeo and B.C. Lee, Equivalence between enhanced assumed strain method and assumed stress hybrid method based on the Hellinger-Reissner principle, Int. J. Numer. Meth. Engng., 39, 3083–3099, 1996.
W. Zhang. Further study of the identity of incompatible displacement element and generalized hybrid element. Acta Mechanica Sinica, 23, 564–570, 1991.
P. J. Zhao; T. H. H. Pian and Y. Sheng. A new formulation of isoparametric finite elements and the relationship between hybrid stress element and incom-patible element. Int. J. Numer. Meth. Engng., 40, 15–27, 1997.
B. W. Char; K. O. Geddes; G. H. Gonnet; B. L. Leong; M. B. Monagan and S. M. Watt. Maple V language reference manualSpringer-Verlag, 1991.
S. Wolfram. The Mathematica Book, 3rd edition. Wolfram Media, Inc., Cam-bridge, 1996.
E. D. L. Pugh; E. Hinton and O. C. Zienkiewicz. A study of quadrilateral plate bending elements with ‘reduced’ integration. Int. J. Numer. Meth. Engng., 12, 1059–1079, 1978.
T. J. R. Hughes; R. L. Taylor and W. Kanoknukulchai. A simple and efficient finite element for plate bending. Int. J. Numer. Meth. Engng., 11, 1529–1543, 1977.
R. D. Cook. More reduced integration and isoparametric elements. Int. J. Numer. Meth. Engng., 5, 141–142, 1972.
H. Stolarski and T. Belytschko. Membrane locking and reduced integration for curved elements. Transactions of the ASME. J. Appl. Mech., 49, 172–176, 1982.
E. L. Wilson; R. L. Taylor; W. P. Doherty and J. Ghaboussi. Incompatible displacement modes. In Numerical and Computers Models in Structural Mechanics, ed. by S. J. Fenves et al. Academic Press, New York, 1973.
T. H. H. Pian. Derivation of element of stiffness matrices by assumed stress distribution. AIAA J., 2, 1333–1336, 1964.
T. H. H. Pian and C. C. Wu. A rational approach for choosing stress terms for hybrid finite element formulations. Int. J. Numer. Meth. Engng., 26, 2331–2343, 1988.
Y. H. Luo and A. Eriksson. Extension of field consistence approach into de-veloping plane stress elements. Comput. Methods Appl. Mech. Engrg., 1998. Accepted.
Y. H. Luo and A. Eriksson. An alternative assumed strain method. Comput. Methods Appl. Mech. Engrg., 1998. Accepted.
Y. H. Luo . Two hybrid mixed variational principles for developing high perfor-mance finite elements. Int. J. Numer. Meth. Eng., 1998. Submitted.
Y. H. Luo and A. Eriksson. Investigation of equivalence between mixed field formulation and reduced integration with symbolic computation software. Int. J. Numer. Meth. Engng., 1998. Submitted.
MATLAB Reference Guide, Math Works Inc., Natick, 1993.
Y. H. Luo. On Some Finite Element Formulations in Structural Mechanics. Doctoral Thesis, Stockholm, 1998.
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Eriksson, A., Luo, Y., Pacoste, C. (1999). Computer Algebra Investigation of Equivalence in 4-node Plane Stress/Strain Finite Elements. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60218-4_5
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DOI: https://doi.org/10.1007/978-3-642-60218-4_5
Publisher Name: Springer, Berlin, Heidelberg
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