×

Relations between enhanced strain methods and the HR method. (English) Zbl 1054.74055

Summary: The three-field Hu-Washizu functional is the starting point to develop a comparative analysis of approximate linear elastostatic problems. In particular, we analyse the limitation phenomena concerning two mixed methods, respectively denoted the strain gap method and the mixed enhanced strain method, and the two-field Hellinger-Reissner (HR) finite element formulation. Accordingly, two original limitation principles are contributed. We provide the theoretical motivation of numerical observations, provided in computational literature, on the coincidence of displacement and stress solutions between the enhanced assumed strain method and the HR method if different choices of shape functions are made. It is further shown that the HR method provides exact stress solution and exact nodal displacements with one element. The related analytical motivation is given. The theoretical analyses presented in the paper are supported by numerical examples on typical benchmarks.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B10 Linear elasticity with initial stresses
74K20 Plates
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alfano, G.; Marotti de Sciarra, F., Mixed finite element formulations and related limitation principles: a general treatment, Comp. meth. appl. mech. engrg., 138, 105-130, (1996) · Zbl 0881.73121
[2] Andelfinger, U.; Ramm, E., EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, Int. J. numer. meth. engrg., 36, 1311-1337, (1993) · Zbl 0772.73071
[3] Andelfinger, U.; Ramm, E.; Roehl, D., 2D and 3D enhanced assumed strain elements and their application in plasticity, (), 1997-2007
[4] Arunakirinathar, K.; Reddy, B.D., Further results for enhanced strain methods with isoparametric elements, Comp. meth. appl. mech. engrg., 127, 127-143, (1995) · Zbl 0862.73056
[5] Bischoff, M.; Ramm, E.; Braess, D., A class of equivalent enhanced assumed strain and hybrid stress finite elements, Computat. mech., 22, 443-449, (1999) · Zbl 0958.74058
[6] Brezis, H., Analyse fonctionnelle, théorie et applications, (1983), Masson Editeur Paris · Zbl 0511.46001
[7] Cesar de Sa, J.; Natal Jorge, R., Enhanced displacement field on low order elements for incompressible problems, (), 565-570
[8] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[9] Fraeijs de Veubeke, B., Displacement and equilibrium models in the finite element method, (), 145-196 · Zbl 0359.76021
[10] Hiriart-Urruty, J.B.; Lemaréchal, C., Convex analysis and minimization algorithms, vol. 1, (1993), Springer New York · Zbl 0795.49001
[11] Hughes, T.J.R., The finite element method—linear static and dynamic finite element analysis, (2000), Dover New York
[12] E.P. Kasper, R.L. Taylor, A mixed-enhanced strain method problems, University of California at Berkeley, Report UCB/SEMM-97/02, 1997
[13] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Clifs, NJ · Zbl 0545.73031
[14] Perego, U., Variationally consistent stress recovery for enhanced strain finite elements, Commun. numer. math. engrg., 16, 151-163, (2000) · Zbl 0969.74066
[15] Pian, T.H.H.; Sumihara, K., Rational approach for assumed stress finite elements, Int. J. num. meth. engrg., 20, 1685-1695, (1985) · Zbl 0544.73095
[16] Reddy, B.D.; Simo, J.C., Stability and convergence of a class of enhanced strain methods, SIAM J. numer. anal., 32, 1705-1728, (1995) · Zbl 0855.73073
[17] Romano, G.; Rosati, L.; Marotti de Sciarra, F., A variational theory for finite-step elastoplastic problems, Int. J. solids struct., 30, 2317-2334, (1993) · Zbl 0781.73081
[18] G. Romano, F. Marotti de Sciarra, M. Diaco, Well-posedness of three field methods with enhanced strains, in: Proceedings of ECCM’ 99, Munich, Germany, 1999 · Zbl 1049.74052
[19] Romano, G.; Rosati, L.; Diaco, M., Well-posedness of mixed formulations in elasticity, Zamm, 79, 435-454, (1999) · Zbl 0956.74005
[20] G. Romano, Theory of structural models, Part I, Elements of functional analysis, Doctoral Lectures (in Italian), vol. II, University of Napoli Federico, 2001
[21] Romano, G.; Marotti de Sciarra, F.; Diaco, M., Well-posedness and numerical performances of the strain gap method, Int. J. numer. meth. engrg., 51, 103-126, (2001) · Zbl 1049.74052
[22] Simo, J.C.; Rifai, M.S., A class of mixed assumed strain methods and the method of incompatible modes, Int. J. numer. meth. engrg., 29, 1595-1638, (1990) · Zbl 0724.73222
[23] Taylor, R.L.; Beresford, P.J.; Wilson, E.L., A non-conforming element for stress analysis, Int. J. numer. meth. engrg., 22, 39-62, (1986)
[24] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Press New York · Zbl 0164.26001
[25] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, vol. 1, (1989), McGraw-Hill London · Zbl 0991.74002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.