Reddy, B. D.; Volpi, M. B. Mixed finite element methods for the circular arch problem. (English) Zbl 0763.73056 Comput. Methods Appl. Mech. Eng. 97, No. 1, 125-145 (1992). Summary: The boundary-value problem for linear elastic circular arches is studied. The governing equations are based on the Timoshenko-Mindlin-Reissner assumptions. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed problem are shown to be stable and convergent, and selective reduced integration applied to the standard discrete problem renders it equivalent to the mixed problem. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem. For the standard problem with full integration, convergence is suboptimal or nonexistent for small values of the thickness parameter, while for the mixed or reduced integration problem, the numerical rates of convergence coincide with those predicted by the theory. Cited in 13 Documents MSC: 74S05 Finite element methods applied to problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) Keywords:boundary-value problem; linear elastic circular arches; Timoshenko- Mindlin-Reissner assumptions; existence; uniqueness; selective reduced integration; convergence PDFBibTeX XMLCite \textit{B. D. Reddy} and \textit{M. B. Volpi}, Comput. Methods Appl. Mech. Eng. 97, No. 1, 125--145 (1992; Zbl 0763.73056) Full Text: DOI References: [1] Ashwell, D. G.; Sabir, A. B., Limitations of certain curved finite elements when applied to arches, Internat. J. Mech. Sci., 13, 133-139 (1971) [2] Arnold, D. N., Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37, 405-421 (1981) · Zbl 0446.73066 [3] Stolarski, H.; Belytchko, T., Shear and membrane locking in curved ((C^0\) elements, Comput. Methods Appl. Mech. Engrg., 41, 279-296 (1983) · Zbl 0509.73072 [4] Zienkiewicz, O. C.; Taylor, R. L.; Too, J. M., Reduced integration techniques in general analysis of plates and shells, Internat. J. Numer. Methods Engrg., 5, 275-290 (1971) · Zbl 0253.73048 [5] Kikuchi, F., Accuracy of some finite element models for arch problems, Comput. Methods Appl. Mech. Engrg., 35, 315-345 (1982) · Zbl 0499.73068 [6] Loula, A. F.D.; Franca, L. P.; Hughes, T. J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. Methods Appl. Mech. Engrg., 63, 281-303 (1987) · Zbl 0607.73077 [7] Loula, A. F.D.; Hughes, T. J.R.; Franca, L. P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. Methods Appl. Mech. Engrg., 63, 133-154 (1987) · Zbl 0607.73076 [8] Reddy, B. D., Convergence of finite element approximations for the shallow arch problem, Numer. Math., 53, 687-699 (1988) · Zbl 0628.73084 [9] Kikuchi, F., An abstract analysis of parameter dependent problems and its applications to mixed finite element methods, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 32, 499-538 (1985) · Zbl 0596.65091 [10] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal. Numér., 8, 129-151 (1974) · Zbl 0338.90047 [11] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 [12] Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics (1976), Springer: Springer Berlin · Zbl 0331.35002 [13] Reddy, B. D., Functional Analysis and Boundary-Value Problems: An Introductory Treatment (1986), Longman: Longman London · Zbl 0653.46003 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.