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Petrov-Galerkin formulations of the Timoshenko beam problem. (English) Zbl 0645.73030

The paper presents the Petrov-Galerkin formulation as an alternative to the Reissner-Mindlin formulation in order to obtain an approximation of Timoshenko’s beam solution by finite elements for straight bars under bending moments and shear stresses. The well known Petrov-Galerkin formulation represents in fact an optimization of Galerkin’s classical residual formulation - weighted residual method in FEM formulation - using a much optimal way of weighted functions. The authors have introduced the notion of “vector interpolation” whose elements are interpolation functions corresponding to the degrees of freedom - translations and rotations - for a straight bar under bending moments and shear stresses of finite elements constructed by means of optimal weighting functions.
As a result, the extension of the “energy best approximation criterion” is obtained for all available domains of finite elements which results in the optimal rate of convergence in the energy norm in Hilbert space. The Petrov-Galerkin formulation appears as a better performance compared with other solutions by finite elements because it eliminates the distorsion effect of the rate between the transversal section high of bar and its length that permits to obtain nodally exact values of the problem without taking count of beam load variations.
In order to extend the Petrov-Galerkin formulation for obtaining interpolation functions to bidimensional problems, the authors suggest discontinuous weighting functions. With these functions it is possible to obtain nodally exact solution for the case of constant forces corresponding to structures discretized by finite elements.
The paper is very interesting from the theoretical point of view, but we appreciate that the conclusions would be much conclusive if they should be sustained by comparative numerical examples.
Reviewer: V.Gioncu

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
49M15 Newton-type methods
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[1] Arnold, D. N., Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37, 405-421 (1981) · Zbl 0446.73066
[2] Barrett, J. W.; Morton, K. W., Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Meths. Appl. Mech. Engrg., 45, 97-122 (1984) · Zbl 0562.76086
[3] Bercovier, M., On \(C^0\) beam elements with shear and their corresponding penalty function formulation, Comput. Math. Appl., 8, 4, 245-256 (1982) · Zbl 0486.73070
[4] 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
[5] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Meths. Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[6] Demkowicz, L.; Oden, J. T., An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in one space variable, (TICOM Rept. 85-3 (1985), The University of Texas at Austin: The University of Texas at Austin Austin, TX) · Zbl 0601.65081
[7] Demkowicz, L.; Oden, J. T., An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables, Comput. Meths. Appl. Mech. Engrg., 55, 63-87 (1986) · Zbl 0602.76097
[8] Fried, I., Shear in \(C^0\) and \(C^1\) bending finite elements, Internat. J. Solids and Structures, 9, 449-460 (1973)
[9] Hemker, P. W., A numerical study of stiff two-point boundary problems, (Ph.D. Thesis (1977), Mathematisch Centrum: Mathematisch Centrum Amsterdam) · Zbl 0426.65043
[10] Hughes, T. J.R.; Cohen, M.; Haroun, M., Reduced and selective integration techniques in the finite element analysis of plates, Nucl. Engrg. Design, 46, 203-222 (1978)
[11] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advection-diffusion systems, Comput. Meths. Appl. Mech. Engrg., 58, 305-328 (1986) · Zbl 0622.76075
[12] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Meths. Appl. Mech. Engrg., 58, 329-336 (1986) · Zbl 0587.76120
[13] Hughes, T. J.R.; Taylor, R. L.; Kanoknukulchai, W., A simple and efficient finite element for plate bending, Internat. J. Numer. Meths. Engrg., 11, 1529-1543 (1977) · Zbl 0363.73067
[14] Kikuchi, F., On a finite element scheme based on the discrete Kirchhoff assumption, Numer. Math., 24, 211-231 (1975) · Zbl 0295.73067
[15] MacNeal, R. H., A simple quadrilateral shell element, Comput. & Structures, 8, 175-183 (1975) · Zbl 0369.73085
[16] Oden, J. T.; Kikuchi, N., Finite element methods for constrained problems in elasticity, Internat. J. Numer. Meths. Engrg., 18, 701-725 (1982) · Zbl 0486.73068
[17] Oden, J. T.; Kikuchi, N.; Song, Y. J., Penalty finite element methods for the analysis of Stokesian flows, Comput. Meths. Appl. Mech. Engrg., 31, 297-329 (1982) · Zbl 0478.76041
[18] Stolarski, H.; Belytschko, T., Shear and membrane locking in curved \(C^0\) elements, Comput. Meths. Appl. Mech. Engrg., 41, 279-296 (1983) · Zbl 0509.73072
[19] Tessler, A.; Dong, S. B., On a hierarchy of conforming Timoshenko beam elements, Comput. & Structures, 14, 3-4, 335-344 (1981)
[20] Wempner, G. A.; Oden, J. T.; Kross, D. A., Finite element analysis of thin shells, ASCE J. Engrg. Mech. Div., 6, 1273-1294 (1968)
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