# zbMATH — the first resource for mathematics

Unlocking with residual-free bubbles. (English) Zbl 0890.73064
Summary: Residual-free bubbles are derived for the Timoshenko beam problem. After eliminating these bubbles, the resulting formulation is form-identical in using the following tricks to the standard variational formulation: (i) one-point reduced integration of the shear energy term; (ii) replace its coefficient $$1/ \varepsilon^2$$ by $$1/ (\varepsilon^2 + (h^2_K /12))$$ in each element; (iii) modify consistently the right-hand side. This final formulation is ‘legally’ obtained in that the Galerkin method enriched with residual-free bubbles is developed using full integration throughout. Furthermore, this method is nodally exact by construction.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text:
##### References:
 [1] Arnold, D.N., Discretization by finite elements of a model parameter dependent problem, Numer. math., 37, 405-421, (1981) · Zbl 0446.73066 [2] Brezzi, F.; Russo, A., Choosing bubbles for advection-diffusion problems, Math. models meths. appl. sci., 4, 571-587, (1994) · Zbl 0819.65128 [3] Franca, L.P.; Russo, A., Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles, Appl. math. lett., 9, 83-88, (1996) · Zbl 0903.65082 [4] Franca, L.P.; Russo, A., Mass lumping emanating from residual-free bubbles, Comput. methods appl. mech. eng., 142, 353-360, (1997) · Zbl 0883.65086 [5] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ [6] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P., Petrov-Galerkin formulations of the Timoshenko beam problem, Comput. methods appl. mech. eng., (1987) · Zbl 0645.73030 [7] MacNeal, R.H., Finite elements: their design and performance, (1994), Marcel Dekker Inc New York · Zbl 0053.26304 [8] Malkus, D.S.; Hughes, T.J.R., Mixed finite element methodsâ€”reduced and selective integration techniques: a unification of concepts, Comput. meth. appl. mech. engrg., 15, 63-81, (1978) · Zbl 0381.73075 [9] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (1989), McGraw-Hill London, New York · 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.