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On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. (English) Zbl 0589.73068
The mathematics of finite elements and applications V, MAFELAP 1984, Proc. 5th Conf., Uxbridge/Engl. 1984, 491-503 (1985).
[For the entire collection see Zbl 0566.00028.]
The paper aims at analysing, from a mathematical point of view, the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. A sequence of problems (for different values of thickness parameter t) is constructed and the behaviour of the corresponding solutions is studied as thickness becomes ”too small”, to test for the stability of discretization of the four-node plate element. It is proved that, at least for particular cases (like a uniform rectangular mesh considered by the authors), the element is uniformly stable with respect to the thickness parameter and convergent with optimal rate. The paper does not establish the stability for shear strains nor the uniform convergence. The authors suggest the application of filtering procedure for shear strains to have an \(L^ 2\)- stability and optimal rate of convergence uniformly in t.
The proofs are based on sound mathematical analysis and involve the use of several theorems and lemmas. The paper is a good contribution to the mathematical theory of finite elements.
Reviewer: V.K.Arya

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates