# Article

 Title: An equilibrium finite element method in three-dimensional elasticity (English) Author: Křížek, Michal Language: English Journal: Aplikace matematiky ISSN: 0373-6725 Volume: 27 Issue: 1 Year: 1982 Pages: 46-75 Summary lang: English Summary lang: Czech Summary lang: Russian . Category: math . Summary: The tetrahedral stress element is introduced and two different types of a finite piecewise linear approximation of the dual elasticity problem are investigated on a polyhedral domain. Fot both types a priori error estimates $O(h^2)$ in $L_2$-norm and $O(h^{1/2})$ in $L_\infty$-norm are established, provided the solution is smooth enough. These estimates are based on the fact that for any polyhedron there exists a strongly regular family of decomprositions into tetrahedra, which is proved in the paper, too. (English) Keyword: composite tetrahedral equilibrium element Keyword: two types of finite approximation Keyword: three-dimensional problem Keyword: polyhedral domain Keyword: Castigliano-Menabrea’s principle Keyword: minimum complementary energy Keyword: a priori error estimates Keyword: existence of strongly regular family of decompositions MSC: 65N15 MSC: 65N30 MSC: 73K25 MSC: 74B99 MSC: 74H99 MSC: 74P99 MSC: 74S05 idZBL: Zbl 0488.73072 idMR: MR0640139 DOI: 10.21136/AM.1982.103944 . Date available: 2008-05-20T18:18:22Z Last updated: 2020-07-28 Stable URL: http://hdl.handle.net/10338.dmlcz/103944 . Reference: [1] Л. Д. Александров: Выпуклые многогранники.H. - Л., Гостехиздат, Москва, 1950. Zbl 1157.76305 Reference: [2] J. H. Bramble M. Zlámal: Triangular elements in the finite element method.Math. Соmр. 24 (1970), 809-820. MR 0282540 Reference: [3] P. G. Ciarlet P. A. Raviart: General Lagrange and Hermite interpolation in $R^n$ with applications to finite element methods.Arch. Rational Mech. Anal. 46 (1972), 177-199. MR 0336957, 10.1007/BF00252458 Reference: [4] P. G. Ciarlet: The finite element method for elliptic problems.North-Holland publishing company, Amsterdam, New York, Oxford, 1978. Zbl 0383.65058, MR 0520174 Reference: [5] G. Duvaut J. L. Lions: Inequalities in mechanics and physics.Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 0521262 Reference: [6] B. J. Hartz V. B. Watwood: An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems.Internat. J. Solids and Struct. 4 (1968), 857-873. 10.1016/0020-7683(68)90083-8 Reference: [7] I. Hlaváček J. Nečas: Mathematical theory of elastic and elasto-plastic bodies.SNTL, Praha, Elsevier, Amsterdam, 1980. Reference: [8] I. Hlaváček: Convergence of an equilibrium finite element model for plane elastostatics.Apl. Mat. 24 (1979), 427-457. MR 0547046 Reference: [9] C. Johnson B. Mercier: Some equilibrium finite element methods for two-dimensional elasticity problems.Numer. Math. 30 (1978), 103-116. MR 0483904, 10.1007/BF01403910 Reference: [10] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Academia, Praha, 1967. MR 0227584 Reference: [11] : Энциклопедия элементарной математики - Геометрия книга 4, 5.Наука, Москва, 1966. Zbl 0156.18206 .

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