An equilibrium finite element method in three-dimensional elasticity

Michal Křížek

Aplikace matematiky (1982)

  • Volume: 27, Issue: 1, page 46-75
  • ISSN: 0862-7940

Abstract

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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 -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.

How to cite

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Křížek, Michal. "An equilibrium finite element method in three-dimensional elasticity." Aplikace matematiky 27.1 (1982): 46-75. <http://eudml.org/doc/15223>.

@article{Křížek1982,
abstract = {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.},
author = {Křížek, Michal},
journal = {Aplikace matematiky},
keywords = {composite tetrahedral equilibrium element; two types of finite approximation; three-dimensional problem; polyhedral domain; Castigliano-Menabrea’s principle; minimum complementary energy; a priori error estimates; existence of strongly regular family of decompositions; composite tetrahedral equilibrium element; two types of finite approximation; three-dimensional problem; polyhedral domain; Castigliano- Menabrea's principle; minimum complementary energy; a priori error estimates; existence of strongly regular family of decompositions},
language = {eng},
number = {1},
pages = {46-75},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An equilibrium finite element method in three-dimensional elasticity},
url = {http://eudml.org/doc/15223},
volume = {27},
year = {1982},
}

TY - JOUR
AU - Křížek, Michal
TI - An equilibrium finite element method in three-dimensional elasticity
JO - Aplikace matematiky
PY - 1982
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 27
IS - 1
SP - 46
EP - 75
AB - 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.
LA - eng
KW - composite tetrahedral equilibrium element; two types of finite approximation; three-dimensional problem; polyhedral domain; Castigliano-Menabrea’s principle; minimum complementary energy; a priori error estimates; existence of strongly regular family of decompositions; composite tetrahedral equilibrium element; two types of finite approximation; three-dimensional problem; polyhedral domain; Castigliano- Menabrea's principle; minimum complementary energy; a priori error estimates; existence of strongly regular family of decompositions
UR - http://eudml.org/doc/15223
ER -

References

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  2. J. H. Bramble M. Zlámal, Triangular elements in the finite element method, Math. Соmр. 24 (1970), 809-820. (1970) MR0282540
  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. (1972) MR0336957
  4. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland publishing company, Amsterdam, New York, Oxford, 1978. (1978) Zbl0383.65058MR0520174
  5. G. Duvaut J. L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin, Heidelberg, New York, 1976. (1976) MR0521262
  6. B. J. Hartz V. B. Watwood, 10.1016/0020-7683(68)90083-8, Internat. J. Solids and Struct. 4 (1968), 857-873. (1968) DOI10.1016/0020-7683(68)90083-8
  7. I. Hlaváček J. Nečas, Mathematical theory of elastic and elasto-plastic bodies, SNTL, Praha, Elsevier, Amsterdam, 1980. (1980) 
  8. I. Hlaváček, Convergence of an equilibrium finite element model for plane elastostatics, Apl. Mat. 24 (1979), 427-457. (1979) MR0547046
  9. C. Johnson B. Mercier, 10.1007/BF01403910, Numer. Math. 30 (1978), 103-116. (1978) MR0483904DOI10.1007/BF01403910
  10. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Praha, 1967. (1967) MR0227584
  11. Энциклопедия элементарной математики - Геометрия книга 4, 5, Наука, Москва, 1966. (1966) Zbl0156.18206

Citations in EuDML Documents

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  1. Alexander Ženíšek, Green's theorem from the viewpoint of applications
  2. Alexander Ženíšek, Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications
  3. Michal Křížek, Colouring polytopic partitions in d
  4. Zdeněk Kestřánek, Variational inequalities in plasticity with strain-hardening - equilibrium finite element approach
  5. Ivan Hlaváček, Michal Křížek, Vladislav Pištora, How to recover the gradient of linear elements on nonuniform triangulations
  6. Radim Hošek, Strongly regular family of boundary-fitted tetrahedral meshes of bounded C 2 domains
  7. Ivan Hlaváček, A finite element solution for plasticity with strain-hardening
  8. Ivan Hlaváček, A finite element analysis for elastoplastic bodies obeying Hencky's law
  9. Michal Křížek, Conforming equilibrium finite element methods for some elliptic plane problems
  10. Miroslav Vondrák, Slab analogy in theory and practice of conforming equilibrium stress models for finite element analysis of plane elastostatics

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