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Title: Shape optimization of an elasto-perfectly plastic body (English)
Author: Hlaváček, Ivan
Language: English
Journal: Aplikace matematiky
ISSN: 0373-6725
Volume: 32
Issue: 5
Year: 1987
Pages: 381-400
Summary lang: English
Summary lang: Russian
Summary lang: Czech
Category: math
Summary: Within the range of Prandtl-Reuss model of elasto-plasticity the following optimal design problem is solved. Given body forces and surface tractions, a part of the boundary, where the (two-dimensional) body is fixed, is to be found, so as to minimize an integral of the squared yield function. The state problem is formulated in terms of stresses by means of a time-dependent variational inequality. For approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular finite elements for stress and backward differences in time are used. Convergence of the approximations to a solution of the optimal design problem is proven. As a consequance, the existence of an optimal boudary is verified. (English)
Keyword: optimal design
Keyword: model of Prandtl-Reuss
Keyword: variational inequality of evolution
Keyword: piecewise linear approximation of the unknown boundary
Keyword: piecewise constant triangular elements for stress
Keyword: backward differences in time
Keyword: convergence
Keyword: elasto-plasticity
Keyword: finite elements
MSC: 65K10
MSC: 65N30
MSC: 73E99
MSC: 73k40
MSC: 74P99
MSC: 74S05
MSC: 74S30
idZBL: Zbl 0632.73082
idMR: MR0909545
DOI: 10.21136/AM.1987.104269
Date available: 2008-05-20T18:33:07Z
Last updated: 2020-07-28
Stable URL:
Reference: [1] I. Hlaváček: Shape optimization in two-dimensional elasticity by the dual finite element method.Math. Model. and Numerical Anal. 21 (1987), 63-92, MR 0882687, 10.1051/m2an/1987210100631
Reference: [2] I. Hlaváček: Shape optimization of elasto-plastic bodies obeying Hencky's law.Appl. Mat. 31 (1986), 486-499. Zbl 0616.73081, MR 0870484
Reference: [3] G. Duvaut J. L. Lions: Les inéquations en mécanique et en physique.Paris, Dunod 1972. MR 0464857
Reference: [4] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: An introduction.Elsevier, Amsterdam 1981. (Czech version - SNTL, Praha 1983.) MR 0600655
Reference: [5] C. Johnson: Existence theorems for plasticity problems.J. Math, pures et appl. 55 (1976), 431-444. Zbl 0351.73049, MR 0438867
Reference: [6] C. Johnson: On finite element methods for plasticity problems.Numer. Math. 26 (1976), 79-84. Zbl 0355.73035, MR 0436626, 10.1007/BF01396567
Reference: [7] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Academia, Praha 1967. MR 0227584
Reference: [8] I. Hlaváček: A finite element solution for plasticity with strain-hardening.R.A.I.R.O. Analyse numér. 14 (1980), 347-368. MR 0596540
Reference: [9] D. Begis R. Glowinski: Application de la méthode des éléménts finis à l'approximation d'un problème de domaine optimal.Appl. Math. Optimiz., 2 (1975), 130-169. MR 0443372, 10.1007/BF01447854


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