×
Pištora, Vladislav

Shape optimization of an elasto-plastic body for the model with strain- hardening. (English) Zbl 0717.73054

Apl. Mat. 35, No. 5, 373-404 (1990).
Summary: The state problem of elasto-plasticity (for the model with strain- hardening) is formulated in terms of stresses and hardening parameters by means of a time-dependent variational inequality. The optimal design problem is to find the shape of a part of the boundary such that a given cost functional is minimized. For the approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular elements for the stress and the hardening parameter, and backward differences in time are used. Existence and uniqueness of a solution of the approximate state problem and existence of a solution of the approximate optimal design problem are proved. The main result is the proof of convergence of the approximations to a solution of the original optimal design problem.

Cited in 3 Documents

MSC:

74P99 Optimization problems in solid mechanics
65K10 Numerical optimization and variational techniques
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
49J40 Variational inequalities

Keywords:

domain optimization; existence; uniqueness; state problem; time-dependent variational inequality; optimal design; piecewise linear approximations of the unknown boundary; piecewise constant triangular elements for the stress; hardening parameter; backward differences in time; convergence
PDF BibTeX XML Cite
\textit{V. Pištora}, Apl. Mat. 35, No. 5, 373--404 (1990; Zbl 0717.73054)
Full Text: EuDML

References:

[1] D. Begis R. Glowinski: Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Appl. Math. Optim. 2 (1975), 130-169. · Zbl 0323.90063
[2] J. Céa: Optimization: Théorie et algorithmes. Dunod, Paris, 1971; · Zbl 0211.17402
[3] P. G. Ciarlet: The finite element method for elliptic problems. North Holland Publ. Comp., Amsterdam, 1978; · Zbl 0383.65058
[4] I. Hlaváček: A finite element solution for plasticity with strain-hardening. RAIRO Annal. Numér. 14 (1980), 347-368.
[5] I. Hlaváček: Optimization of the domain in elliptic problems by the dual finite element method. Apl. Mat. 30 (1985), 50-72. · Zbl 0575.65103
[6] I. Hlaváček: Shape optimization of an elastic-perfectly plastic body. Apl. Mat. 32 (1987), 381-400. · Zbl 0632.73082
[7] C. Johnson: Existence theorems for plasticity problems. J. Math. Pures Appl. 55 (1976), 431-444. · Zbl 0351.73049
[8] C. Johnson: A mixed finite element method for plasticity with hardening. SIAM J. Numer. Anal. 14 (1977), 575-583. · Zbl 0374.73039
[9] C. Johnson: On plasticity with hardening. J. Math. Anal. Appl. 62 (1978), 325-336. · Zbl 0373.73049
[10] A. Kufner O. John S. Fučík: Function spaces. Academia, Praha, 1977.
[11] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Praha, 1967. · Zbl 1225.35003
[12] I. Hlaváček: Shape optimization of elasto-plastic bodies obeying Hencky’s law. Apl. Mat. 31 (1986), 486-499. · Zbl 0616.73081
[13] I. Hlaváček J. Haslinger J. Nečas J. Lovíšek: Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York, 1988. · Zbl 0654.73019
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.