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mixed boundary value problem; deformation theory of plasticity; shape optimization; cost functional; finite elements
Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke’s law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.
[r6] J. Haslinger, P. Neittaanmäki: Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons, Chichester, 1988. MR 0982710
[r1] I. Hlaváček, J. Nečas: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam-Oxford-New York, 1981. MR 0600655
[r7] J. Nečas, I.  Hlaváček: Solution of Signorini’s contact problem in the deformation theory of plasticity by secant modules method. Apl. Mat. 28 (1983), 199–214. MR 0701739
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[r5] J. Haslinger, R. Mäkinen: Shape optimization of elasto-plastic bodies under plane strains: sensitivity analysis and numerical implementation. Struct. Optim. 4 (1992), 133–141. DOI 10.1007/BF01742734
[r2] K. Washizu: Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford, 1968. MR 0391679 | Zbl 0164.26001
[r3] I. Hlaváček: Inequalities of Korn’s type, uniform with respect to a class of domains. Appl. Math. 34 (1989), 105–112. MR 0990298
[r4] D. Begis, R.  Glowinski: Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal: Méthodes de résolution des problèmes approchés. Appl. Math. Optim. 2 (1975), 130–169. DOI 10.1007/BF01447854 | MR 0443372
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