Abstract
The vibration of a shell constrained to be in a specific configuration and the vibration of a shell with that shape at its natural reference position are known to be different by physicists. We consider a shell which is under a large displacement and a small deformation, that gives a constrained shell Ω in a static equilibrium. We consider a small vibration of Ω and we are interested in the equation of that vibration. We first investigate a new exact model for constrained shell that is p(d, ∞) which is the counterpart of the model developed by Delfour and Zolésio for shells in the reference configuration. Then we study the regularity of the solution in the interior domain as well as the the boundary regularity. Both regularities were proven to be interesting for shape differentiability (and thus shape control) in hyperbolic equations, we will recall the shape differentiability results for the wave equation.
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References
Jean-Christophe Aguilar and Jean-Paul Zolésio. Coque fluide intrinsèque dans approximation géométrique (intrinsic fluid shell without geometrical approximation). Comptes Rendus de l’Académie des Sciences, Paris, series I, Partial Differential Equations, 326(11):1341–1346, 1998.
Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and S. K. Mitter. Representation and control of infinite dimensional systems, volume 1. Birkhäuser, 1993.
John Cagnol and Jean-Paul Marmorat. Static equilibrium of hyperelastic thin shell: Symbolic and numerical computation. Mathematics and Computers in Simulation, 46(2):103–115, 1998.
John Cagnol and Jean-Paul Zolésio. Hidden shape derivative in the wave equation. In P. Kall, I. Lasiecka, and M. Polis, editors, System Modelling and Optimization, volume 396 of Chapman & Hall/CRC Research Notes in Mathematics, pages 42–52. IFIP, CRC Press LLC, Boca Raton, 1998. To appear.
John Cagnol and Jean-Paul Zolésio. Hidden shape derivative in the wave equation with dirichlet boundary condition. Comptes Rendus de l’Académie des Sciences, Paris, série I, 326(9): 1079–1084, 1998. Partial Differential Equations.
John Cagnol and Jean-Paul Zolésio. The tensor Ashell for a constrained shell: Explicit computation with a symbolic computation system. Technical report, CMA, Ecole des Mines de Paris, France, 1998.
Michel C. Delfour. Intrinsic p(2, 1) thin shell model and naghdi’s models without a priori assumption on the stress tensor. In those proceedings. Birkhäuser, 1998.
Michel C. Delfour and Jean-Paul Zolésio. Differential equations for linear shells: comparison between intrinsic and classical models. In Luc Vinet, editor, Advances in Mathematical Sciences, CRM’s 25 years, pages 42–124, 1997.
Michel C. Delfour and Jean-Paul Zolésio. Hidden boundary smoothness in hyperbolic tangential problems of nonsmooth domains. In P. Kail, I. Lasiecka, and M. Polis, editors, System Modelling and Optimization, volume 396 of Chapman & Hall/CRC Research Notes in Math., pages 53-61. IFIP, CRC Press LLC, 1998. To appear. [10] Paul Germain. Mécanique, volume I. Ellipses, Ecole Polytechnique, 1986.
Irena Lasiecka, Jacques-Louis Lions, and Roberto Triggiani. Non homogeneous boundary value problems for second order hyperbolic operators. Journal de Mathématiques pures et Appliquées, 65(2): 149–192, 1986.
Irena Lasiecka and Roberto Triggiani. Recent advances in regularity of second-order hyperbolic mixed problems, and applications. In C. Jones, editor, Dynamics reported. Expositions in dynamical systems, pages 104–162. Springer-Verlag, 1994.
Jan Sokolowski and Jean-Paul Zolésio. Introduction to Shape Optimization. SCM 16. Springer-Verlag, 1991.
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Cagnol, J., Zolésio, JP. (1999). Shape Control in Hyperbolic Problems. In: Hoffmann, KH., Leugering, G., Tröltzsch, F., Caesar, S. (eds) Optimal Control of Partial Differential Equations. ISNM International Series of Numerical Mathematics, vol 133. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8691-8_7
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DOI: https://doi.org/10.1007/978-3-0348-8691-8_7
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