Summary
The object of this paper is the development of a penalization technique to compute the shape derivative of cost functionals where the state is the solution of a non-linear equation and/or a linear variational inequality. This type of problem is frequently encountered in Shape Sensitivity Analysis.
Résumé
Cet article présente le calcul des dérivées de forme de fonctionnelles définies sur un domaine géométrique par une méthode de pénalisation. On suppose que l'état est la solution d'une équation non-linéaire ou d'une inéquation linéaire. Ce type de problème est fréquemment rencontré en analyse de sensitivité des formes.
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References
J. Céa:Conception optimale ou identification des formes. Calcul rapide de la dérivée directionnelle de la fonction coût, R.A.I.R.O.,20 (1986), pp. 311–402.
G. Da Prato:Some Results on Bellman Equation in Hilbert Spaces, Scuola Normale Superiore, Pisa, 1983.
M. C.Delfour - J.-P.Zolésio:Further developments in Shape Sensitivity Analysis via a penalization method, in »Boundary Control and Boundary Variations », J. P.Zolésio ed., pp. 153–191, Springer-Verlag, Lecture Notes in Control and Information Sciences, no. 100 (1987).
M. C. Delfour -J.-P. Zolésio:Dérivation d'un MinMax et application à la dérivation par rapport au contrôle d'une observation non différentiable de l'état, C.R. Acad. Sc. Paris,302 (1986), pp. 571–574.
M. C. Delfour -J.-P. Zolésio:Differentiability of a MinMax and Application to Optimal Control and Design Problems, Parts I and II, in «Control Problems for Systems described as Partial Differential Equations and Applications»,I. Lasiecka andR. Triggiani eds., pp. 204–219 and pp. 220–229, Springer-Verlag, New York (1987).
M. C. Delfour -J.-P. Zolésio:Shape sensitivity analysis via MinMax differentiability, SIAM J. on Control and Optim.,26, no. 4, (1988), pp. 834–852.
M. C. Delfour -G. Payre -J.-P. Zolésio:Approximation of non-linear problems associated with radiating bodies in space, SIAM J. on Numerical Analysis,24 (1987), pp. 1077–1094.
M. C. Delfour -G. Payre -J.-P. Zolésio:Shape optimal design of a radiating fin, in «Systems Modelling and Optimization»,P. Thoft-Christensen ed., pp. 810–818, Springer-Verlag, Berlin, New-York, 1984.
J. Ekeland -R. Teman:Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, Paris, Bruxelles, Montréal, 1974.
M. Fortin -R. Glowinski:Augmented Lagrangian Methods, in «Applications to the Numerical Solution of Boundary-Value Problems», North-Holland, Amsterdam, New-York, Oxford, 1983.
M. H. Protter -H. F. Weinberger:Maximum principle in differential equations, Englewood Cleffs, N.J., Prentice-Hall, 1967.
Shi Shuz Hong:Optimal control of strongly monotone variational inequalities, Report CRM-1346, Centre de recherches mathématiques, Université de Montréal, Montréal, Canada, February, 1986.
J.Sokolowski:Conical differentiability of projection on convex sets —an application to sensitivity analysis of Signorini variational inequality, Technical Report, Institute of Mathematics of the University of Genova, 1981.
J. Sokolowski:Sensitivity analysis of contact problems with adhesive friction, Technical Report, University of Florida, Gainesville, Florida, 1986.
J. Sokolowski -J.-P. Zolésio:Dérivée par rapport au domaine de la solution d'un problème unilatéral, C.R. Acad. Sci. Paris, Sér. I,301 (1985), pp. 103–106.
J. Sokolowski -J.-P. Zolésio:Shape sensitivity analysis of unilateral problems, SIAM J. on Math. Anal.,18 (1987), pp. 1416–1437.
J. P. Zolésio:Identification de domaine, Thèse de doctorat d'état, Nice, France, 1979.
J. P. Zolésio:The material derivative (or speed method) for shape optimization, in « Optimization of Distributed Parameter Structures », Vol. II,E. J. Haug andJ. Céa eds., pp. 1098–1151, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1981.
J. P.Zolésio:Semiderivatives of repeated eigenvalue, in the same proceedings as [2], pp. 1457–1473.
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This research was supported in part by the National Sciences and Engineering Council of Canada Operating Grant A-8730 and a FCAR Grant from the « Ministère de l'Education du Québec ».
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Delfour, M.C., Zolésio, J.P. Shape sensitivity analysis via a penalization method. Annali di Matematica pura ed applicata 151, 179–212 (1988). https://doi.org/10.1007/BF01762794
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DOI: https://doi.org/10.1007/BF01762794