# Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

Isabel N. Figueiredo; Carlos F. Leal

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 7, page 135-155
- ISSN: 1292-8119

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topFigueiredo, Isabel N., and Leal, Carlos F.. "Sensitivity Analysis of a Nonlinear Obstacle Plate Problem." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 135-155. <http://eudml.org/doc/90616>.

@article{Figueiredo2010,

abstract = {
We analyse the sensitivity of the solution of a nonlinear obstacle plate
problem, with respect to small perturbations of the middle plane
of the plate. This analysis, which generalizes the results of [9,10]
for the linear case,
is done by application of an abstract variational
result [6], where the sensitivity of parameterized variational
inequalities in Banach spaces, without uniqueness of solution,
is quantified in terms of a generalized
derivative, that is the proto-derivative. We prove that the hypotheses
required by this abstract sensitivity result are verified for
the nonlinear obstacle plate problem. Namely, the constraint set defined
by the obstacle is polyhedric and the mapping involved in the definition
of the plate problem, considered as a function of the middle plane
of the plate, is semi-differentiable. The verification of these two conditions
enable to conclude that the sensitivity is
characterized by
the proto-derivative of the solution mapping associated
with the nonlinear obstacle plate problem, in terms of the
solution of a variational inequality.
},

author = {Figueiredo, Isabel N., Leal, Carlos F.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Plate problem; variational inequality; sensitivity analysis.; plate problem; sensitivity analysis; proto-derivative},

language = {eng},

month = {3},

pages = {135-155},

publisher = {EDP Sciences},

title = {Sensitivity Analysis of a Nonlinear Obstacle Plate Problem},

url = {http://eudml.org/doc/90616},

volume = {7},

year = {2010},

}

TY - JOUR

AU - Figueiredo, Isabel N.

AU - Leal, Carlos F.

TI - Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 7

SP - 135

EP - 155

AB -
We analyse the sensitivity of the solution of a nonlinear obstacle plate
problem, with respect to small perturbations of the middle plane
of the plate. This analysis, which generalizes the results of [9,10]
for the linear case,
is done by application of an abstract variational
result [6], where the sensitivity of parameterized variational
inequalities in Banach spaces, without uniqueness of solution,
is quantified in terms of a generalized
derivative, that is the proto-derivative. We prove that the hypotheses
required by this abstract sensitivity result are verified for
the nonlinear obstacle plate problem. Namely, the constraint set defined
by the obstacle is polyhedric and the mapping involved in the definition
of the plate problem, considered as a function of the middle plane
of the plate, is semi-differentiable. The verification of these two conditions
enable to conclude that the sensitivity is
characterized by
the proto-derivative of the solution mapping associated
with the nonlinear obstacle plate problem, in terms of the
solution of a variational inequality.

LA - eng

KW - Plate problem; variational inequality; sensitivity analysis.; plate problem; sensitivity analysis; proto-derivative

UR - http://eudml.org/doc/90616

ER -

## References

top- H. Brézis, Equations et inéquations nonlinéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble18 (1968) 115-175.
- J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape design, theory and applications. Wiley, Chichester (1988).
- J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite element method for hemivariational inequalities. Theory, methods and applications. Kluwer Academic Publishers (1999).
- A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan29 (1977) 615-631.
- N. Kikuchi and J.T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM (1988).
- A.B. Levy, Sensitivity of solutions to variational inequalities on Banach Spaces. SIAM J. Control Optim.38 (1999) 50-60.
- A.B. Levy and R.T. Rockafeller, Sensitivity analysis of solutions to generalized equations. Trans. Amer. Math. Soc.345 (1994) 661-671.
- F. Mignot, Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal.22 (1976) 130-185.
- M. Rao and J. Sokolowski, Sensitivity analysis of Kirchhoff plate with obstacle, Rapports de Recherche, 771. INRIA-France (1987).
- M. Rao and J. Sokolowski, Sensitivity analysis of unilateral problems in ${H}_{0}^{2}\left(\Omega \right)$ and applications. Numer. Funct. Anal. Optim.14 (1993) 125-143.
- R.T. Rockafeller, Proto-differentiability of set-valued mappings and its applications in Optimization. Ann. Inst. H. Poincaré Anal. Non Linéaire6 (1989) 449-482.
- A. Shapiro, On concepts of directional differentiability. J. Optim. Theory Appl.66 (1990) 477-487.
- J. Sokolowski and J.-P. Zolesio, Shape sensitivity analysis of unilateral problems. SIAM J. Math. Anal.18 (1987) 1416-1437.
- J. Sokolowski and J.-P. Zolesio, Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints. J. Optim. Theory Appl.54 (1987) 361-382.
- J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization - shape sensitivity analysis. Springer-Verlag, Springer Ser. Comput. Math. 16 (1992).
- P.W. Ziemer, Weakly differentiable functions. Springer-Verlag, New York (1989).

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