Sensitivity analysis of a nonlinear obstacle plate problem

Figueiredo, Isabel N.; Leal, Carlos F.

doi:10.1051/cocv:2002006

Figueiredo, Isabel N. ; Leal, Carlos F.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 135-155.

Résumé

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.

MR Zbl

DOI : 10.1051/cocv:2002006

Classification : 49A29, 90C31, 74B20, 74K20
Mots clés : plate problem, variational inequality, sensitivity analysis

@article{COCV_2002__7__135_0,
     author = {Figueiredo, Isabel N. and Leal, Carlos F.},
     title = {Sensitivity analysis of a nonlinear obstacle plate problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {135--155},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002006},
     mrnumber = {1925024},
     zbl = {1042.49038},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002006/}
}

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
PY  - 2002
SP  - 135
EP  - 155
VL  - 7
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002006/
DO  - 10.1051/cocv:2002006
LA  - en
ID  - COCV_2002__7__135_0
ER  -

%0 Journal Article
%A Figueiredo, Isabel N.
%A Leal, Carlos F.
%T Sensitivity analysis of a nonlinear obstacle plate problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 135-155
%V 7
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2002006/
%R 10.1051/cocv:2002006
%G en
%F COCV_2002__7__135_0

Figueiredo, Isabel N.; Leal, Carlos F. Sensitivity analysis of a nonlinear obstacle plate problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 135-155. doi : 10.1051/cocv:2002006. http://www.numdam.org/articles/10.1051/cocv:2002006/

Bibliographie
Cité par

[1] H. Brézis, Equations et inéquations nonlinéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble 18 (1968) 115-175. | EuDML | Numdam | MR | Zbl

[2] J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape design, theory and applications. Wiley, Chichester (1988). | MR | Zbl

[3] J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite element method for hemivariational inequalities. Theory, methods and applications. Kluwer Academic Publishers (1999). | MR | Zbl

[4] A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29 (1977) 615-631. | MR | Zbl

[5] N. Kikuchi and J.T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM (1988). | MR | Zbl

[6] A.B. Levy, Sensitivity of solutions to variational inequalities on Banach Spaces. SIAM J. Control Optim. 38 (1999) 50-60. | MR | Zbl

[7] A.B. Levy and R.T. Rockafeller, Sensitivity analysis of solutions to generalized equations. Trans. Amer. Math. Soc. 345 (1994) 661-671. | MR | Zbl

[8] F. Mignot, Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | MR | Zbl

[9] M. Rao and J. Sokolowski, Sensitivity analysis of Kirchhoff plate with obstacle, Rapports de Recherche, 771. INRIA-France (1987).

[10] M. Rao and J. Sokolowski, Sensitivity analysis of unilateral problems in $H_{0}^{2} (Ω)$ and applications. Numer. Funct. Anal. Optim. 14 (1993) 125-143. | MR | Zbl

[11] R.T. Rockafeller, Proto-differentiability of set-valued mappings and its applications in Optimization. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449-482. | Numdam | MR | Zbl

[12] A. Shapiro, On concepts of directional differentiability. J. Optim. Theory Appl. 66 (1990) 477-487. | MR | Zbl

[13] J. Sokolowski and J.-P. Zolesio, Shape sensitivity analysis of unilateral problems. SIAM J. Math. Anal. 18 (1987) 1416-1437. | MR | Zbl

[14] 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. | MR | Zbl

[15] J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization - shape sensitivity analysis. Springer-Verlag, Springer Ser. Comput. Math. 16 (1992). | Zbl

[16] P.W. Ziemer, Weakly differentiable functions. Springer-Verlag, New York (1989). | MR | Zbl

Cité par Sources :