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Estimation of elastic parameters in a nonlinear elliptic model of a plate. (English) Zbl 0765.73042

The author considers the estimation of an elastic parameter in a nonlinear von Kármán model for large deformations of a thin plate with uniform cross section and with variable Young modulus. A model error function is introduced and conditions under which a solution and its differentiability may be analyzed locally are given. These are used to obtain conditions implying convergence of the augmented Lagrangian method. When the conditions are not satisfied, an analysis of the penalty method applied to the problem is provided. Numerical experiments are reported.

MSC:

74K20 Plates
74B20 Nonlinear elasticity
74P99 Optimization problems in solid mechanics
49Q10 Optimization of shapes other than minimal surfaces
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] Adams, R. A., Sobolev Spaces (1975), Academic: Academic New York · Zbl 0186.19101
[2] Agmon, S., Lectures on Elliptic Boundary Value Problems (1965), Van Nostrand: Van Nostrand Princeton · Zbl 0151.20203
[3] Alt, W., Lipshitzian perturbations of infinite optimization problems, (Fiacco, A. V., Mathematical Programming with Data Perturbations II (1983), Marcell Dekker: Marcell Dekker New York), 7-21
[4] Berger, M., Nonlinearity and Functional Analysis (1977), Academic: Academic New York · JFM 20.0557.02
[5] Ciarlet, P., The Finite Element Method for Elliptic Problems (1978), North Holland: North Holland New York
[6] Ciarlet, P.; Rabier, P., Les Equations de von Karman (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0433.73019
[7] Colonius, F.; Kunisch, K., Output Least Squares Stability in Elliptic Systems, Report 86-76 (1986), Technische Univ. Graz · Zbl 0656.93024
[8] Dunford, N.; Schwartz, J. T., Linear Operators. Part I: General Theory (1966), Wiley · Zbl 0146.12601
[9] Grisward, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman: Pitman Boston
[10] Kravaris, C.; Seinfeld, J. H., Identification of parameters in distributed parameter systems by regularization, SIAM J. Control Optim., 23, 217-241 (1985) · Zbl 0563.93018
[11] K. Kunisch and L.W. White, Regularity and complementarity estimated parameters in parabolic problems, Inverse Problems; K. Kunisch and L.W. White, Regularity and complementarity estimated parameters in parabolic problems, Inverse Problems · Zbl 0667.93022
[12] Ladyzhenskaya, O. A.; Ural’tseva, N., Linear and Quasilinear Elliptic Equations (1968), Academic: Academic New York · Zbl 0164.13002
[13] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (1969), Dunod, Gauthier-Villars · Zbl 0189.40603
[14] Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Vol. 1 (1969), Springer-Verlag: Springer-Verlag New York · Zbl 0251.35001
[15] Maurer, H.; Zowe, J., First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Programming, 16, 98-110 (1979) · Zbl 0398.90109
[16] Polyak, V. T.; Tret’yakov, N. Y., The method of penalty estimates for conditional extremum problems, Z. Vychisl. Mat. i Mat. Fiz., 13, 34-46 (1973)
[17] Schultz, M., Spline Analysis (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0333.41009
[18] Schumaker, L., Spline Functions: Basic Theory (1981), Wiley: Wiley New York · Zbl 0449.41004
[19] Timoshenko, S.; Woinowsky-Krieger, S., Theory of Plates and Shells (1959), McGraw-Hill: McGraw-Hill New York · Zbl 0114.40801
[20] L. White, Estimation of flexural rigidity in a Kirchhoff plate model, Appl. Math. Comput.; L. White, Estimation of flexural rigidity in a Kirchhoff plate model, Appl. Math. Comput. · Zbl 0651.73023
[21] L. White, Estimation of elastic parameters in a von Karman model, submitted for publication.; L. White, Estimation of elastic parameters in a von Karman model, submitted for publication.
[22] Wloka, J., Partial Differential Equations (1987), Cambridge U.P: Cambridge U.P New York
[23] Zowe, J.; Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces, J. Appl. Math. Optim., 5, 49-62 (1979) · Zbl 0401.90104
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