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2016 | OriginalPaper | Buchkapitel

On the Weak Solvability and the Optimal Control of a Frictional Contact Problem with Normal Compliance

verfasst von : Andaluzia Matei

Erschienen in: System Modeling and Optimization

Verlag: Springer International Publishing

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Abstract

In the present work we consider a frictional contact model with normal compliance. Firstly, we discuss the weak solvability of the model by means of two variational approaches. In a first approach the weak solution is a solution of a quasivariational inequality. In a second approach the weak solution is a solution of a mixed variational problem with solution-dependent set of Lagrange multipliers. Nextly, the paper focuses on the boundary optimal control of the model. Existence results, an optimality condition and some convergence results are presented.

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Literatur
1.
Zurück zum Zitat Amassad, A., Chenais, D., Fabre, C.: Optimal control of an elastic contact problem involving Tresca friction law. Nonlinear Anal. 48, 1107–1135 (2002)MathSciNetCrossRefMATH Amassad, A., Chenais, D., Fabre, C.: Optimal control of an elastic contact problem involving Tresca friction law. Nonlinear Anal. 48, 1107–1135 (2002)MathSciNetCrossRefMATH
3.
Zurück zum Zitat Barbu, V.: Optimal Control of Variational Inequalities. Pitman Advanced Publishing, Boston (1984)MATH Barbu, V.: Optimal Control of Variational Inequalities. Pitman Advanced Publishing, Boston (1984)MATH
4.
Zurück zum Zitat Bonnans, J.F., Tiba, D.: Pontryagin’s principle in the control of semiliniar elliptic variational inequalities. Appl. Math. Optim. 23(1), 299–312 (1991)MathSciNetCrossRefMATH Bonnans, J.F., Tiba, D.: Pontryagin’s principle in the control of semiliniar elliptic variational inequalities. Appl. Math. Optim. 23(1), 299–312 (1991)MathSciNetCrossRefMATH
5.
Zurück zum Zitat Capatina, A., Timofte, C.: Boundary optimal control for quasistatic bilateral frictional contact problems. Nonlinear Anal. Theory Methods Appl. 94, 84–99 (2014)MathSciNetCrossRefMATH Capatina, A., Timofte, C.: Boundary optimal control for quasistatic bilateral frictional contact problems. Nonlinear Anal. Theory Methods Appl. 94, 84–99 (2014)MathSciNetCrossRefMATH
7.
Zurück zum Zitat Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefMATH Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefMATH
8.
Zurück zum Zitat Klarbring, A., Mikelič, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)MathSciNetCrossRefMATH Klarbring, A., Mikelič, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)MathSciNetCrossRefMATH
9.
Zurück zum Zitat Klarbring, A., Mikelič, A., Shillor, M.: A global existence result for the quasistatic frictional contact problem with normal compliance. In: del Piero, G., Maceri, F. (eds.) Unilateral Problems in Structural Analysis, vol. 4, pp. 85–111. Birkhäuser, Boston (1991)CrossRef Klarbring, A., Mikelič, A., Shillor, M.: A global existence result for the quasistatic frictional contact problem with normal compliance. In: del Piero, G., Maceri, F. (eds.) Unilateral Problems in Structural Analysis, vol. 4, pp. 85–111. Birkhäuser, Boston (1991)CrossRef
10.
Zurück zum Zitat Lions, J.-L.: Contrôle optimale des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968)MATH Lions, J.-L.: Contrôle optimale des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968)MATH
11.
Zurück zum Zitat Matei, A.: On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers. Proc. Royal Soc. Edinburgh Sect. Math. 143(05), 1047–1059 (2013)MathSciNetCrossRefMATH Matei, A.: On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers. Proc. Royal Soc. Edinburgh Sect. Math. 143(05), 1047–1059 (2013)MathSciNetCrossRefMATH
12.
Zurück zum Zitat Matei, A.: Weak solutions via lagrange multipliers for contact models with normal compliance. Konuralp J. Math. 3(2) (2015) Matei, A.: Weak solutions via lagrange multipliers for contact models with normal compliance. Konuralp J. Math. 3(2) (2015)
13.
Zurück zum Zitat Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance, submitted Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance, submitted
15.
Zurück zum Zitat Mignot, R.: Contrôle dans les inéquations variationnelles elliptiques. J. Func. Anal. 22, 130–185 (1976)CrossRefMATH Mignot, R.: Contrôle dans les inéquations variationnelles elliptiques. J. Func. Anal. 22, 130–185 (1976)CrossRefMATH
16.
17.
Zurück zum Zitat Martins, J.A.C., Oden, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. TMA 11, 407–428 (1987)MathSciNetCrossRefMATH Martins, J.A.C., Oden, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. TMA 11, 407–428 (1987)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Neittaanmaki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems: Theory and Applications. Springer Monographs in Mathematics. Springer, New York (2006)MATH Neittaanmaki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems: Theory and Applications. Springer Monographs in Mathematics. Springer, New York (2006)MATH
19.
Zurück zum Zitat Hild, P., Renard, Y.: A stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 115, 101–129 (2010)MathSciNetCrossRefMATH Hild, P., Renard, Y.: A stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 115, 101–129 (2010)MathSciNetCrossRefMATH
20.
Zurück zum Zitat Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)MathSciNetCrossRefMATH Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Advances in Mechanics and Mathematics. Springer, New York (2009)MATH Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Advances in Mechanics and Mathematics. Springer, New York (2009)MATH
22.
Zurück zum Zitat Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics, London Mathematical Society. Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)CrossRefMATH Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics, London Mathematical Society. Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)CrossRefMATH
23.
Zurück zum Zitat Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1991)MATH Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1991)MATH
24.
Zurück zum Zitat Wohlmuth, B.: A mortar finite element method using dual spaces for the lagrange multiplier. SIAM J. Numeri. Anal. 38, 989–1012 (2000)MathSciNetCrossRefMATH Wohlmuth, B.: A mortar finite element method using dual spaces for the lagrange multiplier. SIAM J. Numeri. Anal. 38, 989–1012 (2000)MathSciNetCrossRefMATH
25.
Zurück zum Zitat Wohlmuth, B.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering, vol. 17. Springer, Heidelberg (2001)MATH Wohlmuth, B.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering, vol. 17. Springer, Heidelberg (2001)MATH
Metadaten
Titel
On the Weak Solvability and the Optimal Control of a Frictional Contact Problem with Normal Compliance
verfasst von
Andaluzia Matei
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-55795-3_35

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