Advertisement

Computational Optimization and Applications

, Volume 62, Issue 1, pp 241–270 | Cite as

Optimal control of electrorheological fluids through the action of electric fields

  • Juan Carlos De Los ReyesEmail author
  • Irwin Yousept
Article

Abstract

This paper is concerned with an optimal control problem of steady-state electrorheological fluids based on an extended Bingham model. Our control parameters are given by finite real numbers representing applied direct voltages, which enter in the viscosity of the electrorheological fluid via an electrostatic potential. The corresponding optimization problem belongs to a class of nonlinear optimal control problems of variational inequalities with control in the coefficients. We analyze the associated variational inequality model and the optimal control problem. Thereafter, we introduce a family of Huber-regularized optimal control problems for the approximation of the original one and verify the convergence of the regularized solutions. Differentiability of the solution operator is proved and an optimality system for each regularized problem is established. In the last part of the paper, an algorithm for the numerical solution of the regularized problem is constructed and numerical experiments are carried out.

Keywords

Electrorheological fluids Optimal control Electrostatic potential Variational inequalities Control in coefficients 

Mathematics Subject Classification

49J40 49J20 49J24 

Notes

Acknowledgments

We would like to thank Sergio González-Andrade for providing us the finite element matrices used in the computational experiment. Research partially supported by the Alexander von Humboldt Foundation and by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.

References

  1. 1.
    Casas, E., Fernández, L.A.: Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differ. Equ. 104(1), 20–47 (1993)CrossRefGoogle Scholar
  2. 2.
    Chan, Tony F., Golub, Gene H., Mulet, Pep: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    De los Reyes, J.C., González, S.: Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods. J. Comput. Appl. Math. 235(1), 11–32 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Los Reyes, J.C.: Optimal control of a class of variational inequalities of the second kind. SIAM J. Control Optim. 49, 1629–1658 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De los Reyes, J.C.: Optimization of mixed variational inequalities arising in flows of viscoplastic materials. Comput. Optim. Appl. 52, 757–784 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De los Reyes, J.C., González, S.: Path following methods for steady laminar Bingham flow in cylindrical pipes. ESAIM M2AN 43, 81–117 (2000)CrossRefGoogle Scholar
  7. 7.
    De los Reyes, J.C., Herzog, R., Meyer, C.: Optimal control of static elastoplasticity in primal formulation, vol. 474. Ergebnisberichte des Instituts für Angewandte Mathematik, TU Dortmund (2013)Google Scholar
  8. 8.
    Ekeland, I., Temam, R.: Convex analysis and variational problems. Classics in Applied Mathematics, vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1999). (English edition translated from the French)Google Scholar
  9. 9.
    Engelmann, B., Hiptmair, R., Hoppe, R.H.W., Mazurkevitch, G.: Numerical simulation of electrorheological fluids based on an extended Bingham model. Comput. Vis. Sci. 2, 211–219 (2000)CrossRefzbMATHGoogle Scholar
  10. 10.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  11. 11.
    Gunzburger, M.D.: Navier-Stokes equations for incompressible flows: finite-element methods. In Handbook of computational fluid mechanics, pp. 99–157. Academic Press, San Diego, CA (1996)Google Scholar
  12. 12.
    Halsey, T.C.: Electrorheological fluids. Science 258(5083), 761–766 (1992)CrossRefGoogle Scholar
  13. 13.
    Herzog, R., Meyer, C., Wachsmuth, G.: B-and strong stationarity for optimal control of static plasticity with hardening. SIAM J. Optim. 23(1), 321–352 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: C-and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28(1), 1–23 (2006). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hoppe, R.H.W., Litvinov, W.G.: Problems on electrorheological fluid flows. Commun. Pure Appl. Anal. 3(4), 809–848 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hoppe, R.W., Litvinov, W.G.: Modeling, simulation and optimization of electrorheological fluids. In: Glowinski, R., Xu, J. (eds.) Numerical Methods for Non-Newtonian Fluids. Handbook of Numerical Analysis, pp. 719–793. Elsevier, Amsterdam (2011)Google Scholar
  18. 18.
    Hoppe, R.H.W., Litvinov, W.G., Rahman, T.: Problems of stationary flow of electrorheological fluids in a cylindrical coordinate system. SIAM J. Appl. Math. 65(5), 1633–1656 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N.S.) 4(2), 203–207 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Litvinov, W.G., Hoppe, R.H.W.: Coupled problems on stationary non-isothermal flow of electrorheological fluids. Commun. Pure Appl. Anal. 4(4), 779–803 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Litvinov, W.G., Rahman, T., Hoppe, R.H.W.: Model of an electro-rheological shock absorber and coupled problem for partial and ordinary differential equations with variable unknown domain. European J. Appl. Math. 18(4), 513–536 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer, Berlin (2000)CrossRefGoogle Scholar
  23. 23.
    Somersalo, E., Cheney, M., Isaacson, D.: Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52(4), 1023–1040 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tao, R.: Electro-rheological fluids and magneto-rheological suspensions. In: Proceedings of the 7th International Conference Honolulu, Hawaii, 9–23 July. World Scientific, Singapore (1999)Google Scholar
  25. 25.
    Troianiello, G.M.: Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics. Plenum Press, New York (1987)CrossRefzbMATHGoogle Scholar
  26. 26.
    Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Computational Optimization and Applications 52(2), 559–581 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yousept, I.: Optimal control of quasilinear \(\varvec {H}(\mathbf{curl})\)-elliptic partial differential equations in magnetostatic field problems. SIAM J. Control Optim. 51(5), 3624–3651 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Research Center on Mathematical Modelling (MODEMAT)EPN QuitoQuitoEcuador
  2. 2.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

Personalised recommendations