Optimal control of electrorheological fluids through the action of electric fields
- 304 Downloads
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.
KeywordsElectrorheological fluids Optimal control Electrostatic potential Variational inequalities Control in coefficients
Mathematics Subject Classification49J40 49J20 49J24
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.
- 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.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
- 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
- 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
- 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