Abstract
In this paper we present a new method for designing a nozzle. In fact the problem is to find the optimal domain for the solution of a linear or nonlinear boundary value PDE, where the boundary condition is defined over an unspecified domain.
By an embedding process, the problem is first transformed to a new shape-measure problem, and then this new problem is replaced by another in which we seek to minimize a linear form over a subset of linear equalities. This minimization is global, and the theory allows us to develop a computational method to find the solution by a finite-dimensional linear programming problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. H. Borzabadi, A. V. Kamyad and M. H. Farahi,Optimal control of the heat equation in an inhomogeneous body, J. Appl. Math. & Computing15 (1–2) (2004), 127–145.
R. Butt,Optimal shape design for a nozzle problem, Journal of the Australian Mathematical Society, Ser. B35 (1993), 71–86.
J. Cea,Identification of Domain, Lecture Notes in Computer Sciences,3 (1973), New York: Springer.
J. Cea and E. J. Haug,Lecture on Optimization Theory and Algorithm, Berlin: Springer-Verlag, 1978.
A. Fakharzadeh,Shapes, measure and elliptic equations, Ph.D Thesis Mathematics, University of Leeds, Leeds, England, 1996.
M. H. Farahi, J. E. Rubio and D. A. Wilson,The optimal control of the linear wave equation, International J. of Control63(1) (1996), 833–848.
M. H. Farahi, J. E. Rubio and D. A. Wilson,The global control of a nonlinear wave equation, International J. of Control65(1) (1996), 1–15.
N. Fujii,Necessary conditions for a domain optimization problem in elliptic boundary value problems, SIAM J. of Control and Optimization24 (1986), 346–360.
A. V. Kamyad and A. H. Borzabadi,Strong controllability and optimal control of the heat equation with a thermal source, Korean J. Comput. & Appl. Math.7(3) (2000), 555–568.
A. V. Kamyad, J. E. Rubio and D. A. Wilson,Optimal control of multidimensional diffusion equation, Journal of Optimization Theory and Applications70 (1991), 191–209.
A. V. Kamyad, J. E. Rubio and D. A. Wilson,An Optimal control problem for the multidimensional diffusion equation with a generalized control variable, Journal of Optimization Theory and Applications75 (1992), 101–132.
V. P. Mikhailov,Partial differential equation, Mir, Moscow, 1978.
B. Mohammadi and O. Pironneau,Applied Shape Optimization for Fluids, Oxford Sciences Publications, 2001.
F. Murat and J. Simon,Studies on some optimal shape design problem, In J. Cea, (ed), Lecture Notes in Computer Science107 (1976), Berlin: Springer-Verlag.
A. Myslinski,Minimax shape optimization problem for von kármán systems, Lecture Notes in Control and Information Sciences144 (1990).
O. Pironneau,On optimum design in fluid mechanics, Journal of Fluid Mechanics64 (1974), 97–110.
O. Pironneau,Optimal shape design for elliptic systems, New York: Springer-Verlag, 1984.
O. Pironneau and B. Mohammadi,Techniques for optimal shape design, to appear.
P. C. Rosenbloom,Quelques classes de probléms extrémaux, Bulletin de la Société Mathématique de France80 (1952), 183–216.
J. E. Rubio,Control and optimization; the linear treatment of non-linear problems, Manchester, U. K., Manchester University Press, 1986.
W. Rudin,Real and Complex Analysis, New York: McGraw-Hill, 1974.
D. A. Wilson, and J. E. Rubio,Existence of optimal controls for the diffusion equation, Journal of Optimization Theory and Applications22 (1977), 91–101.
L. C. Young,Calculus of variations and optimal control theory, Philadelphia: Sunders, 1969.
J. P. Zolesio,Identification of domain by deformation, Ph.D Thesis, University of Nice, 1979.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mohammad Hadi Farahi received his B. Sc from Ferdowsi University of Mashhad, Mashhad, Iran, M. Sc from Brunel University, U. K. and Ph. D. at Leeds University, U. K. At the moment he is associate professor at the College of Mathematics, Ferdowsi University of Mashhad, Iran and his research area are optimal control, optimization, approximation theory and numerical analysis.
Akbar Hashemi Borzabadi received his B. Sc from Birjand University, Iran and his M. Sc from Ferdowsi University of Mashhad. He is lecturer in Damghan University of Basic Sciences and at the moment he is Ph. D student in Ferdowsi University of Mashhad. His research interests center on optimal control of distributed parameter systems and optimal path planning.
Hamed Hashemi Mehne received his B. Sc and M. Sc from Ferdowsi University of Mashhad, Iran. He is lecturer in Aerospace Research Inst. of Tehran, Iran and at the moment he is Ph. D student in Ferdowsi University of Mashhad. His research interests center on optimal shape design and optimal control.
Ali Vahidian Kamyad received his B. Sc from Ferdowsi University of Mashhad, Mashhad, Iran, M. Sc from Institute of Mathematics Tehran, Iran and Ph. D at Leeds University, Leeds, England under supervisior of J. E. Rubio. Since 1972 he has been at the Ferdowsi University of Mashhad, he is professor at the College of Mathematics, Ferdowsi University of Mashhad, Iran and his research interests are mainly on optimal control of distributed parameter systems and applications of fuzzy theory.
Rights and permissions
About this article
Cite this article
Farahi, M.H., Borzabadi, A.H., Mehne, H.H. et al. Measure theoretical approach for optimal shape design of a nozzle. JAMC 17, 315–328 (2005). https://doi.org/10.1007/BF02936058
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02936058