×

zbMATH — the first resource for mathematics

An optimum design problem in magnetostatics. (English) Zbl 1054.49030
Summary: We are interested in finding the optimal shape of a magnet. The criterion to maximize is the jump of the electromagnetic field between two different configurations. We prove existence of an optimal shape into a natural class of domains. We introduce a quasi-Newton type algorithm which moves the boundary. This method is very efficient to improve an initial shape. We give some numerical results.

MSC:
49Q10 Optimization of shapes other than minimal surfaces
49J20 Existence theories for optimal control problems involving partial differential equations
65K10 Numerical optimization and variational techniques
78A30 Electro- and magnetostatics
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] S. Agmon , Lectures on Elliptic Boundary Value Problems . Van Nostrand Math Studies ( 1965 ). MR 178246 | Zbl 0142.37401 · Zbl 0142.37401
[2] J. Baranger , Analyse Numérique . Hermann, Paris ( 1991 ). Zbl 0757.65001 · Zbl 0757.65001
[3] D. Chenais , On the existence of a solution in a domain identification problem . J. Math. Anal. Appl. 52 ( 1975 ) 189 - 289 . Zbl 0317.49005 · Zbl 0317.49005
[4] D. Chenais , Sur une famille de variétés à bord lipschitziennes, application à un problème d’identification de domaine . Ann. Inst. Fourier (Grenoble) 4 ( 1977 ) 201 - 231 . Numdam | Zbl 0333.46020 · Zbl 0333.46020
[5] R. Dautray and J.L. Lions (Eds.), Analyse mathématique et calcul numérique , Vol. I and II. Masson, Paris ( 1984 ).
[6] J.E. Dennis and R.B. Schnabel , Numerical Methods for unconstrained optimization . Prentice Hall ( 1983 ). Zbl 0579.65058 · Zbl 0579.65058
[7] E. Durand , Magnétostatique . Masson, Paris ( 1968 ). Zbl 0053.15404 · Zbl 0053.15404
[8] A. Henrot and M. Pierre , Optimisation de forme (to appear).
[9] M. Pierre and J.R. Roche , Computation of free sufaces in the electromagnetic shaping of liquid metals by optimization algorithms . Eur. J. Mech. B Fluids 10 ( 1991 ) 489 - 500 . Zbl 0741.76095 · Zbl 0741.76095
[10] M. Pierre and J.R. Roche , Numerical simulation of tridimensional electromagnetic shaping of liquid metals . Numer. Math. 65 ( 1993 ) 203 - 217 . Article | Zbl 0792.65096 · Zbl 0792.65096
[11] O. Pironneau , Optimal shape design for elliptic systems . Springer Series in Computational Physics. Springer, New York ( 1984 ). MR 725856 | Zbl 0534.49001 · Zbl 0534.49001
[12] J. Simon , Differentiation with respect to the domain in boundary value problems . Numer. Funct. Anal. Optim. 2 ( 1980 ) 649 - 687 . Zbl 0471.35077 · Zbl 0471.35077
[13] J. Simon , Variations with respect to domain for Neumann condition . Proceedings of the 1986 IFAC Congress at Pasadena “Control of Distributed Parameter Systems”.
[14] J. Sokolowski and J.P. Zolesio , Introduction to shape optimization: shape sensitity analysis . Springer Series in Computational Mathematics, Vol. 10, Springer, Berlin ( 1992 ). MR 1215733 | Zbl 0761.73003 · Zbl 0761.73003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.