An optimum design problem in magnetostatics
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 223-239.

In this paper, 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.

DOI : 10.1051/m2an:2002012
Classification : 49J20, 49Q10, 65K10, 78A30
Mots-clés : shape optimization, optimum design, magnet, numerical examples 
@article{M2AN_2002__36_2_223_0,
     author = {Henrot, Antoine and Villemin, Gr\'egory},
     title = {An optimum design problem in magnetostatics},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {223--239},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/m2an:2002012},
     mrnumber = {1906816},
     zbl = {1054.49030},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2002012/}
}
TY  - JOUR
AU  - Henrot, Antoine
AU  - Villemin, Grégory
TI  - An optimum design problem in magnetostatics
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2002
SP  - 223
EP  - 239
VL  - 36
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2002012/
DO  - 10.1051/m2an:2002012
LA  - en
ID  - M2AN_2002__36_2_223_0
ER  - 
%0 Journal Article
%A Henrot, Antoine
%A Villemin, Grégory
%T An optimum design problem in magnetostatics
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2002
%P 223-239
%V 36
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2002012/
%R 10.1051/m2an:2002012
%G en
%F M2AN_2002__36_2_223_0
Henrot, Antoine; Villemin, Grégory. An optimum design problem in magnetostatics. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 223-239. doi : 10.1051/m2an:2002012. http://archive.numdam.org/articles/10.1051/m2an:2002012/

[1] S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand Math Studies (1965). | MR | Zbl

[2] J. Baranger, Analyse Numérique. Hermann, Paris (1991). | Zbl

[3] D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-289. | Zbl

[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

[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

[7] E. Durand, Magnétostatique. Masson, Paris (1968). | Zbl

[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

[10] M. Pierre and J.R. Roche, Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203-217. | Zbl

[11] O. Pironneau, Optimal shape design for elliptic systems. Springer Series in Computational Physics. Springer, New York (1984). | MR | Zbl

[12] J. Simon, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649-687. | Zbl

[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 | Zbl

Cité par Sources :