This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the decay of the solution at infinity and giving a rigidity matrix that can be very efficiently stored. All aspects are described here, from the practical construction of the mesh, the storage of the matrix, the error estimation of the method, the boundary conditions and a simple preconditionning technique. At the end of the paper, a typical computation of a uniformly magnetized ball is done and compared to the analytic solution. This method gives a natural alternatives to boundary elements methods for 3D computations.

Keywords: micromagnetics, finite element method, preconditionning, exterior problems

@article{COCV_2001__6__629_0, author = {Alouges, Fran\c{c}ois}, title = {Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {629--647}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, zbl = {0992.78007}, mrnumber = {1872391}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__629_0/} }

TY - JOUR AU - Alouges, François TI - Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 SP - 629 EP - 647 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__629_0/ UR - https://zbmath.org/?q=an%3A0992.78007 UR - https://www.ams.org/mathscinet-getitem?mr=1872391 LA - en ID - COCV_2001__6__629_0 ER -

Alouges, François. Computation of the demagnetizing potential in micromagnetics using a coupled finite and infinite elements method. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 629-647. http://www.numdam.org/item/COCV_2001__6__629_0/

[1] Boundary conditions for the numerical solution of elliptic equations in exterior domains. SIAM J. Appl. Math. 42 (1982). | MR | Zbl

, and ,[2] Micromagnetics. Interscience Publishers, Wiley & Sons, New-York (1963).

,[3] Finite element modeling of unbounded problems using transformations: A rigorous, powerful and easy solution. IEEE Trans. Mag. 28 (1992).

, and ,[4] Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 6. Masson, Paris (1988). | MR

and ,[5] Non-reflecting boundary conditions. J. Comp. Phys. 94 (1991) 1-29. | Zbl

,[6] The finite element method with nonuniform mesh sizes for unbounded domains. J. Math. Comput. 36 (1981) 387-404. | MR | Zbl

,[7] Classical Electrodynamics1975). | MR | Zbl

,[8] Approximation of exterior problems. Optimal conditions for the Laplacian. Analysis 16 (1996) 305-324. | Zbl

and ,[9] Numerische Simulation von Ummagnetisierungsvorgängen in hartmagnetischen Materialen, Ph.D. Thesis. Technische Universität Wien (1993).

,[10] Phase distribution and computed magnetic properties of high-remanent composite magnets. J. Magnetism and Magnetic Materials 150 (1995) 329-344.

, , and ,[11] Exterior finite elements for 2-dimensional field problems with open boundaries. Proc. IEE 124 (1977).

, , and ,[12] Infinite Elements Method. Peking University Press. | Zbl

,