The two-scalar potentials idea has been used with success for the computation of static magnetic fields in the presence of nonlinear isotropic magnetic materials by the finite element method. In this communication we formulate the two-scalar-potentials method for anisotropic materials and present a computer program and the solution of an example problem. The use of infinite multipolar elements is also discussed. Several advanced methods and ideas are employed by the program: scalar potentials, rather than vector potentials, giving only one unknown quantity; the finite element method, in which the solution is approximated by a continuous function; the Galerkin method to solve the differential equations; accurate infinite elements, which avoid the introduction of an artificial boundary for unbounded problems; automatic mesh generation, which means that the user can construct a large mesh and represent a complicated geometry with little effort; automatic elimination of nodes outside the iron, which restricts the iterations to the nonlinear anisotropic region with economy of computer time; use of sparse matrix technology, which represents a further economy in computer time when assembling the linear equations and solving them by either Gauss elimination or iterative techniques such as the conjugated gradient method, etc. The combination of these techniques is very convenient.

Solution of three-dimensional, anisotropic, nonlinear problems of magnetostatics using two scalar potentials, finite and infinite multipolar elements and automatic mesh generation

. A considerable amount of literature has been published on the magnetic modelling of uniformly magnetized ellipsoids since the second half of the nineteenth century. Ellipsoids have flexibility to represent a wide range of geometrical forms, are the only known bodies which can be uniformly magnetized in the presence of a uniform inducing field and are the only finite bodies for which the self-demagnetization can be treated analytically. This property makes ellipsoids particularly useful for modelling compact orebodies having high susceptibility. In this case, neglecting the self-demagnetization may strongly mislead the interpretation of these bodies by using magnetic methods. A number of previous studies consider that the self-demagnetization can be neglected for the case in which the geological body has an isotropic susceptibility lower than or equal to 0.1 SI. This limiting value, however, seems to be determined empirically and there has been no discussion about how this value was determined. In addition, the geoscientific community lacks an easy-to-use tool to simulate the magnetic field produced by uniformly magnetized ellipsoids. Here, we present an integrated review of the magnetic modelling of arbitrarily oriented triaxial, prolate and oblate ellipsoids. Our review includes ellipsoids with both induced and remanent magnetization, as well as with isotropic or anisotropic susceptibility. We also discuss the ambiguity between confocal ellipsoids with the same magnetic moment and propose a way of determining the isotropic susceptibility above which the self-demagnetization must be taken into consideration. Tests with synthetic data validate our approach. Finally, we provide a set of routines to model the magnetic field produced by ellipsoids. The routines are written in Python language as part of the Fatiando a Terra, which is an open-source library for modelling and inversion in geophysics.

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Ellipsoids (v1.0): 3-D magnetic modelling of ellipsoidal bodies

The equivalence current model is applied to the case of the prolate spheroid to yield a solution for internal B and H values with surprising simplicity.

https://iopscience.iop.org/article/10.1088/0143-0807/6/3/010/pdf

The amperian current model of magnetisation and the prolate spheroid

The magnetization and the magnetic field distribution inside (outside) an anisotropic type II superconducting ellipsoid, with filamentary structure, is formulated. It is shown that the magnetic field in this case is different from that of the general anisotropic one. The nucleations of the flux lines for specimens with large demagnetization factors are theoretically studied. It is shown that the nucleations of the flux lines, for specimens with large demagnetization factor, appears at a field larger than that of ellipsoidal shape.



Es werden die Magnetisierung und Magnetfeldverteilung innerhalb (auserhalb) eines anisotropen Supraleitungsellipsoids zweiter Art mit Fadenstruktur formuliert. Es wird gezeigt, das das Magnetfeld in diesem Fall sich vom allgemeinen anisotropen Fall unterscheidet. Die Bildung der Fluslinien fur Proben mit grosen Demagnetisierungsfaktoren wird theoretisch untersucht. Es wird gezeigt, das die Keime der Fluslinien fur Proben mit grosen Demagnetisierungsfaktoren bei Feldern auftreten, die groser sind als bei der Ellipsoidform.

Magnetostatics of Anisotropic Superconducting Ellipsoid

It is shown that, contrary to the prevailing opinion, the scattering of the magnetic flux, even in ferrites with high magnetic permeability, is very large. A description of the technique for measuring the parameters of ferrites and inductors on ferrite rings is given. The results of experimental measurements of parameters for various grades of ferrites and various values of magnetic permeability are presented. Recommendations on the design of high-frequency transformers and inductors on ferrite rings are given.

http://ijsei.in/index.php/ijsei/article/download/126/78

Magnetic Flow Scattering in Ferrite Rings

The demagnetizing tensor formalism for the evaluation of magnetization and internal field in ellipsoidal bodies of isotropic materials is extended to the case of anisotropic materials and to the calculation of the field outside the body.

C. Soliverez

Papers

Magnetostatics of anisotropic ellipsoidal bodies