Accurate Magnetic Shell Approximations with Magnetostatic Finite Element Formulations by a Subdomain Approach

This paper presents a subproblem approach with hconformal magnetostatic finite element formulations for treating the errors of magnetic shell approximation, by replacing volume thin regions by surfaces with interface conditions. These approximations seem to neglect the curvature effects in the vicinity of corners and edges. The process from the surface-tovolume correction problem is presented as a sequence of several subdomains, which can be composed to the full domain, including inductors and thin magnetic regions. Each step of the process will be separately performed on its own subdomain and submesh instead of solving the problem in the full domain. This allows reducing the size of matrix and time computation. Keywords-magnetostatic finite element formulation; magnetic scalar potential; magnetic field; magnetic shell; subproblem approach


INTRODUCTION
The local fields in magnetic shells are approximated by a priori 1-D analytical distributions across the shell thicknesses [1,2]. This means that the interior of volume thin regions is not meshed and is introduced by surfaces with impedance-type Interface Conditions (ICs) linked to the inner-analytical distributions. This leads to negligible edges and corners of magnetic shells, increasing with thickness. In order to overcome this disadvantage, the Sub-Problem Method (SPM) for the magnetodynamic problem with dual formulation has been proposed for one-way coupling [3][4][5][6][7][8][9][10]. In this development, a subdomain technique based on the SPM is extended for the h-conformal magnetostatic finite element formulation in order to improve the local fields (magnetic scalar potential, magnetic flux density and magnetic field) appearing around the edges and corners of magnetic shells. The idea of the method is to perform subdomain solving in three steps ( Figure 1): • Step 1: A lower subdomain attending with stranded inductors is first considered on a simplified mesh without any magnetic shells.
• Step 2: A shell with the very coarse mesh that does not contain stranded inductors anymore is then added.
• Step 3: A volume correction replacing the magnetic shell Finite Element (FE) by an actual thin region is introduced to improve shell inaccuracies. The relation between steps is constrained by Volume Source (VS) expressed changes of the material propoerties or Surface Sources (SS) presented changes of ICs. In each step, the problem is independently solved in an individual sub-mesh and its surrounding without depending on other meshes, which allow to distinct mesh refinements. The method is applied on a practical problem.

A. Weak Formulation for Inductor Model -Step 1 (SP q)
The magnetostatic weak formulation ( − ߔ) for Step 1 (SP q) is obtained via the magnetic Gauss's law (1b), i.e. [1,2]: the basis functions for ߔ as well as for the test function ߔ ᇱ . Notations of (. , . ) ஐ and 〈. , . 〉 are respectively the volume integral in ߗ the surface integral on Γ , of the product of their vector field arguments. The surface term < • , ߔ ᇱ > Г ,షം in (5) is considered as a natural BC of type (3a), usually zero.

C. Weak Formulation for Volume Correction -Step 3 (SP k)
The weak form of SP k is finally established via a VS given by (2): At the discrete mesh, the source quantities ߔ and ௦, in (11), defined in previous meshes (SP q and SP p) are also projected to the mesh of SP k via a projection method (see Section D). In addition, the discontinuity of SP p in SP k has to be removed by fixing as: D. Transformation of Solutions between Sub-meshes As presented above, the source fields ߔ and obtained from the previous meshes of SPi (e.g. SP q) are transferred to the mesh of SP p, i.e. [15][16][17]: where ∀ ᇱ ∈ ‫ܪ‬ , ଵ ൫ߗ ௦, ൯ is a curl-conform function space for the p-projected source ,ି (the projection of ,ି on mesh of SP p) and the test function ᇱ defined on ߗ ௦, .
In the same way, the magnetic scalar potential ߔ , can project the grad of ߔ in the mesh of SP q to the mesh of SP p, i.e. [9]: (gradߔ ,ି , gradߔ ᇱ ) ఆ ೞ, = (grad ߔ , gradߔ ᇱ ) ఆ ೞ, ∀ߔ ᇱ ∈ ‫ܪ‬ , ଵ ൫ߗ ൯ (14) where ߔ ᇱ ∈ ‫ܪ‬ , ଵ ൫ߗ ൯ is the grad-conform function space for the p-projected source ߔ ,ି (the projection of ߔ on mesh of SP p) and the test function ߔ ᇱ defined on ߗ ௦, .

IV. APPLICATION TEST
The practical test is a shielded problem. It consists of a plate located in the middle of two stranded inductors carrying a magnetomovetive force of 1000 ampere-turns ( Figure 2). The magnetic shields (screen up and down) cover the plate and the stranded inductors, for ߤ ,௦ௗ =1 and ߤ ,௧ =200. The test is performed in the 2-D case. As a sequence, the test at hand is performed in three steps. The solutions on the magnetic scalar potential ߔ of each subdomain are illustrated in Figure 3. An initial problem SP q including the stranded inductors alone is solved in a sub-domain without the shielding plate and screens up and down (Figure 3(a), ߔ ). Geometry of a 2-D shielding problem (d=3÷7.5mm, Lpl=2m, Ls=2m+2d, Hs=0.4m, Hy=0.14m, Cdx=0.8m, Cdy=0.01m, Cy=0.2m, Cx=0.05m). The shell approximation SP p that does not include the stranded inductors anymore is then added (Figure 3(b), ߔ ). The volume improvements covering the shielding plate and screen up and down are finally introduced to overcome the shell approximations [1,2], for d=5mm, ߤ ,௦ௗ =1 and ߤ ,௧ =200 (Figure 3(c), ߔ ). In a similar way, the distribution of magnetic flux density for each subdomain obtained in each step is shown in detail in Figure 4. The sequence from Step 1 (SP q) → Step 2 (SP p) → Step 3 (SP k) is pointed out from top to bottom, for thickness d=5mm. solution is then checked to be similar to the reference solution in the computation from the traditional finite element method (FEM) [12][13][14].   The relative inaccuracy on the magnetic flux densities before making corrections is presented in Figure 6 for various thicknesses. The error can reach 90% at the end regions of the plate, for d=7.5 mm, and 75% for smaller thickness d=3 mm for ߤ ,௦ௗ =1 and ߤ ,௧ =200. Accurate local improvement with volume correction SP k after improving are less than 15% for d=7.5mm near the plate end, or 10% for d=3 mm. It is worth noting that the error is less than 1% in the middle of the plate for both cases. The relative improvement (correction) of the TS magnetic flux along the screen up is presented in Figure  7 for different screen thicknesses. It can reach a percentage up to 47% near the edge of the screen up, for d=7.5mm. It reduces to about 40% for d=5 mm, or 30% for d=3mm, with ߤ ,௦ௗ =1 and ߤ ,௧ =200.
V. DISCUSSION AND CONCLUSION In this research, a subdomain technique for coupling thin magnetic shells has been successfully developed with hconformal magnetostatic finite element formulations for improving the errors of the magnetic scalar potential, magnetic flux density, and magnetic flux around the edges and corners appearing from the magnetic shell approximation [1,2]. The obtained results of the method were found to be quite similar to the reference solution in the computation of the traditional FEM [12][13][14]. The proposed technique has been successfully carried out with a three step sequence. In the future, it could be extended to the case of multilayer-TS with different characteristics.
The source-code of the method has been extended from the one subproblem method that was developed by the author with the help of Patrick Dular and Christophe Geuzaine at the Deparment of Electrical Engineering and Computer Science, University of Liege, Belgium. It will be ran in the background of the Getdp and Gmsh (http://getdp.info and http://gmsh.info) as open source code.