A Hybrid Method BEM-NRM for Magnetostatics Problems

This article proposes a new method to solve non-linear magnetostatic problems applied to the modeling of electromagnetic devices. A reduced scalar potential formulation is presented and solved by a hybrid BEM-NRM. It has been implemented and tested in a software called MaGot. We will show that the computation time of this new method is very low. Thus, it could be easily and efficiently used for the pre-sizing of actuators.


I. INTRODUCTION
Numerical simulations of electromagnetic actuators are an increasing industrial need.In general, the designing time and number of prototypes required to design products are higher during the first step of the whole process.Several methods have been proposed for actuators designing.Two of the most popular methods ones are the Finite Element Method (FEM) [1], [2] and the Reluctance Network Method (NRM) [4], [5].
Based on an equivalent magnetic circuit, NRM is an approach enabling the very quick understood of devices functioning.Computational tools have been developed allowing the automatic assembly of equations [4].Although easy to use, NRM results can be not so accurate in a first step.Thus, the network can be improved by comparing the results with those from the FEM model and rebuilding a more representative NRM.Once the optimal reluctant model is built, it use is very efficient and quick.
Unfortunately, the development of the model can be very extensive, especially in devices with significant leakage flux.
Tools based on FEM provides to the user effective representation of the device.Compared to NRM, the geometric and physical properties are both precisely and quickly described.On the one hand, the FEM is very general and can solve a wide range of problems; on the other hand, resolution times can be prohibitive for pre-sizing actuators.This question is more meaningful in a context of optimization with a large number of parameters.
(2) For a non-linear magnetic material, we have: Without any surface currents, we have the following boundary conditions:

B. Reduced Scalar Potential Formulation
The domain will be split into several simply connected domains in order to verify the existence of scalar potentials [2], [7].
Let us represent the studied domain.From ( 2) and (3), we get in each region: With (4) and ( 5), we get : By integrating the equation above a path ab that belongs to the boundary me  [10], [11], we get: This equation expresses the influence of source terms on the potential.

A. Fundamental Solution and Poles of the Function
The equation ( 9) represents the integral equation deduced from the identity of Green.Where 0 0 y , x represent the pole of the function: The coefficient   0 0 y , x h varies in function of pole's position.There are three possibilities: in the domain, out of domain or at the interface.(11) shows the Green's function used in (9).This function represents an analytical solution of Laplace's equation in a two-dimensional case:

B. Discretization of e C and Matrix System
The boundary e C of the domain e  is discretized with boundary elements. e For each element, we have two determine: the potential  and the magnetic flux  flowing through the element.To discriminate exact solution and numerical approximation, we will use the for element "i".Quantities are constants by element and we use a collocation approach, i.e. the writing of ( 9) at each element centroid.
By discretizing the equation ( 10), we get: This equation can be represented by the following matrix system: where BEM U and BEM Q are vectors of dimension N associated to following expressions: T is an invertible matrix, we can write a relation linking the flux to the potential on the boundary:  H The matrix represented by equation ( 17) is a fully populated matrix.

A. Generalities
In order to take into account the physical behavior of materials and to develop a simple method without decreasing the accuracy of the results, we decompose the magnetic domain into bricks and we introduce a reluctance network inside each of them.This network can optionally contain sources of magnetomotive force.An originality of our approach is that bricks can be subdivised in order to obtain (if needed) a more accurate representation of the magnetic behavior.

B. Development
Let us suppose a region m  , discretized and composed of M bricks.Each brick k contain T flux tubes connecting the central potential to the potentials of facets.kt q is the flow through the facet t of the brick k , so the following relation can be written [12]: The permeance of a tube can be calculated from its permeability kt  , its length kt L and its cross section kt S [4]:

C. Potentials elimination
By isolating the central potential in the equation (18):

D. Flux elimination
Let us consider a facet shared by two flux tubes.The incoming flux comes in from a single tube, and comes out though the other one.This conservation of the flow can also be expressed by ensuring that the sum flux is equal to zero.
For each facet kt : The sum of both equations links potentials of two adjacent bricks.As the number of facets on equation are the same.It is also applicable on a facet shared by a brick and by an element treated with the boundary element method.

E. Calculation of magnetomotive force
According to Ampere's law and equation ( 8) and also assuming a uniform field into a subset of the domain bricks, we can deduce the magnetomotive force of a tube with: where ds represents the section of a flux tube.

F. Coupling between both methods
The equation ( 21) can be rewritten by the following matrix system:

V. CALCULATING FORCES
We use the Maxwell stress tensor approach to compute forces and couples.The force on a simply connected domain is given by [1]:  By discretizing equation ( 12), we get: where kl S is the surface of the segment kl . 1 kl   if the segment is within the boundary of the magnetic region.

VI. RESULTS FEM AND BEM-NRM
In this section, we present the comparisons between the FEM model and the BEM-NRM model of the actuator represented by figure 7. The purpose of this section is to validate the proposed method.
All the results were obtained in static two-dimension case.The idea is to have a sufficiently large number of static results in order to introduce them into a tool where the study the dynamic behavior of the actuator is possible.However, the dynamic simulation is outside the scope of this work.The magnet is represented by the following physical properties: We use the fixed point method to solve the nonlinear system.The nonlinear ferromagnetic material is represented by an initial permeability and a saturation level in equation 13: The figure 8 shows the mesh and the field lines of the model BEM-NRM1.
In order to present the comparisons, we have to consider the variation of the air gap 3 y .The figures 9 and 10 present the force as a function of the errors computed with MaGot (BEM-NRM) and Flux (FEM tool).BEM-NRM2 was obtained by subdivising each brick of BEM-NRM1 in four bricks.In figure 10, FEM1 and FEM2 represent FEM solutions using different mesh size.The idea is to study the evolution of the accuracy and the computation time.The reference solution is obtained by FEM with a very dense mesh.The error is divided by a reference value of force or torque in order to get a relative one.In a similar way, the calculation of the torque has been studied.It shows the advantages of this approach compared to a pure NRM.In fact, the pure NRM model for rotational motion requires many efforts to be built by designers (due to the leakage fluxes).The figures 11 and 12 show the shape of the torque as a function of the errors.The computation time was measured on the same computing server, and same conditions for each 150 different positions.We observe that a very low number of elements and a very low computation time (BEM-NRM1) to get an accurate result.

VII. EVALUATION OF A METHOD'S ROBUSTNESS
The purpose of this section is to evaluate the method's robustness.For the actuator from the previous section, we built four models using the program RelucTool [4], [5] based on NRM.
Depending on the time taken for the design of these models, its accuracy is higher or lower.This fact can be highlighted looking at the design process.With the evolution of the air gap, field lines are distorted, so it is difficult to define these parameters.
For the results presented below we set the parameters observing the results obtained with the FEM, and then adjusting these parameters.A FEM model built using Flux is show in figure 17.During the process of designing, we are often faced with the need to optimize the structures.This means that the models must be robust to parametric variations.So, we changed two parameters of our actuator: . The objective here is to evaluate the robustness of the models.The results are presented in figure 20 and 21, and show that the change of some parameters

Fig. 4 .
Fig. 4. Introducing NRM in the two bricks of the magnetic domain.
figure presents the method to introduce influence of coil into the reluctance network.
matrix, U NRM the potential, Q NRMborder the flux flowing through the border of the device, and F NRM the magnetomotive forces created by sources.Let us notice that NRM P is a sparse matrix and have more unknowns than equation.Thanks to (21) and (16), we can easily eliminate external flux unknowns and build a matrix representing the BEM-NRM

Fig. 16 .
Fig. 16.Model NRM4.Models NRM1 and NRM2 are simple and apparently can be constructed quickly.The reluctances airgap11 and 12 (NRM2) are represented by three parameters: inner radius, outer radius and angle.

Fig. 17 .
Fig. 17.FEM Model.The model NRM2 is closer to the reference (FEM) than the model NRM1 because of the increase of the reluctance airgap11 and airgap12.The NRM2 model represents better the leakages flux.Leakages fluxes around the magnet are also important and this is why the model NRM3 was built.Despite the positive evolution of these results, they are not enough satisfying.Then a new model (NRM4) is constructed by studying the actuator in more detailed and, therefore, by again spending more time of development.According to figure 18 and 19, we get good results with this last model (NRM4).In comparison with this complex procedure, the development of our NRM-BEM approach is very simple and still more accurate.
Table I shows the computation time and number of elements for each test: