Magnetic Field Calculation of a Novel Electromagnetic Clutch Based on an Interpolating Element-Free Galerkin Method

In this paper, the interpolating element-free Galerkin method (IEFGM) is introduced to calculate the magnetic field distribution of a novel permanent magnet bistable electromagnetic clutch (BPMEC), which is used in the in-wheel motor drive system to flexibly connecting the wheel and motor to release the mechanical shock and electromagnetic impulsion. Firstly, the fundamental of the IEFGM is introduced, and a new weight function for magnetic field calculation is proposed. Then the nodes distribution is discussed, and the magnetic field distribution of the clutch is analyzed. Simulation results show the IEFGM can impose the boundary conditions directly and fewer nodes are employed, make the calculation efficiency higher. Finally, some experiments are carried out, the experimental results verified the effectiveness of the IEFGM in magnetic field calculation. The study results provide references for the application of the interpolating element-free Galerkin method in the magnetic field calculation.


I. INTRODUCTION
Because of its simple mathematical structure and great versatility, the finite element method (FEM) is the most widely used solution method for potential problems [1]. However, the FEM is difficult in some complicated problems, such as fracture analysis and moving conductor problems. The domain of these problems needs remesh at every step during the simulation, resulting in loss of accuracy. Thus meshless methods that do not require a mesh are developed, and the most practical of them is the element-free Galerkin method (EFGM) which was first proposed and developed by Belytschko et al. in [2]- [5]. In the field of electromagnetic field calculation, the EFGM has also been applied and improved [6]- [10].
The discrete equations of the EFGM and the FEM are the same, and the only difference between the FEM and the EFGM is the construction methods of the shape function. Compared with the FEM, the EFGM use moving leastsquares (MLS) approximation to obtain the shape function The associate editor coordinating the review of this manuscript and approving it for publication was Su Yan . and it only requires a set of nodes and a description of boundary conditions. However, the shape function obtained by MLS approximation does not satisfy the property of Kronecker delta function, so the essential boundary conditions cannot be imposed directly [11], [12]. Therefore, some additional methods are required to apply the boundary conditions, such as penalty methods and Lagrange multiplier methods.
Authors of [13] presented an interpolating moving least-squares (IMLS) method which can obtain the shape function that satisfied the property of Kronecker delta function, thus the boundary conditions can be applied directly as in FEM, and the error estimates of this method are discussed in [14]. Based on the IMLS method, the interpolating element-free Galerkin method (IEFGM) with higher computational efficiency is derived [13]. The IEFGM has been discussed in two-dimensional and three-dimensional potential problems [13], [15], and also be used to solve elastoplasticity problems [16], [17]. However, the application of the IEFGM in specific magnetic field calculations has not yet been studied.
In this paper, the interpolating element-free Galerkin method is introduced to calculate the magnetic field VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ distribution of a novel permanent magnet bistable electromagnetic clutch (BPMEC). The clutch is used in the in-wheel motor drive system to flexibly connecting the wheel and motor. The mechanical shock and electromagnetic impulsion in the drive system can be significantly reduced through this flexible connection [18]- [21]. As the clutch works in a self-holding steady state most of the time, the holding force in steady state is the most important index, thus the magnetic field analysis needs to be discussed. This paper developed the magnetic field analysis of the clutch based on the IEFGM, and the comparison with the FEM and experimental results verified the effectiveness of the IEFGM in magnetic field calculations. First, the fundamental of the MLS and the IMLS method is introduced. Second, the structure of the electromagnetic clutch is described and the boundary conditions, as well as discrete equations, are presented. Then a new weight function is proposed and the magnetic field distribution is discussed. Finally, experiments are carried out to validate the effectiveness of the IEFGM. The study results provide references for the application of the IEFGM in magnetic field calculation.

II. MLS METHOD
Assume the field function is u(x), where x = (x, y) in twodimensional, and the value of n nodes are given by (1): the vector form is: Define the trial function as: where p(x) is the basis function, a(x) is the coefficients. In 2D, the basis function is defined as follows. Linear basis: Quadratic basis: p T (x) = [1, x, y, x 2 , xy, y 2 ] In order to determine the coefficients a(x), a weighted function is introduced as: where w(x − x i ) is the weight function with compact support property, the coefficients can be obtained by minimizing J , as in (7). where and Substituting (7) into (3), the trail function is derived as: where N i (x) is the shape function and In the MLS, the shape function does not satisfy the property of Kronecker delta function as: so the boundary conditions cannot be applied directly.

III. IMLS METHOD
Compared with the MLS, the IMLS method revised the basis function p(x). p 1 (x) is normalized as: for j = 2, 3, . . . , m, the functions are generated as follows: where Using the new basis functions b (1) x (x), the corresponding trail function is: where Then according to the MLS approximation, the coefficients are expressed by (19), where and Equation (16) can be written as: where the new shape function is: and the partial derivative of the shape function is:  where k is x or y, and The new trail function satisfied: thus by the IMLS method, the boundary conditions can be applied as convenient as in the FEM.

IV. THE NOVEL ELECTROMAGNETIC CLUTCH
As shown in Fig. 1, the main components of BPMEC are base, permanent magnet (PM), coils, slider, and slideway. The clutch relies on the magnetic force of the slider imposed by PM to maintain in steady state without energy consumption. When the clutch needs actuation, applied current to coils appropriately, the magnetic force imposed on the slider will change direction and make the slider move to the other side, then turn off the current, the clutch returns to the steady state. The clutch has a compact structure to match the limited available space in the hubs, as illustrated in Fig. 2. By installing this clutch, the wheel and motor can realize a flexible connection, thus the mechanical shock and  electromagnetic impulsion can be released at the starting and braking process of the electric vehicles.
It can be seen that the clutch structure is symmetric so that the analysis model can be simplified, as shown in Fig. 3, the main parameters are shown in table 1.

V. IEFGM FOR ELECTROMAGNETIC PROBLEMS A. DISCRETE EQUATIONS
In the electromagnetic field, The boundary problem of the magnetic vector potential is: The Galerkin weak form of (30)-(32) is: According to the IMLS method, the magnetic vector potential A z at position x can be expressed as: Substituting (34) into (33), the discrete equations can be derived as: where  Gaussian integration is used to calculate equation (35). Since the value of nodes is obtained, the magnetic flux density in the model can be calculated.

B. THE SELECTION OF WEIGHT FUNCTION
In this paper, a modified weight function for the IEFGM is proposed, as expressed in (39).
where α is an even positive integer, and choose α = 12 in this paper.
The rectangular region is selected as the domain of influence, as in (40): where and where d m is a scalar parameter that determines the size of the influence domain, and x, y is the node spacing.

VI. MAGNETIC FIELD CALCULATION OF THE BPMEC A. BOUNDARY CONDITIONS
As shown in Fig. 4, The magnetic field and boundary conditions of the BPMEC is given by: The node distribution on the rectangular model is shown in Fig. 5, 31 × 41 nodes are employed in this calculation, and the weight function (39) as well as the linear basis function (4) are used. Fig. 6 depicts the magnetic field distribution of the clutch calculated by the interpolating element-free Galerkin method,   which in excellent agreement with the analysis result by FEM shown in Fig. 7. Fig. 8 shows the analysis result where the weight function presented in [13] is used, it can be seen that the magnetic field distribution is deformation. Therefore, the new weight function (39) proposed in this paper is more precise when used in magnetic field calculation by IEFGM. In addition, the calculation result when distributed 61 × 81 nodes is illustrated in Fig. 9. It is observed the magnetic lines of force become smoother, and the magnetic field distribution is the same as the results by FEM.
Further analysis indicated that it is required to distribute 61 × 81 nodes to get a precision result when analyzed by the EFGM, as shown in Fig. 10, and the spent time is 14.8 s. Meanwhile, when calculated by the IEFGM, we can obtain sufficiently accurate results by employ 31 × 41 nodes, and the spent time is 3.2 s. Therefore, the IEFGM has a higher efficiency than EFGM.

C. MAGNETIC FLUX LEAKAGE
The magnetic flux leakage coefficient reflects the holding force of the slider in steady state and the usage of the    permanent magnet. Fig. 11 illustrates the curve of the flux leakage coefficient varies with the air gap between the slider and permanent magnet changes. It is clear that with the increase of the air gap, the flux leakage coefficient increases, which means the force imposed on the slider by the permanent magnet decreases. Table 2 depicts the detailed value of the flux leakage coefficient k σ calculated by the IEFGM and FEM, it is evident that the result of the IEFGM is very close to FEM. Besides, when the air gap changes, the grids are required to be remeshed when using the FEM; as for the IEFGM, the node distribution  shown in Fig. 5 is always employed and no adjustment is required. Fig. 12 shows the flux leakage coefficient when the width of PM changing, it can be seen that increasing the width of PM can improve the usage of PM, however, the cost of the clutch will be higher, thus the selection of PM width w m = 4 mm.

D. IMPROVEMENT OF THE ANALYSIS MODEL
Compared with the 3D model shown in Fig. 1, it is clear that the 2D analysis model shown in Fig. 2 cannot take the flux leakage on the front and back region of the clutch into account, resulting in the smaller flux leakage. Therefore, the 2D model is improved, and the idea is: firstly, enlarge the magnetic leakage of the upper region by symmetric the upper and lower region; then move the edge of the base to align with the edge of the slider to further compensate the flux leakage. The improved 2D model is shown in Fig. 13, it can compensate the flux leakage on the front and back region, besides, the magnetic field calculation is simpler, and the spent time is 1.5 s which less than half of the spent time by original 2D model. Fig. 14 illustrates the comparison of the leakage coefficient between the 3D model and improved 2D model, it can be seen that the calculation results of these two methods are quite close when the gap smaller than 1.5 mm, and the maximum relative error is less than 10%.

E. FORCE ANALYSIS
The equivalent magnetic circuit of the clutch is shown in Fig.15. According to Fig. 15, the magnetic circuit equations VOLUME 8, 2020   are express in (48), The magnetic flux density of the main magnetic circuit is obtained by solve (48), as expressed in (52),   then based on the virtual displacement method, the force on the slider can be defined by (53).
The force analysis model is shown in Fig. 16. Define f * as the per unit value of the force, and the base value is expressed in (54).

VII. EXPERIMENT
In order to validate the magnetic field calculation results by IEFGM, the clutch prototype (without slideway and coils) and experimental platform are built, as shown in Fig. 17.
The holding force of the clutch when the slider at different positions is illustrated in Fig. 18. It is clear that the results calculated by the IEFGM in excellent agreement with the experimental results. According to Fig. 18, the clutch has two steady states: 'touch' state when the slider close to the permanent magnet on the left side; 'detach' state when the slider is close to the permanent magnet on the right. In addition, when the slider at the center of the clutch, the horizontal force imposed on the slider is zero, but a slight movement of the slider will make the clutch transition to the above steady state, so this is an 'unstable' balanced state. The force on the slider when the clutch in steady state is the holding force, i.e., the force when δ = 0 mm and δ = 3mm in Fig. 18.
According to equation (54), the base value of force is related to the remanence of the permanent magnet, thus if a larger holding force is needed, one can consider using rare-earth permanent magnets that have a larger remanence and better resistance to demagnetization.

VIII. CONCLUSION
This paper introduces the interpolating element-free Galerkin method to the application of magnetic field calculation. A modified weight function is proposed and its effectiveness is demonstrated by studying the magnetic field distribution of a novel permanent magnet bistable electromagnetic clutch. Compared with the traditional EFGM, the IEFGM is convenient to impose boundary conditions and only requires to employ fewer nodes to get accurate results, make the calculation efficiency higher. Based on the IEFGM, the magnetic field of the electromagnetic clutch is analyzed, an improved 2D analysis model is proposed. The improved 2D model solves the problem that the original 2D model cannot consider the magnetic flux leakage on the front and back region of the clutch. Then the force analysis of the clutch is carried out, and the holding force when the clutch in steady state is obtained. Finally, a clutch prototype experimental platform is built, and the experimental results verified the accuracy of the IEFGM in magnetic field analysis.
FAN YANG received the B.S. degree in electrical engineering and automation from Hunan University, Changsha, China, in 2011. He is currently pursuing the Ph.D. degree with the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China.
His research interests include electric vehicles, wheel motor design and control, electromagnetic devices, and magnetic field analysis.
CHENGLIN GU was born in Shangrao, Jiangxi, China, in 1954. He received the Ph.D. degree from the Huazhong University of Science and Technology, Wuhan, China, in 1989.
He has been a Visiting Professor with the University of Manchester Institute of Science and Technology (UMIST), U.K., since 1995. He is currently a Professor with the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology. His research interests include new type of electric machines and drives, including synchronous reluctance motor and permanent magnet machines for electric vehicle applications.
Dr. Gu was a Founding Member with the International Society of Computational Magnetism (ICS). He was also a member of the Chinese Society of Electronics. VOLUME 8, 2020