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Multiscale Finite Element Analysis of Linear Magnetic Actuators Using Asymptotic Homogenization Method

  • Jaewook LeeEmail author
Original Research
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Abstract

This work presents the multiscale finite element analysis of linear magnetic actuators. Here, the actuators includes unidirectional fiber reinforced magnetic composites as a back-iron core component. The composite is employed in actuators to enhance the magnetic force. However, the direct computation of actuators including heterogeneous composite structures requires high computation cost. To overcome this problem, multiscale computational technique for the analysis of a magnetic actuator is proposed in this work. First, the effective magnetic permeability of the composite is calculated at various fiber volume fractions and orientation angles. For this, the asymptotic homogenization method is applied to the composite unit cell model in microscopic coordinate system. Next, the obtained homogenized effective permeability is utilized for the macroscopic magnetostatic analysis of actuator model. Subsequently, the magnetic force acting on a actuator plunger is calculated using the Maxwell stress tensor method. To validate the accuracy and computational benefit of the proposed multiscale approach, a actuator numerical example is provided. For the accuracy validation, the magnetic force and magnetic field distribution obtained from the proposed multiscale approach are compared with those from the direct calculation. In addition, the computation time consumed for the mutiscale approach and direct calculation is compared to validate the benefit of the proposed analysis process.

Keywords

Magnetic actuators Magnetic composite Magnetic force Asymptotic homogenization method Finite element method 

Introduction

A magnetic linear actuator is an electromechanical device that produces the linear motion of a moving part using the magnetic field generated by electric currents [1]. The main actuator design goal is a strong movement at a small actuator size. To achieve this design goal, magnetic composite materials can be utilized as an actuator component. Vigorous research activities has been performed in the field of magnetic composites [2, 3, 4, 5, 6, 7, 8, 9, 10]. Soft magnetic composites are a representative type of the magnetic composites, which can be explained as ferromagnetic powder particles surrounded by an electrical insulating film [2]. Due to their advantages such as high magnetic saturation and low eddy current loss [3], the soft magnetic composite materials have been applied to the back-iron core component of electric motors [3, 4]. In [5, 6, 7], various types of polymer matrix hard magnetic composite materials have been investigated for permanent magnets. In [8, 9, 10], a magnetic composite composed of ferromagnetic materials and metallic conductive materials with a periodic ladder structure was proposed and investigated for magnetic field manipulation in an actuator.

The magnetic composite studied in this work is unidirectional fiber reinforced magnetic composites as shown in Fig. 1. The composite is composed of high permeability ferromagnetic (i.e. fiber) material and low permeability ferromagnetic (i.e. matrix) material. Although the composite is composed of isotropic fiber and matrix materials, its homogenized permeability is anisotropic due to the geometric anisotropy of the composite structure (refer to base cell in Fig. 1). The anisotropic magnetic permeability enables us to manipulate the magnetic field distribution, which might be utilized to enhance the actuator performance (i.e. actuator magnetic force) [8, 9, 10]. This advantage of the unidirectional fiber reinforced composites is different with that of soft magnetic material. It is noted again that the soft magnetic material is composed of ferromagnetic materials with an insulating film, which has advantages of high magnetic saturation and low eddy current loss. The use of the unidirectional fiber reinforced magnetic composite does not always guarantee the performance enhancement. In order to achieve the maximized actuator force enhancement, both the macroscopic composite configuration and the composite microstructures should be determined appropriately using the design optimization procedure. To perform the iterative design optimization procedures efficiently, multiscale computational techniques might be a useful analysis tool.

Accordingly, this work presents multiscale analysis of the linear magnetic actuator including unidirectional fiber reinforced magnetic composites. Multiscale analysis is a computational technique based on a philosophy of multilevel methods [11]. It should be noted again that the direct computation of a heterogeneous composite material often requires high computational costs, which cause difficulties in the iterative design optimization process. This difficulty can be overcome by applying the multiscale analysis procedure using the homogenized effective material property. While there has been a tremendous amount of research on multiscale analysis related to structural problems [12, 13, 14, 15], the research on multiscale analysis dealing with magnetic problems is relatively rare.

In this work, multiscale finite element analysis is proposed to calculate the magnetic force \(\mathbf {F}\) of the actuator including unidirectional fiber reinforced magnetic composites. Figure 1 shows the base cell of the composite in the macroscopic and microscopic coordinate systems with the magnification factor \(1/\epsilon\). Here, \(\epsilon\) is a small parameter representing the ratio of the unit vector in two coordinate systems. The effective material property of the composite is calculated using the asymptotic homogenization method [16] in the microscopic \(y_1y_2\) coordinate system. More specifically, the homogenized magnetic permeability \(\mu ^H\) is calculated at various fiber material volume fractions, and fiber orientation angles. Subsequently, the magnetostatic finite element analysis is performed in the macroscopic \(x_1x_2\) coordinate system using the homogenized magnetic permeability \(\mu ^H\). As the magnetostatic analysis result, the magnetic field distribution in an actuator is obtained, and the magnetic force \(\mathbf {F}\) acting on the actuator plunger is calculated using the Maxwell stress tensor method. The accuracy and computational benefit of the proposed multiscale analysis procedures are validated in a actuator numerical example. In the example, two analysis models are built and their analysis results are compared. The first model utilizes actual periodic heterogeneous composite structures in the finite element model. On the other hand, the second model utilizes the proposed multiscale approach with homogenized composite structures. To validate the accuracy of the proposed multiscale approach, the magnetic force and magnetic field distribution obtained from two models are compared. In addition, the computation time consumed for each model is compared to validate the computational benefit of the proposed approach.

The paper is organized as follows. Section 2 explains the proposed multiscale analysis process to calculate the magnetic force of an actuator. Here the asymptotic homogenization method is applied for bridging two scales. In Section 3, a numerical examples are provided to validate the accuracy and computational benefit of the proposed approach. Finally, conclusions are provided in Section 4.
Fig. 1

Base cell of unidirectional fiber reinforced magnetic composites with macroscopic and microscopic coordinate system for asymptotic homogenization

Multiscale Analysis of Magnetic Actuators

In this section, the multiscale analysis procedure to calculate the magnetic force of actuators are explained. The first step is the calculation of effective material properties of magnetic composites using the asymptotic homogenization method [16] in the microscopic coordinate system. Subsequently, the magnetostatic analysis in the macroscopic coordinate system is performed by applying the obtained homogenized effective material properties. From the magnetostatic analysis result, the magnetic force acting on a actuator plunger is finally calculated using the Maxwell stress tensor method. The detailed formulation for each step is as follows.

Asymptotic Homogenization in Microscopic Scale

The effective magnetic permeability tensor \(\varvec{\mu }^H\) of the composite is calculated using the asymptotic homogenization method [16]. Fig. 2 shows the unit cell of the unidirectional fiber reinforced magnetic composite utilized in this work. The composite consists of high permeability ferromagnetic material (i.e. fiber material) and low permeability material (i.e. matrix material). The volume fraction of the fiber material is given as \(l_f\), and the fiber orientation angle between original \(y_1y_2\) and rotated \(y_1^{\prime }y_2^{\prime }\) coordinated systems is set as \(\theta\). The relative magnetic permeability \(\mu _r\) of the fiber and matrix materials is, respectively, set to 20,000, and 500 in this work.
Fig. 2

Base cell of the magnetic composite utilized in this work with parameters for fiber volume fraction \(l_f\) and orientation \(\theta\)

In the derivation of the asymptotic homogenization method for magnetostatic analysis, either scalar or vector potential formulation is available. In this work, the asymptotic homogenization method based on the scalar potential formulation [17] is applied. In this formulation, the cell problem equation in microscopic \(y_1y_2\) coordinate system is derived as:
$$\begin{aligned} \nabla _y\cdot \big \{\mu (y)[\hat{\mathbf {e}}_i+\nabla _y\chi _i(y)]\big \}=0 \end{aligned}$$
(1)
where \(\hat{\mathbf {e}}_i\) means ith component unit vector. The partial differential equation (1) is solved in the unit cell model shown in Fig. 2 with periodic boundary conditions using the finite element method. As a result, the scalar variable \(\chi _i\) can be calculated in the microscopic y coordinate. It is noted that the cell problem Eq. (1) is solved using the finite element commercial software COMSOL V5.3.
Next, the homogenized effective permeability tensor \({\mu _{ij}^H}\) can be obtained from the obtained scalar variable \(\chi (y)\) using the following formulation:
$$\begin{aligned} \mu _{ij}^H=\hat{\mathbf {e}}_i\cdot \frac{1}{|Y|}\int _Y(\mu \hat{\mathbf {e}}_j+\mu \nabla _y\chi _j)dy \end{aligned}$$
(2)
where |Y| is the volume of the base cell. Please refer to [17] for the detailed derivation of (1) and (2) using the asymptotic homogenization method with the scalar potential formulation. It is noted that the Eqs. (1) and (2) for isotropic fiber and matrix materials can be easily extended to equation for anisotropic materials (refer to [17])
Next, the homogenized permeability \({\varvec{\mu }^H}^{\prime }\) in the rotated \(y_1^{\prime }y_2^{\prime }\) coordinate system is obtained as
$$\begin{aligned} {\varvec{\mu }^H}^{\prime }=\mathbf {R}^T \varvec{\mu }^H \mathbf {R} \end{aligned}$$
(3)
where the transformation matrix \(\mathbf {R}\) is given as
$$\begin{aligned} \mathbf {R}= \begin{bmatrix} \cos (\theta )&-\sin (\theta ) \\ \sin (\theta )&\cos (\theta ) \\ \end{bmatrix} \end{aligned}$$
(4)
Figure 3 shows the obtained homogenized relative permeability \(\mu _r^H\). Figure 3a is the plot of the three components of the homogenized relative permeability \(\mu _r^H\) at various fiber material volume fraction \(l_f\) when the orientation angle \(\theta\) is fixed as zero. It is observed that the \(y_1y_1\) component of the homogenized permeability tensor \(\mu _r^H\) increases linearly as the fiber material volume fraction \(l_f\) increases, while the \(y_2y_2\) components suddenly increase to the maximum value when the fiber material volume fraction \(l_f\) reaches near one. The trend of the homogenized permeability tensor is different in \(y_1\) and \(y_2\) directions. The reason of this different trend is due to the geometric anisotropy of the composite structure (refer to base cell structure in Figs. 1 and 2). Regarding the homogenized permeability in the \(y_1\) direction, the magnetic field flow would be obstructed by the low permeability matrix material structure even in the case of small matrix volume fractions (i.e. high fiber volume fractions \(l_f\)). Thus, the homogenized permeability in \(y_1\) direction \(\mu ^{H'}_{r,11}\) stays at low value in wide range of fiber volume fractions \(l_f\). In contrast, the magnetic field can flow through high permeability fiber material structure even in the case of small fiber volume fractions \(l_f\). The extent to which the magnetic field penetrates in \(y_2\) direction might be proportional to the fiber volume fraction. Thus, the homogenized permeability in \(y_2\) direction \(\mu ^{H'}_{r,22}\) is linearly increased with respect to the fiber volume fractions \(l_f\). Figure 3b is the plot at various fiber orientation angle \(\theta\) when the fiber material volume fraction \(l_f\) is fixed as 0.5. The obtained data-set of the homogenized relative permeability \(\mu _r^H\) for a unit cell shown in Fig. 2 is applied for the macroscopic scale analysis of an actuator.
Fig. 3

Homogenized relative permeability \({\varvec{\mu _r^H}}^{\prime }\) of the magnetic composite shown in Fig. 2. a Plot at various fiber material volume fractions \(l_f\), b plot at various fiber orientation angles \(\theta\)

Magnetic Analysis in Macroscopic Scale

The magnetostatic analysis of the actuator model is performed in the macroscopic scale by solving the Maxwell’s equation. For the magnetostatic analysis, the Maxwell’s equation is modified using the vector potential \(\mathbf {A}\) formulation as:
$$\begin{aligned} \nabla \times \Big (\varvec{\mu }_0 {\varvec{\mu _r^H}}^{\prime }\nabla \times \mathbf {A}\Big )=\mathbf {J} \end{aligned}$$
(5)
where \(\varvec{\mu }_0\) is the air permeability, and \(\mathbf {J}\) is the external current density. Here, the homogenized permeability tensor in the rotated coordinated system \({\varvec{\mu _r^H}}^{\prime }\) can be obtained using the data-set prepared from the microscopic analysis. It is noted again that this data-set is obtained by performing the asymptotic homogenization analysis (14) of the magnetic composite unit cell in the microscopic coordinate system.

To solve the above magnetostatic Eq. (5), the system of linear equations is prepared and solved using finite element commercial software COMSOL V5.3. As the analysis result, the magnetic field distribution of an actuator macroscopic model is obtained.

Next, the magnetic force \(\mathbf {F}\) acting on the plunger of an actuator can be calculated using the Maxwell stress tensor formulation
$$\begin{aligned} \mathbf {F}=\Bigg [\oint {\frac{1}{2\varvec{\mu }_0} \big (B_n^2-B_t^2 \big ) dA}\Bigg ]\mathbf {n}+\Bigg [\oint {\frac{1}{2\varvec{\mu }_0} B_nB_t dA}\Bigg ]\mathbf {t} \end{aligned}$$
(6)
where \(B_n\) and \(B_t\) are normal and tangential components of magnetic flux density \(\mathbf {B}\) (= \(\nabla \times \mathbf {A}\)) at the integration path.

Numerical Example

In this section, a numerical example is provided to validate the effectiveness of the proposed multiscale analysis to calculate the magnetic force of an actuator. Figure 4 shows the model of an actuator numerical example. The back iron part around the coil area is set as unidirectional fiber reinforced magnetic composites. Here, the configuration of the magnetic composite unit cell is same with Fig. 2, and its orientation angles varies as provided in Fig. 4. The magnitude of the electric current density J in the coil area is set as 10,000 \(\mathrm{A/m}^2\). The relative permeability of the plunger is set as 500, which is same with that of the matrix material of the composite.
Fig. 4

Actuator Numerical Example Model

To validate the accuracy and computational benefit of the proposed multiscale analysis, two analysis processes applied to the numerical example. In the first analysis process, the actual periodic heterogeneous structure of the magnetic composite is built in a finite element model. In this model, \(\epsilon\) (i.e. the ratio of the unit vector in two coordinate systems) is set to 0.005. This means that 200 unit cells are located in the macroscopic unit vector. In the first process, the macroscopic scale analysis is solely performed to calculate the magnetic force of an actuator. On the contrary, the second analysis process applies the proposed multiscale approach that utilizes the homogenized magnetic permeability prepared in the microscopic scale analysis. In Fig. 5, the two analysis processes are described in detail.
Fig. 5

Two analysis processes as performed in numerical examples

Table 1

Comparison of magnetic force and computation time obtained in two analysis models

 

Magnetic force F in x1 direction

Magnetic force F in x2 direction

Computation time

Model with actual heterogeneous composite structures

18.030 N

290.60 N

64.8 s

Multiscale model using homogenized properties

18.007 N

290.14 N

3.1 s

Fig. 6

Magnetic field contour lines of a model with actual heterogeneous composite structures, and b multiscale model using homogenized effective permeability

The accuracy of the proposed multiscale approach is validated by comparing the analysis results obtained from the aforementioned two analysis processes. More specifically, the magnetic forces \(\mathbf {F}\) and magnetic flux density \(\mathbf {B}\) distributions obtained from two analysis processes are compared for the validation. In addition, the computational benefit of the proposed multiscale approach is confirmed by comparing the computation time consumed for the two analysis processes.

In Table 1, the magnetic forces obtained from two analysis processes are provided. As expected, the magnetic force calculated using the proposed multiscale analysis process is well matched with the force obtained from the model using actual heterogeneous composite structures. The relative errors of the magnetic forces are merely 0.1275\(\%\) in \(x_1\) directional force and 0.1583\(\%\) in \(x_2\) directional force. In addition, Fig. 6 compares the magnetic field distributions obtained from two analysis processes. Here, the direction of magnetic field contour lines represents the direction of magnetic flux density \(\mathbf {B}\), and the density of the contour lines represents the strength of the magnetic flux density \(\mathbf {B}\). The magnetic field contour lines obtained from the multiscale analysis process in Fig. 6b is well matched with the lines obtained from the model using actual heterogeneous composite structures in Fig. 6a. The comparison result of the magnetic forces and field distributions confirms that the analysis result obtained using the proposed multiscale analysis might be trustworthy in terms of accuracy.

Next, the computational costs consumed for two analysis processes are compared in Table 1. The computation time for the proposed multiscale analysis process requires 1/20 less than the time for the analysis of the actual heterogeneous composite structures. It is noted that both analysis processes are performed using a workstation with an eight-core 4.2 GHz processor and 64 GB of RAM. The degree of freedom for the heterogeneous and multiscale models are, respectively, 3,753,785 and 32,882. The comparison result of the computation time confirms that the computational benefit of the proposed multiscale analysis process. The benefit of the multiscale analysis will become more evident in the design optimization process that requires an iterative analysis procedures. To perform the design optimization of fiber reinforced magnetic composites in actuators, topology optimization [18, 19] might be a promising tool, which will be performed in future work.

Conclusion

In this work, the multiscale finite element analysis is performed to calculate the magnetic force of actuators. Here the actuator includes unidirectional fiber reinforced magnetic composites as back-iron core components. To bridge scales, the asymptotic homogenization method is applied in this work. Using the homogenization method, the effective magnetic permeability is calculated at various fiber volume fraction and orientation angles. The obtained effective permeability is applied to calculate the actuator magnetic force in macroscopic coordinate system. To validate the accuracy and computational benefit of the proposed multiscale analysis, two analysis models are built in the actuator numerical example. The first model utilizes the actual heterogeneous composite structures, and directly calculate the magnetic force in macroscopic coordinate system. On the other hand, the second model utilizes the multiscale process with homogenized material properties. By comparing the magnetic force and field distribution obtained from two models, the accuracy of the proposed multiscale analysis procedure were validated. In addition, the computational benefit is also validated by comparing the computation time consumed for two models. The future work will be topology optimization of unidirectional fiber reinforced magnetic composites in actuators.

Notes

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016 R1D1A1B03931138), and Global University Project (GUP) grant funded by the GIST in 2018.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringGwangju Institute of Science and Technology (GIST)GwangjuSouth Korea

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