Topology optimization of magnetic composite microstructures for electropermanent magnet

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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Research articles

Topology optimization of magnetic composite microstructures for electropermanent magnet

T

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Jaewook Leea, , Tsuyoshi Nomurab,c, Ercan M. Dedec a

School of Mechanical Engineering, Gwangju Institute of Science and Technology (GIST), Gwangju 61005, South Korea Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan c Toyota Research Institute of North America, 1555 Woodridge Avenue, Ann Arbor, Michigan 48105 USA b

A R T I C LE I N FO

A B S T R A C T

Keywords: Magnetic composite Permanent magnet Topology optimization Homogenization Pareto fronts Cross-property bounds

This paper presents topology optimization for the design of magnetic composite applicable to electropermanent magnet. Here, the magnetic composite is built from a periodic microstructure consisting of air, iron and permanent magnet (PM) materials. The combination of non-magnetic, soft and hard magnetic materials in a microscopic scale enables to produce its own persistent magnetic field like PM material, and also enables the control of the magnetic field by an external current like iron material. This work aims to find the optimal microstructure unit cell layout of the electropermanent magnet, and estimate its cross-property bounds. Here, the cross-property bounds connect the effective magnetic permeability and residual flux density, which are calculated using the asymptotic homogenization method. The design objectives (i.e. desired effective properties) are theoretically studied with consideration of application to electromechanical devices. Then, an multi-objective optimization problem to achieve desired effective properties is formulated and solved with a multi-material gradient-based topology optimization formulation. As a result, the optimal composite unit cell layouts that constitute the Pareto fronts are successfully obtained. From the Pareto fronts, cross-property bounds of the electropermanent magnet are numerically constructed and discussed.

1. Introduction A permanent magnet (PM) is an object made of a material that can produce its own persistent magnetic field. It is generally difficult to control the magnetic field generated by a PM because the magnetic permeability of a PM is close to that of a non-magnetic material. A electropermanent magnet is a distinctive concept that can overcome this limitation and control its magnetic field by an electric current in a winding around the PM [1,2]. Due to its unique advantage such as high power to weight ratio, an electropermanent magnet has been applied to various engineering applications. Electromechanical devices such as actuators [3,4] and motors [5,6] were designed using an electropermanent magnet. It was also utilized for anchoring of instrumentation in surgery [7], droplet-based microfluidic systems [8], magnetic levitation conveyors [9], actuation of microfluidic valves [10], and nuclear magnetic resonance systems [11]. The concept of the electropermanent magnet in previous studies was implemented through special macroscopic configurations of PM and iron materials. However, this macroscopic design of electropermanent design might be limited in performance tuning due to the a priori selection of material layout and

⁎

device design. To overcome this limitation, the magnetic composite using a periodic microstructure was proposed in [12]. Fig. 1 describes the concept of the electropermanent magnet proposed in [12]. As shown in Fig. 1, the electropermanent magnet is built from a periodic microstructure consisting of air, iron, and PM materials of specific shape and configuration. Due to the combination of the PM and iron materials at the microscopic scale, the composite has an effective residual magnetic flux density like the PM material, together with an effective magnetic permeability like the iron material. In [12], the asymptotic homogenization formulation was derived for the efficient and accurate calculation of the effective material properties (i.e. effective residual magnetic flux density and magnetic permeability) of the electropermanent magnet. However, the study to determine an optimal microstructure design of the electropermanent magnet has not yet been performed. The design optimization of the electropermanent magnet microstructure is required to apply it to electromechanical devices. In order to find the optimal composite microstructure unit cell, an elaborate design methodology is required to be employed. Based on a simple design approach that relies on intuition, it might be difficult to determine the shape and configuration of the composite constituents

Corresponding author. E-mail address: [email protected] (J. Lee).

https://doi.org/10.1016/j.jmmm.2020.166596 Received 25 November 2019; Received in revised form 1 February 2020; Accepted 8 February 2020 Available online 12 February 2020 0304-8853/ © 2020 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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Fig. 1. electropermanent magnet composed of non-magnetic(i.e. air), soft (i.e. iron) and hard (i.e. permanent magnet) magnetic materials; (a) macroscopic structure, (b) periodic microstructure, (c) microstructure unit cell.

optimization. In [32], microstructure design of porous-core sandwich was carried out. In [33], concurrent topology optimization of microstructured cellular composites was presented. Although a tremendous amount of research on microstructure design using topology optimization has been conducted, design optimization of magnetic problem including non-magnetic (i.e. air), soft (i.e. iron) and hard (i.e. PM) magnetic materials has not yet been comprehensively studied. Accordingly, this work aims to design magnetic composite microstructure unit cell applicable to a electropermanent magnet using topology optimization. As shown in Fig. 1, the composite consists of three materials (i.e. air, PM, and iron materials). For the simultaneous design of three composite components, multi-material topology optimization previously developed for the macroscopic design of magnetic devices [34–39] has been properly modified and adapted in this work. The magnetic composite microstructure should be designed to be beneficial from an engineering point of view. For this, the design objective function needs to be determined through careful consideration of main performances of application devices. The magnetic composite in this work aims to enhance the performance (e.g. magnetic force or torque) of electromechanical devices such as actuators and motors. To achieve this, strong magnetic field generation might be the appropriate design goal of the composite microstructure. Thus, the effective material properties that enable this strong magnetic field are theoretically investigated and numerically validated in order to determine the optimization objective function. Here, the effective material properties incorporate the effective magnetic permeability related to the iron material and the residual flux density related to the PM material. The two effective material properties might compete with each other like the fluid-elastic problem in [40], and the thermo-elastic problem in [41]. By finding the Pareto fronts of a multi-objective optimization problem incorporating the two effective properties, the range of possible effective properties is determined. The Pareto front is a set of Pareto optimal design points, each of which is a point satisfying that there is no other point that reduces one objective function without increasing another one [42]. The Pareto front (i.e. cross-property bounds) might have important physical implications for the design of the magnetic composite that is useful from a practical viewpoint [40]. The paper is organized as follows. In Section 2, the design goal for a

(refer to Fig. 1(c)) for given desired effective properties. Thus, topology optimization is applied in this work for the design of the composite unit cell. Topology optimization [13] is a versatile design tool that aims to find an optimal material distribution for a given design goal. Since topology optimization was first proposed for the structural problem in [14], it has been successfully applied for design involving various physics including thermal-fluid [15], vibration [16], and electromagnetic fields [17]. In the first work on topology optimization [14], a microstrucure with holes is defined in each finite element, and its effective material properties are calculated using the asymptotic homogenization method [18,19]. The optimal topological layout of a macroscopic structure is then acquired by determining the hole size and orientation of each microstructure. Topology optimization initially proposed for the design of a macroscopic structure has been extended to the problem of finding an optimal microstructure unit cell layout satisfying target homogenized effective properties, which is referred to as the inverse homogenization problem [20]. Vigorous research on the inverse homogenization problem has been conducted not only for a single material microstructure design [21–27], but also for composite microstructure design [28–33]. With regard to the single material design works, the unit cell of periodic microstructure was designed using topology optimization for mechanical stiffness and fluid transport in [21,22]. In [23], the unit cell composed of iron material was designed to achieve the target magnetic permeability. In [24,25], topology optimization of the metamaterial unit cell layout was conducted to achieve negative effective permeability. In [26], topology optimization of binary structures method was applied to the microstructure design. In [27], the high-stiffness lattice geometry was optimized using topology optimization. Research on topology optimization of composite microstructures has been also actively carried out. The pioneering work in [28] proposed a composite microstructure design methodology to achieve extreme thermoelastic properties. In [29], topology optimization for the design of multifunctional composites considering structural, thermal, and fluid properties was studied. In [30], a porous composite microstructure was designed for tunable thermal expansion, and its effective property was experimentally validated. In [31], a composite microstructure for a structural-acoustic coupled system was designed using topology 2

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results in the strong magnetic flux density B. Thus, it is obvious that a high effective residual magnetic flux density Beff leads to a strong r magnetic flux density B . More careful thoughts are needed to understand the effect of the effective permeability μeff on the magnetic flux density B because the left-hand side of (4) also include the permeability term. Inside the composite, the permeability term in the left-hand side of (4) might cancel out the effect of the permeability on the surface current strength. However, the magnetic permeability μ outside the composite might be generally fixed as an air permeability. Thus, the high effective permeability μeff of the composite results in weak magnetic flux density B because of weakened PM surface current. This tendency can be confirmed from the analytical study [45] that derives the magnetic field generated by a Halbach cylinder with a relative permeability different from one. In [45], it turned out that high permeability of the Halbach cylinder results in a lower magnetic field. Thus, it can be inferred that a low effective permeability μeff is desired to generate a strong magnetic field by the PM remanence. Here, it is interesting that the desired effective permeability μeff is opposite in two cases with different magnetic field generation sources. In the electromechanical devices, either the external current in the winding or the PM remanence can be the dominant source of the magnetic field generation depending on the operating situation. Thus, the desired effective permeability μeff of the composite might depend on which magnetic field source is dominant in the device. To confirm this finding, a numerical test using a finite element model including a electropermanent magnet is performed. Fig. 2 shows the numerical model built to validate the aforementioned finding about the relation between the effective material properties and magnetic field strength. In the model, the composite corresponding to the electropermanent magnet (width: 0.2 m, height:0.3 m) is located between the winding areas (width 0.04 m, height:0.25 m), where the external current density J is applied. At the line 0.005 m above the top of the electropermanent magnet, the magnitude of the magnetic flux density, B , is investigated to confirm how the effective permeability, μeff , and residual magnetic flux density, Beff r , of the composite affect the strength of a magnetic field generated by the electropermanent magnet. Figs. 3(a) and (b) compare the effect of the effective relative permeability μreff on magnetic field strength in two cases with different

electropermanent magnet is discussed to determine the optimization objective function. Section 3 provides formulations utilized for the microstructure design. Specifically, Section 3.1 reviews the asymptotic homogenization formulation [12] derived to calculate the effective material properties of the electropermanent magnet. Section 3.2 explains multi-material topology optimization formulation proposed for design of the optimal microstructure unit cell layout. Section 4 explains the optimization problem formulation, and numerical design results are presented and discussed in Section 5. Finally, conclusions are provided in Section 6. 2. Design objective of magnetic composite for electropermanent magnet This section discusses the design objectives of the electropermanent magnet microstructure that can lead to the performance enhancement of electromechanical devices. In order to achieve the performance improvement, a magnetic field generated from the composite might have to be strong. Thus, the effective material properties facilitating a strong magnetic field generation will be the design goal of the microstructure unit cell. To determine the effective material properties for a strong magnetic field generation, the magnetostatic governing equation is investigated. The governing equation for the magnetostatic field is derived from Maxwell’s equation as:

∇ × H = J,

(1)

∇ ·B = 0,

(2)

where H is the magnetic field intensity, J is the external current density, and B is the magnetic flux density. The constitutive relation of the case including the composite composed of both iron and PM materials can be written as

B = μ H + Br ,

(3)

where μ is the magnetic permeability, and B r is the residual magnetic flux density of the PM material. By substituting H in (3) into (1), and moving the term incorporating B r into the right-hand side, the following equation is derived:

∇ × (μ −1B) = J + ∇ × (μ −1B r).

(4)

By investigating (4), the effective material properties required to generate a strong magnetic field can be found out. In (4), the permeability, μ , and residual magnetic flux density, B r , are a function of position. Inside the composite, they hold effective properties μeff and Beff r . However, the permeability, μ , outside the composite is generally fixed as air permeability, and the residual flux density B r outside the composite is generally zero. As shown in (4), the magnetic flux density B of the electropermanent magnet is produced by two magnetic field sources: (i) external current J in the winding, and (ii) residual magnetic flux density B r by the PM remanence. The magnetic flux density B resulting from the external current J in the winding can be checked by equating the left-hand side of (4) and the first term of the right-hand side (i.e. J ). Here, it is obvious that the magnetic flux density B become strong when the effective magnetic permeability μeff in the composite is high, and vice versa. Thus, a high effective permeability μeff is desired to generate a strong magnetic field by the external current in the winding. Next, the magnetic flux density B resulting from the PM remanence can be checked by equating the left-hand side of (4) and the second term of the right-hand side (i.e. ∇ × (μ −1B r) ). The second term of the right-hand side is non-zero only at a composite surface if its material properties are homogeneous. Thus, this term can be represented as a surface current, as suggested in an analytical study in [43] and numerical study in [44]. The strength of the equivalent surface current is proportional to the residual magnetic flux density B r , and inversely proportional to the magnetic permeability μ . A strong surface current

Fig. 2. Numerical model investigated to confirm the target effective material properties that will be set as a design goal of the microstructure unit cell. 3

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Fig. 3. Effect of the effective relative permeability μreff on the strength of a generated magnetic field, when the dominant magnetic field source is (a) external current in the winding, or (b) PM remanence.

dominant magnetic field sources. In the figures, the magnitude of calculated magnetic flux density B is plotted along the aforementioned line above the electropermanent magnet. For the first case that the dominant field source is a external current (see Fig. 3(a)), the current density J of the winding is set as 4 × 106A/ m2 corresponding to 80 Ampere with 500 winding turns, and the effective residual magnetic flux density, Beff r is set as 0.02 T. For the second case that the dominant source is a PM remanence (Fig. 3(b)), the current density J of the winding is set as 5 × 105A/ m2 corresponding to 10 Ampere with 500 winding turns, and the effective residual magnetic flux density, Beff r is set as 0.6 T. As expected, the magnetic field tends to be stronger when the effective relative permeability μreff is getting higher for the case that the external current in the winding is a main magnetic field source (see Fig. 3(a)). On the contrary, the lower effective relative permeability μreff tends to generate a strong magnetic field when the PM remanence is the dominant source, as shown in Fig. 3(b). In Fig. 4, the effect of the effective residual magnetic flux density eff Beff r is investigated when the effective relative permeability μr is fixed as 5. As expected, a higher effective residual magnetic flux density Beff r results in a stronger magnetic field in both cases where a dominant magnetic field source is the external current in the winding, Fig. 4, and the PM remanence, Fig. 4(b). Through investigating the numerical test result, the design goals (i.e. desired effective material properties of the composite microstructure) can be confirmed as follows. First, the desired effective permeability μeff differs depending on the dominant magnetic field generating source in the electromechanical devices. When the external current in the winding is the dominant source, maximization of the effective permeability μeff will be the design goal of the composite microstructure. On the other hand, minimization of the permeability μeff will be the design goal when the PM remanence is the dominant source of the device. Next, maximization of the effective residual magnetic flux density Beff r will be the second design goal, no matter which magnetic field source is dominant in the device.

3. Formulations This section describes the formulation applied to find the optimal microstructure unit cell satisfying the design goals discussed above. Section 3.1 summarizes an asymptotic homogenization formulation [12] derived to determine two effective material properties of the composite (i.e. the effective magnetic permeability μeff and residual flux density Beff r ). Section 3.2 explains a topology optimization formulation proposed for the multi-material design of the microstructure unit cell composed of air, iron, and PM materials. 3.1. Asymptotic homogenization to determine effective material properties The asymptotic homogenization formulation [46] of the magnetostatic governing Eq. (1) and (2) can be derived using either scalar potential, or vector potential approach, as shown in [12]. In this work, the scalar potential formulation is applied due to its simpler calculation procedure. In the vector potential formulation, the calculation of the effective residual magnetic flux density Beff r requires the prior calculation of the effective magnetic permeability μeff ; refer to [12]. This means that two effective properties (i.e. μeff , and Beff r ) are coupled, which makes the sensitivity analysis for the optimization much more complex. In contrast, the scalar potential formulation does not require the coupling in the calculation procedure of the two effective properties μeff and Beff r . Due to this simplicity, the scalar potential homogenization formulation is chosen for the composite microstructure unit cell design in this work. In the scalar potential formulation, the magnetic field intensity, H , is represented as a function of the scalar potential Ψ :

H = T + ∇Ψ,

(5)

where T is a vector field satisfying the equation ∇ × T = J , which is derived by substituting (5) into (1), and analytically predetermined using Biot-Savart’s law. Next, by substituting (5) with (3) into (2), the equation for the scalar potential Ψ is derived as 4

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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Fig. 4. Effect of the effective residual magnetic flux density Beff r on the strength of a generated magnetic field, when the dominant magnetic field source is (a) external current in the winding, or (b) PM remanence.

∇ ·[μ (T + ∇Ψ) + B r] = 0.

(6)

( j = 1, 2, 3) and ζ can be obtained, respectively, by solving the cell problem Eqs. (8) and (9) in the composite unit cell with periodic boundary conditions. Please refer to [12] for the detailed derivation process of (8)–(11).

To obtain the homogenization formulation, the scalar potential Ψ is approximated using the double-scale asymptotic series as:

Ψ∊ (x ) = Ψ 0 (x , y ) + ∊ Ψ1(x , y ) + ∊2 Ψ 2 (x , y ) + …,

(7) 3.2. Topology optimization for composite microstructure design

where ∊ is a small parameter representing the ratio of the unit vectors in the microscopic coordinates, y, and macroscopic coordinates, x. Substituting (7) into (6), and then equating the terms with the same order of ∊, cell problem equations are obtained as

∇y ·{μ (y )[ eî + ∇y χi (y )]} = 0 (i = 1, 2, 3),

(8)

∇y ·[μ (y ) ∇y ζ (y ) + B r (y )] = 0.

(9)

This section explains topology optimization applied to the design of the composite microstructure unit cell composed of air, iron and PM materials. The method applied in this work is founded on density- and gradient-based multi-material topology optimization with Helmholtz filtering and Heaviside projection [47]. In topology optimization, design variables are assigned at each finite element inside the design domain, and they determine the material properties of each finite element. In this work, two design variables ϕ1 and ϕ2 are assigned to determine the residual magnetic flux density B r , and magnetic permeability μ of the finite element inside the composite microstructure unit cell. The PM density design variables, ϕ1, determines the PM density field ρPM . Likewise, the iron density design variable, ϕ2 , determines the iron density field, ρiron . The strength of PM magnetization and magnetic permeability are then, respectively, interpolated as the function of two density fields ρPM and ρiron . The detailed formulation of material interpolation is explained as follows. The first step is the regularization of design spaces using Helmholtz filtering. The raw design variables may suffer from an oscillating design like a corrugated shape. Thus, the design variables ϕ1 ∈ [−1, 1] and ϕ2 ∈ [−1, 1] are regularized into filtered variables ϕ¯1 and ϕ¯2 by a Helmholtz partial differential equation (PDE) filtering scheme [47]:

The physical meaning of a variable χi in (8) can be understood by comparing each term in (8) with (6). The variable χi and unit vector eî in (8), respectively, correspond to the scalar potential field Ψ and the vector field T in (6). The term corresponding to the residual magnetic flux density B r in (8) does not exist in (6). Thus, the variable χi can be interpreted as a scalar potential field when the external current J is the unit vector eî and the residual magnetic flux density B r is zero in the composite unit cell on microscopic y coordinate. Likewise, a variable ζ in (9) can be interpreted as a scalar potential field due to the residual magnetic flux density B r when the vector field T is zero in the composite unit cell. Next, by comparing the original scalar potential governing Eq. (6) with the equation derived using (7), the formulation of two homogenized magnetic properties are derived in the following form:

μijeff = eî ·

Breff, i

1 = Y

1 Y

∫Y ⎛μ eĵ + μ∇y χj ⎞ dy, ⎜

⎟

⎝

⎠

∫Y (Br + μ∇y ζ )i dy,

ϕ ⎡ ϕ¯ ⎤ ⎡ ϕ¯ ⎤ − R2∇2 ⎢ 1 ⎥ + ⎢ 1 ⎥ = ⎡ 1 ⎤ ⎢ ϕ2 ⎥ ¯2 ¯2 ϕ ϕ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(10)

(12)

where R is the filter radius. Next, the filtered PM density variable, ϕ¯1, and the filtered iron density variable, ϕ¯2 , determine the PM density and iron density field,

(11)

where Y is the area of the composite unit cell, and the variable χj 5

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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respectively. The PM density field, ρPM ∈ (0, 1], and iron density field, ρiron ∈ (0, 1], are respectively defined as

case, the objective function, Φ1, is defined as

ρPM = Hr (ϕ¯1 )

(13)

Φ1 = w

(14)

On the other hand, for the case that a PM remanence is the dominant source, minimization of the effective permeability and maximization of the PM residual flux density are desired to achieve a strong magnetic field (refer to Section 2). For this second case, the objective function, Φ2 , is defined as

ρiron

= Hr (ϕ¯2 ).

Here, Hr is the regularized Heaviside function [47], which is defined as

⎧∊o ⎪ ¯ Hr (ϕ) = 1 + ⎨2 ⎪1 ⎩

(ϕ¯ < − h) 15 ϕ¯ ( ) 16 h

−

5 8

ϕ¯ 3 h

()

+

3 ϕ¯ ( ) 16 h

(h < ϕ¯ ).

p Br = B¯r ρPM (1 − ρiron ) p ,

Φ2 = −w (15)

Br ·cos(θ) ⎤ , B r ⎛⎜ϕ1, ϕ2⎞⎟ = ⎡ ⎢ Br ·sin(θ) ⎥ ⎣ ⎦ ⎝ ⎠

(16)

(17)

where θ is the angle of PM magnetization direction. In this work, the PM magnetization direction is assumed to be fixed to achieve simple unit cell design results. For the simultaneous design of the PM magnetization direction and optimal PM segmentation, the topology optimization formulation proposed in [36,38] could be applied. Next, the magnetic permeability, μ , is also interpolated using two density fields ρPM and ρiron as p μ (ϕ1, ϕ2) = (μiron − μair ) ρiron (1 − ρPM ) p + μair

4. Optimization strategy This section describes optimization strategies applied for the design of the microstructure unit cell. In order to determine the optimal composite microstructure applicable to the electropermanent magnet, the optimization problem is formulated as

μiron

eff

⎛ ⎞ Br ,2 + ⎜1 − w ⎟ B ⎝ ⎠ r

(26)

VPM (ϕ1) =

1 Y

∫Y ρPM dV

(27)

Viron (ϕ2) =

1 Y

∫Y ρiron dV

(28)

(19)

Maximize Φ(χi , ζ , ϕ1, ϕ2)

(20)

Subject to

(21)

K χi χi = f χi

eff μ22

(25)

where Y is the area of the composite unit cell. In this work, both target volumes VPM , and Viron are equally set as 0.35. Fig. 5 shows an example unit cell of the periodic composite microstructure in two dimension. In this work, symmetry boundary conditions are imposed on the microstructure design result, as shown in the example design in Fig. 5. Thus, only a quarter part of the unit cell is set as the design domain for topology optimization. In the entire unit cell, periodic boundary conditions are applied for the cell problem Eqs. (8) and (9). To solve the cell problem equations in the reduced quarter part, appropriate boundary conditions should be determined. In [41], the boundary conditions for a reduced quarter part was investigated for structural and heat conduction problems. Referring to [41], appropriate

(18)

where μiron , and μair are, respectively, the permeability of iron and air materials. Through the material interpolation formulation in (17) and (18) with (12)–(16), the design variables ϕ1 and ϕ2 assigned at each finite element determine possible materials (e.g. air, PM, and iron materials) in the composite microstructure unit cell.

and ϕ2

eff

⎛ ⎞ Br ,2 + ⎜1 − w ⎟ B ⎝ ⎠ r

In (25) and (26), the iron permeability μiron is set as 1000 times that of the air permeability (electrical steel), and the magnitude of PM residual flux density Br is set as 0.5T (ferrite magnet). A weighting parameter w ∈ [0, 1] is introduced to deal with a multi-objective problem. In this work, the second diagonal component of the effective eff and vertical component of the effective residual flux permeability μ22 eff density Br,2 is assumed as the objective function considering the magnetic field generation in the vertical direction. The system of linear Eqs. (21) and (22) are composed of the stiffness matrices, K χi, K ζ , and the force vectors, f χi f ζ , which are, respectively, derived from the cell problem Eqs. (8) and (9) by finite element formulation. As shown in (23) and (24), the volume of PM material VPM , and iron material Viron are, respectively, constrained as target volumes VPM , and Viron . Here, VPM , and Viron are, respectively, calculated using the PM density field ρPM in (13) and iron density field ρiron in (14):

where B¯r is the maximum value of the PM magnetization strength, and p is the density penalty parameter that is set as a positive integer. Finally, the residual magnetic flux density, B r , is represented as

ϕ1,

μiron

(−h ⩽ ϕ¯ ⩽ h)

where ∊o is a infinitesimal number to avoid singularity, and h is a positive parameter representing the bandwidth between zero (h < ϕ¯ ) and one (ϕ¯ < − h ) domain. The strength of PM magnetization, Br , is then interpolated using two density fields ρPM in (13) and ρiron in (14) as

Find

eff μ22

K ζ ζ = fζ

(22)

VPM (ϕ1) ⩽ VPM

(23)

Viron (ϕ2) ⩽ Viron

(24)

The design variables in (19) includes the PM density variable, ϕ1, and iron density variable, ϕ2 ; refer to Section 3.2. The objective function, Φ in (20), is defined as the weighted sum of two effective material properties. As discussed in Section 2, target effective material properties differ depending on the dominant magnetic field generation source. When an external current in a winding is the dominant source, both the effective magnetic permeability and residual magnetic flux density are required to be maximized to achieve a strong magnetic field. For this

Fig. 5. Example unit cell of periodic microstructure, and one-quarter symmetry part of the unit cell used as the design domain for topology optimization. 6

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Fig. 6. Boundary conditions applied to solve cell problem equations for (a) χ1, (b) χ2 , and (c) ζ .

dissimilar unit cell design results with different values of the objective function Φ . Considering the magnitude of objective function values, the initial distributions of the design variable, ϕ1 and ϕ2 are, respectively, set as

boundary conditions are proposed, as shown in Fig. 6. Fig. 6 (a) and (b) show, respectively, boundary conditions for χ1 and χ2 in (8). Fig. 6 (c) shows boundary conditions for ζ in (9). It is noted again that the variables χ1 and χ2 in (8) is associated with the magnetic field due to the external current in the unit cell, and utilized to calculated the effective permeability μeff . On the other hand, the variable ζ in (9) is associated with the magnetic field due to the PM remanence in the unit cell, and utilized to calculated the effective residual flux density Beff r . The proposed boundary conditions in Fig. 6 are validated by comparing the cell problem analysis result in the entire unit cell with periodic boundary conditions, and the result in the reduced quarter domain with the proposed boundary conditions. In topology optimization for the microstructure unit cell design (i.e. inverse homogenization problem), the initial distribution of design variables highly affects the final design result [21,22]. To apply a gradient-based topology optimization, a non-uniform initial distribution is required because a uniform distribution produces zero sensitivity of the objective function [21,22]. Various non-uniform initial distributions such as a bi-linear function, checkerboard-pattern, and random distribution are tried. Different initial distributions produce

ϕ1initial (y1 , y2 ) = −0.6 + 0.4 × cos(20·2π· y12 + y22 )

(29)

ϕ2initial (y1 , y2 ) = −0.6 − 0.4 × cos(20·2π· y12 + y22 )

(30)

where y1 and y2 are the position in the unit cell coordinate system, where the origin is set as the center of the unit cell. Fig. 7 illustrates the initial distribution (29) and (30) of the design variable, ϕ1 and ϕ2 . For finding the optimal topology of the composite unit cell in Fig. 5, the aforementioned optimization problem (19)–(24) is solved iteratively using the Globally Convergent Method of Moving Asymptotes (GCMMA) [48]. To solve the cell problem Eqs. 21,22), and calculate the objective (20) and constraint functions (23)–(24) with their sensitivities, commercial finite element analysis software, COMSOL V5.3 with Optimization module is utilized in this work. For the convergence criteria of the iterative process, the objective function based approach 7

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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5. Numerical design results The unit cell of the periodic microstructure for the electropermanent magnet is designed using the proposed topology optimization approach. To achieve the design goals discussed in Section 2, two multi-objective functions Φ1 in (25) and Φ2 in (26) are, respectively, applied. The first objective function Φ1 in (25) is for the unit cell design that can be utilized when the external current in the winding is the dominant magnetic field generation source of the device. On the contrary, the second objective function Φ2 in (26) is for the design that can be utilized when the PM remanence is the dominant source in the device. The design results for the first objective function Φ1 in (25) are shown in Fig. 8 and 9. Fig. 8 shows a Pareto optimal set obtained by changing the weighting parameter w. In Fig. 8, , a total of 51 design results with different weighting parameters are marked as black-colored tiny squares, and their regression line is plotted as a blue-colored dotted line, which estimates the cross-property bound of effective permeability and residual flux density of the composite. This bound might have important physical implications for the composite design and its application. The utopia point is located on the right top because the design goal is to maximize both effective permeability and residual flux density. It is interesting to note that the bound for the effective residual eff magnetic flux density Br,2 reaches 1.8 times of the PM residual flux eff density (i.e. Br ), while the range of the effective permeability μ22 is restricted in the iron volume fraction (i.e. Viron ) of the iron permeability μiron . Among the optimization results for Φ1, six representative designs are selected as cases A-F, as shown in Fig. 8. The unit cell design results and corresponding 6 × 6 repetitive arrays of the six representative design results are shown in Fig. 9. In the unit cell design result, black color area represents the iron material, and orange color area represents the PM material. The arrow inside the PM material represents the PM

Fig. 7. Initial distribution of design variables, ϕ1 and ϕ2 .

proposed in [36,38] is applied. In addition, a continuation scheme is applied for the bandwidth, h, of the relaxed Heaviside function (15). The bandwidth, h, starts with 1 and updated as h = h × 0.95 for gradual tightening. The mesh size is set as 0.01, and thus the quarter domain of the composite unit cell is discretized into 50 × 50 first-order finite elements. The filter radius, R, in (12) is set as 0.03 (i.e. three times mesh size), and the density penalty parameter, p, in (16) and (18) is set as 3. The angle of the fixed PM magnetization direction (i.e. θ in (17)) is set as 90∘ in this work.

Fig. 8. Pareto optimal set for the first design objective in (25) (i.e. maximization of both effective permeability and magnetic residual flux density). Unit cell microstructure design results of six representative cases are also provided. The weighting parameter w of the six cases are listed in Table 1. In the unit cell design result, black color area represents the iron material, and orange color area represents the PM material. The arrow inside the PM material represents the PM magnetization direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 8

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

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Fig. 9. (continued) Table 1 Parameters and effective material properties of six representative design results obtained for the first objective function Φ1 in (25).

Fig. 9. Composite microstructure design result for the first design objective in (25). Left figure shows unit cell design result, and right figure shows the corresponding 6 × 6 repetitive array; (a) case A, (b) case B, (c) case C, (d) case D, (e) case E, (f) case F. Black color area represents the iron material, and orange color area represents the PM material. The arrow inside the PM material represents the PM magnetization direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

magnetization direction. The design result for case A (i.e. Fig. 9(a)) is obtained by setting the weighting parameter as zero, which corresponds to the case that the design goal is the maximization of the effective residual magnetic flux density without considering the effective permeability. For this case, zigzag shaped PMs and iron connecting them are designed as shown in Fig. 9(a). Whereas, the design result for case F (i.e. Fig. 9(f)) is obtained by setting the weighting parameter as one, which corresponds to the case that the design goal is only the maximization of the effective permeability. For this case, vertical columnshaped iron without PM material is logically obtained as the unit cell design result. Table 1 summarizes the weighting parameter w, PM volume VPM , iron volume Viron , effective relative permeability μreff , and effective residual magnetic flux density Breff of the six representative design results. As the weighting parameter w increases, the first objective term in (25) (i.e. the second diagonal component of the effective

Case

w

VPM

Viron

μreff

A

0

0.3485

0.3455

0 ⎤ 16.3947 ⎦

B

0.5909

0.3499

0.3453

⎡ 8.4198 ⎣0 ⎡ 6.4006 ⎣0

C

0.593

0.3499

0.3479

0 ⎤ 224.974 ⎦

D

0.603

0.3498

0.3483

⎡ 6.8433 ⎣0 ⎡3.2113 ⎣0

E

0.95

0.3499

0.3499

⎡1.6030 0 ⎤ 350.164 ⎦ ⎣0

F

1

0

0.3499

⎡1.5960 0 ⎤ 350.019 ⎦ ⎣0

Breff (Tesla)

0 ⎤ 161.452 ⎦

0 ⎤ 297.323 ⎦

⎡0 ⎤ ⎣ 0.8852 ⎦ 0 ⎡ ⎤ ⎣ 0.5177 ⎦ ⎡0 ⎤ ⎣ 0.4383 ⎦ ⎡0 ⎤ ⎣ 0.3513 ⎦ 0 ⎡ ⎤ ⎣ 0.1745 ⎦ 0 ⎡ ⎤ ⎣0⎦

eff permeability tensor μ22 ) increases at the expense of the second objective term (i.e. vertical component of the residual magnetic flux density eff Br,2 ). The volume of both PM and iron materials are well constrained to less than the target value of 0.35 in all design results. When comparing Figs. 9(a)-(f), it can be noticed that a vertical column-shaped iron tends to become the main feature, as the weighting parameter w increases, eff which leads to the increase of the first objective term μ22 . Figs. 10 and 11 shows the design results for the second objective function Φ2 in (26). Fig. 10 shows a Pareto optimal set obtained by

9

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

J. Lee, et al.

Fig. 10. Pareto optimal set for the second design objective in (26) (i.e. mimization of effective permeability and maximization of magnetic residual flux density). Unit cell microstructure design results of six representative cases are also provided. The weighting parameter w of the six cases are listed in Table 2. In the unit cell design result, black color area represents the iron material, and orange color area represents the PM material. The arrow inside the PM material represents the PM magnetization direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

changing the weighting parameter w. In this figure, a total of 38 design results with different weighting parameters are marked as black-colored tiny squares, and their regression line is plotted as a blue-colored dotted line. This line estimates the cross-property bound for the magnetic permeability and residual flux density of the proposed magnetic composite. The utopia point in this case is located at the left top because the design goal is to minimize the effective permeability and maximize the residual flux density. When the weighting parameter w is zero, the second objective function Φ2 in (26) becomes same with the first objective function Φ1 in (25). Thus, the design result for the second objective function Φ2 with zero weighting parameter is same with the corresponding design result for the first objective function Φ1 (i.e. case A in Fig. 8). This design result can be also viewed as a solution that balance two objective functions, Φ1 in (25) and Φ2 in (26). When summing Φ1 and Φ2 with same weightings (i.e 0.5 × Φ1 + 0.5 × Φ1), the term for the permeability is canceled out, which is same with the case A in Fig. 8. It is interesting to note that the right side of Fig. 10 can be connected with the left side of Fig. 8, and the design result with zero weighting parameter is shared at the interface. It can be identified that the connected Pareto optimal set completes a mountain shape crossproperty bound of the two effective material properties. In this bound, a peak is located at the shared result with zero weighting parameter (i.e. case A in Fig. 8). Among the optimization results for Φ2 , six representative designs are chosen as cases A-F. Their unit cell design results and corresponding 6 × 6 repetitive arrays are shown in Fig. 11. In the unit cell design result, the black color area represents the iron material, and orange color area represents the PM material. The arrows inside the PM material represents the PM magnetization direction. Table 2 summarizes the weighting parameter w, PM volume VPM , iron volume Viron , effective relative permeability μreff , and effective residual magnetic flux density Breff of the six representative design results. When the weighting parameter w is one, void design result is obtained to minimize the effective permeability without considering the effective residual flux density. As the weighting parameter w decreases, both two objective terms in (26) (i.e. the second diagonal component of the effective permeability tensor eff eff μ22 , and vertical component of the residual magnetic flux density Br,2 )

increases. Here, it is noted that the weighting parameters w in Table 2 are much larger than the parameters in Table 1. This is due to a scale eff / μiron ) in Φ1 and Φ2 . The difference of the permeability term (i.e. μ22 permeability in the minimization problem (i.e. Φ2 ) is much smaller than the permeability in the maximization problem (i.e. Φ1). Thus, much larger weighting parameter w is required to attain a balanced result in the permeability minimization problem (i.e. Φ2 ). The volume of both PM and iron materials are again well constrained less than the target value of 0.35 in all design results. When comparing Figs. 11(a)-(f), it can be noticed that iron material tends to become vertically disconnected and the iron volume tend to be reduced, as the weighting parameter w increases, which lead to the decrease of the first objective eff eff term μ22 while maximizing the second objective term Br,2 . 6. Conclusion In this work, a composite microstructure of the electropermanent magnet is designed using topology optimization. The design goal is set to find the optimal microstructure unit cell that can satisfy target effective material properties. Here the effective material properties covers the effective permeability, μeff , and the residual flux density, Beff r , which are computed using the asymptotic homogenization method. The target material properties are determined through consideration of application into electromechanical devices. It is identified that the desired effective permeability, μeff , differs depending on the operating situation of the device. For the situation that the external current in the winding is the dominant source in the device, maximization of the effective permeability, μeff , becomes the unit cell design goal. On the other hand, minimization of the permeability, μeff , becomes the design goal for the situation that the PM remanence is the dominant source in the device. The maximization of the effective residual magnetic flux density, Beff r , becomes the second design goal, no matter which magnetic field source is dominant in the device. The multi-objective optimization problem for two effective properties μeff and Beff r is formulated and solved with a multi-material topology gradient-based optimization approach. As an optimization result, the optimal configuration and shape of composite constituents are successfully obtained for the given design goals. From 10

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

J. Lee, et al.

Fig. 11. (continued) Table 2 Parameters and effective material properties of six representative design results obtained for the second objective function Φ2 in (26).

Fig. 11. Composite microstructure design result for the second design objective in (26). Left figure shows unit cell design result, and right figure shows the corresponding 6 × 6 repetitive array; (a) case A, (b) case B, (c) case C, (d) case D, (e) case E, (f) case F. Black color area represents the iron material, and orange color area represents the PM material. The arrow inside the PM material represents the PM magnetization direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the optimization results, the Pareto optimal set is constructed, which estimates the cross-property bound of the two effective material properties. Future work will include the experimental validation of the microstructure designed for the electropermanent magnet. The fabrication of the designed microstructure layout will be a challenging but interesting topic.In addition, a functionally graded magnetic composite will be an interesting future work. A design for the composite with spatial variation in the effective properties will enable us to address a macroscopic device model and performance directly. In this work, the effect of the finite coercivity in the ferrite magnet is not investigated yet. A demagnetization effect will be a limiting factor on the performance, which will be a topic for further study.

Case

w

VPM

Viron

μreff

A

0.95

0.3498

0.3469

B

0.968

0.3499

0.3076

⎡ 7.4281 ⎣0 ⎡ 4.7739 ⎣0

C

0.981

0.3499

0.2538

D

0.990

0.3452

0.3387

E

0.995

0.3499

0.2959

F

0.999

0.3490

0

Breff (Tesla)

0 ⎤ 13.8567 ⎦ 0 ⎤ 10.7034 ⎦

⎡3.1820 0 ⎤ 7.4103 ⎦ ⎣0 ⎡138.053 0 ⎤ 3.6470 ⎦ ⎣0 2.3109 0 ⎡ ⎤ 2.2926 ⎦ ⎣0 1 0 ⎡ ⎤ ⎣0 1 ⎦

⎡0 ⎤ ⎣ 0.8882 ⎦ 0 ⎡ ⎤ ⎣ 0.8319 ⎦ ⎡0 ⎤ ⎣ 0.6833 ⎦ ⎡0 ⎤ ⎣ 0.5823 ⎦ 0 ⎡ ⎤ ⎣ 0.4214 ⎦ 0 ⎡ ⎤ ⎣ 0.1713 ⎦

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by the National Research Foundation 11

Journal of Magnetism and Magnetic Materials 503 (2020) 166596

J. Lee, et al.

of Korea (NRF) grant funded by the Korea government (NRF2019R1A2C1002808), and Korea Institute for Advancement of Technology(KIAT) grant funded by the Korea Government(MOTIE) (P0008763, The Competency Development Program for Industry Specialist)..

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