Multiple level set method for multi-material shape optimization in electromagnetic system

Abstract

This paper proposes a multiple level set method for multi-material shape optimization in an electromagnetic system. This method enables the handling of multi-material interfaces simultaneously. The sign combination of the level set functions is used for identifying the different material regions and their interfaces. The continuum sensitivity formula for the multi-material interfaces is adopted for the velocity field in the multiple level set equations. Numerical examples are tested to show the usefulness of the proposed method.

Keywords

Multiple level set method continuum shape sensitivity electromagnetic system finite element method shape optimization

1. Introduction

The level set method has been employed for many problems of shape evolution and optimal shape design thanks to its versatility in treating complicated geometry variation. Besides, the system characteristics can accurately be calculated as avoiding the ambiguity of material near design variables [1, 2]. The feasibility of the level set method has been also shown in optimization problems of electromagnetic systems [3, 4, 5, 6, 7]. Since the conventional level set method uses a single level set function, the design space is divided into two regions, which results in a single interface. This fact limits its application to certain design problems because it can be adopted only for two material system, but cannot deal with multiple material system. Many design problems in electromagnetic devices are cases of multi-interfaces in multiple material systems. For example, a brushless DC machine consists of the four material regions such as iron, air, current, and permanent magnet. The design of its shape involves the shape variations of 6 interfaces: iron-air, iron-current, iron-permanent magnet, air-current, air-permanent magnet, and current-permanent magnet.

For structural optimization in mechanical engineering, the technique treating multi-material has been presented in the previous papers [8, 9, 10]. The studies addressed simple problems in which the design interface is one between two elastic materials with different moduli of elasticity. Most structural design problems concern the shape design of outer boundary surrounded by the air and material interface between elastic bodies. However, electromagnetic systems typically consist of various materials such as conductor, dielectric, air, ferromagnetic material, current, or permanent magnet, etc. Since the materials are different in characteristic properties, their design requires a new systematic methodology for dealing with the multi-material interfaces simultaneously.

This paper proposes a multiple level set method to handle the multi-material interfaces of an electromagnetic system. The material regions are identified by the sign combination of the level set functions. The searching direction of the optimization problem is calculated from the general continuum sensitivity analysis for the multi-material interfaces. Using the proposed method, the diverse interfaces of the electromagnetic system can be simultaneously optimized. Three shape design problems are tested using the proposed method to demonstrate its usefulness.

Figure 1.

Partition of material regions. (a) Single level set method. (b) Multiple level set method.

2. Multiple level set method

When the conventional single level set method is used for a shape design problem, a level set function $\phi$ serves to distinguish the regions and their interface. The level set function is in an implicit form, and its iso-contour at zero level indicates the material interface. The material regions are identified by utilizing the sign of the function as shown in Fig. 1a. The shape evolution is expressed by solving a level set equation [11].

$\displaystyle\frac{\partial\phi}{\partial t}+{V}_{n}\left|{\nabla\phi}\right|=0.$ (1)

where $t$ is the time and $V_{n}$ is the normal component of the velocity field. During the optimization, the direction of the shape evolution depends on the value of velocity field $V_{n}$ . When the velocity field value on the part of material interface is positive or negative, its shape evolves toward outer or inner direction, respectively. The conventional method can be used to design only single material interface of two material system because only one level set Eq. (1) is analyzed.

On the other hand, multiple level set functions are adopted to identify more than two material regions in the multiple level set method. The level set functions are defined as signed distance functions whose zero levels cover all the material interfaces. The signs of the functions are the same at every point in each material region, and one or more signs are changed in the adjacent region. The material regions are distinguished by the sign combination of the functions.

$\displaystyle S=\text{sign}\left({\phi_{1},\phi_{2},\cdots,\phi_{m}}\right).$ (2)

As the intersections of the functions also stand for the material regions, just $m$ level set functions are sufficient to represent up to $2^{m}$ materials. When the number of the materials is less than $2^{m}$ , two or more sign combinations represent one material. Figure 1b shows four material regions distinguished by two level set functions. The sign combination of the level set functions is different in each material region. The number of level set equations expressing the evolution of the material interfaces is equivalent to the adopted level set functions.

$\displaystyle\frac{\partial\phi_{i}}{\partial t}+{V}_{ni}\left|{\nabla\phi_{i}% }\right|=0,\quad\text{for}\quad i=1,\cdots,m.$ (3)

Although the computational cost increases, the number of equations to be solved is minimized by using the sign combination of the functions. The multi-material interfaces are simultaneously optimized by analyzing the level set Eq. (3) together.

Figure 2.

Optimization procedure using multiple level set method.

3. Optimization procedure in electromagnetic system

Figure 2 shows the procedure of shape optimization in electromagnetic systems using the multiple level set method. Before shape optimization proceeds, a definite statement of the problem should be given. The objective function, defined as the regional integral of the magnetic vector potential and field, is as follows:

$\displaystyle{F}=\int_{\Omega}g(A_{z},\textbf{\text{B}}(A_{z}))m_{p}d\Omega.$ (4)

where $g$ is the differentiable function, $A_{z}$ is the state variable, B is the magnetic flux density, and $m_{p}$ is the characteristic function for the integration region. The constraints, the design domain, and the field sources are also defined together. Next, the initial designs of the multiple interfaces are determined. The design variables in the problem are the material interfaces. In each iteration of the optimization, the state and the adjoint Eqs (5) and (6) are numerically solved by the finite element method.

$\displaystyle\nabla^{2}A_{z}=-\mu J_{z}.$ (5) $\displaystyle\nabla^{2}\lambda=\mu\left({g_{A}(A_{z})+\nabla\times\bm{g}_{% \textbf{\text{B}}}(A_{z})}\right)m_{p}.$ (6)

where $J_{z}$ is the current density, $\lambda_{z}$ is the adjoint variable, $\mu$ is the magnetic permeability, and the subscripts $A$ and B mean the partial derivatives with respect to the state and field variables. When the objective function is not converged, all the material interfaces are deformed using the velocity field from the continuum shape sensitivity. The shape sensitivity formula is an interface integral form.

$\displaystyle{\dot{F}}=\sum\limits_{i=1}^{m}\int_{\gamma_{i}}G_{i}(A_{z},% \lambda_{z}){V}_{ni}d\Gamma.$ (7)

The velocity field is taken as follows:

$\displaystyle{V}_{ni}=kG_{i}(A_{z},\lambda_{z}),\quad\text{for}\quad i=1,% \cdots,m.$ (8)

where $k$ is the negative constant for the minimization problem. The velocity field (8) has different forms according to the type of materials located on either side of the interface. The multiple level set method and the shape sensitivity analysis are coupled by substituting Eq. (8) into Eq. (3). Then, the deformations of the material interfaces are calculated by solving the level set Eq. (3).

4. Numerical examples

The multiple level set method can be applied to diverse optimization problems in electromagnetic systems. Three numerical examples are tested to show the feasibility and usefulness of the proposed method.

Figure 3.

Initial design of inductor model.

4.1 Inductor design problem

Figure 3 shows the initial state of the inductor design model. The design objective is to maximize the inductance by deforming the surface shapes of the iron and the current. The objective function is the system energy because the inductance is directly proportional to the system energy.

$\displaystyle{F}=W_{m}=\frac{1}{2\mu}\int_{\Omega}\textbf{\text{B}}(A_{z})% \cdot\textbf{\text{B}}(A_{z})d\Omega.$ (9)

When the system energy is the objective function, the adjoint variable equation is the same as the state equation. The searching direction is to maximize the objective function (9). The constraint is the constant volume of the iron and the current.

$\displaystyle\int_{\gamma_{i}}{V}_{ni}d\Gamma=0,\quad\text{for}\quad i=1,2.$ (10)

The shape sensitivity for two material interfaces is as follows:

$\displaystyle{\dot{F}}=\int_{\gamma_{1}}\left({\frac{1}{\mu_{a}}-\frac{1}{\mu_% {i}}}\right){\textbf{\text{B}}}(A_{za})\cdot{\textbf{\text{B}}}(A_{zi})V_{n1}d% \Gamma+\int_{\gamma_{2}}\left[{\left({\frac{1}{\mu_{i}}-\frac{1}{\mu_{c}}}% \right){\textbf{\text{B}}}(A_{zi})\cdot{\textbf{\text{B}}}(A_{zc})+J_{zc}A_{zc% }}\right]V_{n2}d\Gamma.$ (11)

where the subscripts $a$ , $i$ , and $c$ mean that the variables belong to the air, iron, and current regions, respectively. The velocity fields on each material interface are as follows:

$\displaystyle V_{n1}=\left({\frac{1}{\mu_{a}}-\frac{1}{\mu_{i}}}\right){% \textbf{\text{B}}}(A_{za})\cdot{\textbf{\text{B}}}(A_{zi})$ (12) $\displaystyle V_{n2}=\left({\frac{1}{\mu_{i}}-\frac{1}{\mu_{c}}}\right){% \textbf{\text{B}}}(A_{zi})\cdot{\textbf{\text{B}}}(A_{zc})+J_{zc}A_{zc}.$

Figure 4.

Optimization processes of inductor model with different design variables. (a) Single LSM, Iron surface. (b) Single LSM, Current surface. (c) Multiple LSM, Iron and current surfaces.

Figure 5.

Comparison of objective functions in inductor design problem.

Figure 6.

Initial design of transformer model.

Figure 7.

Optimization processes of transformer model with different design variables. (a) Single LSM, Iron surface. (b) Single LSM, Current surface. (c) Multiple LSM, Iron and current surfaces.

Figure 8.

Comparison of objective functions in transformer design problem.

Figure 9.

Equivalent circuit of transformer.

Figure 10.

Comparison of magnetic flux density distributions in transformer cores.

To show the usefulness of the multiple level set method, its result is compared to the ones by the conventional single level set method. Figure 4 shows the shape variations during the optimization processes using the single and multiple level set methods. In Fig. 4a and b, the iron and the current are deformed by the single level set method, respectively. Although the moving surfaces tend to be round, the fixed surfaces limit their shapes. In Fig. 4c, the proposed method deforms both the materials simultaneously. The cross-section of the current becomes the circle, and the shape of the iron gets almost concentric ring in the final design. The shapes of the iron and the current produce the maximum flux with the same current. Figure 5 shows the comparison of the objective functions during the design processes. The inductances increase by 3.96% and 7.16% over the initial shape, when the iron and the current are varied by the single level set method. However, the final design by the proposed method enhances the inductance by 16.8%. The proposed method could provide the better design than the conventional method.

4.2 Transformer design problem

A shell type transformer is optimized using the proposed method. The design objective is to maximize the mutual inductance between the primary and secondary windings. Figure 6 shows the initial design of the transformer model, which is a classical design by the magnetic circuit method. The objective function, the constraint, the shape sensitivity and the velocity fields used in the previous inductor problem are adopted here again. The volumes of each winding are maintained during the optimization.

The result by the multiple level set method is compared to the ones obtained by the single level set method, as in the previous problem. Figure 7a and b show the optimization processes using the single level set method. The shapes of the movable interfaces are rounded for increasing the flux linkage with respect to the fixed interfaces. In Fig. 7c, both the interfaces of the transformer model are optimized by the proposed method. The shape of the windings gradually varies to be the circular cross-section, and the iron is rounded to match the windings. Figure 8 shows the evolution of the objective functions. While the mutual inductances of the final shapes in Fig. 7a and b increase by 5.03% and 12.7%, respectively, the final design by the proposed method increases the mutual inductance by 77.8%.

The optimum design by the proposed method improves the other performances of the transformer. The larger the mutual inductance, the smaller the leakage inductance. These mean that the magnetic flux in the iron core increases with the same current, and so the voltage increases. Therefore, its capacity is enhanced using the same amount of materials. The increased mutual inductance can be expressed as the increased magnetizing inductance $L_{m}$ in the equivalent circuit of Fig. 9. The increase in magnetizing inductance $L_{m}$ makes the current on it $I_{m}$ small. Since it causes the decrease in winding currents $I_{p}$ and $I_{s}$ , the copper loss could be reduced. Figure 10 shows the flux density distributions when the same voltage source excites the initial and final designs in Fig. 7. The flux distribution is quite uniform in the iron core of the optimum design by the proposed method, whereas the partial concentration of the flux is found near the windings of the others. Thus, the partial saturation can be avoided in the core of the optimum design as using the iron core evenly. In addition, the uniform flux distribution leads to the less core loss than the others. The core loss due to the hysteresis characteristic and the eddy current is commonly proportional to about square of the magnetic flux density [12].

$\displaystyle\begin{array}[]{l}P_{h}=K_{h}\nu\left(B\right)^{l},\quad 1.5<l<2.% 5\\ P_{e}=K_{e}\left({B\nu}\right)^{2}\\ \end{array}.$ (13)

Figure 11.

Initial design of solenoid actuator model.

where $K$ is the constant determined by the material properties and dimension of the core, $\nu$ is the frequency, and the subscripts $h$ and $e$ mean the hysteresis and the eddy current, respectively. According to the calculation using the field variable, the final designs of Fig. 7a–c could generate lesser core loss than the initial design by 4.75%, 7.05%, and 36.6%, respectively. Although the results of the conventional method are also the better design than the initial model, the optimal design obtained by the proposed method is more improved one in terms of the performances mentioned above.

4.3 Solenoid actuator design problem

A pull-type solenoid actuator is optimized by the multiple level set method. Figure 11 shows its initial design state. The winding is placed in the E-shaped core and the iron plunger moves between two legs of the core. They initially have rectangular cross-sections. The plunger slides down when the current is applied to the winding, and experiences much increased force as its position gets lower. Since the large force difference between the ends of the stroke is unwanted in industrial applications, the input current is usually controlled by using the external semiconductor circuit. The external input control deteriorates the system efficiency and makes the system bigger. The design objective of this problem is to minimize the force variation during the stroke of the actuator as the surfaces of the plunger and the winding are designed together. The magnetic force on the plunger is expressed as the derivative of the magnetic system energy with respect to the displacement.

$\displaystyle f_{p}=\frac{dW_{m}}{dy}.$ (14)

When the system energy linearly varies with the displacement, the magnetic force is constant. Therefore, the objective function to be minimized is defined as a summation of the energy differences at the plunger positions through the stroke.

$\displaystyle{F}=\sum\limits_{y}{\left[{W_{m}(y)-W_{mt}(y)}\right]}^{2}.$ (15)

where $W_{mt}$ is the target energy value which is a linear function of the displacement. The profile of $W_{mt}$ is determined by the least square method based on the energy values of the initial design. The shape sensitivity for the objective function is as follows:

$\displaystyle{\dot{F}}=2\sum\limits_{y}{\left[{W_{m}(y)-W_{mt}(y)}\right]}\dot% {W}_{m}(y)=2\sum\limits_{y}{\left[{W_{m}-W_{mt}}\right]}\left[{\int_{\gamma_{1% }}{\left({\frac{1}{\mu_{a}}-\frac{1}{\mu_{i}}}\right){\textbf{\text{B}}}(A_{za% })\cdot{\textbf{\text{B}}}(A_{zi})V_{n1}d\Gamma}+\int_{\gamma_{2}}{J_{zc}A_{zc% }V_{n2}d\Gamma}}\right].$ (16)

The velocity fields on each material interface is calculated as

$\displaystyle V_{n1}=-2\sum\limits_{y}{\left[{W_{m}-W_{mt}}\right]}\left({% \frac{1}{\mu_{a}}-\frac{1}{\mu_{i}}}\right){\textbf{\text{B}}}(A_{za})\cdot{% \textbf{\text{B}}}(A_{zi})$ (17) $\displaystyle V_{n2}=-2\sum\limits_{y}{\left[{W_{m}-W_{mt}}\right]}J_{zc}A_{zc}.$

The design domain is limited to prevent any contact among the design variables and the fixed core.

Figure 12.

Design process of solenoid actuator model.

Figure 13.

Evolution of objective function in solenoid actuator design problem.

Figure 14.

Comparison of solenoid actuator characteristics. (a) System energy. (b) Force on plunger.

Figure 12 shows the shape variations of the interfaces during the optimization. The upper part of the plunger widens and its length shortens while the optimization proceeds. The final shape of the plunger at $t=$ 12 s in Fig. 12 makes the air-gap shorter and longer than the initial design at the topmost and bottommost positions, respectively. Meanwhile, the cross-section of the winding is varied to attain the convex shape near the plunger, contrary to the concave middle part of the plunger. After the design process is finished, the plunger and the winding volumes are reduced by 8.6% and 2.4%, respectively. The objective function decreases in the optimization process and converges after $t=$ 12 s as shown in Fig. 13. Figure 14a shows that the system energy profile of the final design is closer to the target values than the initial design. This leads to the decreased force difference. The force profiles of the initial and final designs are compared in Fig. 14b. The force difference between the ends of the stroke is reduced by 53% in the final design. When the force range of the final design is acceptable for a specific application, the external circuit is unnecessary.

5. Conclusion

This paper proposed the multiple level set method for addressing the shape optimization problem of multi-material interfaces in electromagnetic systems. The multiple level set functions are adopted and their sign combination is utilized for identifying the material regions. This method simultaneously optimizes multiple material interfaces by solving corresponding number of level set equations. Since the velocity fields in the equations are obtained from the general continuum shape sensitivity formula, the various types of material interfaces can be treated accurately. Three numerical examples were tested by the proposed method. The comparisons of the results by the conventional and the proposed methods showed that the system characteristics were more improved using the optimum designs by the proposed method. As the design variable is diverse by the proposed method, the design space of the optimization would be broader than the one by the conventional method. The proposed method will be helpful in providing a performance-enhanced design of various electromagnetic systems.

Footnotes

Acknowledgments

This paper was supported by Samsung Research Fund, Sungkyunkwan University, 2013.

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