Shape optimization of interface between ferromagnetic material and air in eddy current system using continuum sensitivity analysis and level-set method

Abstract

This paper presents a method for optimizing the shape of the interface between ferromagnetic materials and air in eddy current systems through a continuum sensitivity analysis and level-set method. The continuum sensitivity formula of the eddy current system is obtained as the surface integral form in terms of the velocity field. The shape evolution of the material interface is represented by solving the level-set equation coupled with the common velocity term in the continuum sensitivity formula. The optimization method is validated by shape design examples.

Keywords

Continuum sensitivity analysis ferromagnetic material eddy current system level-set method shape design

1. Introduction

Electromagnetic devices such as induction motors, induction heaters, and transformers involve a time-varying magnetic field carrying induced eddy current. In this eddy current systems, the shape of the ferromagnetic material has a considerable influence on device performance. The ferromagnetic material is usually surrounded by air or an insulator. Therefore, the shape design of the interface between ferromagnetic material and air is one of the important tasks in the development of the system. The sensitivity analysis has been widely applied to optimize the shape of material interfaces in eddy current systems [1–3]. However, this conventional shape optimization method utilizes classical boundary parameterization methods such as the spline, Bezier curve, and NURBS, which involve a considerable number of design variables and complicated work.

The level-set method has been introduced to express the evolving geometries efficiently in many practical applications by solving a first-order Hamilton–Jacobi equation [4–8]. In shape optimization using sensitivity analysis, the continuum sensitivity formula can be derived with respect to the velocity field, and therefore, it can be directly coupled with the level-set equation. The level-set method has been applied successfully for shape optimization using the continuum sensitivity for the electro- and magneto-static systems in previous studies [9–11]. For instance, state-of-the-art studies are to design the conductor shape in electrostatic systems [12] and multiple material shape in linear and nonlinear magnetostatic systems [13,14]. The magnetostatic system that was dealt with in the previous optimization problems is composed of non-conductive materials. Unfortunately, the optimization method using the continuum sensitivity analysis coupled with the level-set method has not been studied yet in the shape design of the eddy current systems including conductive materials.

This paper proposes an optimization method to overcome the disadvantages of the traditional parameterization method and design the shape of ferromagnetic materials efficiently in the eddy current system. In this optimization method, the continuum sensitivity analysis is used to optimize the objective function, and the level-set method is employed to express the shape evolution of the interface between ferromagnetic materials and air. The continuum sensitivity formula is derived in terms of the velocity field based on the material derivative concept of continuum mechanics by using the Lagrange multiplier method and adjoint variable method. The sensitivity formula is coupled with the level-set equation through the common term of the velocity field. Shape variation during design optimization is represented by solving the level-set equation. Shape design examples for a ferromagnetic material-air interface in an eddy current system are presented to show the usefulness of the proposed design method.

2. Continuum sensitivity on material interface in eddy current system

Figure 1 shows an eddy current system consisting of two materials that belong to domains 𝛺₁ and 𝛺₂ with different values for reluctivity 𝜈, conductivity σ, and current density J. It is assumed that the material properties 𝜈 and σ are constant in this system. The magnetic field is generated by quasi-static frequency induction from current J.

Fig. 1.

Interface design of eddy current system.

In the shape design of the system, a general objective function F is defined as $\begin{eqnarray}\displaystyle F=\int _{{\Omega}}g(\mathbf{A},\mathbf{B}(\mathbf{A}))m_{p}d{\Omega}, & & \displaystyle\end{eqnarray}$ (1) where g is the continuously differentiable function, A is the magnetic vector potential, B() is the curl operator, and m _p is a characteristic function of the integral domain 𝛺_p. g is a function of A and B(A). Therefore, the objective function F can be given as the magnetic field [15], system energy [13,14,16], and so on.

The governing differential equations for the state variable A in eddy current system is given by $\begin{eqnarray}\displaystyle {\nabla}\times {\nu}_{i}{\nabla}\times \mathbf{A}_{i}+j{\omega}{\sigma}_{i}\mathbf{A}_{i}=\mathbf{J}_{i}\quad \text{in }{\Omega}_{i}∼\text{where }i=1,2. & & \displaystyle\end{eqnarray}$ (2) The boundary conditions in the eddy current system are given as follows: $\begin{eqnarray}\displaystyle & \displaystyle \mathbf{A}_{1}=\mathbf{0}\quad \text{on }{\Gamma}^{0}, & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \mathbf{A}_{1}}{\partial n}=\mathbf{0}\quad \text{on }{\Gamma}^{1}. & \displaystyle\end{eqnarray}$ (4) The variational form of the state equations is represented with the bilinear form a (A, A ) and the source linear form l ( A ) as $\begin{eqnarray}\displaystyle a(\mathbf{A},\overline{\mathbf{A}})=l(\overline{\mathbf{A}})\quad \forall \overline{\mathbf{A}}\in {\Phi}, & & \displaystyle\end{eqnarray}$ (5) where $\begin{eqnarray}\displaystyle \displaystyle a(\mathbf{A},\overline{\mathbf{A}}) & \equiv & \displaystyle \int _{{\Omega}_{1}}({\nu}_{1}\mathbf{B}(\mathbf{A}_{1})\cdot \mathbf{B}(\overline{\mathbf{A}}_{1})+j{\omega}{\sigma}_{1}\mathbf{A}_{1}\cdot \overline{\mathbf{A}}_{1})d{\Omega}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \displaystyle +\int _{{\Omega}_{2}}({\nu}_{2}\mathbf{B}(\mathbf{A}_{2})\cdot \mathbf{B}(\overline{\mathbf{A}}_{2})+j{\omega}{\sigma}_{2}\mathbf{A}_{2}\cdot \overline{\mathbf{A}}_{2})d{\Omega},\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle l(\overline{\mathbf{A}})\equiv \int _{{\Omega}_{1}}\mathbf{J}_{1}\cdot \overline{\mathbf{A}}_{1}d{\Omega}+\int _{{\Omega}_{2}}\mathbf{J}_{2}\cdot \overline{\mathbf{A}}_{2}d{\Omega}, & & \displaystyle\end{eqnarray}$ (7) and A is the arbitrary virtual vector potential.

The variational state Eq. ((5)) is treated as an equality constraint in the design problem in the eddy current system. For optimizing the design of the system, the Lagrange multiplier method is employed, and an augmented objective function G is introduced as follows: $\begin{eqnarray}\displaystyle G=F+l(\overline{\mathbf{A}})-a(\mathbf{A},\overline{\mathbf{A}})\quad \forall \overline{\mathbf{A}}\in {\Phi}. & & \displaystyle\end{eqnarray}$ (8) The continuum sensitivity is obtained by taking the material derivative of G. $\begin{eqnarray}\displaystyle {\dot{G}}={\dot{F}}+\dot{l}(\overline{\mathbf{A}})-{\dot{a}}(\mathbf{A},\overline{\mathbf{A}})\quad \forall \overline{\mathbf{A}}\in {\Phi}. & & \displaystyle\end{eqnarray}$ (9)

To express the sensitivity $\dot{G}$ explicitly as a surface integral in terms of the velocity field, a variational adjoint equation is defined as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \int _{{\Omega}_{1}}({\nu}_{1}\mathbf{B}(\boldsymbol{\lambda}_{1})\cdot \mathbf{B}(\overline{\boldsymbol{\lambda}}_{1})+j{\omega}{\sigma}_{1}\boldsymbol{\lambda}_{1}\cdot \overline{\boldsymbol{\lambda}}_{1})d{\Omega}+\int _{{\Omega}_{2}}({\nu}_{2}\mathbf{B}(\boldsymbol{\lambda}_{2})\cdot \mathbf{B}(\overline{\boldsymbol{\lambda}}_{2})+j{\omega}{\sigma}_{2}\boldsymbol{\lambda}_{2}\cdot \overline{\boldsymbol{\lambda}}_{2})d{\Omega}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \displaystyle \quad =\int _{{\Omega}_{1}}(\mathbf{g}_{\mathbf{A}_{1}}\cdot \overline{\boldsymbol{\lambda}}_{1}+\mathbf{g}_{\mathbf{B}_{1}}\cdot \mathbf{B}(\overline{\boldsymbol{\lambda}}_{1}))m_{p}d{\Omega}+\int _{{\Omega}_{2}}(\mathbf{g}_{\mathbf{A}_{2}}\cdot \overline{\boldsymbol{\lambda}}_{2}+\mathbf{g}_{\mathbf{B}_{2}}\cdot \mathbf{B}(\overline{{\lambda}}_{2}))m_{p}d{\Omega}\quad \forall \overline{{\lambda}}_{1},\overline{{\lambda}}_{2}\in {\Phi},\end{eqnarray}$ (10) where $\begin{eqnarray}\displaystyle & \displaystyle \mathbf{g}_{\mathbf{A}}\equiv \frac{\partial g}{\partial \mathbf{A}}=\left[\frac{\partial g}{\partial A_{x}},\frac{\partial g}{\partial A_{y}},\frac{\partial g}{\partial A_{z}}\right]^{T}, & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle & \displaystyle \mathbf{g}_{\mathbf{B}}\equiv \frac{\partial g}{\partial \mathbf{B}}=\left[\frac{\partial g}{\partial B_{x}},\frac{\partial g}{\partial B_{y}},\frac{\partial g}{\partial B_{z}}\right]^{T}, & \displaystyle\end{eqnarray}$ (12) and $\boldsymbol{\lambda}$ and $\overline{\boldsymbol{\lambda}}$ are the adjoint variable and virtual potential, respectively. The differential form of adjoint equations is obtained from Eq. ((10)) as follows: $\begin{eqnarray}\displaystyle {\nabla}\times {\nu}_{i}{\nabla}\times \boldsymbol{\lambda}_{i}+j{\omega}{\sigma}_{i}\boldsymbol{\lambda}_{i}=\mathbf{g}_{\mathbf{A}_{i}}m_{p}+{\nabla}\times \mathbf{g}_{\mathbf{B}_{i}}m_{p}\quad \text{in }{\Omega}_{i}∼\text{where }i=1,2. & & \displaystyle\end{eqnarray}$ (13) with the following boundary conditions: $\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\lambda}_{1}=\mathbf{0}\quad \text{on }{\Gamma}^{0}, & \displaystyle\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \boldsymbol{\lambda}_{1}}{\partial n}=\mathbf{0}\quad \text{on }{\Gamma}^{1}. & \displaystyle\end{eqnarray}$ (15)

Using the governing equations and the boundary conditions for the state and adjoint variables in the eddy current system, the general sensitivity formula for the interface variation is derived from Eq. ((9)) as follows: $\begin{eqnarray}\displaystyle {\dot{G}}=\int _{{\gamma}}[({\nu}_{2}-{\nu}_{1})\mathbf{B}(\mathbf{A}_{1})\cdot \mathbf{B}(\boldsymbol{\lambda}_{2})+j{\omega}({\sigma}_{2}-{\sigma}_{1})\mathbf{A}_{1}\cdot \boldsymbol{\lambda}_{2}-(\mathbf{J}_{2}-\mathbf{J}_{1})\cdot \boldsymbol{\lambda}_{2}]V_{n}d{\Gamma} & & \displaystyle\end{eqnarray}$ (16) where V _n is the outward normal component of the velocity field on the boundary 𝛾. Note that, in an eddy current system, we assume the ferromagnetic material has low conductivity because of a mixture of impurities and lamination of the steel sheet, and the source current is not applied to the ferromagnetic material. Then, the sensitivity formula for the shape variation of the interface between the ferromagnetic material and air can be simplified as $\begin{eqnarray}\displaystyle {\dot{G}}=\int _{{\gamma}}({\nu}_{2}-{\nu}_{1})\mathbf{B}(\mathbf{A}_{1})\cdot \mathbf{B}(\boldsymbol{\lambda}_{2})V_{n}d{\Gamma}. & & \displaystyle\end{eqnarray}$ (17)

3. Shape optimization using level set method

The level-set method provides easy and efficient schemes for the representation of shape variation. The level-set equation is written in an implicit form as $\begin{eqnarray}\displaystyle \frac{\partial {\phi}}{\partial {\tau}}+V_{n}|{\nabla}{\phi}|=0, & & \displaystyle\end{eqnarray}$ (18) where 𝜙 is the level-set function and τ is the time. The shape of material interface varying with time is indicated by an iso-contour of the level-set function.

The level-set method is available for the shape optimization design of the material interface by using the velocity field V _n in the continuum sensitivity formula. The velocity field V _n in the level-set Eq. (18) is determined by using the continuum sensitivity formula of Eq. (17) for the shape optimization of the interface between the ferromagnetic material and air as follows: $\begin{eqnarray}\displaystyle V_{n}=k\left[({\nu}_{2}-{\nu}_{1})\mathbf{B}(\mathbf{A}_{1})\cdot \mathbf{B}(\boldsymbol{\lambda}_{2})-\int _{{\gamma}}({\nu}_{2}-{\nu}_{1})\mathbf{B}(\mathbf{A}_{1})\cdot \mathbf{B}(\boldsymbol{\lambda}_{2})d{\Gamma}\left/\int _{{\gamma}}\right.d{\Gamma}\right]. & & \displaystyle\end{eqnarray}$ (19) The velocity field V _n indicates the magnitude of the shape change perpendicular to the surface. The first term on the right-hand side in (19) is the velocity field derived from the sensitivity formula (17) to optimize the value of the objective function. The second term is the average value of the sensitivity along the surface used to maintain an initial material volume. Coefficient k determines the sign of V _n and is assigned 1 or −1, respectively, to maximize or minimize the objective function (1).

4. Design examples

The optimization method using the continuum sensitivity analysis and level-set method is applied to two shape design problems in eddy current systems. Shape design I is intended to validate the proposed optimization method. Shape design II shows the application of the optimization method to a practical design problem. In these problems, the design variable is the shape of the interface between the ferromagnetic material and air. The state and adjoint equations are numerically calculated using the finite element method. During the optimization process, shape variation of the interface is represented by solving the level-set Eq. (18) along with the velocity field represented by Eq. (19).

4.1. Ferromagnetic material–air interface shape design I

The shape design of a ferromagnetic material in an axi-symmetric eddy current system is performed to validate the optimization method. The eddy current system consists of a cylindrical conductor at the center of the system, a sinusoidal source current flowing in the 𝜙 direction, a ferromagnetic material, and air. The initial shape of the design model is shown in Fig. 2. The relative permeability of the ferromagnetic material is 500, and that of conductor and air is 1. The conductivity of the conductor is 6 × 10⁷ S∕m, and that of the ferromagnetic material and air is neglected in this system. Current density and frequency are set to 5 A∕mm² and 60 Hz, respectively.

Fig. 2.

Initial design of the eddy current system I.

The design objective is to minimize the magnetic field B in the conductor region. In this design problem, the zero magnetic vector potential A applies to the outer and symmetry boundaries. Therefore, the objective function F to be minimized can be defined as $\begin{eqnarray}\displaystyle F=\int _{{\Omega}}A_{{\phi}}m_{p}d{\Omega}, & & \displaystyle\end{eqnarray}$ (20) where A _𝜙 is the 𝜙 component of the magnetic vector potential A. This objective function is used to calculate the adjoint equation and adjoint variable $\boldsymbol{\lambda}$ is obtained numerically.

Level-set equation (18) is calculated using the numerically obtained state and adjoint variables to represent the variation in the material shape during optimization. The shape evolution of the ferromagnetic material is shown in Fig. 3. The value of time t describes the shape evolution time when the velocity field of Eq. (19) with k = −1 is applied to the level-set equation. As expected, the shape of the ferromagnetic material eventually becomes a hollow cylinder at t = 500 μs. Figure 4 shows the evolution of the volume-averaged magnetic field B in the conductor with time. The magnetic field decreases by 16.2% from 0.751 mT to 0.629 mT. These design results demonstrate the applicability of the proposed optimization method to the design of ferromagnetic material in eddy current system.

Fig. 3.

Shape evolution during the shape optimization process for the eddy current system I.

Fig. 4.

Magnetic field in the conductor during the optimization of eddy current system I.

4.2. Ferromagnetic material–air interface shape design II

The optimization method is applied to the shape design of a two-dimensional eddy current system, which consists of a coil with sinusoidal current, a hollow cylindrical conductor, a ferromagnetic material, and air. The initial shape of the system is shown in Fig. 5. The material properties and current source conditions are the same as in the previous problem. The shape of the outer surface 𝛾 on the ferromagnetic material is the design variable in this design problem. The design variable is constrained to design domain 𝛺_d shown in Fig. 5.

Fig. 5.

Initial design of the eddy current system II.

The design objective is to obtain the minimum magnetic field B in the region of the conductor as well as the previous design problem. The objective function is determined as $\begin{eqnarray}\displaystyle F=\int _{{\Omega}}A_{z}m_{p}d{\Omega}, & & \displaystyle\end{eqnarray}$ (21) where A _z is the z component of the magnetic vector potential A.

Figure 6 shows the shape evolution of the ferromagnetic material during the optimization process by solving level-set equation (18). The magnetic flux distributions are also represented in Fig. 6. The ferromagnetic material is deformed gradually and separated into two parts. It is noted that the existence of the ferromagnetic material on the y-axis does not have an effect on confining the magnetic field generated by the coil in it. The evolution of the volume-averaged magnetic field in the conductor is shown in Fig. 7. The magnetic field in the conductor is 0.112 mT in the initial model. This decreases by 12.4% in the final model, where the value of the magnetic field is 0.098 mT.

Fig. 6.

Shape and magnetic field evolution during the shape optimization process for the eddy current system II.

Fig. 7.

Magnetic field in the conductor during the optimization of eddy current system II.

5. Conclusions

This paper proposed an optimization method for the shape design of a ferromagnetic material–air interface in an eddy current system. The optimization method was based on the continuum sensitivity analysis, and the level-set method was used to express the shape variation more effectively than the conventional boundary parameterization method. The continuum sensitivity formula is derived based on the material derivative concept of continuum mechanics. Shape variation during the optimization process is represented by solving a level-set equation, where the velocity field in the level-set equation is obtained from the continuum sensitivity formula. The optimization method was applied to numerical examples of eddy current systems to show its usefulness. The design results demonstrated that the proposed method provides an optimal shape of the ferromagnetic material efficiently by solving a simple equation without lots of design variables and complicated calculations.

Footnotes

Acknowledgements

This work was supported by “Human Resources Program in Energy Technology” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea. (No. 20184030202190).

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