Home Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
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Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation

  • Hiroshi Masuda EMAIL logo , Yutaro Kanda , Yoshifumi Okamoto , Kazuki Hirono , Reona Hoshino , Shinji Wakao and Tomonori Tsuburaya
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

It is very important to design electrical machineries with high efficiency from the point of view of saving energy. Therefore, topology optimization (TO) is occasionally used as a design method for improving the performance of electrical machinery under the reasonable constraints. Because TO can achieve a design with much higher degree of freedom in terms of structure, there is a possibility for deriving the novel structure which would be quite different from the conventional structure. In this paper, topology optimization using sequential linear programming using move limit based on adaptive relaxation is applied to two models. The magnetic shielding, in which there are many local minima, is firstly employed as firstly benchmarking for the performance evaluation among several mathematical programming methods. Secondly, induction heating model is defined in 2-D axisymmetric field. In this model, the magnetic energy stored in the magnetic body is maximized under the constraint on the volume of magnetic body. Furthermore, the influence of the location of the design domain on the solutions is investigated.

Keywords: Heaviside function; IH equipment; method of moving asymptotes; sequential linear programming; topology optimization
PACS: 02.60.-x; 02.60.Pn

1 Introduction

The induction heating (IH) using eddy current is suitable for molding various papers and printing. The heating performance is determined by the heating speed in the heated area and the uniformity of temperature on the surface of heated area [1]. In the induction heating model in this paper, the secondary iron core is heated by the eddy current, which is induced by the change of magnetic flux derived from primary side. Therefore, magnetic circuit has to be designed so that the magnetic flux excited from the coil on the primary side reasonably pass through the secondary side. The design for the structure of the iron core in primary side is investigated in reference [2]. However, there are few studies that optimize the iron core structure of the secondary side.

One of the most predominant methods to efficiently design the induction heating model is the numerical optimization, which type is mainly classified into size-shape optimization and topology optimization (TO) [3]. While the size-shape optimization is strongly dependent on the initial structure, TO is completely free from the preparation of initial structure. Consequently, TO has the possibility to propose new structure of magnetic circuit in IH model. The procedure of topology optimization is constructed by two segments; the first is finite element method for the field computation, and the second is mathematical programming to solve optimization problem. Moreover, the method to model the structure in design domain is important from the viewpoint of the practical manufacturing. Then, smoothed Heaviside function [4], which is able to effectively binarizing the design domain, is applied to the function controlling the material distribution.

It has been seen that the performance of method of moving asymptotes (MMA) is higher than that of level-set method in [5]. However, the performance comparison between MMA and sequential linear programming (SLP) is not investigated. Therefore, the performance evaluation of mathematical programming among MMA and SLP is achieved in the 2-D magnetic shielding. It is well known that the oscillating behavior of convergence history for objective function is frequently confirmed in SLP iteration. To suppress the oscillating behavior, the fixed move-limit, which is normally required to protect the over correction of design variables, is extended to the value which adaptively changes. Therefore, the proposed method supported by SLP with adaptive move-limit is applied to the IH model, and the structure to maximize the magnetic energy stored in the specified region of the secondary side is investigated.

2 Method of topology optimization

2.1 Magnetic reluctivity using Heaviside function

In this paper, the nonlinear magnetic reluctivity ν (ψ, B2) in the design domain is formulated as follows:

ν(ψ,B2)={1−H(ψ)}ν0+H(ψ)νe(B2),(1)

where ν0 is the reluctivity in vacuum, νe (B2) is the function describing the nonlinear magnetic property. When νe (B2) is replaced with ν0νr, ν (ψ, B2) becomes the linear magnetic reluctivity. The parameter νr is the reciprocal of relative permeability on the magnetic body. Then, ψ is the design variables, and H(ψ) is the smoothened Heaviside function which determines the structure of magnetic body. Then, H(ψ) can be shown as follows:

H(ψ)=316ψh5−58ψh3+1516ψh+12(−h≤ψ≤h),(2)

where h is the transition width that continuously connects the interval between H(ψ) = 0 and H(ψ) = 1.

2.2 MMA based formulation of TO

The nonlinear optimization problem can be formulated as follows:

min.f0(ψ)s.t.fi(ψ)≤fi0,(i=1,2,⋯,m)−h≤ψj≤h,(j=1,2,⋯,n)(3)

where f0(ψ) is the objective function, fi(ψ) ≤ fi0 are constraint condition, m is the number of constraints, and n is number of design variables. The expanded function with convexity is generated every optimization step. The subproblem, which is derived from the expansion using fractional function, is given as follows:

min.F0(k)(ψ)s.t.Fi(k)(ψ)≤fi0,(i=1,2,⋯,m)Lj(k)<αj(k)≤ψj≤βj(k)<Uj(k),(j=1,2,⋯,n)(4)

where k is the number of iteration, αj(k) and βj(k) are parameters that limit the variation of design variables, Uj(k) and Lj(k) are parameters for adjusting the degree of convexity in function, and Fi(ψ) is expanded functions, which is as follows:

Fi(k)(ψ)=ri(k)+∑j=1npij(k)Uj(k)−ψj+qij(k)ψj−L(k),(5)

where the constants ri(k),pij(k), and qij(k) are

ri(k)=fi(ψ(k))−∑j=1npij(k)Uj(k)−ψj(k)+qij(k)ψj(k)−L(k)(6)
pij(k)=Uj(k)−ψj(k)2∂fi/∂ψjψj(k)∂fi/∂ψjψj(k)>00∂fi/∂ψjψj(k)≤0(7)
qij(k)=0∂fi/∂ψjψj(k)≥0−ψj(k)−Lj(k)2∂fi/∂ψjψj(k)∂fi/∂ψjψj(k)<0.(8)

In this paper, the finite element method is used for the evaluation of the objective function, and the adjoint variable method [6] is used for the sensitivity analysis. Then, subproblem shown in (4) is solved using dual method according to reference [7].

2.3 SLP with move limit

When Uj(k)andLj(k) are made to asymptotically approach ±∞, then Fi(k) becomes linear function with respect to ψ. For example, when ∂fi/∂ψ > 0, substituting (6)~ (8) into (5), then Fi(k) can be expressed as follows:

Fi(k)(ψ)=fi(ψ(k))+∑j=1npij(k)Uj(k)−ψj−pij(k)Uj(k)−ψj(k).(9)

Here, expanding pij(k), and transforming Fi(k),(9) can be shown as follows:

Fi(k)(ψ)=fi(ψ(k))+∑j=1n∂fi(ψ(k))∂ψ1−ψj/Uj(k)1−ψj(k)/Uj(k)ψj−ψj(k).(10)

Then, taking the limit Uj(k)→∞,Fi(k) is as follows:

Fi(k)(ψ)=fi(ψ(k))+∑j=1n∂fi(ψ(k))∂ψ(ψj−ψj(k)),(11)

where this formulation is exactly identical to the expanded function for SLP. when ∂fi/∂ψ ≤ 0, taking the limit Lj(k)→−∞, also (11) is obtained.

In order to suppress oscillation of convergence characteristics at the end of the search by the sequential linear programming, move-limit is imposed on the correction of the design variable δψj(k). In reference [4], when the number of iterations exceed an arbitrary value, move-limit is damped. However, this method is not efficient because the number of iterations until convergence differs for each problem. Therefore, in this paper, when the oscillation of the objective function is detected, move-limit is damped. Then, move-limit ζk can be defined as follows:

ζ(0)=0.03hζ(k)=ζ(k−1){(f0(k)−f0(k−1))(f0(k−1)−f0(k−2))≥0}cNζ(k−1){(f0(k)−f0(k−1))(f0(k−1)−f0(k−2))<0},(12)

where N is total number of times the oscillation in objective function is detected, and c is a parameter (0 < c < 1). δψj(k) is defined as follows:

−ζ(k)≤δψj(k)≤ζ(k).(13)

In this paper, two methods are defined: F-SLP (c = 1) where ζ is fixed during optimization process, and R-SLP (c < 1) where ζ is relaxed when the objective function oscillates.

3 Analysis model

3.1 2-D magnetic shielding model

Figure 1 shows a magnetic shield model for preventing the invasion of magnetic flux derived from the source current of 2000 AT. The optimization target of this problem is to determine the topology that minimizes the magnetic energy stored in the target area Ωt. Furthermore, an area constraint condition is imposed so that the area S(ψ) of the magnetic body in Ωd is less than the area constraint value S0. The optimization problem is formulated as follows:

Figure 1 Optimization model
Figure 1

Optimization model

min.f0=12∫ΩtBTνBdSs.t.S(ψ)=∫ΩdH(ψ)dS≤S0−h≤ψj≤hj=1,2,⋯n,(14)

where S0 is set to the 3.5×10−3 m2 which corresponds to 45% of the area in the entire region. Due to the symmetry, the analysis region is divided into 1/4, and the first order triangular element is used as the discretization element. The number of nodes and the number of elements in entire domain are 4,101 and 8,038.

3.2 Induction heating model

Figure 2 shows the axisymmetric analysis model of the IH equipment. The optimization target of this model is to maximize magnetic energy stored in target domain Ωt. The constraint condition is imposed so that the volume V(ψ) of the magnetic body in design domain is less than the volume constraint value V0. Therefore, the optimization problem can be formulated as follows:

Figure 2 Optimization model of IH equipment
Figure 2

Optimization model of IH equipment

min.−f0f0=12∫ΩtBTνBdSs.t.V(ψ)=2π∫ΩdrH(ψ)dS≤V0−h≤ψj≤hj=1,2,⋯n.(15)

The magnetic material 30Z120 and S45C are applied to the primary design domain Ωp and secondary design domain Ωs, respectively. First order triangular element is adopted, and the number of nodes and the number of elements in entire domain are 33,292 and 77,838.

4 Optimization results

4.1 2-D magnetic shielding model

Table 1 lists the optimization parameters. ϵsub and ϵopt are convergence criterions of the subproblem and the main problem, respectively. As shown in Figure 3(a), the initial value of the design variables is set to ψ(0) = 0. The maximum iteration number was set to 500.

Tabel 1

Optimization Parameter for Magnetic Shielding Model

modelmagnetic propertyh[mm]nϵsubϵopt
Magnetic shieldinglinear (μr =1,000)21,97610−410−8
Figure 3 Optimized magnetic shielding structure
Figure 3

Optimized magnetic shielding structure

Figure 3(b), 3(c), and 3(d) show the optimization results of MMA, F-SNP, and R-SLP, respectively. When MMA was applied, the structure of the initial topology is changed into the shielding with two layers. When F-SLP and R-SLP are used, the structure of the initial topology is changed into the shielding with four layers.

Table 2 lists the optimization results. The parameter kopt is the elapsed iteration for topology optimization. When F-SLP was used, the convergence value of the objective function was the best among the three methods. On the other hand, the constraint condition is not satisfied when F-SLP was used.

Table 2

Optimization Results for Magnetic Shielding Model.

methodkoptf0 [nJ]elapsed time [s]S(ψ)/Sall
MMA9638.04.850.45
F-SLP5004.5132.30.47
R-SLP42913.728.10.45

  1. CPU: Intel Core i7 4790K 4.0 GHz & 16.0 GB RAM

Figure 4 shows the convergence characteristics of objective function. When F-SLP was used, the objective function does not converge due to the oscillation at the end of the search. However, the oscillation could be suppressed by using R-SLP.

Figure 4 Convergence characteristics of objective function
Figure 4

Convergence characteristics of objective function

4.2 Induction heating model

Based on the results in the previous section, R-SLP is applied to this problem.

Table 3 lists the optimization parameters. In this problem, two types of design domains are defined. First, the design domain is limited to the primary domain Ωp. Secondly, the whole of the primary and secondary domain Ωp & Ωs is set to design domain. The initial value of the design variables is set to ψ(0) = 0.

Table 3

Optimization Parameter for IH Model.

modelmagnetic propertyh [mm]nϵsubϵopt
IH equipmentnonlinear11,311Ωp3,064Ωp&Ωs10−410−3

Figure 5 shows the optimized structure of IH model. Case 1 is the result of the optimized structure in Ωp. In case 1, V0 is set to the value which corresponds to 40% of the volume in entire design domain VΩp. Case 2 is also the result of the optimized structure in Ωp. V0 is set to the value which corresponds to 20% of VΩp.

Figure 5 Optimized structures of Ωp in IH model
Figure 5

Optimized structures of Ωp in IH model

Figure 6 shows the optimized structure of Ωp and Ωs in IH model. In case 3, from the viewpoint of the practical manufacturing, Ωs must be structurally connected to Ωc as shown in Figure 2. However, these are partially separated. To overcome this unreality of structure, in case 4, the initial structure which partially connect Ωc and Ωs is defined as shown in Figure 6(c). In Figures 6(b), 6(d), the optimized structure of the case 3 is quite similar to that of case 4, except for the structure of the support.

Figure 6 Optimized structures of Ωp and Ωs in IH model
Figure 6

Optimized structures of Ωp and Ωs in IH model

Table 4 lists the optimization results. The elapsed iteration and elapsed time in case 3 increased more than those in case 1. On the other hand, the convergence value of the objective function in case 3 was noticeably improved in comparison with that in case 1.

Table 4

Optimization Results in IH Model.

caseVallV0/Vallkoptf0 [mJ]elapsed time [s]V (ψ)/Vall
1VΩp0.49876.085.20.39
2VΩp0.213376.11960.20
3VΩp + VΩp0.42404642980.40
4VΩp + VΩp0.53253774290.44

  1. Intel Core i7 4790K 4.0 GHz & 16.0 GB RAM

Figure 7 shows the convergence characteristics of objective function. Comparing case 3 with case 4, there is a difference in the convergence characteristics of the objective function, because putting support affected to reduce the magnetic energy stored in the target domain.

Figure 7 Convergence characteristics of objective function
Figure 7

Convergence characteristics of objective function

5 Conclusions

In this paper the topology optimization method, based on the move limit with adaptive relaxation, is proposed and successfully applied to the induction heating problem to maximize magnetic energy in the 2-D axisymmetric field. It is clarified that the proposed method can effectively suppress the oscillation of the objective function. Furthermore, it is also clarified that the simultaneous topology optimization of iron core in primary and secondary side is effective for improvement of energy amount stored in the heated region.

References

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Received: 2017-11-2
Accepted: 2017-11-12
Published Online: 2017-12-29

© 2017 H. Masuda et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2017-0100
Keywords for this article
Heaviside function; IH equipment; method of moving asymptotes; sequential linear programming; topology optimization
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