﻿ 基于智能优化算法和有限元法的多线圈均匀磁场优化设计<sup>*</sup>
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Optimal design of multi-coil system for generating uniform magnetic field based on intelligent optimization algorithm and finite element method
LYU Zhifeng, ZHANG Jinsheng, WANG Shicheng, ZHAO Xin, LI Ting
Precision Guidance and Simulation Lab, Rocket Force University of Engineering, Xi'an 710025, China
Received: 2018-09-05; Accepted: 2018-11-23; Published online: 2018-12-27 16:28
Foundation item: National Natural Science Foundation of China (11602296, 61503393)
Abstract: To solve the problem of high-order derivation and reliability evaluation of optimization results in the multi-coil system magnetic field uniformity optimization design, an optimization design method based on intelligent optimization algorithm and finite element method is proposed. First, the parameters to be optimized are determined, and the magnetic field deviation rate is taken as the objective function. Then, the objective function is optimized by the intelligent optimization algorithm. Finally, based on the optimized structural parameters, the corresponding finite element numerical simulation model is established to verify the reliability of the optimization results. The structural parameters of two sets of Helmholtz coils are optimized. The simulation results show that the optimal parameters obtained by the proposed method are superior to the parameters obtained by traditional derivation method. And the reliability of the optimization results is confirmed by the finite element numerical method.
Keywords: magnetic field uniformity     optimization design     intelligent optimization algorithm     finite element method     credibility assessment

1 方形线圈均匀磁场的基本理论

1.1 方形线圈轴线上的磁场计算

 (1)
 图 1 单个方形载流线圈 Fig. 1 Single square current-carrying coil

1.2 方形线圈均匀磁场的传统优化设计方法

 图 2 方形亥姆霍兹线圈示意图 Fig. 2 Schematic diagram of square Helmholtz coil

 (2)

B(z)在中心点O附近进行泰勒展开，得

 (3)

 (4)
 (5)

 (6)

 图 3 两组方形亥姆霍兹线圈示意图 Fig. 3 Schematic diagram of two sets of square Helmholtz coils

 (7)

2 多线圈均匀磁场优化设计方法 2.1 智能优化算法基本理论

2.2 磁场的有限元分析方法

2.3 线圈均匀磁场优化设计流程

 图 4 参数优化流程图 Fig. 4 Flowchart of parameter optimization
3 仿真分析 3.1 有效性验证

 轴线长度/m 磁场偏差率 本文方法 传统求导方法 0.2 0.001 03 0.001 26 0.4 0.017 02 0.017 85 0.6 0.072 79 0.074 35 0.8 0.177 05 0.179 13 1 0.311 80 0.313 98

3.2 优势性验证

3.2.1 两组亥姆霍兹线圈均匀磁场优化设计

 结构参数 本文方法 传统求导方法 l/m 0.5 0.5 a1/m 0.147 1 0.128 1 a2/m 0.551 5 0.505 5 N1 70 64 N2 150 150 I/A 0.1 0.097 48

 图 5 两种参数情况磁场分布均匀性对比 Fig. 5 Comparison of magnetic field distribution uniformity between two different parameters

 轴线长度/m 磁场偏差率 本文方法 传统求导方法 0.2 1.847 1×10-5 1.261 6×10-4 0.4 6.552 9×10-5 6.298 4×10-4 0.6 2.685 1×10-4 0.003 9 0.8 0.006 4 0.020 7 1 0.035 6 0.070 2

3.2.2 最优参数的可信度检验与评估

 图 6 两组亥姆霍兹线圈三维数值仿真模型 Fig. 6 Three-dimensional numerical simulation models of two sets of Helmholtz coils
 图 7 不同平面磁场分布 Fig. 7 Magnetic field distribution in different planes

 图 8 z轴磁场分布 Fig. 8 Magnetic field distribution in z axis

 轴线长度/m 磁场偏差率 本文方法 传统求导方法 0.2 8.326 8×10-5 9.240 8×10-5 0.4 3.236 3×10-4 3.740 1×10-4 0.6 3.964 2×10-4 0.001 7 0.8 0.005 6 0.017 4 1 0.035 9 0.067 8

4 结论

1) 较传统的亥姆霍兹线圈，多线圈系统能够扩大磁场的均匀区域，但是其参数优化设计如果仍采用传统的求导优化方法，会增加问题求解的复杂度。

2) 在多线圈系统的参数优化问题中，本文方法要优于传统的求导寻优，对于2组方形亥姆霍兹线圈，在其中心轴线长度为线圈边长的0.2倍、0.4倍和0.6倍的范围内，传统求导方法与本文方法的磁场偏差率不在同一数量级，二者的比值分别为6.83、9.61、14.52，说明由中点沿轴线向两侧延伸，本文方法的优势越发明显。

3) 通过引入有限元法进行验证，一方面能够对本文方法求得的参数的可信度进行检验与评估，另一方面也验证了本文方法优于传统的求导寻优。

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#### 文章信息

LYU Zhifeng, ZHANG Jinsheng, WANG Shicheng, ZHAO Xin, LI Ting

Optimal design of multi-coil system for generating uniform magnetic field based on intelligent optimization algorithm and finite element method

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(5): 980-988
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0524