标量磁位积分方程法计算舰艇三维静磁场

doi:10.12382/bgxb.2022.0622

摘要/Abstract

摘要：

掌握舰艇磁场分布规律是实施舰艇磁防护技术的重要前提,采用积分方程法计算舰艇三维静磁场是有效手段之一。积分方程法通常建立以剖分单元内磁化强度或者磁感应强度为待求量的方程组,基于所求得的源区值获得整个场域的解。若想进一步提高磁场计算精度,需要增加网格密度,代价则是计算机的内存需求和计算量急剧增加。针对这一问题,采用一种以标量磁位为待求量的积分方程。将铁磁源区划分为四面体单元,根据变分原理将未知函数分片展开,按照单元和节点的相对位置关系寻找奇异积分,采用参数替换的方法转化为非奇异积分。球体解析模型表明:使用标量磁位进行建模仅需将源区离散为粗糙的网格单元即可得到高精度的磁场计算结果;网格加密后,标量建模方法在保持计算精度的同时,内存需求和CPU执行时间大幅小于矢量建模方法。开展了三维舰船磁场的虚拟验证实验,对于大型船体等需要精细划分网格的铁磁物体,标量磁位积分方程法计算得到的磁场与有限元商业软件的计算结果吻合较好,并且与矢量建模方法相比更加高效和节约内存。

关键词: 舰艇磁场, 积分方程法, 标量磁位, 四面体单元, 奇异积分

Abstract:

Mastering the magnetic field distribution of ship is an important prerequisite for implementing the ship magnetic protection technology. Using the integral equation method to calculate the three-dimensional static magnetic field of the ship is one of the effective means.The integral equation method usually establishes the equation sets with the magnetization or magnetic flux density in the discrete elements as quantity to be calculated, and therefore obtains the solution of entire field based on the source region values. To further improve the calculated accuracy of magnetic field, it is necessary to increase the grid density at the cost of a sharp increase in computer memory requirement and computation. An integral equation method with scalar magnetic potential as quantity to be solved is used to address this problem. The ferromagnetic source is divided into tetrahedral elements, the unknown function is expanded in pieces according to the variational principle, and the singular integrals are found based on the relative positions of elements and nodes. The singular integrals are transformed into non-singular integrals by parameter substitution. The spherical analytical model shows that the modeling method using scalar magnetic potential only needs to discretize the source region into coarse grid cells to obtain the high-accuracycalculation results ofmagnetic field. After grids encryption, the scalar modeling method can maintain the computational accuracy, while its memory requirement and CPU execution time are significantly lower than those of vector modeling method. A virtual verification experiment is made for the three-dimensional ship magnetic field. For ferromagnetic objects such as ship hulls that need to be finely divided into elements, the static magnetic field calculated by the integral equation method based on the scalar magnetic potential agrees well with the results calculated by the finite element commercial software, and the integral equation methodis more efficient and save memory compared with the vector modeling method.

Key words: ship’s magnetic field, integral equation method, scalar magnetic potential, tetrahedral element, singular integral

中图分类号:

TM153.1

何保委, 周国华, 刘胜道, 宗敬文, 唐烈峥. 标量磁位积分方程法计算舰艇三维静磁场[J]. 兵工学报, 2024, 45(3): 948-956.

HE Baowei, ZHOU Guohua, LIU Shengdao, ZONG Jingwen, TANG Liezheng. Computation of Ship’s 3D Magnetostatic Field Utilizing Integral Equation Method of Scalar Magnetic Potential[J]. Acta Armamentarii, 2024, 45(3): 948-956.

图/表 9

图1 舰艇的四面体离散单元模型

Fig.1 Tetrahedral discrete elements model of ship

图2 四面体单元示意图

Fig.2 Schematic diagram of tetrahedral element

图3 三角形表面局部坐标系

Fig.3 Local coordinate system of triangular surface

图4 场点P位于源动点Q所在平面时的积分区域

Fig.4 Integral area when P is located on the plane of Q

图5 局部坐标系与全局坐标系的关系

Fig.5 The relationship between the local coordinate system and the global coordinate system

图6 铁质实心球体模型和测量点示意图

Fig.6 Schematic diagram of iron solid sphere model and measuring points

图7 球体模型磁场计算结果和离散单元数量的关系

Fig.7 The relationship between the sphere’s magnetic field calculation result and the number of discrete elements

表1 不同方法计算舰艇感应磁场的区别

Table 1 Differences between different methods for calculating ship induced magnetic field

感应磁场分量	标量磁位建模方法			磁感应强度建模方法
感应磁场分量	相对误差/%	计算时间/h	内存消耗/Mb	相对误差/%	计算时间/h	内存消耗/Mb
X_i_x	11.0	0.8	26	20.0	1.6	3533
Z_i_x	8.6	0.8	26	19.4	1.6	3533
X_i_z	4.9	0.9	26	14.2	1.7	3533
Z_i_z	2.7	0.9	26	7.3	1.7	3533

图8 不同方法计算得到的舰艇感应磁场

Fig.8 Ship’s induced magnetic fields calculated by different methods

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