A New Simple Method to Design Degaussing Coils Using Magnetic Dipoles

Abstract

Since submarines are mostly made of ferromagnetic materials, anti-submarine warfare aircraft detect submarines using all kinds of magnetometers. In order to make the submarine magnetically silent, it is usually equipped with degaussing coils to neutralize the magnetic anomaly. However, with the increased size of the submarine, more coils are needed by conventional degaussing methods, and the degaussing system becomes complex to design and implement. To simplify the design of submarine degaussing coils and improve their degaussing accuracies, this paper presents a novel and efficient method of the degaussing coil design, which is based on the simplest equivalent model of multiple magnetic dipoles. First, the influence of the magnetic moment and spatial distribution of multi-magnetic dipoles on the equivalent effects with different spatial scales were studied. Then the simplest model of multiple magnetic dipoles was proved to capably model complex ferromagnets. We simulated the degaussing coils of a submarine by COMSOL Multiphysics software to verify the validity of the simplest modeling method and the design of the coils. The simulation results show that the magnetic anomaly induced by the submarine was reduced by at least 99% at different ranges.

1. Introduction

With the continuous progress of submarine silencing technology, most of the submarine’s underwater noise has been close to (or even lower than) the level of the ocean background noise. Acoustic detection alone cannot address the need for modern anti-submarine warfare and optical methods have also been severely challenged [1]. Aeromagnetic exploration technology has the advantage of a large detection range, short execution time, and not being affected by a complex marine environment [2,3], so it is considered by the west as the most reliable non-acoustic detection method to improve the anti-submarine warfare [4]. Represented by P-3C and P-8A anti-submarine aircraft of the US Army, they are equipped with high-precision magnetic anomaly detection equipment, and magnetic anomaly detection has become an important means of underwater target detection [5].

The main sources of a submarine’s magnetic field are, firstly, the main part, called the fixed magnetic field, which is induced by ferromagnetic materials magnetized by the Earth’s magnetic field in long-term accumulation, secondly, the so-called induced magnetic field caused by a submarine’s operation in the Earth’s magnetic field, thirdly, eddy currents by rolling, and the last is the stray field due to various onboard equipment [6,7,8]. The fixed magnetic field is acquired during the submarine’s construction. The stray field is the least contributor, which is mainly due to the onboard generator, distribution cable, communication system, and so on [9].

Compared with the induced magnetic field, eddy currents (by rolling), and the stray field, the fixed magnetic field contributes the most, and the induced magnetic field changes according to the position, course, and azimuth of the submarine under the Earth’s magnetic field [10]. Therefore, in order to make the submarine magnetically silent, the countermeasure system should be installed aboard a submarine to reduce its effect on the Earth’s magnetic field disruption and makes the submarine virtually undetectable by magnetic mines or other devices [11]. The system design is mainly aimed at the influence of the fixed magnetic field [12]. There are two main ways to construct a degaussing system—deperming stations (so as to temporarily eliminate the magnetization), and degaussing coils installed onboard to generate an opposing field, which is comparatively more efficient and, thus, more widely used [13,14]. Precisely speaking, the purpose of degaussing is to minimize the effect of the submarine’s fixed magnetic field to a level that is undetectable at different altitudes [15], as shown in Figure 1. So, in the design of degaussing coils, the most important thing is to decide the optimum degaussing currents and installation positions individually allotted to degaussing coils [16].

In order to accurately determine degaussing currents and the installation positions of degaussing coils, a distributed equivalent model of the submarine is needed. For vessel targets, high-precision magnetic field models commonly used include uniformly magnetized rotating ellipsoid array models, uniformly magnetized rotating ellipsoid and magnetic dipole array hybrid models [17], numerical models based on the three-dimensional finite element method, etc. [18,19,20]. In engineering applications, the finite element method has some problems, such as a large number of calculations and difficulty in obtaining real-time distributions of spatial magnetic fields. The other models mentioned above are based on the idea of magnet simulation, which uses magnetic fields generated by magnetic sources with specific magnetic moments to simulate the magnetic field of vessels [21]. However, there are also some problems with the magnet simulation method: (1) it is difficult to obtain magnetic field distribution data of large vessels by means of field measurements; (2) in order to establish a high-precision magnetic field model, the simulation accuracy is generally improved by increasing the number of magnetic sources, and the increase in the number of magnetic sources also increases the amount of computer fitting calculations, which brings great complexity to the solution of the equivalent equation. In addition, for the determination of the direction, magnitude, and distribution of the magnetic moment in the magnet simulation method, there is no clear and effective method system at present. The traditional submarine equivalent degaussing coils have to consider vertical, longitudinal, and athwartship coil installation at the same time [22], resulting in a high implementation–complexity installation.

To simplify the design of submarine degaussing coils and improve the degaussing accuracy, a coil design method based on a multi-magnetic dipole simplest-equivalent model is proposed in this paper. Given the problems existing in the magnet simulation method, the influence of the number, distribution, and magnetic moment direction of the equivalent magnetic dipoles on the equivalent effect of the magnetic field distribution on the measuring plane under different spatial scales was studied, and the basis for establishing the simplest equivalent model of multiple magnetic dipoles was given. A submarine model was taken as an example. COMSOL Multiphysics simulation software was used to evaluate the equivalent effect of the model. COMSOL Multiphysics is a commercial finite element package that allows users to build complex models simply using a GUI. Further advantages of COMSOL are its ability to couple different physical effects in the same model and that it has a large built-in library of meshing tools, numerical solvers, and post-processing tools [23]. On this basis, the simplest design method of submarine degaussing coils was proposed. According to the simplest equivalent model of the multi-magnetic dipole, the design requirements of the coil currents and installation positions were clearly proposed, which greatly simplified the existing design process of degaussing coils. Finally, the degaussing effect was verified by COMSOL Multiphysics.

2. Theoretical Basis of the Magnet Simulation Method

From the point of view of the atomic structure of matter, the spin-induced interaction between electrons in ferromagnetic materials is very strong, and under this interaction, small spontaneously magnetized regions called magnetic domains are formed in ferromagnetic materials [24]. Magnetic domains are small magnetized areas with different directions in order to reduce magnetostatic energy during the spontaneous magnetization of ferromagnetic materials. Domain structure is a basic component of the ferromagnet [25].

When the ferromagnet is in the neutral state of original demagnetization, the magnetization vector of each domain in the ferromagnet is taken in different directions, so the magnetism is not displayed externally at the macro level [26,27]. When ferromagnets are in an external magnetic field, domains with small angles between the direction of spontaneous magnetization and the direction of the external magnetic field expand with the increase of the external magnetic field and further shift the direction of magnetization of the domain towards the direction of the external magnetic field. Some domains with large angles between spontaneous magnetization and external magnetic field gradually decrease in volume, and ferromagnetism appears macroscopic. As the external magnetic field increases, these effects increase until all magnetic domains are saturated along the external magnetic field. Because the magnetic moments of each unit in each magnetic domain have been neatly arranged, they have strong macroscopic magnetism [28].

According to the domain theory above, the magnetic induction field distribution of ferromagnets in magnetized space is formed by the superposition of a series of magnetic domains in the ferromagnet. The volume shape distribution of the magnetic domain is closely related to the microscopic properties of the material. For the microstructure magnetic domain, the magnetic field distribution can be equivalent to that of the magnetic dipole outside a certain height range. The superimposed magnetic fields of multiple magnetic domains can be equivalent to magnetic objects with known magnetic field distribution, which is called the magnet simulation method or equivalent magnetic source method. For the vessel magnetic induction field, the magnetic field models of magnetic targets are commonly used, including the rotating ellipsoid model, magnetic dipole array model, and the hybrid model of the rotating ellipsoid and magnetic dipole array [29].

3. Effect of Magnetic Moment Distribution of the Equivalent Magnetic Dipole Array on an Equivalent Effect

3.1. Definition of the Spatial Scale

In practical engineering applications, compared with the other two models, the magnetic dipole array model greatly reduces the computational complexity of the model, and at the same time, it has good equivalent accuracy when the magnetic field distribution is equivalent at a long distance. At present, the influence of the number of magnetic dipoles, the size and direction of the magnetic moment, and the distribution of magnetic dipoles on the equivalent effect of ferromagnet magnetic field distribution in magnetic dipole array model has not been systematically studied.

The study and analysis of the equivalent effect of magnetic field distribution of the ferromagnet is based on the specific measurement plane and the structural size of the ferromagnet. Therefore, the concept of the spatial scale is proposed in this paper, i.e., the spatial scale is defined as $\alpha =H/L_T$, where H is the height of the measurement plane and $L_T$ is the maximum structural size of the ferromagnet or magnetic dipole array.

3.2. Determination of the Magnetic Moment Direction for Multi-Magnetic Dipole Equivalent Model

Submarines and other vessel targets are generally made of high-strength alloy steel, and their shells are ferromagnetic materials. In the process of processing and manufacturing, repeated changes in residual stress of the material, the rise and fall of temperature, and the influence of the local geomagnetic field, will cause the formation of non-hysteresis magnetization in ferromagnetic materials, which is attributed to the permanent magnetic field of the target [30]. The formation of the permanent magnetic field is the static magnetization process of ferromagnetic materials, and the magnetization of internal magnetic domains mainly includes the displacement magnetization process and the rotational magnetization process of magnetic domains. Domain wall displacement magnetization refers to the spontaneous magnetization direction close to the direction that the external magnetic field grows (under the action of the external magnetic field); the adjacent domain with a large deviation from the direction of the spontaneous magnetization direction and the direction of the external field are correspondingly compressed, so that the position of the domain wall changes; the process of magnetic domain rotation magnetization is the process in which all magnetic moments in the magnetic domain rotate uniformly toward the direction of the external magnetic field under the action of the external magnetic field [31].

Therefore, when using the equivalent model of multiple magnetic dipoles to figure out the equivalent magnetic field of targets, such as vessels, it can be seen from the magnetic domain theory that most of the internal magnetic domains will eventually tend to be consistent with the direction of the external magnetic field, i.e., the direction of the equivalent magnetic dipole moment can be determined in the same arrangement of magnetization directions as the external field.

By studying the equivalent effect of two magnetic dipoles with different directions of the magnetic moments to a single magnetic dipole, it can be further confirmed that the directions of the magnetic moments of the multiple magnetic dipoles must belong to the same directions. The magnetic induction intensity generated by a single magnetic dipole at a measuring point in space is

$$\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi r^5}}(3(\mathit{M}·\mathit{r})\mathit{r}-r^2\mathit{M})$$

(1)

where ${\mu }_0$ is the relative permeability, $\mathit{M}$ is the magnetic moment vector of the magnetic dipole, and $\mathit{r}$ is the relative position vector of the measuring point to the magnetic dipole. As shown in Figure 2, the magnetic induction intensity ${\mathit{B}}_1$, ${\mathit{B}}_2$ generated by two magnetic dipoles ${\mathit{M}}_1$, ${\mathit{M}}_2$, which is equivalent to the magnetic induction intensity ${\mathit{B}}_0$ generated by a single magnetic dipole ${\mathit{M}}_0$, should satisfy the vector superposition principle: ${\mathit{B}}_0={\mathit{B}}_1+{\mathit{B}}_2$.

(1) Considering extreme cases, when the measurement point is far away from the magnetic source: ${\mathit{r}}_0={\mathit{r}}_1={\mathit{r}}_2$, where there is

$${\mathit{B}}_0={\displaystyle \frac{{\mu }_0}{4\pi r^5}}(3({\mathit{M}}_0·{\mathit{r}}_0){\mathit{r}}_0-r_0^2{\mathit{M}}_0)$$

$$\begin{array}{c}\hfill {\mathit{B}}_1+{\mathit{B}}_2={\displaystyle \frac{{\mu }_0}{4\pi r^5}}(3({\mathit{M}}_1·{\mathit{r}}_0){\mathit{r}}_0-r_0^2{\mathit{M}}_1+3({\mathit{M}}_2·{\mathit{r}}_0){\mathit{r}}_0-r_0^2{\mathit{M}}_2)\\ \hfill ={\displaystyle \frac{{\mu }_0}{4\pi r^5}}(3({\mathit{M}}_1·{\mathit{r}}_0+{\mathit{M}}_2·{\mathit{r}}_0){\mathit{r}}_0-r_0^2({\mathit{M}}_1+{\mathit{M}}_2))\end{array}$$

(2)

i.e.,

$$3({\mathit{M}}_1·{\mathit{r}}_0+{\mathit{M}}_2·{\mathit{r}}_0){\mathit{r}}_0-r_0^2({\mathit{M}}_1+{\mathit{M}}_2)=3({\mathit{M}}_0·{\mathit{r}}_0){\mathit{r}}_0-r_0^2{\mathit{M}}_0$$

Therefore,

$$3(({\mathit{M}}_1+{\mathit{M}}_2-{\mathit{M}}_0)·{\mathit{r}}_0){\mathit{r}}_0=r_0^2({\mathit{M}}_1+{\mathit{M}}_2-{\mathit{M}}_0)$$

(3)

i.e., condition 1: ${\mathit{M}}_1+{\mathit{M}}_2={\mathit{M}}_0$ or condition 2: $\left\{\begin{array}{c}3(({\mathit{M}}_1+{\mathit{M}}_2-{\mathit{M}}_0)·{\mathit{r}}_0)=r_0^2\hfill \\ {\mathit{r}}_0={\mathit{M}}_1+{\mathit{M}}_2-{\mathit{M}}_0\hfill \end{array}\right.$ must be satisfied. From condition 2: $3r_0^2=r_0^2$, so obviously condition 2 is untenable. Therefore, the necessary condition for two magnetic dipoles to be equivalent to a single magnetic dipole is: ${\mathit{M}}_1+{\mathit{M}}_2={\mathit{M}}_0$.

(2) We consider two magnetic dipoles with different magnetic moment directions, and verify that there is no equivalent single magnetic dipole solution under the above magnetic moment equivalent conditions.

The mean absolute percentage error of the total field value of the magnetic induction field at each measurement point is defined for the equivalent effect assessment (mean absolute percentage error, MAPE). Furthermore, we define the equivalent consistency (EC) as: EC = 1 − MAPE. The equivalent effect satisfies when the equivalent consistency is higher than 99%.

The equivalent effects of two magnetic dipoles with the same directions and different directions of the magnetic moment are compared on the measurement plane with a spatial scale of 3, wherein the equivalent single magnetic dipole position is spatially traversed close to two magnetic dipoles. As shown in Figure 3, it can be seen that when the magnetic moment direction is inconsistent, it cannot meet the equivalent consistency of higher than 99%. Meanwhile, when the magnetic moment direction is consistent, it can always search out the equivalent single magnetic position that satisfies the equivalent effect.

To summarize, when using the equivalent model of multi-magnetic dipoles to carry out the equivalent magnetic fields of targets, such as vessels, it can be seen from the theory of magnetization of magnetic domains that most of the internal magnetic domains will eventually coincide with the direction of the external magnetic field. Furthermore, it is confirmed by simulation results that the magnetic moment direction of equivalent magnetic dipoles belongs to the same magnetization direction.

3.3. The Simplest Equivalent Condition of the Magnetic Moment Magnitude and Distribution of Magnetic Dipoles

The simplest equivalent condition of multi-magnetic dipoles is to find out the equivalent spatial magnetic field distribution of the target ferromagnet with the minimum number of magnetic dipoles under a specific spatial scale condition.

Firstly, the equivalent condition of two magnetic dipoles equivalent to a single magnetic dipole is discussed. As mentioned above, the magnetic moment conditions of two magnetic dipoles equivalent to a single magnetic dipole are as follows: the direction of the magnetic moment of two magnetic dipoles is consistent with that of the single magnetic dipole, and the sum of magnetic moments is equal to that of the single magnetic dipole. The distribution of the magnetic moment of two magnetic dipoles determines the position of the equivalent single magnetic dipole. The mean square error of each component between the superposition magnetic induction field of two magnetic dipoles and the magnetic induction field of a single magnetic dipole on the measurement line is used to evaluate the equivalent effect quantitatively. Assuming that the ratio of the magnetic moments of two magnetic dipoles is ${\mathit{M}}_2/{\mathit{M}}_1=3$. Magnetic dipoles are located at (−1, 0, 0) m and (1, 0, 0) m, respectively, and the spatial scale is taken as 10. Then, the measurement line is set as $(x,0,z),x\in (-\infty ,+\infty )$. The equivalent effects of the three components of the magnetic induction field are calculated as:

$$\left\{\begin{array}{c}{E}_x={\int }_{-\infty }^{+\infty }(\frac{1}{{({(x-x_1)}^2+z^2)}^{\frac{5}{2}}}(3(m_{1x}(x-x_1)+m_{1z}z)·(x-x_1)-m_{1x}({(x-x_1)}^2+z^2))+\hfill \\ \frac{1}{{({(x-x_2)}^2+z^2)}^{\frac{5}{2}}}(3(m_{2x}(x-x_2)+m_{2z}z)·(x-x_2)-m_{2x}({(x-x_2)}^2+z^2))-\hfill \\ \frac{1}{{({(x-x_0)}^2+z^2)}^{\frac{5}{2}}}(3(m_{0x}(x-x_0)+m_{0z}z)·(x-x_0)-m_{0x}({(x-x_0)}^2+z^2)){)}^{2}dx\hfill \\ E_y={\int }_{-\infty }^{+\infty }{(\frac{m_{1y}}{{({(x-x_1)}^2+z^2)}^{\frac{3}{2}}}+\frac{m_{2y}}{{({(x-x_2)}^2+z^2)}^{\frac{3}{2}}}-\frac{m_{0y}}{{({(x-x_0)}^2+z^2)}^{\frac{3}{2}}})}^{2}dx\hfill \\ E_z={\int }_{-\infty }^{+\infty }(\frac{1}{{({(x-x_1)}^2+z^2)}^{\frac{5}{2}}}(3(m_{1x}(x-x_1)+m_{1z}z)·z-m_{1z}({(x-x_1)}^2+z^2))+\hfill \\ \frac{1}{{({(x-x_2)}^2+z^2)}^{\frac{5}{2}}}(3(m_{2x}(x-x_2)+m_{2z}z)·z-m_{2z}({(x-x_2)}^2+z^2))-\hfill \\ \frac{1}{{({(x-x_0)}^2+z^2)}^{\frac{5}{2}}}(3(m_{0x}(x-x_0)+m_{0z}z)·z-m_{0z}({(x-x_0)}^2+z^2)){)}^{2}dx\hfill \end{array}\right.$$

The relationship between the equivalent single magnetic dipole position and the equivalent mean square error of the magnetic induction intensity is shown in Figure 4. The optimal position distribution of the equivalent single magnetic dipole is proportional to the magnitude distribution of magnetic moments of two magnetic dipoles (the ratio of magnetic moment is 3:1, and the equivalent error tends to be 0 when the equivalent single magnetic dipole is located at 3/4 of the connection line).

In order to further determine the distribution law of equivalent single magnetic dipole position under the condition of different ratios of magnetic moments of two magnetic dipoles, assume that the ratio of magnetic moment magnitude increases from 0.1 to 10, and we obtain the equivalent single magnetic dipole position, i.e., the minimum equivalent error position, as shown in Figure 5. The fitting curve is $x_0=(p-1)/(p+1)$, where $p=M_2/M_1$. Thus, the equivalent single magnetic dipole position satisfies the proportional relationship with the magnitude distribution of the magnetic moments of two magnetic dipoles.

3.4. Magnetic Field Distribution Equivalent Consistency of Multiple Magnetic Dipoles with the Same Magnetic Moment to Single Magnetic Dipole

In order to determine the simplest equivalent effectiveness of two magnetic dipoles at a specific spatial scale, the equivalent effects of two and three magnetic dipoles to a single magnetic dipole on the measurement plane are calculated and evaluated at different spatial scales.

3.4.1. Equivalent Consistency of Two Magnetic Dipoles to a Single Magnetic Dipole

Magnitudes of magnetic moments of two magnetic dipoles are set, respectively, to half of that of an equivalent single magnetic dipole, and directions of magnetic moments are consistent with that of the single magnetic dipole. The single magnetic dipole is located in the middle of two magnetic dipoles. Spatial scales are 0.5, 1, 3, and 4, respectively (i.e., the measurement height is respectively 0.5, 1, 3, and 4 times the distance between two magnetic dipoles). The total field distribution generated by two magnetic dipoles and that generated by the single magnetic dipole are shown in Figure 6.

3.4.2. Equivalent Consistency of Three Magnetic Dipoles to Single Magnetic Dipole

Magnitudes of magnetic moments of three magnetic dipoles are set as 1/3 that of an equivalent single magnetic dipole, and the directions of the magnetic moments are consistent with the single magnetic dipole. The single magnetic dipole is located in the distribution center of three magnetic dipoles and spatial scales are 0.5, 1, 3, and 4, respectively. The equivalent consistency of 2 and 3 magnetic dipoles under different spatial scales are shown in

Table 1. Comparison of the equivalent consistency of the total magnetic field between two or three magnetic dipoles and a single magnetic dipole with different spatial scales.

Spatial Scale	0.5	1	3	4
EC of two magnetic dipoles	72.45%	92.61%	99.15%	99.52%
EC of three magnetic dipoles	78.07%	95.11%	99.43%	99.68%

.

The following conclusions can be drawn from the above results: (1) Under the condition of large spatial scale, the direction of the magnetic moment should be consistent with that of the equivalent single magnetic dipole when the distribution of magnetic induction field of the multiple magnetic dipole array is equivalent to that of a single magnetic dipole; (2) When two magnetic dipoles are equivalent to a single magnetic dipole, the sum of the magnetic moments of two magnetic dipoles is equal to the magnetic moment of the single magnetic dipole. Moreover, the position distribution of the equivalent single magnetic dipole is affected by magnitudes of magnetic moments of two magnetic dipoles; (3) The larger the spatial scale, the higher the equivalent consistency. Two magnetic dipoles already have a satisfying equivalent consistency higher than 99%.

3.5. Simplest Equivalent Conditions of Multi-Magnetic Dipole Arrays

When using the magnet simulation method to perform ferromagnetic equivalence, a large number of ellipsoids or magnetic dipole arrays are generally used to improve the equivalent accuracy. From the above simulation results, it can be seen that under relatively large spatial scale conditions, a smaller number of magnetic dipole arrays have achieved satisfying equivalent consistency. Therefore, it is necessary to study the simplest equivalent combination of multiple magnetic dipole arrays under different spatial scales to reduce the computational complexity. Assuming a uniformly distributed multi-magnetic dipole array with a magnetic dipole number of $2^N$ and the spacing between magnetic dipoles is L, the array line length is $(2^N-1)\ast L$. The measurement height is equivalent to the line length. The array is shown in Figure 7.

Assuming that the number of multiple magnetic dipoles is 4, as shown in Figure 8, the measurement height $H=(4-1)\ast L=3L$ meets the required equivalent condition. After one equivalent into two magnetic dipoles, the maximum distance between the two magnetic dipoles is $3L$. Thus, the measurement height and spacing are equal, and it cannot meet the equivalent condition. Similarly, it can be seen that the simplest equivalent condition of the multiple magnetic dipole array under a specific spatial scale condition involves finding out the number of magnetic dipoles contained in the last equivalent when the multiple magnetic dipole array cannot meet the condition of continuing the next equivalent.

It can be deduced from the above example that when the spatial scale is 1, the maximum spacing between magnetic dipoles can be $(2^M-1)\ast L$ after equivalent M times. Thus, the simplest equivalent conditions of multiple magnetic dipole arrays must satisfy both $(2^N-1)\ast L\ge 3\ast (2^M-1)\ast L$ and $(2^N-1)\ast L\le 3\ast (2^{M+1}-1)\ast L$. M can be obtained by rounding $M=N-2$. Therefore, when the spatial scale is 1, no matter how large N is, the array’s simplest equivalent model is the four-magnetic dipole array.

To summarize, it can be seen that under the condition of a small spatial scale, the simplest equivalent conditions for multiple magnetic dipoles are as follows: when the spatial scale is from 1 to 3, the number of the simplest equivalent magnetic dipoles is 4. When the spatial scale is from 1/3 to 1, the number of the simplest equivalent magnetic dipoles is 8. Moreover, the rest can be done in the same manner. The directions of magnetic moments are consistent with those of the equivalent magnetic dipole moments in the large spatial scale, and the sum of magnetic moments is equal to that of the equivalent magnetic dipole.

4. The Spatial Distribution of the Submarine’s Magnetic Induction Field Obtained by COMSOL Simulation Software

As mentioned above, another problem with the magnet simulation method is that the magnetic field distribution of large vessels is difficult to obtain by means of onsite measurements. To solve the problem of the submarine magnetic induction field, COMSOL Multiphysics AC/DC module is based on the finite element method. By solving the following basic equations for the static magnetic field and establishing the boundary conditions using magnetic shielding characteristics and external insulation against the magnetization field, the spatial distribution of the submarine magnetic induction field under the known geomagnetic background field can be obtained.

In the region without the current, the basic equation of the static magnetic field is as follows:

$$\nabla \times \mathit{H}=0(\mathrm{Maxwell}-\mathrm{Ampere}\mathrm{Law})$$

(4)

The magnetic scalar potential $V_m$ satisfies the following relationship:

$$\mathit{H}=\nabla V_m$$

(5)

The constitutive relationship between the magnetic induction and magnetic field strength is:

$$\mathit{B}={\mu }_0{\mu }_r\mathit{H}$$

(6)

and $\nabla \times \mathit{B}=0$ (Gauss’s Law of magnetism). The equation for the magnetic scalar potential can be obtained as:

$$-\nabla ·({\mu }_0{\mu }_r\nabla V_m)=0$$

(7)

For the submarine magnetic induction field, the background field is known as the geomagnetic field, then the differential equation to be solved is:

$$-\nabla ·({\mu }_0{\mu }_r\nabla V_m+{\mathit{B}}_{\mathit{e}})=0$$

(8)

where ${\mathit{B}}_{\mathit{e}}$ is the background geomagnetic field. Taking a large submarine as the research object, we establish its geometric model. The background geomagnetic field vector, shell thickness, and material relative permeability are given according to the real situation as far as possible. The submarine model and its total magnetic field distribution on different measurement planes are shown in Figure 1 and Figure 9.

As can be seen from the COMSOL simulation results, with the decrease of the spatial scale, the distribution characteristics of the submarine magnetic moment are gradually prominent. The results are consistent with the above conclusion, indicating that the smaller the spatial scale, the more magnetic dipoles are needed for the equivalent of the submarine magnetic induction field.

5. Verification of the Multi-Magnetic Dipole Simplest Equivalent Model of the Submarine Magnetic Induction Field Distribution by COMSOL

The simplest multi-magnetic dipole equivalent modeling method for ferromagnet in a specific spatial scale involves the following steps. (1) The magnetic moment vector of the far-field single magnetic dipole model is solved by using the magnetic induction field data of the large spatial scale measurement plane. (2) Determine the three-dimensional (length, width, and height) spatial scale of the ferromagnet by considering the measurement height and the size of the ferromagnet. (3) According to the simplest equivalent condition of the multi-magnetic dipoles, the magnitude and distribution of the magnetic moment of each magnetic dipole are solved by using the data of the magnetic induction field of the measurement plane.

As shown in Figure 10, for the submarine, a ferromagnet with an incomplete regular shape, the center point along the long axis is taken as the origin according to its shape characteristics. Its far-field equivalent magnetic dipole position is assumed as ($x_0$,0,$z_0$), and the magnetic moment is assumed as $\mathit{M}$ = ($M_x$,$M_y$,$M_z$). Thus, according to Equation (1), by obtaining more than two known magnetic induction field values of measurement points, the equation set can be established as Equation (9). We use the L–M (Levenberg–Marquardt) algorithm to solve the above five unknown parameters. The 1stOpt optimization fitting software is applied for the parameter solution.

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mu }_0}{4\pi r_i^5}}((3r_{ix}^2-r_i^2)M_x+3r_{ix}{r}_{iy}{M}_y+3r_{ix}{r}_{iz}{M}_z)=B_{ix}\hfill \\ {\displaystyle \frac{{\mu }_0}{4\pi r_i^5}}((3r_{iy}^2-r_i^2)M_x+3r_{ix}{r}_{iy}{M}_y+3r_{iy}{r}_{iz}{M}_z)=B_{iy}\hfill \\ {\displaystyle \frac{{\mu }_0}{4\pi r_i^5}}((3r_{iz}^2-r_i^2)M_x+3r_{ix}{r}_{iz}{M}_y+3r_{iy}{r}_{iz}{M}_z)=B_{iz}\hfill \end{array}\right.$$

(9)

where, ${\mathit{r}}_{\mathit{i}}=(r_{ix},r_{iy},r_{iz})$ is the displacement vector of each measurement point to the single magnetic dipole and ${\mathit{B}}_{\mathit{i}}=(B_{ix},B_{iy},B_{iz}$) is the magnetic induction intensity vector measured at each measurement point. By solving the COMSOL submarine model, the magnetic moment vector of the far-field single magnetic dipole is obtained as $\mathit{M}=(1.5161\ast {10}^6,-3.5530\ast {10}^3,6.4737\ast {10}^3)$${\mathrm{Am}}^2$.

The magnetic induction intensity distribution of the submarine model is obtained on the measurement plane with a spatial scale of 1. Since the other two dimensions of the submarine are much smaller than the measurement height, the simplest four-magnetic dipole model is established as shown in Figure 10. The direction of the magnetic moment of each magnetic dipole is the same as that of the far-field single-magnetic dipole model. Therefore, the magnetic moment vectors of the four magnetic dipoles must satisfy

$$\left\{\begin{array}{c}{\mathit{M}}_1=p_1\mathit{M}\hfill \\ {\mathit{M}}_2=p_2\mathit{M}\hfill \\ {\mathit{M}}_3=p_3\mathit{M}\hfill \\ {\mathit{M}}_4=p_4\mathit{M}\hfill \\ p_1+p_2+p_3+p_4=1\hfill \\ p_1,p_2,p_3,p_4>0\hfill \end{array}\right.$$

(10)

By solving seven independent unknowns, including the magnetic dipole position, the simplest multi-magnetic dipole model of the submarine can be obtained when the spatial scale is greater than or equal to 1.

Assuming that the magnetic induction intensity vector at the measurement point $Q_j(x_j,y_j,z_j)$ on the measurement plane of spatial scale 1 is ${\mathit{B}}_{\mathit{j}}=(B_{jx},B_{jy},B_{jz})$, the equivalent equations are established according to the superposition of magnetic vectors as follows, where ${\mathit{r}}_{\mathit{ij}}=(r_{ijx},r_{ijy},r_{ijz})$ is the displacement between the measurement point $Q_j$ and the ith magnetic dipole $D_i$.

$$\left\{\begin{array}{c}\sum _{i=1}^4{\displaystyle \frac{{\mu }_0}{4\pi r_{ij}^5}}((3r_{ijx}^2-r_{ij}^2)M_{ix}+3r_{ijx}{r}_{ijy}{M}_{iy}+3r_{ijx}{r}_{ijz}{M}_{iz})=B_{ix}\hfill \\ \sum _{i=1}^4{\displaystyle \frac{{\mu }_0}{4\pi r_{ij}^5}}((3r_{ijy}^2-r_{ij}^2)M_{iy}+3r_{ijx}{r}_{ijy}{M}_{ix}+3r_{ijy}{r}_{ijz}{M}_{iz})=B_{iy}\hfill \\ \sum _{i=1}^4{\displaystyle \frac{{\mu }_0}{4\pi r_{ij}^5}}((3r_{ijz}^2-r_{ij}^2)M_{iz}+3r_{ijx}{r}_{ijz}{M}_{ix}+3r_{ijy}{r}_{ijz}{M}_{iy})=B_{iz}\hfill \end{array}\right.$$

(11)

In order to evaluate the equivalent effect of this simplest multi-magnetic dipole model, we compare the field data of multiple measurement planes with a spatial scale greater than or equal to 1 with the calculated data of the model, as shown in Figure 11. Moreover, we calculate its equivalent consistency under multiple spatial scale conditions, as shown in Figure 12.

It can be seen that the simplest model of four magnetic dipoles established with the spatial scale of 1 has a high equivalent consistency when the spatial scale is greater than or equal to 1, and the equivalent consistency increases with the increase of the spatial scale.

6. Design and Verification of the Degaussing Coils

Since the traditional degaussing coil design involves coil windings in three directions, the distributions of the windings and current size calculations are complicated; the installation is also extremely complicated. Based on the simplest multi-magnetic dipole model proposed in this paper, the degaussing coil design only needs to calculate the installation direction of the coil winding, according to the magnetic moment direction of the multi-magnetic dipole, which greatly simplifies the installation difficulty. In addition, since the measurement height is generally more than three times the size of the coil, each degaussing coil can be regarded as a magnetic dipole. According to the coil magnetic moment formula $\mathit{M}=NI\mathit{S}$ (N denotes the number of winding turns, I denotes the current size, and $\mathit{S}$ denotes the area vector), the degaussing coil design can be fulfilled simply by the calculation of the number of winding turns and the current size.

According to the calculation of the above multi-magnetic dipole model, the magnetic moments of four magnetic dipoles are 0.0774, 0.2953, 0.5196, and 0.1078 times the magnetic moments of the equivalent single magnetic dipole, respectively, and the installation positions are −12.97, −44.56, 21.17, and 27.55 m, respectively. The degaussing coils are set in COMSOL simulation software as shown in Figure 1.

Comparing the magnetic induction intensity of multiple measurement planes before and after degaussing, as shown in Figure 13 and Figure 14, the following conclusions can be drawn:

(1) On the measurement plane with a spatial scale greater than or equal to 1, the magnetic induction field can be reduced by more than 99% after installing the degaussing coils. (2) With the increase of the spatial scale, the degaussing effect becomes more significant, since the equivalent consistency increases therewith.

7. Discussions and Conclusions

There are some disadvantages (e.g., a large number of calculations and complex configurations) of to the degaussing coil system in the methods used for degaussing submarines [22]. The simplest method proposed in this paper only requires the arrangement of degaussing coils magnetized in one direction, so that the operational difficulty of the degaussing coil system is reduced. In addition, the simplest multi-magnetic dipole model can effectively reduce the number of calculations of the equivalent coil current, since the method largely reduces the equivalent optimization parameters. Furthermore, COMSOL simulation results verified the equivalent accuracy of the method and the accuracy of the proposed method is much higher compared with the accuracy of traditional demagnetization methods [15,16].

In this paper, by studying the influence of the number, distribution, size, and direction of the magnetic moments of the magnetic dipoles on the equivalence at different spatial scales, the simplest equivalent modeling method for multi-magnetic dipoles is systematically proposed. The magnetic field distribution of the submarine was obtained by COMSOL Multiphysics simulation software and the modeling method was verified. Furthermore, given the high installation complexity problem of three-dimensional degaussing coils, based on the simplest multi-magnetic dipole model, a simplified method for degaussing the coil design is proposed, and the effectiveness of the degaussing coils was verified by COMSOL. The method can hopefully improve the deperming efficiency for vessels in practice.