Symmetric magnetic field calculation method

Abstract

The invention discloses a symmetric magnetic field calculation method, which comprises the following steps of: analyzing the symmetry of magnetic field source distribution, and selecting a reference source in a group of symmetric sources; calculating the magnetic field intensity of the reference source at any field point in the space; based on symmetry, mapping the field points to equivalent field points corresponding to the symmetric sources through a rotation matrix, and calculating magnetic fields of the symmetric sources at the equivalent field points; restoring the magnetic field at the equivalent field point to an original field point direction through an inverse matrix of the rotating matrix; and finally, superposing the magnetic field contributions of all symmetrical sources at the original field point to obtain the total magnetic field of the point. According to the method, the symmetry of a magnetic field source is fully utilized, a rotating group processing method in the group theory is adopted, accurate calculation is performed in combination with the Bios-Savart law, the symmetry and direction correctness of the magnetic field can be strictly ensured, and high-precision calculation of the extremely low-intensity magnetic field is realized; the problems of errors and direction errors existing in symmetrical field and weak field calculation of a traditional finite element method are solved.

Description

Technical Field

[0001] This invention belongs to the field of finite element simulation technology, specifically relating to a method for calculating symmetrical magnetic fields. Background Technology

[0002] Finite element method (FEM) simulation technology uses mathematical approximations to simulate real physical systems. By using simple, interacting elements, a finite number of unknowns can approximate an infinite number of unknowns in a real system. In electromagnetic field analysis, the FEM is used to calculate the distribution characteristics of electromagnetic fields, such as electromagnetic induction and electromagnetic wave propagation, and to predict electric and magnetic field strengths and electromagnetic forces. However, in some magnetic field calculations, the magnetic field distribution exhibits strong symmetry, and the field strength is very weak in certain regions. Current FEM methods are less effective at calculating extremely weak and symmetric fields.

[0003] Traditional finite element analysis (FEM) primarily solves magnetic fields by combining boundary conditions with differential equations. This method has the advantage of broad applicability, theoretically capable of solving all electromagnetic problems. However, in situations requiring symmetry and weak field distributions, the general approach of using differential equations and boundary conditions has limitations. For example, mesh quality can degrade during mesh generation; insufficient mesh precision can lead to divergent or infinitesimal singularities in some solutions. Even worse, under boundary condition constraints, serious errors violating fundamental laws of physics can still occur, which is extremely detrimental in magnetic field solutions. Summary of the Invention

[0004] To address the aforementioned problems, the purpose of this invention is to provide a symmetric magnetic field calculation method that solves the problem that existing finite element analysis performs poorly in calculating ultraweak and symmetric fields.

[0005] To achieve the above objectives, the technical solution adopted by the present invention includes: A method for calculating a symmetrical magnetic field specifically includes the following steps: S1. Analyze the symmetry of the magnetic field source distribution, determine the symmetry group and its corresponding rotation angle, and obtain the rotation matrix describing the symmetry; S2, Select one source from the symmetry group as the reference source, and calculate the first magnetic flux density of the reference source at the target field point in space; S3, based on symmetry, maps the target field point to the equivalent field point corresponding to other sources in the symmetry group through a rotation matrix; S4, calculate the second magnetic flux density of each symmetric source at the equivalent field point; S5. By using the inverse of the rotation matrix, the direction of the second magnetic flux density at each equivalent field point is restored to the direction of the target field point, thus obtaining the third magnetic flux density of each other source at the target field point. S6. Vector superposition of all third magnetic flux densities and first magnetic flux densities of all symmetrical sources at the target field point to obtain the total magnetic field at the target field point.

[0006] Preferably, the magnetic field source is a current-carrying conductor or a permanent magnet.

[0007] Preferably, the magnetic field source is a current-carrying conductor, and the Biot-Savart law is used to calculate the first magnetic flux density in S2.

[0008] Preferably, the magnetic field source is a permanent magnet, and the calculation of the first magnetic flux density in S2 adopts a magnetic vector potential calculation model based on magnetization intensity or magnetic dipole moment density.

[0009] Preferably, the rotation matrix is ​​a rotation matrix in three-dimensional space, and its rotation axis is any one of the x-axis, y-axis, and z-axis.

[0010] Preferably, the rotation matrix when the rotation axis is the x-axis is as shown in equation (1), the rotation matrix when the rotation axis is the y-axis is as shown in equation (2), and the rotation matrix when the rotation axis is the Z-axis is as shown in equation (3). (1) (2) (3) in, It represents the rotation angle in the clockwise direction, and the value is positive.

[0011] Preferably, the symmetry of the magnetic field source distribution includes at least one of rotational symmetry or mirror symmetry.

[0012] Preferably, the magnetic field source is a magnetic device consisting of six hexagonally distributed current-carrying conductor plates.

[0013] A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the symmetrical magnetic field calculation method disclosed in this application.

[0014] A computer program product includes a computer program / instructions that, when executed by a processor, implement the symmetrical magnetic field calculation method disclosed in this application.

[0015] Compared with the prior art, the advantages of the present invention are: (1) The symmetric magnetic field calculation method of the present invention focuses on the magnetic field source itself. When the source distribution is known, the problem of setting boundary conditions does not need to be considered. Since the field source distribution analysis takes into account symmetry and the symmetry group is used in the calculation, the symmetry of the field distribution is strictly maintained, and the symmetry breaking introduced by numerical discretization is avoided.

[0016] (2) The symmetrical magnetic field calculation method of the present invention is based directly on the basic laws of electromagnetic fields (such as the Biot-Savart law), avoiding the truncation error and iteration error caused by numerical solution of complex differential equations, so that reliable and accurate numerical results can be obtained even in extremely weak regions where the magnetic field is almost canceled.

[0017] (3) The symmetric magnetic field calculation method of the present invention starts from the magnetic field source, considers the characteristics of the source distribution, and uses its symmetry to calculate the magnetic field. The key step adopts the rotation group processing method, which fully demonstrates and ensures the symmetry of the magnetic field. At the same time, the magnetic field strength is directly calculated based on the magnetic field source using Biot-Savart's law, which can fundamentally ensure the correctness of the magnetic field distribution. The accurate numerical calculation can ensure that strict numerical results can be given for no matter how weak the field is, and there is no approximate solution.

[0018] (3) The symmetric magnetic field calculation method of the present invention only requires a complete field calculation for a reference source, and the contributions of other symmetric sources are quickly obtained through transformation, which greatly reduces the amount of calculation of direct integration, improves the calculation efficiency while maintaining high accuracy, and is especially suitable for complex array structures with high symmetry, taking into account both calculation efficiency and accuracy. Attached Figure Description

[0019] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the following detailed description to explain the invention, but do not constitute a limitation thereof. In the drawings: Figure 1 This is a schematic diagram of the current-carrying conductor plate structure with symmetrical hexagonal arrangement in the embodiment; Figure 2 This is a spatial distribution cloud map of the magnetic field strength calculated using the method of this invention; Figure 3 This is a vector diagram showing the distribution of the magnetic field on the transverse section (XoY plane); Figure 4 This is a graph showing the distribution of the magnetic field along the central axis (Z-axis). Figure 5 A schematic diagram showing the calculation results of the same model using the traditional finite element method. Detailed Implementation

[0020] The invention is not limited to the specific embodiments described below. All equivalent modifications made based on the technical solutions of this application fall within the protection scope of this invention. Unless otherwise specified, all components and devices in this invention utilize components and devices known in the prior art.

[0021] Example To demonstrate the accuracy of magnetic field calculations and the correctness of magnetic field directions, this embodiment provides a magnetic device composed of six current-carrying conductor plates distributed in a hexagonal pattern, and calculates its field distribution. This device has an obvious rotational symmetry feature in terms of symmetry, and the magnetic fields inside the device cancel each other out, so a very weak field appears in the main axis part, which can fully reflect the advantages of the present invention.

[0022] As Figure 1 shown, the magnetic field source in this embodiment has six symmetric regions, and the included angle between each adjacent plate is 60°, which is in radian representation. Selecting the topmost conductor plate as the reference to determine its integration interval, our design parameters are a length of 0.5 m, a width of 0.027713 m, a thickness of 0.01 m, the inner wall is 0.03 m away from the central axis, and the current passing through the cross-section is set to 40000 A.

[0023] The magnetic field source in this embodiment is a current-carrying conductor. If we take the line where the central axis is located as x = 0 m, y = 0 m parallel to the z-axis, then the integration domain of the reference plate we selected is -0.0138565 m < x < 0.0138565 m, 0.03 m < y < 0.04 m, 0 < z < 0.5 m. The current density in the Biot-Savart law is , where the current density is a constant because the current is a constant current without change and will be evenly distributed on the cross-section. We select 0.001 m as the distribution interval of the field points, that is, the field points are selected at a step of 0.001 m in the x, y, and z directions for calculation.

[0024] This embodiment discloses a symmetric magnetic field calculation method for the above magnetic field source, which specifically includes the following steps: S1, analyze the symmetry of the magnetic field source distribution, determine the symmetry group and its corresponding rotation angle, and obtain the rotation matrix describing this symmetry; The rotation matrix when the rotation axis is the x-axis is as shown in Equation (1), the rotation matrix when the rotation axis is the y-axis is as shown in Equation (2), and the rotation matrix when the rotation axis is the Z-axis is as shown in Equation (3); (1) (2) (3) Among them, represents the rotation angle rotating in the clockwise direction and takes a positive value.

[0025] Specifically, the symmetry of the magnetic field source distribution includes at least one of rotational symmetry or mirror symmetry.

[0026] Specifically, the magnetic field source is either a current-carrying conductor or a permanent magnet. When the magnetic field source is a current-carrying conductor, the Biot-Savart law is used to calculate the first magnetic flux density in S2; when the magnetic field source is a permanent magnet, the magnetic vector potential calculation model based on magnetization or magnetic dipole moment density is used to calculate the first magnetic flux density in S2.

[0027] The Biot-Savart law is calculated as follows:

[0028] in, For the field points to be calculated The magnetic flux density at that point The absolute permeability at that point is It is pi. It is the source point Current density at that point This represents the tensor product (also known as the cross product). It represents the region where the selected source exists.

[0029] The calculation formula for the magnetic vector potential based on the magnetization intensity or magnetic dipole moment density is as follows:

[0030]

[0031] in, Represents the magnetic vector potential. Represents the magnetic dipole moment density. It is the vector differential symbol, defined as ; It is a unit vector of length 1 pointing in the x, y, and z directions.

[0032] S2, Select one source from the symmetry group as the reference source, and calculate the first magnetic flux density of the reference source at the target field point in space; S3, based on symmetry, maps the target field point to the equivalent field point corresponding to other sources in the symmetry group through a rotation matrix; This embodiment simplifies the calculation by leveraging the symmetry of the source, transforming the source symmetry into field symmetry, and calculating the symmetric points of the field points about the source's axis of symmetry.

[0033]

[0034] ...

[0035] Here It is a symmetric field point position vector that transforms symmetry from the source point to the field point. In this embodiment, n takes the values ​​1, 2, ..., 5.

[0036] S4, calculate the second magnetic flux density of each symmetric source at the equivalent field point; This embodiment does not change the integration region of the source, and calculates the magnitude and direction of the magnetic flux density at the corresponding field points of all sources with symmetry. .

[0037] S5. By using the inverse of the rotation matrix, the direction of the second magnetic flux density at each equivalent field point is restored to the direction of the target field point, thus obtaining the third magnetic flux density of each other source at the target field point. S6, perform vector superposition of all third magnetic flux densities and first magnetic flux densities of all symmetric sources at the target field point. The inverse matrix is ​​represented as follows: The total magnetic field at the target point is obtained. The calculation results are as follows: Figure 2 As shown in Figure 4, the magnetic field distribution strictly maintains sixfold rotational symmetry, and the extremely weak field region near the central axis still has high precision and correct orientation.

[0038] This embodiment also discloses a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the symmetrical magnetic field calculation method of the present invention.

[0039] This embodiment also discloses a computer program product, including a computer program / instructions, which, when executed by a processor, implement the symmetrical magnetic field calculation method of the present invention.

[0040] Comparative Example This comparative example uses the finite element method, which combines traditional differential equations with boundary conditions, to calculate the same model. The results are as follows: Figure 5 As shown, the calculation results of this method have serious problems, the most important being the direction issue. According to Biot-Savart's law, the magnetic induction intensity... The direction is the current density The vector pointing from the source point to the field point The result is obtained through cross product; here we assume the direction of the current is... Direction, if the field point is chosen within the "loop" surrounded by the magnet structure, then The direction after the cross product is This result is related to whether the field point is "inside the loop" or not. That is, even if the z-direction has left the interior of the magnet structure, there is still a change at the boundary of the magnet structure in the xoy plane. The magnetic field "inside the loop" and the magnetic field "outside the loop" are opposite in direction. However, as can be seen from the figure, the calculation results of the differential equation with boundary conditions do not show this change; they are all counterclockwise. This indicates that the calculation results of this method have serious problems. Furthermore, as can be seen from the figure, the extremely weak field near the axis is so small that even a slight error will produce a strong directional deviation, and it is impossible to guarantee the same good symmetry as the strong field part outside. This also demonstrates the superiority of this invention.

[0041] In summary, the traditional method exhibits directional errors in extremely weak field regions and suffers significant symmetry degradation, thus verifying the superiority of the method disclosed in the embodiments.

[0042] The preferred embodiments of this disclosure have been described in detail above with reference to the accompanying drawings. However, this disclosure is not limited to the specific details of the above embodiments. Within the scope of the technical concept of this disclosure, various simple modifications can be made to the technical solutions of this disclosure, and these simple modifications all fall within the protection scope of this disclosure.

[0043] It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable manner without contradiction. In order to avoid unnecessary repetition, this disclosure will not describe the various possible combinations separately.

[0044] Furthermore, the various implementation methods disclosed in this solution can be combined in any way, as long as they do not violate the spirit of this disclosure, they should also be regarded as the content invented by this disclosure.

Claims

1. A method for calculating a symmetrical magnetic field, characterized in that, Specifically, the steps include the following: S1. Analyze the symmetry of the magnetic field source distribution, determine the symmetry group and its corresponding rotation angle, and obtain the rotation matrix describing the symmetry; S2, Select one source from the symmetry group as the reference source, and calculate the first magnetic flux density of the reference source at the target field point in space; S3, based on the symmetry, the target field point is mapped to the equivalent field point corresponding to other sources in the symmetry group by a rotation matrix; S4, calculate the second magnetic flux density of each symmetric source at the equivalent field point; S5, by using the inverse matrix of the rotation matrix, the direction of the second magnetic flux density at each equivalent field point is restored to the direction of the target field point, and the third magnetic flux density of each other source at the target field point is obtained. S6. Vector superposition of all third magnetic flux densities and first magnetic flux densities of all symmetrical sources at the target field point to obtain the total magnetic field of the target field point.

2. The method for calculating symmetrical magnetic fields as described in claim 1, characterized in that, The magnetic field source is a current-carrying conductor or a permanent magnet.

3. The method for calculating symmetrical magnetic fields as described in claim 2, characterized in that, The magnetic field source is a current-carrying conductor, and the Biot-Savart law is used to calculate the first magnetic flux density in S2.

4. The method for calculating symmetrical magnetic fields as described in claim 3, characterized in that, The magnetic field source is a permanent magnet. In S2, the calculation of the first magnetic flux density adopts a magnetic vector potential calculation model based on magnetization intensity or magnetic dipole moment density.

5. The method for calculating symmetrical magnetic fields as described in any one of claims 1-4, characterized in that, The rotation matrix is ​​a three-dimensional rotation matrix, and its rotation axis is any one of the x-axis, y-axis, and z-axis.

6. The method for calculating symmetrical magnetic fields as described in claim 5, characterized in that, The rotation matrix when the rotation axis is the x-axis is as shown in equation (1), the rotation matrix when the rotation axis is the y-axis is as shown in equation (2), and the rotation matrix when the rotation axis is the Z-axis is as shown in equation (3). （1） （2） （3） in, It represents the rotation angle in the clockwise direction, and the value is positive.

7. The method for calculating symmetrical magnetic fields as described in claim 6, characterized in that, The symmetry of the magnetic field source distribution includes at least one of rotational symmetry or mirror symmetry.

8. The method for calculating symmetrical magnetic fields as described in claim 6, characterized in that, The magnetic field source is a magnetic device consisting of six hexagonally distributed current-carrying conductor plates.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the symmetrical magnetic field calculation method according to any one of claims 1-8.

10. A computer program product, characterized in that, Includes a computer program / instruction, which, when executed by a processor, implements the symmetrical magnetic field calculation method according to any one of claims 1-8.

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