One-dimensional magnetic field alignment grid division and integration method
Through the one-dimensional magnetic field alignment grid division and integration method, the magnetic problem in the numerical calculation of electrons in the Hall thruster was solved, and technical means were realized. The magnetic field direction was aligned with the grid through the magnetic line tracking method, which reduced the calculation cost and time and improved the calculation accuracy and area coverage.
Patent Information
- Application Number
- CN202510778795.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-11
- Publication Date
- 2025-09-19
AI Technical Summary
In the existing electronic numerical simulation of Hall thrusters, the numerical diffusion phenomenon caused by the inconsistency between the magnetic field direction and the grid is serious, the calculation results are inaccurate and the calculation cost is high. The traditional particle simulation method takes a long time to calculate, and the fluid simulation method has limited accuracy under boundary conditions.
A one-dimensional magnetic field aligned grid division and integration method is adopted, and the grid direction is aligned with the magnetic field direction through the magnetic line tracing method, which reduces the difficulty of grid division, avoids the influence of solving Poisson's equation, improves grid accuracy, and uses fluid method to simulate electron motion.
It reduces numerical diffusion, improves the accuracy and speed of calculation results, reduces calculation costs, expands the calculation area, and improves grid division accuracy.
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Figure CN120671386A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of numerical calculation of Hall thrusters, and in particular to a one-dimensional magnetic field aligned grid division and integration method applied to numerical simulation calculation of Hall thruster electronics. Background Art
[0002] As a future-oriented advanced space electric propulsion device, the Hall thruster has become the most technologically mature and widely used space electric propulsion device due to its advantages such as simple structure, stable discharge and high reliability. It is mainly used in a series of on-orbit missions such as position maintenance, satellite orbit change and deep space exploration.
[0003] Research on Hall thrusters primarily involves two approaches: experimental and numerical. Numerical calculation techniques for Hall thrusters can delve into the complex physical mechanisms of plasmas and reveal plasma parameters that are difficult to measure experimentally. Furthermore, they can be used in conjunction with experiments to theoretically interpret experimental results, providing theoretical guidance for subsequent experiments and thruster design. Therefore, the development of numerical calculation techniques is crucial for studying the physical mechanisms within Hall thrusters and for their engineering design. Because electrons participate in multiple important physical processes within Hall thrusters, including plasma generation, electric field organization, and energy transfer, accurately simulating electron behavior is a prerequisite for numerical calculations of Hall thrusters.
[0004] Currently, there are two main methods for numerically simulating electrons in Hall thrusters: particle simulation and fluid simulation. Traditional particle simulation methods, such as PIC (Particle-in-cell), treat electrons as discrete particles and obtain a more accurate electron distribution by numerically counting a large number of particles. However, since statistical particle methods can cause numerical noise and the simulation process uses particle tracking, the computational cost is high and the calculation time is long. The use of acceleration technology can also lead to computational distortion. Fluid simulation methods, on the other hand, use a set of fluid dynamics equations to process electrons, offering faster computation speeds and a larger computational area. They have become one of the mainstream simulation methods for studying electron motion in Hall thrusters.
[0005] Since the Hall parameter in the Hall thruster is much larger than 1, that is, the electrons are fully magnetized. Under the constraints of the electric and magnetic fields, the electrons have sufficient collisions, so it can be regarded as a quasi-Maxwell fluid, suitable for fluid simulation methods. However, classical plasma theory points out that due to the high anisotropy of electrons, such as the electron transport coefficient perpendicular to the magnetic field is inversely proportional to the square of the magnetic induction intensity, and is much smaller than the electron transport coefficient parallel to the magnetic field. This anisotropy of the electron transport coefficient will lead to numerical errors when solving the transport equation. This error is usually called numerical diffusion, and the numerical diffusion is particularly obvious when the computational grid is inconsistent with the magnetic field direction.
[0006] Typically, using a grid aligned with the magnetic field can effectively reduce this numerical diffusion phenomenon. NASA's Jet Propulsion Laboratory (JPL) developed the fluid simulation program Hall2De based on the HPHALL2 hybrid simulation program, using the magnetic field aligned mesh (MFAM), effectively reducing numerical diffusion. Since then, magnetic field aligned meshes have been widely used in Hall thruster simulations. The Ahedo team in Spain developed the Hall thruster hybrid simulation program HYPHEN, based on the Hall2De program and the PIC method. It also uses a grid aligned with the magnetic field for electronic simulations.
[0007] The existing method for generating a magnetic field-aligned grid is as follows: First, based on the relationship between the magnetic flow function and the magnetic scalar potential function and the magnetic induction intensity, the distribution of the magnetic flow function and the magnetic scalar potential function is obtained by solving two Poisson equations respectively. Then, the contour lines of the magnetic flow function and the magnetic scalar potential function are screened, and the intersection points of the contour lines are indexed, and then the quadrilateral elements formed are indexed. Finally, the grid quality is optimized using relevant methods. Since this method requires solving the Poisson equation, the boundary conditions have a great influence on the distribution of the magnetic flow function and the magnetic scalar potential function. In addition, the grid quality of the magnetic field-aligned grid also has a certain impact on the calculation accuracy. Summary of the Invention
[0008] To overcome the shortcomings of the existing technology, the present invention proposes a one-dimensional magnetic field alignment grid division and integration method, which is applied to the numerical simulation calculation of Hall thruster electronics. This method aligns the grid direction with the magnetic field direction, avoids directional inconsistency, adopts the magnetic line tracking method, reduces the difficulty of magnetic field grid division, avoids the influence of boundary conditions when solving Poisson's equation, and improves the grid division accuracy.
[0009] The one-dimensional magnetic field alignment grid division and integration method includes the following steps: S1: Calculate the magnetic induction intensity and scalar function of the Hall thruster and its plume area; S2: Use the magnetic field line tracking method to obtain the distribution of magnetic field lines in the calculation area, divide the calculation area using magnetic field lines, and obtain a one-dimensional magnetic field alignment grid; according to the equivalence characteristics of the scalar function along the magnetic field lines, obtain the value of the corresponding scalar function on the grid line; S3: By the method of differentiation first and then summation, the magnetic lines of force and the one-dimensional magnetic field alignment grid are differentiated into multiple infinitesimal curves and infinitesimal control volumes, and the surface integral and volume integral are performed on the infinitesimal curves and infinitesimal control volumes respectively. Finally, the entire scalar function contour line and the entire magnetic field alignment grid are summed respectively to obtain the surface integral of the physical quantity with respect to the scalar function contour line and the volume integral of the physical quantity with respect to the magnetic field alignment grid.
[0010] The beneficial effects of the present invention compared to the prior art are: This application method is based on the magnetic field distribution of a Hall thruster and can be used for numerical simulation calculations of Hall thruster electronics. Compared to structured orthogonal grids, this application aligns the grid direction with the magnetic field direction, avoiding directional inconsistencies. This significantly reduces the problem of excessive differences in electron transport coefficients in different directions during the calculation process, making the calculation results more accurate. Compared to traditional particle methods for simulating electrons, this application uses a fluid method to simulate electrons, greatly saving computing costs, increasing computing speed, and expanding the calculation area. Compared to existing magnetic field alignment grid division methods, this application uses a magnetic line tracking method, which reduces the difficulty of magnetic field grid division, avoids the influence of boundary conditions when solving Poisson's equation, and improves grid division accuracy.
[0011] The following is a further description of the scheme of the application in conjunction with the accompanying drawings and embodiments: BRIEF DESCRIPTION OF THE DRAWINGS
[0012] Figure 1 is the scalar function distribution diagram of the Hall thruster and plume area; Figure 2 Schematic diagram of the structured orthogonal grid of the Hall thruster; Figure 3 The contour lines of scalar functions and the physical boundary map drawn using the tracking method; Figure 4 The diagram for calculating the principle of the contour area integral of a physical quantity over a scalar function; Figure 5 Schematic diagram of the calculation principle of the magnetic field aligned grid volume integration for physical quantities. DETAILED DESCRIPTION
[0013] The embodiments of the technical solution of the present invention will be described in detail below with reference to the accompanying drawings. Unless otherwise specified, the technical terms or scientific terms used in this application have the common meanings understood by those skilled in the art.
[0014] Because important physical processes such as ionization and acceleration of the working fluid in a Hall thruster primarily occur in the axial direction, the internal magnetic field is primarily distributed radially, and the electron temperature and potential barely change along the magnetic field lines. To improve the computational accuracy of the one-dimensional electron model of a Hall thruster, this embodiment proposes a one-dimensional axial magnetic field alignment grid division and grid parameter integration method. The method comprises the following steps: S1: Calculate the magnetic induction intensity and scalar function of the Hall thruster and its plume area; S2: Use the magnetic field line tracking method to obtain the distribution of magnetic field lines in the calculation area, divide the calculation area using magnetic field lines, and obtain a one-dimensional magnetic field alignment grid; according to the equivalence characteristics of the scalar function along the magnetic field lines, obtain the value of the corresponding scalar function on the grid line; S3: By the method of differentiation first and then summation, the magnetic lines of force and the one-dimensional magnetic field alignment grid are differentiated into multiple infinitesimal curves and infinitesimal control volumes, and the surface integral and volume integral are performed on the infinitesimal curves and infinitesimal control volumes respectively. Finally, the entire scalar function contour line and the entire magnetic field alignment grid are summed respectively to obtain the surface integral of the physical quantity with respect to the scalar function contour line and the volume integral of the physical quantity with respect to the magnetic field alignment grid.
[0015] Furthermore, in step 1, the circumferential component of the magnetic potential inside the Hall thruster and the plume region is obtained by solving the Biot-Savart law equation in the two-dimensional axisymmetric coordinate system as follows: , (1) in is the relative magnetic permeability, is the circumferential component of the magnetic vector potential, is the vacuum permeability, , is the circumferential component of the current density, is the radial position of the point where the circumferential component of the magnetic vector potential lies; Obtain the circumferential component of the magnetic potential inside the Hall thruster and the plume area ,Will Substitute into equations (2) and (3); (2) (3) Obtain the magnetic induction intensity B and the scalar function λ, where represents the radial unit vector of the Hall thruster, represents the axial unit vector of the Hall thruster, represents the radial partial derivative, represents the partial derivative about the axial direction, represents the gradient operator.
[0016] At this time, the dot product of the gradient of the scalar function and the magnetic induction intensity satisfies;
[0017] That is, the scalar function gradient and the direction of magnetic induction intensity are always the same, and the scalar function contours coincide with the magnetic field lines. By selecting a series of equidistant scalar function contours, the two-dimensional axisymmetric computational domain can be discretized into a one-dimensional axisymmetric computational domain composed of a one-dimensional magnetic field grid. In this case, the scalar function contours are the one-dimensional magnetic field aligned grid lines.
[0018] There are relatively mature numerical methods for solving the Biot-Savart law equation. In addition, commercial software and open source software can be used to model and solve it. In this embodiment, the open source finite element method electromagnetic software FEMM (Finite Element Method Magnetics) is used to obtain the distribution of the magnetic field and scalar function. The specific steps are as follows: the two-dimensional axisymmetric model of the Hall thruster is imported into FEMM to establish the magnetic circuit model of the Hall thruster; FEMM is used to calculate the magnetic field distribution of the Hall thruster and its plume area as well as the scalar function distribution. The calculated scalar function distribution is as follows: Figure 1 As shown in the figure, xpos represents the axial position and ypos represents the radial position, and the unit is m.
[0019] Furthermore, the process of using the magnetic field line tracking method to obtain the distribution of magnetic field lines within the calculation area in step S2 is as follows: in step S2, the distribution of magnetic field lines is obtained by the magnetic field line tracking method. Based on the magnetic field distribution of the magnetic induction intensity obtained in step S1, the magnetic field lines are tracked from each equidistant point on the center line of the Hall thruster discharge channel. The magnetic field and spatial position of each point on any magnetic field line satisfy the following formula: (4) It is an empirical parameter related to the spatial distribution of the magnetic field. It is used to prevent the acquisition points from being too dense when tracing magnetic field lines in weak magnetic field areas. The value is generally the square root of the ratio of the maximum magnetic induction intensity in the calculation area to the magnetic induction intensity at the starting point: （5） in, and They are the axial and radial magnetic field induction intensities of the current tracking point, dz and dr are the tracking steps used to determine the next tracking point, respectively. is the step coefficient. On the same magnetic line, the step coefficient is a constant value and can be written as a fixed step coefficient. With empirical parameters The product of , where the fixed step size coefficient for: (6) Where, and are the axial and radial spatial steps of the structured orthogonal grid of the Hall thruster, respectively. Figure 2 The Hall thruster structured orthogonal grid shown, It is an empirical parameter that controls the global magnetic field line index density, and is generally set to 0.1-0.5.
[0020] During the tracking process, first determine the location of the starting point, and determine the location of the next tracking point starting from the starting point through the tracking method; and so on to obtain the location of each tracking point in the calculation domain, connect these tracking points with a smooth curve to obtain the distribution of magnetic lines of force. According to the equivalence of the scalar function along the magnetic lines of force, the distribution of magnetic lines of force is the distribution of the scalar function contour lines. At this point, the division of the one-dimensional magnetic field alignment grid is completed. The value of the corresponding scalar function on the grid line is obtained through the distribution of the scalar function calculated in step S1. Define the closed area enclosed by any two adjacent scalar function contour lines and the boundary of the calculation area as a one-dimensional magnetic field alignment grid. The drawn one-dimensional magnetic field alignment grid and the Hall thruster calculation boundary division are as follows: Figure 3 shown.
[0021] Furthermore, in step S3, a physical quantity in the fluid equation In terms of calculation, the finite volume method is usually used for solution. According to the Gaussian divergence theorem, the volume integral of the physical quantity on the control volume can be converted into the surface integral on the closed surface surrounding the control volume. That is, when using the finite volume method for calculation on a one-dimensional magnetic field aligned grid, it is necessary to solve the surface integral of the physical quantity on the scalar function contour line and the volume integral on the magnetic field aligned grid. This embodiment uses the method of first differentiating and then summing the one-dimensional magnetic field aligned grid, and the specific steps are: First, solve the surface integral of the physical quantity on the scalar function contour line: Figure 4 As shown, from a point on the center line of the Hall thruster discharge channel Start by using the tracking method to obtain the scalar function contour line , and a series of tracking points. Use these tracking points to contour the scalar function Differentiation is divided into many differential curves. Define any two consecutive tracking points on the contour line of the scalar function (such as , The differential curve between the two tracking points is the differential surface element j. The center of the differential surface element is determined by the positions of these two tracking points. Location and the infinitesimal area , represents the position of the infinitesimal surface center in the two-dimensional axisymmetric coordinate system, where and are the center points of the differential surface elements The radial and axial positions of the face are calculated using the area weight method. Physical quantity at . The physical quantity at the center of the surface and the infinitesimal area Multiply and add the contour lines of the scalar function Sum all the differential elements on the surface to obtain the physical quantity The surface integral on the contour line of a scalar function is expressed as: (7) Among them, is a physical quantity in fluid calculations. is the unit normal vector on the magnetic field line, is the area of the surface enclosed by the magnetic lines of force, j is the infinitesimal surface between any two consecutive tracking points on the contour line of the scalar function, is the physical quantity at the center of the infinitesimal surface, is the normal vector of the infinitesimal surface, is the area of the infinitesimal surface.
[0022] Solve the volume integral of the physical quantity on the magnetic field aligned grid: Figure 5 As shown, any two scalar function contour lines obtained by the tracking method are selected , The starting point of these two scalar function contour lines is linearly interpolated to obtain the starting point . For the starting point Use the tracking method to obtain the contour lines of the scalar function And a series of tracking points. On the scalar function contour Select any two adjacent tracking points , , use the least squares method to obtain the scalar function gradient at these two tracking points 、 At this point, through the tracking point The distance between two scalar function contour lines It can be written as the absolute value of the scalar function difference between these two scalar function contours and Absolute value of the gradient of a point scalar function The ratio of:
[0023] Similarly, we can obtain The distance between the contour lines of two scalar functions of a point , and then obtain the coordinates of the four vertices of the infinitesimal control body and calculate the volume of the infinitesimal control body . Obtain the microelement curve by linear interpolation Center At this point, it is considered that the point is the center of the infinitesimal control volume. The center of the volume is calculated using the area weight method. Physical quantity at a position , and then with the infinitesimal volume Multiply and sum the entire magnetic field alignment grid to finally get the physical quantity of the entire magnetic field grid Volume fraction: (8) in: is a physical quantity in fluid calculation, c is the infinitesimal control volume, is the volume of a one-dimensional magnetic field aligned grid, is the physical quantity at the center of the infinitesimal control body, is the volume of the infinitesimal control volume.
[0024] (9) represents the distance between the contour lines of two scalar functions, Represents the absolute value of the scalar function difference between two scalar function contour lines, Represents the absolute value of the gradient of the scalar function at the tracking point; the above method can realize the division of the one-dimensional magnetic field into aligned grids, and has the function of calculating the integral of the physical quantity over the magnetic field grid volume and the integral of the scalar function contour line.
[0025] This application has been disclosed as above with preferred implementation cases, but it is not intended to limit the present invention. Any technician familiar with this profession can make slight changes or modifications to equivalent implementation cases using the above-disclosed structures and technical contents without departing from the scope of the technical solution of the present invention, and all of these changes still fall within the scope of the technical solution of the present invention.
Claims
1. One-dimensional magnetic field alignment grid division and integration method, characterized by: The following steps are included: S1: Calculate the magnetic induction intensity and scalar function of the Hall thruster and its plume area; S2: Use the magnetic field line tracking method to obtain the distribution of magnetic field lines in the calculation area, divide the calculation area using magnetic field lines, and obtain a one-dimensional magnetic field alignment grid; according to the equivalence characteristics of the scalar function along the magnetic field lines, obtain the value of the corresponding scalar function on the grid line; S3: By the method of differentiation first and then summation, the magnetic lines of force and the one-dimensional magnetic field alignment grid are differentiated into multiple infinitesimal curves and infinitesimal control volumes, and the surface integral and volume integral are performed on the infinitesimal curves and infinitesimal control volumes respectively. Finally, the entire scalar function contour line and the entire magnetic field alignment grid are summed respectively to obtain the surface integral of the physical quantity with respect to the scalar function contour line and the volume integral of the physical quantity with respect to the magnetic field alignment grid.
2. The one-dimensional magnetic field alignment grid division and integration method according to claim 1, characterized in that: The magnetic induction intensity B and scalar function λ in step S1 are: Solve the Biot-Savart law equation in the two-dimensional axisymmetric coordinate system shown below, （1） in is the relative magnetic permeability, is the circumferential component of the magnetic vector potential, is the vacuum permeability, , is the circumferential component of the current density, is the radial position of the point where the circumferential component of the magnetic vector potential lies; Obtain the circumferential component of the magnetic potential inside the Hall thruster and the plume area ,Will Substitute into equations (2) and (3); （2） （3） Obtain the magnetic induction intensity B and the scalar function λ, where represents the radial unit vector of the Hall thruster, represents the axial unit vector of the Hall thruster.
3. The one-dimensional magnetic field alignment grid division and integration method according to claim 1, characterized in that: The process of using the magnetic field line tracking method to obtain the magnetic field line distribution within the calculation area in step S2 is as follows: According to the magnetic field distribution of the magnetic induction intensity obtained in step S1, the discharge channel center line of the Hall thruster The magnetic lines of force are traced from equidistant points on the magnetic field. The magnetic field and spatial position of each point on any magnetic line of force satisfy the following formula: （4） The value of is the square root of the ratio of the maximum magnetic induction intensity in the calculation area to the magnetic induction intensity at the starting point; （5） in and They are the axial and radial magnetic field induction intensities of the current tracking point, dz and dr are the tracking steps used to determine the next tracking point, respectively. is the step coefficient. On the same magnetic line, the step coefficient is a constant value and can be written as a fixed step coefficient. With empirical parameters The product of ; where the fixed step size coefficient for: （6） Where, and are the axial and radial spatial steps of the Hall thruster structured orthogonal grid, is a parameter that controls the global magnetic field line index density.
4. The one-dimensional magnetic field alignment grid division and integration method according to claim 1, characterized in that: In step S2, the calculation area is divided using magnetic lines of force to obtain a one-dimensional magnetic field aligned grid. The process is as follows: 在追踪过程中，首先确定起始点所在位置， The position of the next tracking point starting from the starting point is determined by the tracking method, and the position of each tracking point is obtained in the calculation domain by analogy. These tracking points are connected with a smooth curve to obtain the distribution of magnetic lines of force. According to the isovalue of the scalar function along the magnetic lines of force, the distribution of magnetic lines of force is the isovalue distribution of the scalar function. At this point, the division of the one-dimensional magnetic field alignment grid is completed.
5. The one-dimensional magnetic field alignment grid division and integration method according to claim 1, characterized in that: In step S3, the surface integral of the physical quantity over the contour line of the scalar function is expressed as: （7） Among them, is a physical quantity in fluid calculations. is the unit normal vector on the magnetic field line, is the area of the surface enclosed by the magnetic lines of force, j is the infinitesimal surface between any two consecutive tracking points on the contour line of the scalar function, is the physical quantity at the center of the infinitesimal surface, is the normal vector of the infinitesimal surface, is the area of the infinitesimal surface.
6. The one-dimensional magnetic field alignment grid division and integration method according to claim 5, characterized in that: The process of establishing the surface integral of the physical quantity with respect to the scalar function contour line in step S3 is as follows: starting from a point on the center line of the Hall thruster discharge channel, using the tracking method to obtain the scalar function contour line and a series of tracking points, using these tracking points to differentiate the scalar function contour line into multiple differential curves; defining the differential curve between any two consecutive tracking points on the scalar function contour line as a differential surface element, and determining the position of the surface center and the differential surface element area by the positions of the two tracking points, using the area weight method to calculate the physical quantity at the surface center, multiplying the physical quantity at the surface center by the differential surface element area, and summing all the differential surface elements on the scalar function contour line to obtain the surface integral of the physical quantity on the scalar function contour line.
7. The one-dimensional magnetic field alignment grid division and integration method according to claim 1, characterized in that: The volume integral of the physical quantity to the magnetic field aligned grid in step S3 is expressed as: （8） in: is a physical quantity in fluid calculation, c is the infinitesimal control volume, is the volume of a one-dimensional magnetic field aligned grid, is the physical quantity at the center of the infinitesimal control body, is the volume of the infinitesimal control volume.
8. The one-dimensional magnetic field alignment grid division and integration method according to claim 7, characterized in that: The process of establishing the volume integral of the physical quantity to the magnetic field aligned grid in step S3 is as follows: Select any two scalar function contour lines obtained by the tracking method, perform linear interpolation on the starting point positions of the two scalar function contour lines to obtain the starting point, use the tracking method on the starting point to obtain the scalar function contour line and a series of tracking points, select any two adjacent tracking points on the scalar function contour line, and use the least squares method to obtain the scalar function gradient at the two tracking points. At this time, the distance between the two scalar function contour lines passing through the tracking points is expressed as the ratio of the absolute value of the scalar function difference between the two scalar function contour lines to the absolute value of the scalar function gradient at the tracking point. （9） represents the distance between the contour lines of two scalar functions, Represents the absolute value of the scalar function difference between two scalar function contour lines, Represents the absolute value of the scalar function gradient at the tracking point; The distances between the contour lines of the two scalar functions passing through the two tracking points are obtained respectively, and then the coordinates of the four vertices of the infinitesimal control body are obtained, and the volume of the infinitesimal control body is calculated. The center of the differential surface element is obtained by linear interpolation, which is the body center of the infinitesimal control body. The physical quantity at the body center position is calculated using the area weight method, and then multiplied by the infinitesimal volume and summed over the entire magnetic field alignment grid to finally obtain the volume integral of the physical quantity over the entire magnetic field grid.