Precise magnetism numerical modeling method

Abstract

The present invention relates to a precise magnetism numerical modeling method, which is based on the establishment of equation equations by special designed points, i.e. equilibrium points, in the unit. In this new calculation framework, the grid dependence characteristics of the unit matrix can be significantly reduced, thereby allowing more efficient convergence to the theoretical value with minimal difference. The method can handle various extreme physical conditions in three-dimensional and two-dimensional scenarios and can provide highly accurate calculation solutions. The effectiveness of the method is verified by the passive shielding design of a 9.4T whole-body magnetic resonance imaging superconducting magnet and the conceptual design of a very weak magnetic field scientific facility with a cross-scale geometry.

Description

Technical Field

[0001] This invention belongs to the field of electrical engineering theory and new technology, specifically relating to a precise magnetic numerical modeling method for calculating the magnetization induction magnetic field of magnetic materials. Background Technology

[0002] Many modern scientific instruments and medical equipment operate based on magnetic fields, such as magnetic resonance imaging (MRI), nuclear magnetic resonance spectrometers, gravimeters, and magnetic shielding devices. The distribution characteristics of the magnetic field significantly affect the performance of these instruments and equipment. Especially with the rise of interdisciplinary studies, magnetic fields play a crucial role in multidisciplinary applications. Under certain complex and extreme conditions, the accurate analysis and prediction of magnetic fields provide vital support for the manufacture of instruments and equipment and the discovery of important scientific phenomena.

[0003] Mesh-based numerical modeling methods have become important tools for calculating complex magnetic field models. Traditional mesh matrix computation frameworks establish equations based on nodes or cells, and then build basis function models within the cell mesh, representing approximate solutions within the mesh. This framework exhibits good convergence when dealing with complex models; however, it is always accompanied by computational errors. Within traditional frameworks, reducing computational errors typically requires increasing the mesh density, which significantly increases the computational burden, and the corresponding computational errors can only be reduced, not eliminated.

[0004] Chinese invention patent CN116911090A discloses a modeling method for electromagnetic models of high-frequency transformers based on the magnetic permeability-capacitance analogy. Although the method innovates in the modeling steps, it is still based on the traditional finite element framework. Chinese patent CN116306163A discloses a finite element electromagnetic numerical method for port cross-section transmission matching in waveguide design. The method innovates in basis functions and boundary conditions, but it is still based on the traditional finite element framework. Summary of the Invention

[0005] To achieve accurate numerical modeling of problems under complex limiting conditions, this invention proposes a novel finite element and boundary element theory framework based on equilibrium point equations, which is applied to the field of magnetic numerical modeling.

[0006] The technical solution adopted in this invention is as follows:

[0007] A precise magnetic numerical modeling method, comprising the following steps:

[0008] Step 1: Discretize the solid model using a mesh generation method;

[0009] Step 2: Magnetization calculation is performed using magnetic scalar potential based on the mesh model;

[0010] Step 3: Set an equilibrium point within the grid between the grid nodes and the centroid, and establish an equation with the magnetic scalar potential at the equilibrium point as the target.

[0011] Step 4: Establish the error iterative equation of the equation and solve for the magnetic scalar potential;

[0012] Step 5: Using the standard analytical model as a calibration method, the magnetic scalar potential at the grid nodes is calculated by adjusting the position of the equilibrium point. Then, the physical parameters used for calibration are calculated and compared with the analytical solution until the two results are completely consistent.

[0013] Step 6: Fix the position of the equilibrium point and generalize to other arbitrary calculation models.

[0014] Furthermore, the meshing method in step 1 includes tetrahedral finite element meshing and triangular boundary element meshing. The solid model includes a three-dimensional solid model and a two-dimensional shell model. The tetrahedral finite element meshing is applied to the three-dimensional solid model, and the triangular boundary element meshing is applied to the two-dimensional shell model. The meshing region only includes the solid model and does not include the air region.

[0015] Furthermore, the method for calculating the magnetic scalar potential in step 2 is as follows:

[0016] The total magnetic scalar potential is equal to the sum of the source scalar potential and the magnetization scalar potential:

[0017]

[0018] The scalar potential of the magnetic source is calculated by integrating the magnetic field strength along the path vector dot product from the source point to the target point:

[0019]

[0020] The magnetization scalar potential under tetrahedral finite element mesh conditions is calculated by the following formula:

[0021]

[0022] The magnetization scalar potential under triangular boundary element mesh conditions is calculated by the following formula:

[0023]

[0024] in, The total magnetic scalar potential is given. Let be the scalar potential of the magnetic source. For the magnetization scalar potential, H sLet be the magnetic field strength of the magnetic source, l be the integration path from source point a to target point b, M be the magnetization intensity, S' be the boundary surface of the tetrahedral finite element mesh, r be the observation point, r' be the integration point, l' be the boundary line of the triangular boundary element mesh, and t be the thickness of the two-dimensional shell model.

[0025] Furthermore, the method for calculating the magnetic scalar potential in step 3 is as follows:

[0026] Four equilibrium points are established within the tetrahedral finite element mesh, each located on one of the four median lines of the corresponding tetrahedron; three equilibrium points are established within the triangular boundary element mesh, each located on one of the three median lines of the triangle. The ratio of the distance between an equilibrium point and its corresponding node to the median line of that node is defined as γ. The γ values ​​for the four equilibrium points in the tetrahedral finite element mesh are consistent, as are the γ values ​​for the three equilibrium points in the triangular boundary element mesh.

[0027] Within a tetrahedral finite element mesh, the following equation is established with the magnetic scalar potential at the equilibrium point as the objective:

[0028]

[0029] Based on the magnetic scalar potentials at the four nodes of the tetrahedral finite element mesh, the magnetic scalar potential at the equilibrium point, calculated using linear basis functions, is as follows:

[0030]

[0031] Within the triangular boundary element mesh, the following equation is established with the magnetic scalar potential at the equilibrium point as the objective:

[0032]

[0033] Based on the magnetic scalar potentials at the three nodes of the triangular boundary element mesh, the magnetic scalar potential at the equilibrium point, calculated using linear basis functions, is as follows:

[0034]

[0035] in, Let be the magnetic scalar potential at the grid node. Let A be the magnetic scalar potential at the equilibrium point, and let A be the magnetization scalar potential transfer matrix calculated according to equation (3) or equation (4). This represents the functional relationship between the total magnetic scalar potential and the magnetization.

[0036] Furthermore, the error iteration equation in step 4 is as follows:

[0037]

[0038] Where k is the number of iterations. This represents the initial magnetic scalar potential.

[0039] For the magnetic scalar potential at step k, the gradient is calculated to obtain the magnetic field strength H at step k. There is a functional relationship g(H) between the magnetization M and the magnetic field strength H, and the ratio of magnetization to magnetic field strength is χ. The transformation matrix Z is obtained by multiplying the magnetization scalar potential transfer matrix and the above ratio, and then subtracting the unit diagonal matrix eye(n). The inverse of the transformation matrix is ​​then combined with the source magnetic scalar potential to establish an iterative relationship between the magnetic scalar potentials at steps k and k+1. After each iteration, the residual e of the equation is calculated to evaluate the convergence of the iterative calculation.

[0040] Furthermore, the standard analytical model in step 5 uses a spherical magnetic shielding shell, and the physical parameter used for calibration is the shielding coefficient, which is calculated as follows:

[0041]

[0042] Where, μ r R is the relative permeability, R is the inner radius of the spherical magnetic shielding shell, and t is the thickness of the spherical magnetic shielding shell.

[0043] Using the same spherical magnetic shielding shell model, the shielding coefficient calculated according to the proposed method and the analytical solution are compared.

[0044] Furthermore, the location of the equilibrium point in step 6 is as follows:

[0045] Within the tetrahedral finite element mesh, the γ value at the equilibrium point is -0.583; within the triangular boundary element mesh, the γ value at the equilibrium point is 0.27325. Based on these γ values, the magnetization numerical model can be established using steps 4 and 5.

[0046] This method can be extended to other numerical modeling fields. The steps performed in different modeling fields are the same, but the corresponding basic theories need to be adapted to the field of application. At the same time, the corresponding equilibrium point position also changes accordingly.

[0047] The present invention has the following beneficial technical effects:

[0048] This invention eliminates the need for mesh generation in the air domain, making it easier to create high-quality meshes in complex models. It exhibits low dependence on mesh density, eliminating the need to artificially increase mesh density to improve computational accuracy and thus reducing computational burden. This invention can be applied to extreme physical models across scales, where large-scale and micro-scale models can be meshed separately, ensuring computational feasibility. Using this method, computational accuracy is improved by more than one order of magnitude compared to traditional theoretical frameworks. Attached Figure Description

[0049] Figure 1This is a schematic diagram of the mesh model and equilibrium point of the present invention;

[0050] Figure 2a -c represents the numerical simulation model of a 9.4T human magnetic resonance imaging superconducting magnet with passive shielding. Figure 2a It is a superconducting magnet coil. Figure 2b For the tetrahedral finite element mesh of the shielded room, Figure 2c The relationship between the magnetization M and the magnetic field strength H of the magnetic material in the shielded room is shown.

[0051] Figure 3a -c represents the numerical simulation results of passive shielding of a 9.4T human magnetic resonance imaging superconducting magnet, where... Figure 3a The convergence curve for the iterative calculation process. Figure 3b The stray magnetic field distribution in the XOZ plane. Figure 3c The stray magnetic field distribution in the YOZ plane;

[0052] Figure 4a -c represents the numerical simulation model of the geomagnetic field shielding device, where... Figure 4a For active shielding coils, Figure 4b The boundary element mesh is a shielded shell triangle. Figure 4c The relationship between the magnetization M and the magnetic field strength H of the magnetic material of the shielding shell is a function.

[0053] Figure 5a -c represents the numerical simulation results of the geomagnetic field shielding device, where... Figure 5a The convergence curve for the iterative calculation process. Figure 5b To shield the distribution of stray magnetic fields inside the shell, Figure 5c The distribution of stray magnetic fields in the central region of the shielding shell.

[0054] Explanation of the attached diagram labels: 1 is a grid node, 2 is the grid centroid, and 3 is the equilibrium point. Detailed Implementation

[0055] The technical solutions of the present invention will now be clearly and completely described with reference to the accompanying drawings in the embodiments of the present invention.

[0056] The technical solution adopted in this invention is as follows:

[0057] A precise numerical modeling method for magnetism, characterized by the following steps:

[0058] Step 1: Discretize the solid model using a mesh generation method;

[0059] Step 2: Magnetization calculation is performed using magnetic scalar potential based on the mesh model;

[0060] Step 3: Set an equilibrium point 3 between grid node 1 and centroid 2 within the grid, and establish an equation with the magnetic scalar potential at equilibrium point 3 as the target.

[0061] Step 4: Establish the error iterative equation of the equation and solve for the magnetic scalar potential;

[0062] Step 5: Using the standard analytical model as a calibration method, the magnetic scalar potential at grid node 1 is calculated by adjusting the position of equilibrium point 3. Then, the physical parameters used for calibration are calculated and compared with the analytical solution until the two results are completely consistent.

[0063] Step 6: Fix the position of equilibrium point 3 and generalize to other arbitrary calculation models.

[0064] Furthermore, the meshing method in step 1 includes tetrahedral finite element meshing and triangular boundary element meshing. The solid model includes a three-dimensional solid model and a two-dimensional shell model. The tetrahedral finite element meshing is applied to the three-dimensional solid model, and the triangular boundary element meshing is applied to the two-dimensional shell model. The meshing region only includes the solid model and does not include the air region.

[0065] Furthermore, the method for calculating the magnetic scalar potential in step 2 is as follows:

[0066] The total magnetic scalar potential is equal to the sum of the source scalar potential and the magnetization scalar potential:

[0067]

[0068] The scalar potential of the magnetic source is calculated by integrating the magnetic field strength along the path vector dot product from the source point to the target point:

[0069]

[0070] The magnetization scalar potential under tetrahedral finite element mesh conditions is calculated by the following formula:

[0071]

[0072] The magnetization scalar potential under triangular boundary element mesh conditions is calculated by the following formula:

[0073]

[0074] in, The total magnetic scalar potential is given. Let be the scalar potential of the magnetic source. For the magnetization scalar potential, H sLet be the magnetic field strength of the magnetic source, l be the integration path from source point a to target point b, M be the magnetization intensity, S' be the boundary surface of the tetrahedral finite element mesh, r be the observation point, r' be the integration point, l' be the boundary line of the triangular boundary element mesh, and t be the thickness of the two-dimensional shell model.

[0075] Furthermore, the method for calculating the magnetic scalar potential in step 3 is as follows:

[0076] Four equilibrium points 3 are established within the tetrahedral finite element mesh, each located on one of the four median lines of the corresponding tetrahedron; three equilibrium points 3 are established within the triangular boundary element mesh, each located on one of the three median lines of the triangle. The ratio of the distance between an equilibrium point 3 and its corresponding node 1 to the median line of that node 1 is defined as γ. The γ values ​​are consistent across the four equilibrium points in the tetrahedral finite element mesh and across the three equilibrium points in the triangular boundary element mesh. Figure 1 As shown.

[0077] Within a tetrahedral finite element mesh, the following equation is established with the magnetic scalar potential at equilibrium point 3 as the objective:

[0078]

[0079] Based on the magnetic scalar potentials at the four nodes 1 of the tetrahedral finite element mesh, the magnetic scalar potential at the equilibrium point 3, calculated using linear basis functions, is as follows:

[0080]

[0081] Within the triangular boundary element mesh, the following equation is established with the magnetic scalar potential at equilibrium point 3 as the objective:

[0082]

[0083] Based on the magnetic scalar potentials at the three nodes 1 of the triangular boundary element mesh, the magnetic scalar potential at the equilibrium point 3, calculated using linear basis functions, is as follows:

[0084]

[0085] in, Let be the magnetic scalar potential at grid node 1. Let A be the magnetic scalar potential at equilibrium point 3, and let A be the magnetization scalar potential transfer matrix calculated according to equation (3) or equation (4). This represents the functional relationship between the total magnetic scalar potential and the magnetization.

[0086] Furthermore, the error iteration equation in step 4 is as follows:

[0087]

[0088] Where k is the number of iterations. This represents the initial magnetic scalar potential.

[0089] For the magnetic scalar potential at step k, the gradient is calculated to obtain the magnetic field strength H at step k. There is a functional relationship g(H) between the magnetization M and the magnetic field strength H, and the ratio of magnetization to magnetic field strength is χ. The transformation matrix Z is obtained by multiplying the magnetization scalar potential transfer matrix and the above ratio, and then subtracting the unit diagonal matrix eye(n). The inverse of the transformation matrix is ​​then combined with the source magnetic scalar potential to establish an iterative relationship between the magnetic scalar potentials at steps k and k+1. After each iteration, the residual e of the equation is calculated to evaluate the convergence of the iterative calculation.

[0090] Furthermore, the standard analytical model in step 5 uses a spherical magnetic shielding shell, and the physical parameter used for calibration is the shielding coefficient, which is calculated as follows:

[0091]

[0092] Where, μ r R is the relative permeability, R is the inner radius of the spherical magnetic shielding shell, and t is the thickness of the spherical magnetic shielding shell.

[0093] Using the same spherical magnetic shielding shell model, the shielding coefficient calculated according to the proposed method and the analytical solution are compared.

[0094] Furthermore, the position of equilibrium point 3 in step 6 is as follows:

[0095] Within the tetrahedral finite element mesh, the γ value at equilibrium point 3 is -0.583; within the triangular boundary element mesh, the γ value at equilibrium point 3 is 0.27325. Based on these γ values, the magnetization numerical model can be established using steps 4 and 5.

[0096] This method can be extended to other numerical modeling fields. The steps performed in different modeling fields are the same, but the corresponding basic theories need to be adapted to the field of application. At the same time, the position of the corresponding equilibrium point 3 also changes accordingly.

[0097] This invention is applied in two typical numerical modeling cases: one is a numerical simulation model of a passive shielding superconducting magnet for 9.4T human magnetic resonance imaging, which is divided into tetrahedral finite element meshes; the other is a numerical simulation model of a geomagnetic field shielding device, which is divided into triangular boundary element meshes.

[0098] Figure 2a -c represents the numerical simulation model of a 9.4T human magnetic resonance imaging superconducting magnet with passive shielding. Figure 2a It is a superconducting magnet coil. Figure 2b For the tetrahedral finite element mesh of the shielded room, Figure 2cThe relationship between the magnetization M and the magnetic field strength H of the magnetic material in the shielded room is shown.

[0099] Figure 3a -c represents the numerical simulation results of passive shielding of a 9.4T human magnetic resonance imaging superconducting magnet, where... Figure 3a The convergence curve for the iterative calculation process. Figure 3b The stray magnetic field distribution in the XOZ plane. Figure 3c The stray magnetic field distribution in the YOZ plane;

[0100] Figure 4a -c represents the numerical simulation model of the geomagnetic field shielding device, where... Figure 4a For active shielding coils, Figure 4b The boundary element mesh is a shielded shell triangle. Figure 4c The relationship between the magnetization M and the magnetic field strength H of the magnetic material of the shielding shell is a function.

[0101] Figure 5a -c represents the numerical simulation results of the geomagnetic field shielding device, where... Figure 5a The convergence curve for the iterative calculation process. Figure 5b To shield the distribution of stray magnetic fields inside the shell, Figure 5c The distribution of stray magnetic fields in the central region of the shielding shell.

[0102] Those skilled in the art will readily understand that the above description is merely a typical embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A precise magnetic numerical modeling method, characterized in that, This method consists of the following steps: Step 1: Discretize the solid model using a mesh generation method to obtain a mesh model; Step 2: Magnetization calculation is performed using magnetic scalar potential based on the mesh model; Step 3: Set an equilibrium point within the grid between the grid nodes and the centroid, and establish an equation with the magnetic scalar potential at the equilibrium point as the target. Step 4: Establish the error iterative equation of the equation and solve for the magnetic scalar potential; Step 5: Using the standard analytical model as a calibration method, the magnetic scalar potential at the grid nodes is calculated by adjusting the position of the equilibrium point. Then, the physical parameters used for calibration are calculated and compared with the analytical solution until the two results are completely consistent. Step 6: Fix the position of the equilibrium point and generalize to other arbitrary calculation models; other arbitrary calculation models include: numerical simulation model of passive shielding of superconducting magnet for human magnetic resonance imaging and numerical simulation model of geomagnetic field shielding device; The method for calculating the magnetic scalar potential in step 2 is as follows: The total magnetic scalar potential is equal to the sum of the source scalar potential and the magnetization scalar potential: （1） The scalar potential of the magnetic source is calculated by integrating the magnetic field strength along the path vector dot product from the source point to the target point: （2） The magnetization scalar potential under tetrahedral finite element mesh conditions is calculated by the following formula: （3） The magnetization scalar potential under triangular boundary element mesh conditions is calculated by the following formula: （4） in, The total magnetic scalar potential is given. Let be the scalar potential of the magnetic source. For the magnetization scalar potential, H s denoted as , where is the magnetic field strength of the magnetic source, l is the integration path from source point a to target point b, M is the magnetization intensity, S' is the boundary surface of the tetrahedral finite element mesh, r is the observation point, r' is the integration point, l' is the boundary line of the triangular boundary element mesh, and t is the thickness of the two-dimensional shell model. The method for calculating the magnetic scalar potential at the equilibrium point in step 3 is as follows: Four equilibrium points are established within the tetrahedral finite element mesh, each located on one of the four median lines of the corresponding tetrahedron; three equilibrium points are established within the triangular boundary element mesh, each located on one of the three median lines of the triangle; the ratio of the distance between the equilibrium point and the corresponding node to the median line of the corresponding node is defined as γ, the γ values ​​of the four equilibrium points within the tetrahedral finite element mesh are consistent, and the γ values ​​of the three equilibrium points within the triangular boundary element mesh are consistent. Within a tetrahedral finite element mesh, the following equation is established with the magnetic scalar potential at the equilibrium point as the objective: （5） Based on the magnetic scalar potentials at the four nodes of the tetrahedral finite element mesh, the magnetic scalar potential at the equilibrium point, calculated using linear basis functions, is as follows: （6） Within the triangular boundary element mesh, the following equation is established with the magnetic scalar potential at the equilibrium point as the objective: （7） Based on the magnetic scalar potentials at the three nodes of the triangular boundary element mesh, the magnetic scalar potential at the equilibrium point, calculated using linear basis functions, is as follows: （8） in, , , , Let be the magnetic scalar potential at the grid node. , , , Let f(φ) be the magnetic scalar potential at the equilibrium point, A be the magnetization scalar potential transfer matrix calculated according to equation (3) or equation (4), and f(φ) be the functional relationship between the total magnetic scalar potential and the magnetization.

2. The precise magnetic numerical modeling method according to claim 1, characterized in that, The meshing methods in step 1 include tetrahedral finite element meshing and triangular boundary element meshing. The solid model includes a three-dimensional solid model and a two-dimensional shell model. The tetrahedral finite element meshing is applied to the three-dimensional solid model, and the triangular boundary element meshing is applied to the two-dimensional shell model. The meshed region only includes the solid model and does not include the air region.

3. The precise magnetic numerical modeling method according to claim 1, characterized in that, The error iteration equation in step 4 is as follows: （9） Where k is the number of iterations. This is the initial magnetic scalar potential; φ k For the magnetic scalar potential at step k, the gradient is calculated to obtain the magnetic field strength H at step k. There is a functional relationship g(H) between the magnetization M and the magnetic field strength H. The ratio of the magnetization M to the magnetic field strength is χ. By multiplying the magnetization scalar potential transfer matrix and the above ratio and subtracting the unit diagonal matrix eye(n), the transformation matrix Z is obtained. Then, the inverse of the transformation matrix is ​​combined with the source magnetic scalar potential to establish the iterative relationship between the magnetic scalar potentials at steps k and k+1. After each iteration, the residual e of the equation is calculated to evaluate the convergence effect of the iterative calculation.

4. The precise magnetic numerical modeling method according to claim 3, characterized in that, The standard analytical model in step 5 uses a spherical magnetic shielding shell, and the physical parameter used for calibration is the shielding coefficient, which is calculated as follows: （10） Where, μ r Where is the relative permeability, R is the inner radius of the spherical magnetic shielding shell, and t is the thickness of the spherical magnetic shielding shell; Using the same spherical magnetic shielding shell model, the shielding coefficient calculated according to the proposed method and the analytical solution are compared.

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