A high-precision calculation method for edge element static magnetic field based on MVP formula and edge smoothing

Abstract

The application relates to a high-precision calculation method of an edge-smoothed edge element static magnetic field based on an MVP formula, which divides grid data into a plurality of polyhedral smoothing domains with corresponding edges as centers according to each edge in the grid data, and the polyhedral smoothing domain is formed by nodes, body centers and face centers of tetrahedral units in the grid data around the corresponding edge; the vector function gradient of the corresponding edge is calculated based on the vector function of each tetrahedral unit in the polyhedral smoothing domain corresponding to the edge; the static magnetic field equation is constructed based on the MVP formula and the Coulomb norm, and the static magnetic field equation is rewritten according to the vector function gradient of each edge and the Galerkin method, so that the system equation of the static magnetic field is obtained; the system equation of the static magnetic field is further rewritten into Newton iteration linear equations; the Newton iteration method is used to iteratively solve the Newton iteration linear equations until convergence, so that the magnetic vector potential in the static magnetic field is obtained. The method realizes the high calculation precision and stability requirement of an engineering static magnetic field problem.

Description

Technical Field

[0001] The present application relates to the technical field of static magnetic field calculation, and in particular to a high-precision static magnetic field calculation method for edge elements with smooth edges based on the MVP formula. Background Art

[0002] Numerical calculations of static magnetic fields primarily rely on finite element methods (FEM), which typically rely on two core formulas: the magnetic scalar potential (MSP) and the magnetic vector potential (MVP). Because the MSP requires specialized processing when dealing with multiply connected domains and lacks versatility, MVP-based static magnetic field calculations are more popular.

[0003] Based on the discretization method of the MVP formula, the finite element method can be divided into nodal elements and edge elements. The edge element method is highly favored in static magnetic field calculations because it naturally meets the physical properties of electromagnetic fields (allowing normal discontinuities). However, the edge element method currently faces two major challenges: First, because the divergence of the edge element basis function is always zero, the edge element method based on the MVP formula cannot eliminate the non-uniqueness of the magnetic vector potential by applying the Coulomb gauge like the nodal element method, resulting in a singular assembled system matrix and the inability to directly solve the linear sparse equations. Second, with the increasingly complex geometric structures of electromagnetic devices, mesh generation is usually limited to unstructured grids. However, the edge element method based on unstructured grids often lacks accuracy in static magnetic field calculations. These challenges severely limit the application of the edge element method in static magnetic fields. Therefore, the development of a stable and high-precision edge element numerical algorithm is particularly urgent and necessary. Summary of the Invention

[0004] Based on this, it is necessary to provide a high-precision calculation method for static magnetometry of edge elements with smooth edges based on the MVP formula, which includes:

[0005] S1: Import the mesh data of electromagnetic materials;

[0006] S2: Divide the mesh data into several polyhedral smooth domains centered on the corresponding edges according to the edges in the mesh data. The polyhedral smooth domains are composed of the nodes, body centers, and face centers of the tetrahedral elements in the mesh data surrounding the corresponding edges. Calculate the vector shape function gradient of the corresponding edge based on the vector shape function of each tetrahedral element in the polyhedral smooth domain corresponding to the edge.

[0007] S3: Construct the static magnetic field equation based on the MVP formula and Coulomb gauge, and rewrite the static magnetic field equation according to the vector shape function gradient of each edge and the Galerkin method to obtain the static magnetic field system equation; further rewrite the static magnetic field system equation into Newton iterative linear equation;

[0008] S4: Newton iteration method is used to iteratively solve the Newton iteration linear equation until convergence, and the magnetic vector potential in the static magnetic field is obtained.

[0009] Preferably, the vector shape function gradient of the edge is calculated as:

[0010] ;

[0011] in, Represents a smooth polyhedron k Middle edge e The vector shape function gradient of ; Represents a smooth polyhedron k Middle edge e Vector shape function of ; represents partial derivative; express r direction, r Directions include direction, direction, direction; Indicates the k A polyhedral smooth domain; Represents the edges in the tetrahedral element e Vector shape function of ; Indicates the k Smooth functions of polyhedral smooth domains; Indicates the k The number of tetrahedral elements in a polyhedral smooth domain; Indicates the n The edges of the tetrahedral element e Vector shape function of ; Indicates the n The volume of a tetrahedral unit.

[0012] Preferably, the smooth function is the inverse of the volume of the corresponding polyhedron smooth domain, and the smooth function satisfies the following conditions:

[0013] ;

[0014] in, Indicates the k A polyhedral smooth domain; Indicates the k Smooth functions of a polyhedral smooth domain.

[0015] Preferably, the static magnetic field equation is expressed as:

[0016] ;

[0017] ;

[0018] in, represents the curl; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field.

[0019] Preferably, the process of obtaining the system equation of the static magnetic field includes:

[0020] The static magnetic field equation is rewritten according to the vector shape function gradient of each edge and the Galerkin method, and the weak form of the static magnetic field equation is obtained.

[0021] According to the vector shape function gradient of each edge and the difference theorem, the magnetic vector potential and virtual scalar potential are expanded;

[0022] By coupling the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential, we obtain the system equations of the static magnetic field.

[0023] Preferably, the weak form of the static magnetic field equation is expressed as:

[0024] ;

[0025] ;

[0026] in, Indicates the k A polyhedral smooth domain; represents the curl; Represents a smooth polyhedron Middle edge i Vector shape function of ; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field; Represents a smooth polyhedron midpoint j The shape function at .

[0027] Preferably, the expanded magnetic vector potential and the expanded virtual scalar potential are respectively expressed as:

[0028] ;

[0029] ;

[0030] in, represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; Represents a smooth polyhedron The number of edges; Represents a smooth polyhedron The number of midpoints; Represents a smooth polyhedron Middle edge i Vector shape function of ; Indicates edges i The tangential magnetic vector value on ; Represents a smooth polyhedron midpoint j The shape function at ; Indicates edges i The virtual scalar potential value on .

[0031] Preferably, the Newton iterative linear equation is expressed as:

[0032] ;

[0033] ;

[0034] ;

[0035] in, represents the system Jacobian matrix; represents the residual of the magnetic vector potential; represents the residual of the system equations; Indicates the h +1 iteration step of magnetic vector potential; Indicates the h Magnetic vector potential at the iteration step; represents the system stiffness matrix; Represents a known external force vector.

[0036] Preferably, the iterative solution process of the Newton iteration method includes:

[0037] Step 1: Initialize the iteration step;

[0038] Step 2: Solve the Newton iterative linear equation under the current iteration step to obtain the magnetic vector potential of the current iteration step;

[0039] Step 3: Add the magnetic vector potential of the current iteration step to the residual of the magnetic vector potential to obtain the magnetic vector potential of the next iteration step;

[0040] Step 4: Update the iteration step;

[0041] Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the static magnetic field is obtained.

[0042] Preferably, it also includes: taking the curl of the magnetic vector potential in the static magnetic field as the magnetic induction intensity, and dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material as the magnetic field intensity, and the magnetic induction intensity and the magnetic field intensity are used to provide quantitative indicators for the optimization of static magnetic properties.

[0043] Beneficial effects: First, the grid data is divided into several polyhedral smooth domains centered on the corresponding edges according to the edges in the grid data. The polyhedral smooth domains are composed of the nodes, body centers, and face centers of the tetrahedral elements in the grid data around the corresponding edges; and the vector shape function gradient of the corresponding edge is calculated based on the vector shape function of each tetrahedral element in the polyhedral smooth domain corresponding to the edge; secondly, the static magnetic field equation is constructed based on the MVP formula and Coulomb norm, and the static magnetic field equation is rewritten based on the vector shape function gradient of each edge and the Galerkin method to obtain the system equation of the static magnetic field; the system equation of the static magnetic field is further rewritten as a Newton iterative linear equation; finally, the Newton iterative linear equation is iteratively solved using the Newton iterative method until convergence to obtain the magnetic vector potential in the static magnetic field. This method couples the edge element with the edge element smoothing technology, and at the same time constructs the static magnetic field equation based on the MVP formula and Coulomb norm, achieving the high computational accuracy and stability requirements of engineering static magnetic field problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0045] Figure 1 This is a flow chart of a high-precision calculation method for static magnetometry of smooth-edged edge elements based on the MVP formula in an embodiment of the present application.

[0046] Figure 2 Schematic diagram of the structure of the polyhedral smooth domain in the embodiment of the present application. DETAILED DESCRIPTION

[0047] To make the above-mentioned objects, features, and advantages of the present application more clearly understood, the specific embodiments of the present application are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present application. However, the present application can be implemented in many other ways than those described herein, and those skilled in the art can make similar improvements without violating the scope of the present application. Therefore, the present application is not limited to the specific embodiments disclosed below.

[0048] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of the technical features being referred to. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of such features. Throughout the description of this application, "plurality" means at least two, for example, two, three, etc., unless otherwise specifically defined.

[0049] like Figure 1 As shown, this embodiment provides a high-precision calculation method for static magnetometry of smooth-edged edge elements based on the MVP formula, the method comprising:

[0050] S1: Import the mesh data of electromagnetic materials.

[0051] S2: Divide the mesh data into several polyhedral smooth domains centered on the corresponding edges according to the edges in the mesh data. The polyhedral smooth domains do not overlap with each other. The polyhedral smooth domains are composed of the nodes, body centers (tetrahedron midpoints), and face centers (face points) of the tetrahedral units in the mesh data around the corresponding edges, such as Figure 2 As shown; the vector shape function gradient of the corresponding edge is calculated based on the vector shape function of each tetrahedral unit in the polyhedral smooth domain corresponding to the edge.

[0052] Specifically, the core idea of ​​edge smoothing technology is to smooth the vector shape function gradient of the geometric unit. For the edge element, the vector shape function gradient of the edge is calculated as:

[0053] ;

[0054] in, Represents a smooth polyhedron k Middle edge e The vector shape function gradient of ; Represents a smooth polyhedron k Middle edge e Vector shape function of ; represents partial derivative; express r direction, r Directions include direction, direction, direction; Indicates the k A polyhedral smooth domain; Represents the edges in the tetrahedral element e Vector shape function of ; Indicates the k Smooth functions of polyhedral smooth domains; Indicates the k The number of tetrahedral elements in a polyhedral smooth domain; Indicates the n The edges of the tetrahedral element e Vector shape function of ; Indicates the n The volume of a tetrahedral unit.

[0055] This method improves the accuracy of static magnetic field calculations on unstructured grids by combining the edge element method with edge smoothing. Furthermore, since both edge smoothing and the edge element method operate on edges, they are naturally compatible. Therefore, coupling edge smoothing with edge elements eliminates the need for geometric construction overhead, enabling the implementation of this method and its application in static magnetic fields without additional geometric overhead.

[0056] Furthermore, the smooth function is the inverse of the volume of the corresponding polyhedron smooth domain, and the smooth function satisfies the following conditions:

[0057] ;

[0058] in, Indicates the k A polyhedral smooth domain; Indicates the k Smooth functions of a polyhedral smooth domain.

[0059] S3: The static magnetic field equations are constructed based on the MVP formula and Coulomb gauge, and the static magnetic field equations are rewritten according to the vector shape function gradients of each edge and the Galerkin method to obtain the system equations of the static magnetic field; the system equations of the static magnetic field are further rewritten as Newton iterative linear equations.

[0060] Specifically, the static magnetic field equation based on the MVP formula is:

[0061] ;

[0062] in, represents the curl; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; Represents the source current density of the static magnetic field. This formula is the MVP formula of the static magnetic field without the Coulomb norm. It only specifies the curl of A, but does not specify the divergence of A in the equation. Therefore, the uniqueness of A cannot be guaranteed when solving the above equation. Although the non-uniqueness of A does not affect the uniqueness of the magnetic induction intensity, the direct solver cannot solve it because the system matrix is ​​singular. Even if the iterative solver is used for solution, the convergence of the solution is poor and may even diverge. Therefore, in order to ensure that the direct solver can solve robustly, this embodiment considers the Coulomb norm and introduces a virtual scalar potential in the entire solution domain. , thereby ensuring that the divergence of A is 0, forming the static magnetic field equation, the static magnetic field equation expression is:

[0063] ;

[0064] ;

[0065] in, represents the curl; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field. In the static magnetic field equation, the virtual scalar potential Dirichlet boundary conditions are imposed on the boundaries of the solution domain, for which Satisfy the Laplace equation in the solution domain, while ensuring the uniqueness of A, The solution in the solution domain is 0.

[0066] In the framework of coupled edge elements and edge smoothing technology, the Coulomb gauge of virtual scalar potential is introduced to solve the uniqueness problem of the coupling method and make the method have reliable stability and convergence.

[0067] Furthermore, the process of obtaining the system equation of the static magnetic field includes:

[0068] The static magnetic field equation is rewritten according to the vector shape function gradient of each edge and the Galerkin method to obtain the weak form of the static magnetic field equation; the weak form of the static magnetic field equation is expressed as:

[0069] ;

[0070] ;

[0071] in, Indicates the k A polyhedral smooth domain; represents the curl; Represents a smooth polyhedron Middle edge i Vector shape function of ; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field; Represents a smooth polyhedron midpoint j The shape function at .

[0072] According to the vector shape function gradient of each edge and the difference theorem, the magnetic vector potential and virtual scalar potential are expanded; the expanded magnetic vector potential and the expanded virtual scalar potential are expressed as:

[0073] ;

[0074] ;

[0075] in, represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; Represents a smooth polyhedron The number of edges; Represents a smooth polyhedron The number of midpoints; Represents a smooth polyhedron Middle edge i Vector shape function of ; Indicates edges i The tangential magnetic vector value on ; Represents a smooth polyhedron midpoint j The shape function at ; Indicates edges i The virtual scalar potential value on .

[0076] By coupling the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential, we obtain the system equation of the static magnetic field, which is expressed as:

[0077] ;

[0078] in, represents the system stiffness matrix; represents the magnetic vector potential; Represents a known external force vector.

[0079] When there is a nonlinear material in the solution domain, that is, the magnetic permeability is nonlinear, then the above system equation is nonlinear. In order to deal with nonlinear magnetostatic problems, the Newton iteration method is a good choice because of its quadratic convergence rate. Therefore, the system equation is rewritten as the Newton linear iteration equation, which is expressed as:

[0080] ;

[0081] ;

[0082] ;

[0083] in, represents the system Jacobian matrix; represents the residual of the magnetic vector potential; represents the residual of the system equations; Indicates the h +1 iteration step of magnetic vector potential; Indicates the h Magnetic vector potential at the iteration step; represents the system stiffness matrix; Represents a known external force vector.

[0084] S4: Newton iteration method is used to iteratively solve the Newton iteration linear equation until convergence, and the magnetic vector potential in the static magnetic field is obtained.

[0085] Specifically, the iterative solution process of the Newton iteration method includes:

[0086] Step 1: Initialize the iteration step;

[0087] Step 2: Solve the Newton iterative linear equation under the current iteration step to obtain the magnetic vector potential of the current iteration step;

[0088] Step 3: Add the magnetic vector potential of the current iteration step to the residual of the magnetic vector potential to obtain the magnetic vector potential of the next iteration step;

[0089] Step 4: Update the iteration step;

[0090] Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the static magnetic field is obtained.

[0091] In this embodiment, the curl of the magnetic vector potential in the static magnetic field is used as the magnetic induction intensity, and the quotient obtained by dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material is used as the magnetic field intensity. The magnetic induction intensity and the magnetic field intensity are used to provide key quantitative indicators for optimizing the static magnetic properties of electromagnetic equipment such as motors, transformers, and brakes.

[0092] The high-precision calculation method for static magnetometry of edge elements with smoothed edges based on the MVP formula provided in this embodiment has the following beneficial effects: the method combines the edge element algorithm based on the MVP formula with the edge smoothing technology, and at the same time couples the Coulomb specification based on the virtual scalar potential, thereby achieving the high calculation accuracy and stability requirements of engineering static magnetic field problems. At the same time, the edge smoothing technology and the edge element have natural compatibility, and can be quickly improved on the basis of traditional edge elements. Finally, under the premise of ensuring the calculation accuracy, the unstructured grid is directly adopted to realize the wide application of the new edge element method in complex engineering electromagnetic fields such as transformers and motors. In addition, since the method is compatible with low-order unstructured grids, and the unstructured grid can handle complex engineering models, the method provides a new solution for the performance optimization of complex electromagnetic equipment such as motors, transformers, and brakes.

[0093] The technical features of the above-mentioned embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above-mentioned embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0094] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present patent application shall be determined by the appended claims.

Claims

1. A high-precision calculation method for static magnetometry of smooth-edged edge elements based on the MVP formula, characterized in that: include: S1: Import the mesh data of electromagnetic materials; S2: Divide the mesh data into several polyhedral smooth domains centered on the corresponding edges according to the edges in the mesh data. The polyhedral smooth domains are composed of the nodes, body centers, and face centers of the tetrahedral elements in the mesh data surrounding the corresponding edges. Calculate the vector shape function gradient of the corresponding edge based on the vector shape function of each tetrahedral element in the polyhedral smooth domain corresponding to the edge. S3: Construct the static magnetic field equation based on the MVP formula and Coulomb gauge, and rewrite the static magnetic field equation according to the vector shape function gradient of each edge and the Galerkin method to obtain the static magnetic field system equation; further rewrite the static magnetic field system equation into Newton iterative linear equation; S4: Newton iteration method is used to iteratively solve the Newton iteration linear equation until convergence, and the magnetic vector potential in the static magnetic field is obtained.

2. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 1 is characterized in that: The calculation formula of the vector shape function gradient of the edge is: ； in, Represents a smooth polyhedron k Middle edge e The vector shape function gradient of ; Represents a smooth polyhedron k Middle edge e Vector shape function of ; represents partial derivative; express r direction, r Directions include direction, direction, direction; Indicates the k A polyhedral smooth domain; Represents the edges in the tetrahedral element e Vector shape function of ; Indicates the k Smooth functions of polyhedral smooth domains; Indicates the k The number of tetrahedral elements in a polyhedral smooth domain; Indicates the n The edges of the tetrahedral element e Vector shape function of ; Indicates the n The volume of a tetrahedral unit.

3. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 2 is characterized in that: The smooth function is the inverse of the volume of the corresponding polyhedron smooth domain, and the smooth function satisfies the following conditions: ； in, Indicates the k A polyhedral smooth domain; Indicates the k Smooth functions of a polyhedral smooth domain.

4. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 1 is characterized in that: The static magnetic field equation is expressed as: ； ； in, represents the curl; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field.

5. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 4 is characterized in that: The process of obtaining the system equations of the static magnetic field includes: The static magnetic field equation is rewritten according to the vector shape function gradient of each edge and the Galerkin method, and the weak form of the static magnetic field equation is obtained. According to the vector shape function gradient of each edge and the difference theorem, the magnetic vector potential and virtual scalar potential are expanded; By coupling the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential, we obtain the system equations of the static magnetic field.

6. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 5 is characterized in that: The weak form of the static magnetic field equation is expressed as: ； ； in, Indicates the k A polyhedral smooth domain; represents the curl; Represents a smooth polyhedron Middle edge i Vector shape function of ; represents the magnetoresistance of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; represents the source current density of the static magnetic field; Represents a smooth polyhedron midpoint j The shape function at .

7. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 5 is characterized in that: The expanded magnetic vector potential and the expanded virtual scalar potential are expressed as: ； ； in, represents the magnetic vector potential of the static magnetic field; represents a virtual scalar potential; Represents a smooth polyhedron The number of edges; Represents a smooth polyhedron The number of midpoints; Represents a smooth polyhedron Middle edge i Vector shape function of ; Indicates edges i The tangential magnetic vector value on ; Represents a smooth polyhedron midpoint j The shape function at ; Indicates edges i The virtual scalar potential value on .

8. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 1 is characterized in that: Newton iterative linear equation is expressed as: ； ； ； in, represents the system Jacobian matrix; represents the residual of the magnetic vector potential; represents the residual of the system equations; Indicates the h +1 iteration step of magnetic vector potential; Indicates the h Magnetic vector potential at the iteration step; represents the system stiffness matrix; Represents a known external force vector.

9. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 8, characterized in that: The iterative solution process of the Newton iteration method includes: Step 1: Initialize the iteration step; Step 2: Solve the Newton iterative linear equation under the current iteration step to obtain the magnetic vector potential of the current iteration step; Step 3: Add the magnetic vector potential of the current iteration step to the residual of the magnetic vector potential to obtain the magnetic vector potential of the next iteration step; Step 4: Update the iteration step; Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the static magnetic field is obtained.

10. The high-precision calculation method for static magnetometry with smooth edges based on the MVP formula according to claim 1, characterized in that: Also includes: The curl of the magnetic vector potential in the static magnetic field is taken as the magnetic induction intensity, and the quotient obtained by dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material is taken as the magnetic field intensity. The magnetic induction intensity and the magnetic field intensity are used to provide quantitative indicators for optimizing static magnetic properties.

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