MVP formula-based edge element magnetostatic high-precision calculation method with smooth edges

Abstract

The invention relates to an MVP formula-based edge element magnetostatic high-precision calculation method for smooth edges, and the method comprises the steps: dividing grid data into a plurality of polyhedral smooth regions with corresponding edges as centers according to the edges in the grid data; the polyhedral smooth domain is formed by surrounding corresponding edges by nodes, body centers and face centers of tetrahedron units in the grid data; calculating the vector shape function gradient of the corresponding edge based on the vector shape function of each tetrahedron unit in the polyhedral smooth domain corresponding to the edge; constructing a static magnetic field equation based on an MVP formula and coulomb specifications, and rewriting the static magnetic field equation according to the vector shape function gradient of each edge and a Galerkin method to obtain a system equation of a static magnetic field; further rewriting the system equation of the static magnetic field into a Newton iteration linear equation; and iteratively solving the Newton iteration linear equation by adopting a Newton iteration method until convergence to obtain the magnetic vector position in the static magnetic field. According to the method, the requirements of high calculation precision and stability of the engineering static magnetic field problem are met.

Description

Technical Field

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

[0002] The numerical calculation of static magnetic fields mainly relies on the finite element method, and the static magnetic field calculation of the finite element method usually depends on two core formulas: magnetic scalar potential (MSP) and magnetic vector potential (MVP). Since the magnetic scalar potential requires special treatment when dealing with multi-connected domain problems and lacks generality, the static magnetic field calculation based on MVP is more popular.

[0003] According to the discrete method of the MVP formula, the finite element method can be divided into nodal elements and edge elements. The edge element method is favored in static magnetic field calculation because it naturally satisfies the physical properties of electromagnetic fields (allowing normal discontinuities). However, the edge element method still faces two major challenges: First, since 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 like the nodal element method by applying the Coulomb gauge, resulting in a singular assembled system matrix and the inability to directly solve the linear sparse equations; Second, as the geometric structure of electromagnetic devices becomes increasingly complex, the mesh generation is usually limited to unstructured meshes, but the accuracy of the edge element method based on unstructured meshes is often insufficient in static magnetic field calculation. These challenges severely limit the application of the edge element method in static magnetism. Therefore, it is particularly urgent and necessary to develop a stable and high-precision edge element numerical algorithm. Summary of the Invention

[0004] Based on this, it is necessary to provide a high-precision calculation method for edge elements of static magnetism with smooth edges based on the MVP formula. The method includes: S1: Import the mesh data of electromagnetic materials; S2: Divide the mesh data into several polyhedron smooth domains centered on the corresponding edges according to each edge in the mesh data. The polyhedron smooth domain is composed of the nodes, the centroid of the body, and the centroid of the face of the tetrahedral elements in the mesh data around the corresponding edge; calculate the gradient of the vector shape function of the corresponding edge based on the vector shape functions of the tetrahedral elements in the polyhedron smooth domain corresponding to the edge. S3: Construct the static magnetic field equation based on the MVP formula and the Coulomb gauge, and rewrite the static magnetic field equation according to the gradient of the vector shape function of each edge and the Galerkin method to obtain the system equation of the static magnetic field; further rewrite the system equation of the static magnetic field into a Newton iterative linear equation. S4: Use the Newton iterative method to iteratively solve the Newton iterative linear equation until convergence to obtain the magnetic vector potential in the static magnetic field.

[0005] Preferably, the calculation formula for the gradient of the vector shape function of the edge is: ; Among them, represents the gradient of the vector shape function of the edge k in the polyhedral smooth domain e ; represents the vector shape function of the edge k in the polyhedral smooth domain e ; represents the partial derivative; represents r direction, r The direction includes direction, direction, direction; represents the k th polyhedral smooth domain; represents the vector shape function of the edge e in the tetrahedral element represents the k th smooth function of the polyhedral smooth domain; represents the k th number of tetrahedral elements in the polyhedral smooth domain; represents the n th vector shape function of the edge e in the tetrahedral element represents the n th volume of the tetrahedral element.

[0006] Preferably, the smooth function is the reciprocal of the volume of the corresponding polyhedral smooth domain, and the conditions satisfied by the smooth function are: ; Among them, represents the k th polyhedral smooth domain; represents the k th smooth function of the polyhedral smooth domain.

[0007] Preferably, the expression of the static magnetic field equation is: ; ; Among them, represents the curl; represents the magnetic resistivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field.

[0008] Preferably, the process of obtaining the system equation of the static magnetic field includes: Rewrite the static magnetic field equation according to the gradients of the vector shape functions of each edge and the Galerkin method to obtain the weak form of the static magnetic field equation; Expand the magnetic vector potential and the virtual scalar potential according to the gradients of the vector shape functions of each edge and the difference theorem; Couple the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential to obtain the system equation of the static magnetic field.

[0009] Preferably, the weak form of the static magnetic field equation is expressed as: ; ; where, represents the k th polyhedral smooth domain; represents curl; represents the polyhedral smooth domain the vector shape function of edge i in; represents the magnetic resistivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field; represents the polyhedral smooth domain the shape function at node j in;

[0010] Preferably, the expanded magnetic vector potential and the expanded virtual scalar potential are respectively expressed as: ; ; where, represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the number of edges in the polyhedral smooth domain ; represents the polyhedral smooth domain the number of nodes in; represents the polyhedral smooth domain the vector shape function of edge i in; represents the tangential magnetic vector value on edge i ; represents the polyhedral smooth domain the shape function at node j in; represents the virtual scalar potential value on edge i ;

[0011] Preferably, the Newton iteration linear equation is expressed as: ; ; ; Among them, represents the system Jacobian matrix; represents the residual of the magnetic vector potential; represents the residual of the system equation; represents the h +1 magnetic vector potential at the iteration step; represents the h magnetic vector potential at the iteration step; represents the system stiffness matrix; represents the known external force vector.

[0012] Preferably, the iterative solution process of the Newton iteration method includes: Step 1: Initialize the iteration step; Step 2: Solve the Newton iteration linear equation at the current iteration step to obtain the magnetic vector potential at the current iteration step; Step 3: Add the magnetic vector potential at the current iteration step to the residual of the magnetic vector potential to obtain the magnetic vector potential at the next iteration step; Step 4: Update the iteration step; Step 5: Repeat Steps 2 - 4 until the Newton iteration linear equation converges, and the magnetic vector potential in the static magnetic field.

[0013] Preferably, it further includes: taking the curl of the magnetic vector potential in the static magnetic field as the magnetic induction intensity, and taking the quotient obtained by dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material as the magnetic field intensity. The magnetic induction intensity and the magnetic field intensity are used to provide a quantitative index for optimizing the static magnetic properties.

[0014] Beneficial effects: First, several polyhedral smooth domains centered on the corresponding edges are divided from the mesh data according to each edge in the mesh data. The polyhedral smooth domain is composed of the nodes, the centroid of the body, and the centroid of the face of the tetrahedral elements in the mesh data around the corresponding edge; and the gradient of the vector shape function corresponding to the edge is calculated based on the vector shape functions of each tetrahedral element in the polyhedral smooth domain corresponding to the edge. Secondly, a static magnetic field equation is constructed based on the MVP formula and the Coulomb gauge, and the static magnetic field equation is rewritten according to the gradient of the vector shape function 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 iteration linear equation; finally, the Newton iteration method is used to iteratively solve the Newton iteration linear equation 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 a static magnetic field equation based on the MVP formula and the Coulomb gauge, meeting the high calculation accuracy and stability requirements of engineering static magnetic field problems. Description of the Drawings

[0015] To more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the accompanying drawings required for the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0016] Figure 1 It is a flowchart of the high-precision edge element static magnetic calculation method with smooth edges based on the MVP formula in the embodiments of the present application.

[0017] Figure 2 It is a schematic diagram of the construction of the smooth domain of the polyhedron in the embodiments of the present application. Detailed implementation manners

[0018] To make the above objects, features, and advantages of the present application more obvious and understandable, the following will make a detailed description of the specific implementation manners of the present application with reference to the accompanying drawings. Many specific details are elaborated in the following description to fully understand the present application. However, the present application can be implemented in many other ways different from those described herein. Those skilled in the art can make similar improvements without departing from the connotation of the present application. Therefore, the present application is not limited by the specific embodiments disclosed below.

[0019] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of such features. In the description of the present application, "a plurality of" means at least two, such as two, three, etc., unless otherwise specifically and clearly defined.

[0020] As Figure 1 shown, this embodiment provides a high-precision edge element static magnetic calculation method with smooth edges based on the MVP formula. The method includes: S1: Import the grid data of the electromagnetic material.

[0021] S2: Divide the grid data into several polyhedron smooth domains centered on the corresponding edges according to the edges in the grid data. The polyhedron smooth domains do not overlap with each other. The polyhedron smooth domain is composed of the nodes, the centroid (the midpoint of the tetrahedron), and the face centroid (the midpoint of the face) of the tetrahedron elements in the grid data around the corresponding edge. As Figure 2 shown; calculate the gradient of the vector shape function of the corresponding edge based on the vector shape functions of the tetrahedron elements in the polyhedron smooth domain corresponding to the edge.

[0022] Specifically, the core idea of the edge smoothing technique is to smooth the gradient of the vector shape function of geometric elements. For edge elements, the calculation formula for the gradient of the vector shape function of an edge is: ; where, represents the vector shape function gradient of edge k in the polyhedron smoothing domain e ; represents the vector shape function of edge k in the polyhedron smoothing domain e ; represents partial derivative; represents r direction, r The direction includes direction, direction, direction; represents the k th polyhedron smoothing domain; represents the vector shape function of edge e in the tetrahedron element; represents the k th smoothing function of the polyhedron smoothing domain; represents the k th number of tetrahedron elements in the polyhedron smoothing domain; represents the n th vector shape function of edge e in the th tetrahedron element; n represents the volume of the

[0023] By combining the edge element method with the edge smoothing technique, the calculation accuracy of the static magnetic field under unstructured grids is improved. In addition, both the edge smoothing technique and the edge element method act on the edges, and they have natural compatibility. Therefore, there is no need to worry about the geometric construction cost of the edges when coupling the edge smoothing technique in the edge elements, and the implementation of this method and its application in the static magnetic field can be completed without additional geometric cost.

[0024] Furthermore, the smoothing function is the reciprocal of the volume of the corresponding polyhedron smoothing domain, and the conditions satisfied by the smoothing function are: ; where, represents the k th polyhedron smoothing domain; represents the k th smoothing function of the polyhedron smoothing domain.

[0025] S3: Construct the static magnetic field equation based on the MVP formula and the Coulomb gauge, and rewrite the static magnetic field equation according to the gradients of the vector shape functions of each edge and the Galerkin method to obtain the system equation of the static magnetic field; further rewrite the system equation of the static magnetic field into a Newton iteration linear equation.

[0026] Specifically, the static magnetic field equation based on the MVP formula is: ; where represents the curl; represents the magnetic resistivity 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 gauge, which only specifies the curl of A and 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, since the system matrix is singular, the direct solver cannot solve it. Even if the iterative solver is used to solve it, its convergence during the solution is poor or even divergent. Therefore, in order to ensure that the direct solver can solve robustly, this embodiment considers the Coulomb gauge and introduces a virtual scalar potential in the entire solution domain, so as to ensure that the divergence of A is 0 and form the static magnetic field equation. The expression of the static magnetic field equation is: ; ; where represents the curl; represents the magnetic resistivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field. In the static magnetic field equation, the virtual scalar potential is applied with the Dirichlet boundary condition on the boundary of the solution domain. Therefore, satisfies the Laplace equation in the solution domain. While ensuring the uniqueness of A, the solution in the solution domain is 0.

[0027] In the framework of coupling edge elements and edge smoothing techniques, the Coulomb gauge of the virtual scalar potential is introduced, thus solving the uniqueness problem of this coupling method and making the method have reliable stability and convergence.

[0028] Furthermore, the process of obtaining the system equation of the static magnetic field includes: Rewrite the static magnetic field equation according to the gradients of the vector shape functions 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: ; ; wherein, represents the k th polyhedral smooth domain; represents curl; represents the vector shape function of the edge in the polyhedral smooth domain i ; represents the reluctivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field; represents the shape function at the node in the polyhedral smooth domain j ;

[0029] Expand the magnetic vector potential and the virtual scalar potential according to the gradient of the vector shape function of each edge and the difference theorem; the expanded magnetic vector potential and the expanded virtual scalar potential are respectively expressed as: ; ; wherein, represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the number of edges in the polyhedral smooth domain ; represents the number of nodes in the polyhedral smooth domain ; represents the vector shape function of the edge in the polyhedral smooth domain i ; represents the tangential magnetic vector value on the edge i ; represents the shape function at the node in the polyhedral smooth domain j ; represents the virtual scalar potential value on the edge i ;

[0030] Couple the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential to obtain the system equation of the static magnetic field, which is expressed as: ; wherein, represents the system stiffness matrix; represents the magnetic vector potential; represents the known external force vector.

[0031] When there are nonlinear materials in the solution domain, that is, the magnetic permeability is nonlinear, the above system equations are nonlinear. To handle the nonlinear static magnetic problem, the Newton iteration method has a quadratic convergence rate and is thus a good choice. Therefore, the system equations are rewritten as Newton linear iteration equations, and the Newton iteration linear equations are expressed as: ; ; ; where, represents the system Jacobian matrix; represents the residual of the magnetic vector potential; represents the residual of the system equations; represents the h +1-th iteration step of the magnetic vector potential; represents the h -th iteration step of the magnetic vector potential; represents the system stiffness matrix; represents the known external force vector.

[0032] S4: Use the Newton iteration method to iteratively solve the Newton iteration linear equations until convergence to obtain the magnetic vector potential in the static magnetic field.

[0033] Specifically, the iterative solution process of the Newton iteration method includes: Step 1: Initialize the iteration step; Step 2: Solve the Newton iteration linear equations at the current iteration step to obtain the magnetic vector potential at the current iteration step; Step 3: Add the magnetic vector potential at the current iteration step to the residual of the magnetic vector potential to obtain the magnetic vector potential at the next iteration step; Step 4: Update the iteration step; Step 5: Repeat steps 2-4 until the Newton iteration linear equations converge to the magnetic vector potential in the static magnetic field.

[0034] In this embodiment, it further includes: taking the curl of the magnetic vector potential in the static magnetic field as the magnetic induction intensity, and taking the quotient obtained by dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material as the magnetic field intensity. The magnetic induction intensity and the magnetic field intensity provide key quantitative indicators for optimizing the static magnetic performance of electromagnetic devices such as motors, transformers, and brakes.

[0035] The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula provided in this embodiment has the following beneficial effects: This method combines the edge element algorithm based on the MVP formula and the edge smoothing technology, and at the same time couples the Coulomb gauge based on the virtual scalar potential, realizing the high calculation accuracy and stability requirements for 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 the traditional edge element. Finally, on the premise of ensuring the calculation accuracy, directly using unstructured grids can realize the wide application of the new edge element method in complex engineering electromagnetic fields such as transformers and motors. In addition, since this method is adaptable to low-order unstructured grids, and unstructured grids can handle complex engineering models, this method provides a new solution idea for the performance optimization of complex electromagnetic devices such as motors, transformers, and brakes.

[0036] The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope described in this specification.

[0037] The above-described embodiments only represent several implementation manners of the present application, and the description thereof is relatively specific and detailed, but it should not be construed as a limitation on the scope of the patent application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the appended claims.

Claims

1. A high-precision calculation method for the edge element of static magnetism with smooth edges based on the MVP formula, characterized in that Including: S1: Import the grid data of the electromagnetic material; S2: Divide the grid data into several polyhedron smooth domains centered on the corresponding edges according to each edge in the grid data. The polyhedron smooth domain is composed of the nodes, the centroid of the body, and the centroid of the face of the tetrahedral elements in the grid data around the corresponding edge; calculate the vector shape function gradient of the corresponding edge based on the vector shape functions of the tetrahedral elements in the polyhedron smooth domain corresponding to the edge; S3: Construct the static magnetic field equation based on the MVP formula and the Coulomb gauge, and rewrite the static magnetic field equation according to the vector shape function gradients of each edge and the Galerkin method to obtain the system equation of the static magnetic field; further rewrite the system equation of the static magnetic field into a Newton iterative linear equation; S4: Use the Newton iterative method to iteratively solve the Newton iterative linear equation until convergence to obtain the magnetic vector potential in the static magnetic field.

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

3. The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula according to claim 2, characterized in that, The smoothing function is the reciprocal of the volume of the corresponding polyhedron smooth domain, and the conditions satisfied by the smoothing function are: ； Among them, represents the k th polyhedral smooth domain; represents the smooth function of the k th polyhedral smooth domain.

4. The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula according to claim 1, characterized in that The expression of the static magnetic field equation is: ； ； Among them, represents curl; represents the magnetic resistivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field.

5. The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula according to claim 4, characterized in that The process of obtaining the system equation of the static magnetic field includes: Rewrite the static magnetic field equation according to the vector shape function gradients of each edge and the Galerkin method to obtain the weak form of the static magnetic field equation; Expand the magnetic vector potential and the virtual scalar potential according to the vector shape function gradients of each edge and the difference theorem; Couple the weak form of the static magnetic field equation, the expanded magnetic vector potential, and the expanded virtual scalar potential to obtain the system equation of the static magnetic field.

6. The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula according to claim 5, characterized in that The weak form of the static magnetic field equation is expressed as: ； ； Among them, represents the k th polyhedral smooth domain; represents curl; represents the polyhedral smooth domain in which the edge i vector shape function; represents the magnetic reluctivity of the static magnetic field; represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the source current density of the static magnetic field; represents the polyhedral smooth domain in which the node j shape function at.

7. The method for high-precision calculation of the edge element of magnetostatic field with smooth edges based on the MVP formula according to claim 1, characterized in that The expanded magnetic vector potential and the expanded virtual scalar potential are respectively expressed as: ； ； Among them, represents the magnetic vector potential of the static magnetic field; represents the virtual scalar potential; represents the number of edges in the smooth domain of the polyhedron ; represents the number of nodes in the smooth domain of the polyhedron ; represents the vector shape function of the edges in the smooth domain of the polyhedron ; i represents the tangential magnetic vector value on the edge ; i represents the shape function at the node in the smooth domain of the polyhedron ; represents the virtual scalar potential value on the edge j ; represents the virtual scalar potential value on the edge i .

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

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

10. The edge element static magnetic high-precision calculation method with smooth edges based on the MVP formula according to claim 1, wherein, It also includes: Take the curl of the magnetic vector potential in the static magnetic field as the magnetic induction intensity, and take the quotient obtained by dividing the magnetic induction intensity by the magnetic permeability of the electromagnetic material as the magnetic field intensity. The magnetic induction intensity and the magnetic field intensity are used to provide a quantitative index for optimizing the static magnetic properties.

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