Novel self-adaptive mesh refining method based on finite element boundary magnetic field conservation

Abstract

The invention discloses a novel self-adaptive mesh refinement method based on finite element boundary magnetic field conservation, and belongs to the technical field of self-adaptive mesh division, and the refinement method comprises the following specific steps: I, building a control equation according to a magnetostatic force problem, and converting the control equation into an integral form through a weighted residual method, integral terms are processed through segmented integral; iI, on the element interface, obtaining an error estimation value by calculating the integral difference of the product of the magnetic field and the interpolation function, and refining each element through a non-uniform grid division method; according to the method, a smaller unit can be generated in an area with a larger magnetic flux line curvature, so that the precision of a solution is improved, and in addition, the method is based on a non-consistent finite element, so that the generation of a flat unit or a unit with a poor shape is effectively avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of adaptive mesh generation, and particularly to a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary. Background Art

[0002] The FEM is one of the successful numerical simulation methods in electromagnetics. However, one problem is how to generate an appropriate mesh to improve the simulation accuracy and reduce the computing time and memory used. Therefore, some adaptive mesh generation techniques have been studied. To obtain a good adaptive mesh generation method, two main techniques are required: 1) a good error estimation method; 2) an appropriate mesh refinement technique. Among the techniques mainly applied in the currently studied adaptive mesh generation methods, the ZZ method is a very good error estimation method that is widely used at present. However, in the air region far from the object, this method often unnecessarily subdivides into smaller elements, thus more necessarily increasing the number of elements and the computing time, and at the same time not improving the computing accuracy. The mesh improvement process is an important task for obtaining a high-precision solution. Currently, the Delaunay triangulation method is usually adopted for mesh improvement. It is a well-known and powerful method for generating triangular or tetrahedral meshes. However, when it is used for mesh refinement in adaptive finite element analysis, many elements with better quality will be generated. Even if the shape of the parent element is good, the newly generated sub-elements after subdivision will become worse in shape; or if the parent element is flat, after subdivision, the generated sub-elements will be flatter. With each further subdivision, the quality of the elements will decrease once. Therefore, the Delaunay triangulation method is not suitable for mesh improvement in adaptive finite element analysis.

[0003] After retrieval, Chinese Patent No. CN116912453A discloses a two-dimensional triangular mesh generation method for adaptive mesh density adjustment. Although this invention can adapt to complex model structures and generate high-quality triangular meshes, it cannot generate smaller elements in the region with a large curvature of magnetic flux lines, reducing the accuracy of the solution, and it is easy to generate flat or poorly shaped elements. For this reason, we propose a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary. Summary of the Invention

[0004] The purpose of the present invention is to solve the defects existing in the prior art, and to propose a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] A new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary, and the specific steps of this refinement method are as follows:

[0007] Ⅰ. Based on the magnetostatic problem, establish the governing equations, and by the weighted residual method, transform the governing equations into integral form, and deal with the integral terms through integration by parts;

[0008] Ⅱ. On the element interface, obtain the error estimate value by calculating the integral difference of the product of the magnetic field and the interpolation function, and refine each element by the non-uniform mesh division method;

[0009] Ⅲ. Establish the original finite element equations, and construct the finite element equation set by correlating the vector potentials of the master side and the slave side through the constitutive matrix;

[0010] Ⅳ. Solve the finite element equation set, refine the mesh through multiple iterations until the error estimate values of all element interfaces are lower than the predetermined threshold.

[0011] As a further solution of the present invention, the specific steps of transforming the governing equations into integral form by the weighted residual method in step Ⅰ are as follows:

[0012] S1.1: Based on the magnetostatic problem, establish the governing equations, and by the weighted residual method, construct the residual expression;

[0013] S1.2: Select the weighting function, perform integral processing on the entire calculation domain, and then perform integration by parts on the high-order differential terms in the integral formula to transform the governing equations into integral form.

[0014] As a further solution of the present invention, the specific calculation formula of the governing equations in S1.1 is as follows:

[0015]

[0016] In the formula, A and J0 respectively represent the magnetic vector potential and the source current; applying the weighted residual method to formula (1) to obtain the following formula:

[0017]

[0018] In the formula, w and v respectively represent the vector interpolation function and the integration volume; then transform the left term of formula (2) through integration by parts, and its specific transformation form is as follows:

[0019]

[0020] In the formulation of the edge-based finite element model, since the tangential component value of the magnetic field H is considered the same on the interface between two adjacent elements, formula (3) can be transformed into the following form:

[0021]

[0022] In the formula, S and n respectively represent the integration surface and the unit normal vector of the integration surface S.

[0023] As a further solution of the present invention, the specific steps of refining each element by the non-uniform grid division method in step II are as follows:

[0024] S2.1: Let elements i and j have a common interface S, and this interface S is composed of edges k, l, and m. Then each edge corresponds to its vector interpolation function w k 、w l and w m . Then, formula (4) is transformed, and its specific transformation form is as follows:

[0025] d e,k =∫ S (H e ×w k )·n e dS (e = i, j) (5)

[0026] And all three edges satisfy the following equation:

[0027] D f =d i,f +d j,f =0 (f = k, l, m) (6)

[0028] In the formula, D k 、D l and D m respectively represent the error amounts calculated on the three edges k, l, and m on the interface S;

[0029] S2.2: Take the values of D k 、D l and D m calculated in formula (6) as the error estimation value E ij . If E ij exceeds the preset threshold ε, elements i and j will be adaptively subdivided into smaller elements, where the error estimation value E ij is specifically expressed by the following formula:

[0030] E ij =max(|D k |,|D l |,|D m |)>ε (7)

[0031] In the formula, ε represents the preset threshold.

[0032] As a further solution of the present invention, the specific expression form of associating the vector potential of the main edge and the slave edge through the constitutive matrix in step III is as follows:

[0033] Ka = b (8)

[0034] In the formula, K, a, and b represent the stiffness matrix, vector potential, and source vector, respectively; among them, the vector potential a on the slave edge and the vector potential on the master edge The relationship between them is specifically as follows:

[0035]

[0036] In the formula, C is the constitutive matrix derived according to the relationship between the master edge and the slave edge; among them, the constitutive matrix C:

[0037]

[0038] Among them, t represents the transpose, and a symmetric stiffness matrix is generated through C t

[0039] As a further solution of the present invention, the specific expression formula for solving the finite element equations in step IV is as follows:

[0040]

[0041] In the formula, C i is the constitutive vector between the master edge and the slave edge in the i-th adaptive subdivision step.

[0042] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0043] The new adaptive mesh refinement method based on the conservation of the magnetic field at the finite element boundary establishes a control equation according to the magnetostatic problem, and then converts the control equation into an integral form through the weighted residual method. After using integration by parts to process the integral terms, the error estimate value is obtained by calculating the integral difference of the product of the magnetic field and the interpolation function. If it exceeds the predetermined threshold, it is considered that the element needs to be refined. The original finite element equation is established, the vector potentials of the master edge and the slave edge are correlated through the constitutive matrix, and the introduced transpose is used to generate a symmetric stiffness matrix, and the system of equations is solved, and the mesh is refined through multiple iterations. After each iteration, the stiffness matrix and the source vector are updated according to the new mesh topology, and the constitutive vector in the previous iteration step is reused, and the refinement process is repeated until the error estimate values of all element interfaces are lower than the predetermined threshold. It can generate smaller units in the region with a large curvature of the magnetic flux line, thereby improving the accuracy of the solution. In addition, this method is based on non-conforming finite elements, effectively avoiding the generation of flat or poorly shaped units. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The drawings are used to provide a further understanding of the present invention and constitute a part of the specification. They are used together with the embodiments of the present invention to explain the present invention and do not constitute a limitation to the present invention.

[0045] Figure 1 ​Flow chart of a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary proposed by the present invention;

[0046] Figure 2 Algorithm flow chart of a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary proposed by the present invention. Detailed implementation manners

[0047] Example, referring to Figure 1-2 a new adaptive mesh refinement method based on the conservation of magnetic field at the finite element boundary, the specific steps of the refinement method are as follows:

[0048] According to the magnetostatic problem, establish a control equation, and transform the control equation into an integral form by the weighted residual method, and process the integral terms by integration by parts.

[0049] Specifically, according to the magnetostatic problem, establish a control equation, and by the weighted residual method, construct a residual expression, select a weighting function, and perform an integral process on the entire calculation domain, and then perform integration by parts on the high-order differential terms in the integral formula to transform the control equation into an integral form.

[0050] It should be further noted that the specific calculation formula of the control equation is as follows:

[0051]

[0052] In the formula, A and J0 respectively represent the magnetic vector potential and the source current; apply the weighted residual method to formula (1) to obtain the following formula:

[0053]

[0054] In the formula, w and v respectively represent the vector interpolation function and the integration volume; then transform the left term of formula (2) by integration by parts, and its specific transformation form is as follows:

[0055]

[0056] In the formulation of the edge-based finite element model, since the tangential component value of the magnetic field H is considered the same on the interface between two adjacent elements, formula (3) can be transformed into the following form:

[0057]

[0058] In the formula, S and n respectively represent the integration surface and the unit normal vector of the integration surface S; then transform formula (4), and its specific transformation form is as follows:

[0059] d e,k = ∫ S (H e × wk )·n e dS(e = i, j)

[0060] (5).

[0061] On the element interface, the error estimate value is obtained by calculating the integral difference of the product of the magnetic field and the interpolation function, and each element is refined by the non-uniform grid division method.

[0062] Specifically, let elements i and j have a common interface S, and this interface S is composed of edges k, l, and m. Then each edge corresponds to its vector interpolation function w k 、w l and w m , and the error amounts D k 、D l and D m are calculated respectively on the three edges k, l, and m. Then the calculated D k 、D l and D m values are used as the error estimate value E ij . If E ij exceeds the preset threshold ε, elements i and j will be adaptively subdivided into smaller elements.

[0063] It should be further noted that the specific transformation form of formula (4) is as follows:

[0064] d e,k =∫ S (H e ×w k )·n e dS (e = i, j) (5)

[0065] And all three edges satisfy the following equation:

[0066] D f =d i,f +d j,f =0 (f = k, l, m) (6)

[0067] In the formula, D k 、D l and D m respectively represent the error amounts calculated on the three edges k, l, and m on the interface S;

[0068] The error estimate value E ij is specifically expressed by the following formula:

[0069] E ij =max(|D k |,|D l |,|D m |)>ε (7)

[0070] In the formula, ε represents a preset threshold value.

[0071] Establish the original finite element equation, and construct the finite element equation set by associating the vector potentials of the master side and the slave side through the constitutive matrix.

[0072] It should be further noted that the specific expression form of associating the vector potentials of the master side and the slave side through the constitutive matrix is as follows:

[0073] Ka = b (8)

[0074] In the formula, K, a, and b represent the stiffness matrix, vector potential, and source vector respectively; among them, the vector potential a on the slave side and the vector potential on the master side have the following specific relationship:

[0075]

[0076] In the formula, C is the constitutive matrix derived according to the relationship between the master side and the slave side; among them, the constitutive matrix C:

[0077]

[0078] Among them, t represents the transpose, and a symmetric stiffness matrix is generated through C t

[0079] Solve the finite element equation set, and refine the mesh through multiple iterations until the error estimation values of all element interfaces are lower than the predetermined threshold.

[0080] It should be further noted that the specific expression formula for solving the finite element equation set is as follows:

[0081]

[0082] In the formula, C i is the constitutive vector between the master side and the slave side in the i-th adaptive subdivision step.

[0083] In addition, in this embodiment, it is necessary to solve the linear equation set (11) in the i-th adaptive subdivision step, and the constitutive vector C j (j = 1,..., i - 1) of the previous step can be reused to reduce the workload. For the stiffness matrix K i and the source vector b i , only the elements related to the newly subdivided elements need to be added to K i-1 and b i-1 of the previous step.​

Claims

1. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field in the finite element, characterized in that, The specific steps of the refinement method are as follows: Ⅰ. According to the magnetostatic problem, establish the control equation, and by the weighted residual method, transform the control equation into an integral form, and process the integral terms through integration by parts; Ⅱ. On the element interface, obtain the error estimate value by calculating the integral difference of the product of the magnetic field and the interpolation function, and refine each element by the non-uniform mesh division method; Ⅲ. Establish the original finite element equation, and construct the finite element equation set by correlating the vector potentials of the main side and the slave side through the constitutive matrix; Ⅳ. Solve the finite element equation set, refine the mesh through multiple iterations until the error estimate values of all element interfaces are lower than the predetermined threshold.

2. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field of the finite element, as claimed in claim 1, wherein The specific steps of transforming the control equation into an integral form by the weighted residual method described in step Ⅰ are as follows: S1.1: According to the magnetostatic problem, establish the control equation, and by the weighted residual method, construct the residual expression; S1.2: Select the weighting function, perform integral processing on the entire calculation domain, and then perform integration by parts on the high-order differential terms in the integral expression to transform the control equation into an integral form.

3. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field of the finite element, as described in claim 2, wherein The specific calculation formula of the control equation described in S1.1 is as follows: In the formula, A and J0 represent the magnetic vector potential and the source current respectively; apply the weighted residual method to formula (1) to obtain the following formula: In the formula, w and v represent the vector interpolation function and the integral volume respectively; then transform the left term of formula (2) through integration by parts, and its specific transformation form is as follows: In the formulation of the edge-based finite element model, since the tangential component value of the magnetic field H is considered the same on the interface between two adjacent elements, formula (3) can be transformed into the following form: In the formula, S and n represent the integration surface and the unit normal vector of the integration surface S respectively.

4. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field of the finite element, as claimed in claim 3, wherein The specific steps of refining each element by the non-uniform mesh division method described in step Ⅱ are as follows: S2.1: Make elements i and j have a common interface S, and this interface S is composed of edges k, l, and m. After that, each edge corresponds to its vector interpolation function w k , w l and w m . After that, transform formula (4), and its specific transformation form is as follows: d e,k = ∫ S (H e × w k )· n e dS(e = i,j) (5) And all three sides satisfy the following equation: D f = d i,f + d j,f = 0 (f = k, l, m) (6) where D k , D l and D m represent the error amounts calculated on three sides k, l, and m on the interface S, respectively; S2.2: Take the values of D k , D l and D m calculated in formula (6) as the error estimate value E ij . If E ij exceeds the preset threshold ε, elements i and j will be adaptively subdivided into smaller elements, where the error estimate value E ij is specifically expressed by the following formula: E ij = max(|D k |, |D l |, |D m |) > ε (7) In the formula, ε represents the preset threshold.

5. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field of the finite element, as claimed in claim 4, characterized in that The specific expression form of correlating the vector potentials of the main side and the slave side through the constitutive matrix described in step Ⅲ is as follows: Ka = b (8) In the formula, K, a, and b represent the stiffness matrix, vector potential, and source vector, respectively; among them, the relationship between the vector potential a on the side and the vector potential on the main side is specifically as follows: In the formula, C is the constitutive matrix derived according to the relationship between the main side and the slave side; among them, the constitutive matrix C: C t KCà = C t b (10) where t represents transpose, and a symmetric stiffness matrix is generated through C t ​ 6. A new adaptive mesh refinement method based on the conservation of the boundary magnetic field in the finite element, as claimed in claim 5, wherein, The specific expression formula of solving the finite element equation set described in step Ⅳ is as follows: where C i is the constitutive vector between the primary edge and the secondary edge in the i-th adaptive refinement step.

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