JPH0778522B2 - 3D magnetic field analyzer - Google Patents

Description

TECHNICAL FIELD The present invention relates to a three-dimensional magnetic field analyzer using the finite element method.

[Problems to be Solved by Conventional Techniques and Inventions] With the development of electronic computers, magnetic field analysis by the finite element method has been extensively researched, and a method using a magnetic vector potential has already been applied to a two-dimensional problem. Has been established. However, although various methods have been proposed for the three-dimensional magnetic field analysis problem, there is currently no definite method.

The magnetic vector potential method was formulated by the variational method, but there is a problem in handling gauge conditions.
^{(1) to (3)} .

(1) MVKChari, P.Silvester, A, Konrad, ZJCsendes
and MAPalmo, “Three-Dimensional Magnetostatic F
ield Analysis of Electrical Machinery by the Finit
e-Element Method ", IEEE Trans.on Power App.Syst., vo
l.PAS-100, pp.4007-4019, Aug. (1981) (2) NADemerdash, TWNehl, FAFouad and OAMoh
ammed, “Three Dimensional Finite Element Vector Po
tential Formulation of Magnetic Fields in Electric
al Apparatus ", IEEE Trans.on Power App.Syst.vol.PAS
-100.pp4104-4111.Aug. (1981) (3) JLCoulomb, “Finite Element Three Dimension
al Magnetic Field Computation ", IEEE Trans.on Mag
n., vol.MAG-17, pp.3241-3246, Nov. (1981) Also, since there are 3 unknowns per node,
There is a problem that it requires a lot of calculation time and memory.

Therefore, research is currently centered on the scalar potential methods ^{(4) to (5)} or the combined scalar-vector potential method ⁽⁶⁾ .

(4) J. Simkin and CWTrowbridge, “Three-dimensio
nal Nonlinear Electromagnetic Field Computations, u
sing Scalar Potentials ", IEE Proc., vol.127, Pt.B, pp.
368-374, Nov. (1980) (5) ID Mayergoyz, MVKChari and J.D'Angelo, “A
New Scalar Potential Formulation for Three-Dimens
ional Magnetostatic Problems ", IEEE Trans.on Magn.,
vol.MAG-23.pp.3889-3894, Nov. (1987) (6) CJ Carpenter, “Comparison of Alternative Fo
rmulation of 3-Dimensional Magnetic-Field and Eddy
-Current Problems at Power Frequencies ", Proc.IEE, vol.124, pp.1026-1034, Nov. (1
977) However, in these methods, it is unclear how to treat the current region, the magnetic substance, and the boundary condition between different magnetic substances [ ^7] .

(7) A.Krawczyk and J.Turowski, “Recent Developme
nt in Eddy Current Analysis ", IEEE Trans.on Magn., p
p.3032-3037, vol.MAG-23, Sep. (1987) From these facts, it can be said that the method using the magnetic vector potential is the best way to deal with problems involving multiple media in a unified manner.

The present invention has been made under such a background, and an object of the present invention is to provide a three-dimensional magnetic field analysis apparatus capable of easily performing a three-dimensional magnetic field analysis using the finite element method.

[Means for Solving the Problems] In order to solve the above problems, the present invention is based on Poisson's equation of a three-dimensional static magnetic field, and analyzes a three-dimensional magnetic field including different magnetic bodies by using the finite element method. In a three-dimensional magnetic field analyzer, the magnetic permeability μ _{0 of} vacuum and the magnetic field strength Magnetized to the product of Magnetic flux density The first calculation means for obtaining the magnetic vector potential Under Coulomb's gauge condition with zero divergence of Is used to form a Poisson's equation of a three-dimensional static magnetic field, and a vector potential is calculated from the boundary condition between the different magnetic bodies. And the vector potential by performing the discretization by the Galerkin method using the Poisson's equation and the interpolation function. Fourth computing means for obtaining three finite element formulas separated for each three-dimensional component using the boundary condition of, and a vector potential for each finite element based on the finite element formula Fifth, obtaining a coefficient matrix composed of the coefficients of each component of
And the vector potential by solving the determinant using the coefficient matrix Sixth computing means for obtaining From the magnetic flux density Seeking and magnetizing And a seventh arithmetic means for correcting

[Operation] According to the above configuration, the finite element expression can be separated for each three-dimensional component.

That is, as will be described later in detail, (1) (2) Vector potential Make the divergence of zero By imposing the Coulomb gauge condition, it becomes possible to eliminate the surface integral term of the finite element formula. This allows the finite element formula to be separated for each three-dimensional component.

As a result, the vector potential Since the three components can be analyzed independently, the memory capacity and calculation time of the analyzer can be significantly reduced.

[Examples] Examples of the present invention will be described below.

FIG. 1 is a functional block diagram showing the configuration of a three-dimensional magnetic field analysis apparatus according to an embodiment of the present invention. A description will be given with reference to this figure.

[1] Derivation of Poisson's equation in a three-dimensional static magnetic field The basic equation in a static magnetic field is given by the following equation.

The first calculation means 1 shown in FIG. 1 calculates the equation (3).

From the above equation (2), the magnetic vector potential shown in the following equation Can be defined.

From equations (1), (3) and (4), Rot-rot expression is obtained, but in this form the boundary conditions are unclear mathematically.

By the way, in equation (4) Is allowed to have an indeterminacy of grad φ, but by imposing the Coulomb gauge given by The uniqueness of can be guaranteed.

Substituting equation (6) into equation (5), We obtain the Poisson's equation of the three-dimensional static magnetic field. The second arithmetic means shown in FIG. 1 executes the above-mentioned arithmetic operation and outputs Poisson's equation.

Equation (7) has clear boundary conditions when it is solved as a boundary value problem, and includes Coulomb's gauge conditional expression (6). Is guaranteed to be unique.

[2] Boundary condition between different magnetic materials As shown in Fig. 2, when there is a boundary between different magnetic materials in the analysis region, it is necessary to examine the boundary condition ⁽⁸⁾ .

(8) Hasebe, Kano: "Handling of boundary conditions in 3D magnetic field analysis", Japan Society for Simulation Technology
From the 7th Symposium on Application of Finite Element Method to Electrical and Electronic Engineering, pp.133-138 (SHO 61) Equations (1) and (2), the boundary conditions between different magnetic materials are as follows: .

Now consider the Cartesian coordinates (x, y, z), set the boundary plane on the yz plane, and define the unit vector in the y and z directions. Then, equations (8) and (9) can be written as the following equations.

Equations (10) and (11) are Is a boundary condition equation for, but using the equations (3) and (4) It can be rewritten as the boundary condition expression for.

First, the magnetic field strength Since the y component H _y and the z component H _{z of} are continuous, Is obtained. Also, the magnetic flux density Since the x component of is continuous, the following equation is obtained.

However, the subscripts (I) and (II) on the right shoulder are shown on the magnetic body (I) side and the magnetic body (II) side, respectively.

Where the magnetic vector potential at the interface Assuming continuity of and using Coulomb gauge condition,
The following equation is obtained as a sufficient condition for satisfying equations (12) to (14).

Equations (15) to (17) are sufficient conditions of equations (12) to (14) and are complete as boundary conditions for Poisson's equation (7).

The third computing means 3 shown in FIG. 1 calculates these boundary condition expressions.

The above calculation is compared with a conventional method which does not use the Coulomb gauge condition. That is, the magnetic vector potential on the boundary surface The continuity of is assumed, but Coulomb's gauge conditional expression (6)
When is not used, the following equation is obtained as a sufficient condition for satisfying the following (12) to (14).

If these three formulas (18) to (20) are satisfied, formula (12)
~ (14) is fully satisfied. However, ∂A _x / ∂x
The conditions for are insufficient. By the way, magnetic vector potential To solve Poisson's equation (7) using A boundary condition for the value of and its derivative is required. Therefore, it is understood that the boundary conditions shown in Expressions (18) to (20) are insufficient, and Expression (7) cannot be uniquely solved.

Although the boundary surface is assumed to be the yz plane here, z-
The same argument holds when the x plane and the xy plane are boundary surfaces.

[3] Discretization by Galerkin method and examination of boundary integral term Next, the calculation means 4 shown in FIG. 1 will be described. This computing means 4 discretizes Poisson's equation (7) element by element. The following description focuses on one element. That is, the symbols V and S below the integration symbol indicate the volume integral and area integral of one element, respectively.

(A) Discretization by the Galerkin method If the norm of Poisson's equation (7) and the interpolation function is taken and discretized by the Galerkin method, the following equation is obtained.
⁽⁹⁾ .

(9) KHHeubner (Translated by Yamada): “Finite Element Method”, pp.11
6-117, Science and Technology Publishing, (53) However, in equation (21), (B) Examination of boundary integral term If the boundary integral term in the equation (21) is taken out, it becomes the following equation.

If the boundary surface is selected as the y-z plane, the equation (22) becomes the following equation.

∂A _x / ∂x, (∂A _{y in} the integrand of the boundary integral equation (23)
/ ∂x−M _z ) and (∂A _z / ∂X + M _y ) are respectively expressed by the formula (1
As shown in 5), (16) and (17), it is continuous on both sides of the boundary surface where different magnetic materials come into contact. And the interpolation function Since B is selected continuously, the boundary integral equation (23) is canceled with an opposite sign on both sides of the boundary surface, and the entire boundary can be erased.

In addition, on the boundary that gives the Dirichlet condition, the interpolation function Is zero and ∂Ax / x∂ on the boundary that gives Neumann condition.
= 0, and the above equations (16), (1
By using 7), the whole integrand in equation (23) can be set to zero. Therefore, the boundary integral term shown in equation (23) can be eliminated even under the conditions of Dirichlet and Neumann.

On the other hand, as can be seen from the comparison between the equations (15) and (18), the continuity of ∂Ax / ∂x cannot be guaranteed between the elements of different magnetic materials when the Coulomb gauge is not imposed. , ∂Ax / ∂x in the integrand of equation (23) cannot be deleted.

As described above, the discussion has been made here on the assumption that the boundary surface is the yz plane, but the same argument holds when the boundary surface is the zx, yz plane.

Thus, by imposing Coulomb's gauge conditional expression (6), not only the boundary condition of Poisson's equation (7) can be completed, but also weakening by the Galerkin method is possible.

[4] Finite Element Formula The finite element formula shown below can be obtained from the discussion described in the above [3].

By the way, equation (24) is an arbitrary interpolation function that satisfies the boundary condition. Need to hold for. In other words, the following formula needs to hold.

By using the equations (25) to (27), the three components of the magnetic vector potential can be treated independently, so that the memory and calculation time of the computer can be saved.

[5] Creation of Data for Numerical Calculation In order to perform a three-dimensional magnetic field analysis, the three-dimensional space containing the analysis target is divided into, for example, fine rectangular parallelepiped elements, and the magnetic characteristics are (In this case, the magnetization obtained by the measurement And magnetic flux density Relational expression), exciting current density (Size and direction), magnetization Initial value of and vector potential Gives the initial value of.

At this time, do not create an element that crosses boundaries where materials change. In addition, the vector potential is on the outer periphery of the three-dimensional space. Give the boundary conditions for. In addition, when the analysis target has symmetry, 1/2, 1/4,
The area of …… is the analysis space, and the vector potential is also on the symmetry plane. Give the boundary conditions for. Data of the finite element method is created by the above procedure.

[6] Numerical calculation Input the data created above and enter the vector potential. The coefficient matrix of is calculated (calculating means 5). In addition, The effect of is obtained as a column vector, and the determinant of the whole system is obtained. Solving this coefficient matrix, the vector potential Each component of is calculated (calculation means 6), and the calculated vector potential From magnetic flux density Etc. are calculated (calculation means 7).

Hereinafter, a specific numerical calculation will be started. This numerical calculation is carried out by applying the method obtained in [5] above using the values obtained in advance as input data, except that the above equations (25) to (27) are used. The procedure is the same as the conventional one.

For each element, Substituting the numerical value of into equation (25) and performing Gauss integration, vector potential at each node i (i = 1 to total number of unknown nodes n) The coefficient of the x component A _xi of is calculated (calculation means 5).

The coefficients of the unknowns A _xi obtained for each element are added to create a coefficient matrix for the entire unknowns A _xi . As a result, an n × n coefficient matrix is created (operation means 5). In addition, The effect of is also obtained as a column vector at the same time, and the determinant of the whole system is completed.

The n × n matrix calculation is performed on the unknown A _xi , and the vector potential at each node i is calculated. The value A _{xi of} is calculated (calculation means 6).

Using equations (26) and (27), vector potential The values of y component A _yi and z component A _zi of are also obtained in the same manner (calculation means 5 and 6).

By the magnetic flux density in the element Is calculated (calculation means 7).

The magnetization in the element depends on the relationship with To fix.

The convergence is determined, and if it is converged, the obtained value is output. If the convergence is insufficient, the above operations 1 to 3 are repeated.

Numerical calculation example 1 Perform static magnetic field analysis using the model shown in FIG. I checked the value of. The region 10 is equally divided using eight-node primary elements, and a permanent magnet 11 with M _x = 1 [T] and M _y = M _z = 0 [T] is arranged at the center, and Neumann condition is used as a boundary condition. Gave.
This model includes boundaries between different magnetic bodies (permanent magnet 11 and vacuum 12) and is suitable for confirming the theory and formulation of this embodiment.

Fig. 4 shows the magnetic flux density Shows the distribution of. Magnetic flux density in the direction and length of the arrow Shows the direction and size of the magnetic flux density at the position where the permanent magnet 11 is placed. The maximum values of B _x = 0.3709 [T] and B _y = B _z = 0 [T].

As I said above, the magnetic vector potential In the three-dimensional static magnetic field analysis using, the problem is whether the boundary conditions between different magnetic materials are satisfied. In order to investigate this, at the boundary node between the permanent magnet 11 and the vacuum 12, The value of was calculated. In this example, the boundary nodes between different magnetic materials share eight elements, so the boundary nodes are shared first.
At Gaussian integration points in The value of is calculated and shown in Table 1.

From this table, If the values of are averaged and obtained as the value at the node on the boundary between different magnetic materials, the value becomes zero. The value is zero at other boundary nodes and any nodes in the region, which shows the validity of the above theory and formulation. Here, the boundary condition is the Neumann condition, but in the case of the Dirichlet condition, similarly, at any node in the region. Met.

Numerical Calculation Example 2 FIGS. 5 and 6 are views showing numerical calculation examples using the Institute of Electrical Engineers model.

In this model, an excitation winding 22 is wound around a 100 × 100 × 200 mm columnar iron core 21 (relative permeability μ _r = 1000).
This is a standard model for accuracy verification established by the "Technical Committee for Three-dimensional Electromagnetic Field Calculation Technology Investigation of the Institute of Electrical Engineers of Japan". The magnetic flux densities in the x, y, and z directions are measured in the space portions of the upper surface and the lateral surface of the iron core 21 and used as standard data for verification. The calculation result can be evaluated based on this standard data.

FIG. 5 shows the results of three-dimensional magnetic field analysis using the above-described apparatus according to the embodiment, using this model, with the position of the center of gravity as the origin, and for 1/8 volume. Each arrow indicates the magnitude and direction of the magnetic flux density. Table 2 and FIG. 6 show measured values and calculated values obtained along the height direction (z direction) at the position of x = y = 6.25 mm. As can be seen from this table and the figure, good calculation results are obtained. The current passed through the winding wire 22 at this time was 3000 AT.

Numerical Calculation Example 3 FIGS. 7 to 8 are diagrams in which the apparatus of the above-described embodiment is applied to the magnetic field analysis of the magnetic head.

A coil 32 is wound around the core 31 of the magnetic head 30,
A gap 33 is formed in 31. This magnetic head 30
Therefore, using the symmetry, the dashed-dotted frame VI in Fig. 7
As shown in II, the 1/4 region near the gap 33 was analyzed.

Figure 8 shows the result, and the arrow shows the magnetic flux density. Shows the size and direction of the.

Comparison between this example and conventional method Conventional magnetic vector potential Formulation of three-dimensional static magnetic field analysis by the variational method using is shown (the above-mentioned documents (1) to (3)). These deal with gauge conditions by the Lagrange constant method. However, it is reported that the introduction of gauge conditions by this method is extremely poor in terms of numerical calculation and difficult to handle.
^{(10) ~ (11)} (10) Yokoi, Aratani, Konuki: "Magnetic field analysis of a model for studying an open region static magnetic field using the three-dimensional finite / boundary element combined method", The 8th Japan Society for Simulation Engineering finite Proceedings of the Symposium on Application of Element Method, pp.99-104, (62) (11) Hasebe, Kano, "Relationship between Boundary Condition and Gauge in 3D Magnetic Field Analysis-By Verification Model-" Machinery Research Institute, SA-87-1, RM-87-12, pp.1-9, (Sho 6
2) Therefore, analyzes have been conducted ignoring gauge conditions.
Mohammed et al. Have shown that the indeterminacy of the magnetic vector potential of grad φ can be suppressed by the Dirichlet condition, so there is no need to impose a gauge condition ⁽¹²⁾ .

(12) OAMohammed, WADavis, BDPopovic, TWNehl
and NADemerdash, “On the Uniqueness of Solution
of Magnetostatic Vector-Potential Problems by Thre
e-Dimensional Finite-Element Methods ", Journal of A
pplied Physics, vol.53 (11), pp.8402-8404, Nov. (198
2) Among them, the theory is developed that the value of gradφ can be suppressed at the node, so that the value of gradφ at the boundary between elements is continuous. With the same magnetic material, the continuity of grad φ between elements is guaranteed. However, even though the bonding conditions of the basic equations seen from each medium are not imposed at the element boundaries between different magnetic materials, they are single nodes, so gr
It is incorrect to assume that the values of adφ are continuous. Therefore,
In the obtained calculation results The value of is not zero.

There are also other cases where discussions are being conducted in exactly the same way ⁽¹³⁾ .

(13) Ikeda: “Gauge problems in three-dimensional magnetic field analysis by the finite element method”, IEICE static and rotary machine joint research fund, SA
-87-20, RM-87-57, pp.21-29 (Sho 62) On the other hand, in this example, (1) in the three-dimensional static magnetic field analysis by the finite element method,
We show that it is necessary to introduce the gauge condition to meet the boundary condition at the boundary between different magnetic materials. (2) Formulate a new finite element formula including the gauge condition. And confirmed that the formulation is correct.

According to this formulation, the magnetic vector potential Since each component of can be analyzed independently, it is possible to save computer memory and calculation time.

[Effects of the Invention] As described above, the present invention is Instead of And the vector potential By imposing the Coulomb gauge condition with zero divergence of, it becomes possible to eliminate the surface integral term of the finite element formula. This allows the finite element formula to be separated for each three-dimensional component.

As a result, the vector potential Since the three components can be analyzed independently, the memory capacity and calculation time of the analyzer can be significantly reduced.

[Brief description of drawings]

FIG. 1 is a functional block diagram showing the configuration of a three-dimensional magnetic field analyzer according to an embodiment of the present invention, FIG. 2 is a diagram showing boundary conditions between different magnetic materials, and FIGS. 3 and 4 are numerical calculation examples. FIG. 5, FIG. 6 and FIG. 6 are views showing other numerical calculation examples,
FIGS. 7 and 8 are diagrams showing an example of numerical calculation of the three-dimensional magnetic field of the magnetic head, FIG. 7 is a diagram showing the structure of the magnetic head and a position VIII to be subjected to magnetic field analysis, and FIG. 8 is a magnetic field. It is a figure which shows the result of an analysis. 1-7 ... 1st-7th arithmetic means.

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Taizo Iijima 3-27-9 Shibakubo-cho, Tanashi City, Tokyo (72) Inventor Koichi Takegasa 5-9-1-901 (56) Aso-cho, Kita-ku, Sapporo, Hokkaido References Enoki et al .: Some Improvements of Boundary Element Method in Magnetic Field Analysis, Simulation, Volume 6, No. 1, PP. 42-49 “Numerical Simulation in Engineering” PP. 151-178, Maruzen, issued January 30, 1988

Claims (1)

[Claims] [Claim 1] and based on the Poisson equation of three-dimensional static magnetic field, the three-dimensional magnetic field analysis apparatus for analyzing a three-dimensional magnetic field comprising different magnetic body using the finite element method, the magnetic permeability mu ₀ and the magnetic field of a vacuum Strength Magnetized to the product of Magnetic flux density And a magnetic vector potential Under Coulomb's gauge condition with zero divergence of Is used to form a Poisson's equation of a three-dimensional static magnetic field, and a vector potential is calculated from the boundary condition between the different magnetic bodies. And the vector potential by performing the discretization by the Galerkin method using the Poisson's equation and the interpolation function. Fourth computing means for obtaining three finite element formulas separated for each three-dimensional component using the boundary condition of, and a vector potential for each finite element based on the finite element formula Fifth, obtaining a coefficient matrix composed of the coefficients of each component of
And the vector potential by solving the determinant using the coefficient matrix Sixth computing means for obtaining From the magnetic flux density Seeking and magnetizing And a seventh arithmetic means for correcting the three-dimensional magnetic field analysis apparatus.