JPH11296504A - Electromagnetic field analyzing method by combination of finite-element method and boundary element method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain a partial differential equation analysis of satisfactory calculating precision by partially solving concerning a boundary element method are by a leaving a joining face with a finite element method, adding the matrix of this part to the matrix of a finite element method area, then solving the part of the finite element method, returning this solution to the boundary element method area to obtain the whole resolution of the differential equation so as to save a memory to increase the number of dividing. SOLUTION: In the process (c) of matrix preparation, a matrix is prepared individually concerning the boundary element method area and the finite element method area and the matrix is partially solved by leaving the joining part with the finite element method area concerning the boundary element method area to prepare a coefficient matrix so as to collect known numbers to the joining part. In a process (d) solving simultaneous equations, this coefficient matrix is added to the matrix of the finite element method area. Then, the size of the matrix is reduced by the portion of not being added with a part excepting for the joining face of the boundary element method area as compared with what is obtained by simultaneously superposing the matrixes of the boundary element method area and the finite element method area.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for numerically analyzing a partial differential equation used as a means for predicting the performance of a product at the time of product design, and particularly to a case where a plurality of complicated structures exist in a space. The present invention relates to a method for solving a partial differential equation by combining a finite element method and a boundary element method, which can easily create input data and have high analysis accuracy.

[0002]

2. Description of the Related Art As a conventional joint solution of the finite element method and the boundary element method, there is known a method of simply superposing and solving individual matrices (CABrebia).
Author, Masataka Tanaka translation, Application of boundary element method 1, 1983, Planning Center). However, this method has the drawback that the size of the matrix becomes large, and in particular, there are many matrix elements in which components related to the finite element method become zero, so that the memory use efficiency of the computer is low and the calculation time is long.

[0003]

SUMMARY OF THE INVENTION An object of the present invention is to eliminate the above-mentioned disadvantages of the conventional combined solution of the finite element method and the boundary element method, to save computer memory and increase the number of divisions. To provide a PDE analysis method with high calculation accuracy.

[0004]

As a means for numerically analyzing a partial differential equation representing a physical field such as an electromagnetic field by using a computer, a method for solving the problem by combining a finite element method and a boundary element method is described. Create separate matrices for the element method area and the finite element method area.First, for the boundary element method area, leave the junction with the finite element method area and partially solve the matrix and aggregate the unknowns at the junction. Create a coefficient matrix. This matrix is added to the matrix in the finite element method domain. By doing so, the matrix size is smaller than that obtained by superimposing the matrices of the boundary element method area and the finite element method area at the same time, except that the joint surface of the boundary element method area is not added. The memory required by the computer is reduced accordingly.

[0005]

FIG. 1 shows an embodiment in which the present invention is applied to electric field analysis. Before explaining FIG. 1, the basic principle of the present invention will be described.

Considering the material anisotropy nonlinearity, the governing equation of the electric field is as follows.

[0007]

[Equation 1] ( _{εαβφ , β} ) _{, α} = 0 (Equation 1) The subscripts in the above equation follow Einstein's rules. The comma means differentiation, and the suffix that appears repeatedly takes the sum in the x, y, and z directions. _εαβ represents a dielectric constant or a conductivity tensor.

In the boundary element region, if the anisotropy and non-linearity of the material are not taken into consideration, the expression (1) is simplified as follows.

[0009]

Εφ _, αα = 0 (Equation 2) The above equation is discretized by the boundary element method.

When the equation (2) is converted to a boundary integral equation using the Green function as φ *, the following equation is obtained.

[0011]

(Equation 3)

Here, C _i is a constant depending on the solid angle in anticipation of the analysis area, and is １ ／ when the observation point is on a smooth boundary. D _Γ indicates a boundary integral. q can be written as follows using a differential value of the potential when the components of the outward normal vectors of the boundary flux and n _alpha.

[0013]

Equation 4 q = εφ _{, α} n _α (Equation 4) q * can be written in the same manner.

[0014]

Q * = φ * _{, α} n _α (Equation 5) In the finite element method, when Expression (1) is discretized, the following is obtained.

[0015]

(Equation 6)

In the above equation, dΩ indicates a domain integral. Further, N _i denotes the shape function that approximates the physical quantity of the element.
Next, the combined solution of the boundary element method and the finite element method is described in a matrix display.

When the discretization formula of the boundary element region is represented by a matrix based on the formula (3), it is as follows.

[0018]

(Equation 7)

Here, the suffixes Γ ₁ , Γ ₂ , 示 す_in indicate the first and second type boundary conditions and the joining boundary.

The matrix of the finite element method is written as follows.

[0021]

(Equation 8)

Here, the superscript b indicates the boundary element method, and f indicates the value of the finite element area.

The joining conditions are

[0024]

(Equation 9)

[0025]

(Equation 10)

## EQU1 ##

Equation (7) is transformed into a known term and an unknown term and transformed as follows.

[0028]

[Equation 11]

Equation (11) is solved formally as follows.

[0030]

(Equation 12)

This is rewritten as follows.

[0032]

(Equation 13)

The above equation is further divided into a joint and other parts.

[0034]

[Equation 14]

[0035]

(Equation 15)

In equation (15), if the number of joint nodes is m, [A ₂ ] is an m × m matrix. The equation (15) and the equations (8), (9), and (10) are solved simultaneously. Substituting equation (15) into equation (8) taking into account equations (9) and (10),

[0037]

(Equation 16)

In the above equation, since the first two terms are the matrix itself by the finite element method, the matrix creation algorithm itself of the finite element method may be used.
FIG. 1 shows an algorithm for realizing this idea.

After inputting the element data and the node data in the process a, the data belonging to the finite element region and the data belonging to the boundary element region are separated from the data by the index creation process b. Further, in the boundary element region, a connection relationship between regions having different physical property values is searched (process b).
2) In process b3, it is determined whether the area is divided by the finite element method area. Element / node numbers are renumbered for each of the divided areas (processes b4 and b5), and at this time, elements / nodes included in the connection boundary with the finite element method are registered (process b6). Next, a process c for matrix creation is started.

In this process, the integration of the equation (3) is executed, a coefficient matrix of the equation (7) is created for each area of the boundary element method area (process c2), and the coefficient matrix of the connection boundary portion is superimposed. Match (process c
3). At this time, a matrix relating to the joint with the finite element method region is also created (process c4, (Equation 7)).
H _in , G _{in in} the formula). Next, a solution is formally obtained for the connection boundary with the finite element method according to the equation (13) (process c5), and coefficient matrices of the equations (14) and (15) are created and stored. The above is the end of the matrix operation for the boundary element method. If there is no connection boundary with the finite element method, the solution obtained here becomes the entire solution as it is.

If there is an area of the finite element method, the integration of the equation (6) is executed to create a coefficient matrix of the equation (8) (process c6). Next, the matrix of the boundary element method part of the equation (15) is added to the matrix of the finite element method according to the equation (16) (process c9), and further a boundary condition is added (process c8). By solving the simultaneous equations thus obtained in the process d1, the potential on each node can be calculated. By substituting this value into the right side of equation (14), the potential and electric field at the remaining nodes on the boundary element method are obtained (process d2).

The data flow of the present invention is shown using the configuration of the computer shown in FIG. The data input in the process a is stored in the storage device via the central processing unit. Based on this data, various indexes required for the calculation are created by the process b and stored in the storage device. In the process c, various matrices are created based on the index and transferred to the storage device. Using this value, the central processing unit solves the matrix and stores the result (process d). FIG. 4 shows the matrix generation and the relationship between the respective regions when the above steps are applied to the GIS spacer portion shown in FIG.

For example, the part of the spacer 2 is the finite element method area 2 of 4 and the GIS space is the boundary element area 1 of 3 and 5
And 3. FIG. 4 shows a matrix generated for each region. The areas of the boundary element method are full matrices 8 and 10 (processes c2 and c2).
3) The domain of the finite element method is sparse matrix 9
(Process c6). The matrices of the equations (14) and (15) for the joints of the boundary element method are shown.
(Processes c4 and c5). When this is added to the matrix in the area of the finite element method, the matrix 13 is finally synthesized (process c9). When this matrix is solved and the solution of the finite element method area (process d1) is reflected on the boundary element method area using equation (14) (process d2), the entire solution as shown in FIG. 5 is obtained.

According to the present embodiment, since the matrix is formed separately in the finite element method area and the boundary element method area, the matrix size has the effect of being smaller than when the above two areas are simultaneously synthesized.

FIG. 6 shows a second embodiment in which the present invention is applied to a nonlinear electric field analysis. The structure of the procedure is almost the same as that of the first embodiment, but the following points are different in a part of the matrix creation and the stage of solving the simultaneous equations.

In the matrix creation stage, only the element matrix is created, and the matrix of the entire finite element region is assembled in the stage of solving the following simultaneous equations. Process c of FIG.
In FIG. 6, processes 7 to c9 are newly added to processes d4 to d6 and become processes d3 to d8. That is, in the step of solving the simultaneous equations, there is a process of convergence calculation (process d3), and when the conductivity (or permittivity) depends on the electric field, an electric field calculation in a finite element is performed (process d4). Is calculated (process d5). A matrix in a finite element region is assembled based on the electric conductivity (process d6). Next, the joining matrix with the boundary element method is added (process d
8) A boundary condition is given (process d7). After solving the matrix, it is the same as in the first embodiment.

According to the present embodiment, in addition to the effect that nonlinear calculation can be performed with a small matrix size, the number of operations per one repetition calculation is reduced, so that the calculation time can be shortened.

FIG. 7 shows a third embodiment in which the present invention is applied to linear magnetic field analysis. The basic configuration is the same as that of the first embodiment, except that a procedure for generating a magnetic field by a coil is added. Specifically, the index creation part (b10 and b1) for the coil is added to the index creation part.
1) In the matrix creation part, this is a part (c10) in which the magnetic potential due to the coil is added to the boundary condition.
In the stage of obtaining the solution, the calculation part of the magnetic field in each region (d9)
It is. According to the present embodiment, there is an effect that the calculation can be performed with a small matrix size even in the magnetic field calculation.

FIG. 8 shows a fourth embodiment in which the present invention is applied to a nonlinear magnetic field analysis. The basic configuration is the same as that of the second embodiment, except that a procedure for generating a magnetic field by a coil is added. Specifically, the index creation part (b10 and b1) for the coil is added to the index creation part.
1) In the matrix creation part, this is a part (c10) in which the magnetic potential due to the coil is added to the boundary condition.
Further, in the iterative calculation process, the calculation of the magnetic permeability is added from the magnetic field instead of the conductivity (or the dielectric constant) (d
5). According to the present embodiment, in addition to the effects of the third embodiment, the number of operations per one repetition calculation is reduced in the magnetic field calculation, so that the calculation time can be shortened.

FIG. 9 shows a fifth embodiment in which the present invention is applied to stress analysis. The configuration is almost the same as that of the first embodiment, except that the unknown variable is not a potential but a displacement vector. For this reason, the unknown increases three times in the case of an electric field. Thus, the matrix size is further increased, and the effect of reducing the matrix size according to the present invention is further increased.

[0051]

According to the first aspect, in order to form a matrix separately into a finite element method area and a boundary element method area,
The matrix size is smaller than the case where the above two regions are simultaneously synthesized, thereby reducing the memory capacity of the computer.

According to the second aspect, there is an effect that the memory capacity of the computer can be reduced even when the non-linear area is partially present.

According to the third aspect, there is an effect that the memory capacity of the computer can be reduced even when the unknown is a vector.

[Brief description of the drawings]

FIG. 1 is a flowchart of a linear electric field analysis according to an embodiment of the present invention.

FIG. 2 is a block diagram showing a configuration of a computer for executing the present invention.

FIG. 3 is a diagram showing a program applied to a GIS spacer using the present invention.

FIG. 4 is a view showing a process of generating a matrix in each procedure of the present invention.

FIG. 5 is a diagram showing an analysis result of a program applied to a GIS spacer using the present invention.

FIG. 6 is a flowchart of a nonlinear electric field analysis according to a second embodiment of the present invention.

FIG. 7 is a flowchart of a non-linear electric field analysis according to a third embodiment of the present invention.

FIG. 8 is a flowchart of a non-linear electric field analysis according to a fourth embodiment of the present invention.

FIG. 9 is a flowchart of a non-linear electric field analysis according to a fifth embodiment of the present invention.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Tank wall, 2 ... Spacer, 3 ... Boundary element area 1, 4
... finite element area 2, 5 ... boundary element area 3, 6 ... joining boundary 1, 7 ... joining boundary 2, 8 ... matrix of boundary element area 1, 9 ... matrix of finite element area 2, 10 ... boundary element area 3 Matrix, 11: Matrix of joining boundary 1, 12: Matrix of joining boundary 2, 13: Matrix finally synthesized, 14: Conductor

Claims (3)

[Claims] As a method of numerically analyzing a partial differential equation representing a physical field such as an electromagnetic field by using a computer, a method of solving by combining a finite element method and a boundary element method is described. After partially solving the area of 接合, leaving the interface with the finite element method, adding the matrix of this part to the matrix of the finite element method area, solving the finite element method part, and converting this solution to the boundary element method area An electromagnetic field analysis method combining the finite element method and the boundary element method, characterized by returning and solving the whole. 2. The electromagnetic field analysis method according to claim 1, wherein the boundary element method and the finite element method are combined with a region showing nonlinearity in a region of the finite element method. 3. The electromagnetic field analysis method according to claim 1, wherein a finite element method using a vector quantity as an unknown and a boundary element method are combined.