JPH09229727A - Method and apparatus for estimating physical amount - Google Patents

Abstract

PROBLEM TO BE SOLVED: To estimate the physical amount with high accuracy in a smaller calculation number of times without dividing a subject space excessively minutely, by dividing the object space into a plurality of areas, simplifying each area, and using a polynominal of transformation functions for every area. SOLUTION: A space-setting/space-dividing part 3 divides an object space into a plurality of areas on the basis of an inequality input through a man- machine(M/M) device 1. An area boundary condition-setting part 5 extends boundaries of divided areas and simplifies the outside of the areas. In the meantime, a transformation function-setting part 4 obtains a solution of the physical amount of each position within each simplified area to simplified boundary conditions of the area, and sets the solution as a transformation function. A polynominal-generating part 6 approximates the solution of the physical amount within each area by a polynominal of transformation functions having an unknown coefficient, and forms simultaneous linear equations from the generated polynominal with the use of mutual boundary conditions of areas input from the setting part 5. An equation-operating part 7 solves the simultaneous linear equations and obtains the unknown in each polynominal, thereby calculating the physical amount of each position.

Description

Description: BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to various machine designs.
Electric field characteristics, fluid characteristics, stress characteristics in the required predetermined space
Quantity prediction method and quantity prediction method for predicting physical quantity such as sex
Related to the device. [0002] Electric field characteristics and fluid characteristics in a predetermined space
The physical quantities required for equipment design such as
This physical law is governed by the corresponding physical law
Can be obtained by calculating Each obtained
Skills are used for equipment design, phenomenon prediction, product inspection, etc.
it can. Many of these physical laws are based on partial differential equations.
expressed. Then, as this partial differential equation, for example,
Plus equation (static magnetic field, electrostatic field, ideal fluid), Helmho
Rutz equation (acoustic, wave), Maxwell equation (electromagnetic
Boundary), Hamilton-Jacobi equation (dynamic body),
Iconar equation (optical), Navier-Stokes method
Equation (fluid), wave equation, Schrodinger equation (electric
Child, atom, molecule), heat conduction equation (heat, temperature), diffusion method
Equation (substance concentration), stress-strain equation (mechanical structure,
Structural analysis of buildings) is known. For each of these partial differential equations, the prediction pair
The elephant space is defined, and the initial physical quantity in this target space
If conditions and boundary conditions are given, numerical values can be calculated using a computer.
Solve the relevant partial differential method by calculation and calculate these physical quantities.
I can know. As a physical law using this computer
Finite element method and boundary element method are practical methods for numerical solution of
Has been Furthermore, most of the above PDEs
As a simple Laplace equation solution, conformal mapping method,
Charge method, finite element method and other analytical methods have been proposed
I have. Next, the conformal mapping conventionally adopted
The method, the substitute charge method and the finite element method will be described. (1) Conformal mapping method FIG. 8 defines the electrostatic field distribution as a physical law, for example.
Processor for solving the Laplace equation using the conformal mapping method
It is a flowchart which shows order. In this conformal mapping method, step S8
In -1, set the area of the target space to be calculated
I do. That is, this region is represented by x and y coordinates, and (x.
y) is expressed as z = x + jy j: imaginary unit, and the complex potential w is w = u + jv as a physical quantity at each position, the real part u indicates the electric flux and the imaginary part v indicates the potential.
You. Therefore, the solution of this Laplace equation is
The complex potential w is a complex number z (= x +
jy) function is shown below. w = f (z) Next, in step S8-2, the set area is paired.
Set the boundary conditions to be set. This boundary condition is the boundary position
And the like. Then, in step S8-3, the area
, The complex potato
Ask for the online w. Is this conformal mapping method old?
It is a known method of solving the Laplace equation, for example
Details are described in the literature. Henrici, P., "Applied and Computational
 Complex Analysis Vol.3 "John Wiley & Sons. New Yor
k. (1986) Nehari.Z "Conformal Mapping" McGraw-Hill. New Y
ork. (1952) (2) Alternative charge method Figure 9 shows the solution of the Laplace equation using the alternative charge method.
It is a flow chart which shows a processing procedure in a case. In step S9-1, the area of the target space is set.
Then, the boundary condition is set in S9-2. And S9
-3, the complex potential w is converted into the coefficient q in the region._i Not yet
Create an electric charge relational expression to be known. And this charge function
Create a system of linear equations to satisfy the boundary condition
(S9-4), and the simultaneous linear equations are not solved yet.
The knowledge number is determined and the complex potential w is calculated (S9-
5). The details of this substitute charge method are described below.
In the literature. Murashima Sadayuki. Substitute charge method and its application Morikita Publishing, Tokyo.
(1983) H. Okamoto and Y. Katsuta. Fast solution of potential problems
Applied Mathematics Vol. 2, No. 3, P2-20. (1992) (3) Finite element method FIG. 10 shows the solution of the Laplace equation using the finite element method.
It is a flow chart showing a processing procedure in the case. In step S10-1, the area of the target space is
Then, the boundary condition is set in S10-2. And
The target space area is divided into a plurality of small areas in S10-3.
(S10-3), the complex potential w in each region
The function value is an unknown number (S10-4). And each area
The partial differential equation is discretized in (S10-5), and the boundary
A simultaneous linear equation for satisfying the condition is created (S10-
6), solve the simultaneous linear equations to determine unknowns,
The complex potential w is obtained (S10-7). In addition to the above solution methods, the boundary element method
Has been put to practical use. This boundary element method
When applied to the equation
It is. However, in the substitute charge method, the charge is
It is placed near the outside, but in the boundary element method, the charge is
Put on the line. Therefore, there is a direct influence of the singular point,
There is a place where the error becomes infinite in the world. Boundary element method is
It can be applied to equations, but the characteristics are similar.
You. However, the above-mentioned problems have been solved.
In each method, there are still the following issues that need to be resolved.
Was. In the conformal mapping method, no region division is performed
So easy and accurate results. However, for example, S-C
Use the method of transformation (Schwarz-Christopher transformation)
And the problem that integration is difficult to solve by algebraic calculation
Many. For example, to explain an example of S-C conversion theory, if the shape of the boundary is a straight line, the solution is a linear function and it is extremely easy. If the right angle is one, the square root function is required (easy). Necessary (easy) 3 to 4 right angles Required elliptic function (somewhat difficult) 5 or more right angles impossible Impossible areas with complicated shapes and complicated boundary lines
It is almost useless for actual mechanical design with
Absent. The substitute charge method is different from the conformal mapping method.
Although it can be used for any problem, it requires a large amount of calculation
There is a problem that the accuracy of the result is poor. Accuracy especially near singular points
Is bad. This changes the functional form of the substitute charge depending on the singularity.
Because there is no. Further, in the finite element method, what question is
There is an advantage that you can solve even the subject. However, the finite element method is
There is a drawback in terms of increased physical quantity and accuracy. One is a mesh
If you do not subdivide, the accuracy will not increase and the amount of calculation processing will increase.
Is Rukoto. Especially in the vicinity of the singular point, this defect is serious.
Yes. That is, this finite element method
Because it assumes a uniform space without considering the shape of
You. Further, the high-order expansion formula is not often used, and the differential
It is common to stop at the lowest order needed to calculate the coefficient
You. Special ingenuity is required to use higher-order equations.
You. The present invention has been made in view of such circumstances.
Divides the target space into multiple areas, simplifying each area
Then, by using the polynomial of the conversion function for each area,
Therefore, there is no need to subdivide the target space excessively,
Physical quantity prediction that can predict physical quantity with high accuracy in arithmetic
Aiming to provide a measuring method and a physical quantity prediction device
You. The present invention is directed to a position space and time.
Each position and each position in the target space consisting of interspaces etc.
The physical law that defines the relationship with the physical quantity
The boundary conditions of the applicable target space and the initial conditions of physical quantities or
Target from the applicable physical laws by setting end conditions
Physical quantity prediction method for predicting physical quantity at each position in space
Applied to the law. And, in order to solve the above-mentioned problem
In 1, the space component that divides the target space into multiple regions
Split process and extend the boundary of each divided area
Area simplification process to simplify the outside, and each of these simplified
Objects at each position in the region for the simplified boundary conditions of the region
A conversion function setting step of obtaining a solution of a physical quantity and using it as a conversion function,
The solution of the physical quantity at each position in each area is transformed with the coefficient as an unknown number.
Polynomial setting process of approximating with the polynomial of the transfer function and each region phase
A system of linear equations is created from each polynomial using the mutual boundary conditions.
The simultaneous linear equation creation process to be generated and the created simultaneous equations
Obtain each unknown from the linear equation and calculate the physical quantity at each position.
And a physical quantity calculation step of outputting. According to claim 2, the object of claim 1
The physical law in the physical quantity prediction method is the two-dimensional Laplace equation
Or a partial differential equation consisting of two-dimensional Poisson equation
And the simplified conversion function of each area is a two-dimensional
This is the solution of the plus equation. Further, according to claim 3, in claim 2
The conversion function setting process in the physical quantity prediction method is a physical law.
To have all or part of the PDE as
This includes finding a solution to the equation. Another invention is position space, time space, etc.
At each position in the target space consisting of
The physical laws that define the relationship with
Boundary condition of elephant space and initial condition or end condition of physical quantity
By setting it, the applicable physical law can be set in the target space.
Applied to a physical quantity prediction device that predicts the physical quantity at each position
It is. And, in order to solve the above-mentioned problem
In 4, the space for dividing the target space into a plurality of regions
The dividing means and the boundary of each divided area are extended to extend outside the area.
Areas to be simplified
Solution of physical quantity at each position in the region for singularized boundary condition
And the conversion function setting means for obtaining
The solution of the physical quantity at each position is converted into
Boundary between polynomial setting means approximated by polynomial and each area
Generate a system of linear equations from each polynomial using field conditions
From the simultaneous linear equations creating means and the simultaneous linear equations described above,
Physical quantity calculator that calculates unknown quantity and calculates physical quantity at each position
It is provided with steps and. Further, the invention of claim 5 is the physics of claim 4.
Each quantity obtained by the physical quantity calculation means for the quantity prediction device
Graphic output that graphs and outputs physical quantity of position
Means are added. According to the present invention, the above-mentioned structure is provided.
In the physical quantity prediction method and physical quantity prediction device of
Divide the space into multiple areas and simplify each divided area
The boundary line that simplifies the solution of the physical quantity at each position in each area.
The conversion function is obtained by using the condition. Then, the solution of the desired physical quantity is obtained for each area.
Is defined by the polynomial of the conversion function. Note that this polynomial is
Leave the coefficients of the terms of the conversion function of each order
You. Therefore, depending on the boundary conditions between the areas,
Create a system of linear equations to find
If you solve the equation and determine the unknowns, you will end up in the target space.
The physical quantity at each position in is obtained. Therefore, the target space is divided into a plurality of regions.
By doing so, each area can be easily simplified,
The conversion function in each area can be simplified. Also, the polynomial
By using finite element method, boundary element method, substitute
Fewer unknowns than conventional methods such as the charge method
Less computational effort than conventional methods to achieve the same accuracy
It also reduces the amount of computer storage required. Further, by using the conversion function, each region is
Eliminates the influence of singular points in the region, making calculation easier and more accurate
Becomes possible. The physical quantity predicting device according to claim 5
The obtained physical quantity at each position is graphed and output.
This makes the device very easy for the operator to use.
You. Embodiments of the present invention will be described below with reference to the drawings.
Will be described. FIG. 1 is a physical quantity prediction of an embodiment of the present invention.
A block diagram showing a schematic configuration of a physical quantity prediction device adopting the method
FIG. The physical quantity prediction apparatus of this embodiment is a computer
It is composed of a kind of information processing device such as a computer.
M or application written in the storage unit such as main storage
Processing unit formed on the application program, and M / M
(Man-machine) device 1 and print output device 2 such as a printer
It is composed of The application program
As shown in FIG. 1, the processing unit formed on top of the target space
Setting / space division unit 3, conversion function setting unit 4, area boundary
Boundary condition setting unit 5, polynomial generation unit 6, and equation calculation processing
It is composed of a unit 7 and a calculation result graphic processing unit 8.
I have. Next, the operation of each section will be described in order. M /
The M (man-machine) device 1 is, for example, a CRT display device.
It consists of a keyboard and a mouse, and the operator can predict physical quantities.
Coordinate information that identifies the target space to be
Dividing information when dividing, easy when simplifying each area
It is a device for inputting generalization information and various boundary conditions. Also,
If necessary, the object at each position in the area simplified by the operator
It is used when inputting an already known solution of the physical quantity. In order to solve the given problem, we need to query the object space.
Divide into areas according to the subject and number each area. Sa
In addition, set some boundary points appropriately on the boundary of each area.
Number them. For example, at the boundary between area A and area B,
At some point, the operator can control the area A through the M / M device 1.
Enter the number and position coordinates of zone B. In addition, the range of each area
Enter the inequalities that determine the bounds. The operator operates the target space setting / space division unit 3.
The physical quantity can be predicted from the coordinate values input via the M / M device 1.
Until the end of physical quantity calculation by specifying the target space that indicates the measurement range
Retain. Furthermore, this target space setting / space division unit
3 is a boundary line of the target space input via the M / M device 1.
Items are additionally set in the set target space. This target sky
Boundary condition between is a physical quantity at the boundary position of the target space
Etc. to obtain the physical quantity at each position in this target space.
This is a requirement. Further, the target space setting / space dividing unit 3 is
Based on inequalities entered by the operator via the M / M device 1.
Then, given the position coordinate z in the target space, which region
Determine whether it belongs to the area. Regarding the conversion function setting section 4, the operator
After processing (1) and (2), the conversion function setting unit 4
Execute the process. (1) For each area that was divided into multiple parts by the division process,
Extend the boundary of the area to simplify outside the area. this
The purpose of area simplification is to extend the boundary of the area
Singularity does not occur in the area,
In order to easily derive the solution of the physical quantity
You. (2) Each region simplified by the above method
Solution of physical quantity in_i (Z_i ) Is derived. these
Is known and is stored and held in the conversion function setting unit 4.
If so, specify the solution number, move the origin, and coordinate
Specify the rotation number. Also, it is not stored
If, the derived solution g_i (Z_i ) Function value
The subroutine is set in the conversion function setting unit 4. (3) Input by the operator via the M / M device 1
Conversion function setting unit based on the specified instruction and subroutine
4 is, when the position coordinate z and the region number i are given,
Using the conversion function corresponding to the area number i, r_i = G
_i (Z_i ) Value is calculated. The polynomial generating unit 6 calculates the solution of the physical quantity in each area.
Is a polynomial of a conversion function in which the coefficient is an unknown number w = P_i (R
_i ) Approximates. The inter-region boundary condition setting unit 5 uses the M / M device.
On the boundary line between areas based on the instruction from
The condition that the values of the polynomials in the mutual areas are equal
Input to the polynomial generator 6. For example, the boundary between the continuity of electric potential and the continuity of electric flux
As for the conditions, as described above, the boundary point number m and the coordinates
z_m And the region number ia ib, the equation P_ia(Z_m ) -P_ib(Z_m ) = 0 is obtained. When there is a fixed potential condition, the position
Mark z_m And the region number i and the potential v, the equation I [P_i (Z_m )] = V is obtained. However, I [] is a symbol representing an imaginary part.
You. Further, the electric flux condition is set inside or at the boundary of the region i.
If there is a coordinate z_m And coordinate z_nThe electric flux connects the line connecting
Assuming that it does not cross (is blocked), the equation R [P_i (Z_m ) -P_i (Z_n )] = 0 is obtained. However, R [] is a symbol representing a real part.
You. The polynomial generator 6 is an inter-region boundary condition setting unit.
Using the boundary conditions input from 5, the polynomial generated earlier
Formula w = P_i (R_i ), A simultaneous equation consisting of multiple equations
Create the following equation. The equation calculation processing unit 7 is the polynomial generation unit 6.
Solve the created system of linear equations to obtain the unknowns of each polynomial.
Get the value of each coefficient that is Then, the obtained coefficients are
Value of physical quantity at each position in each area by substituting in polynomial
Get. The calculation result graphic processing unit 8
The physical quantity of each position in the
Obtain a physical quantity that can be displayed in a graph so that the operator can easily see it.
The print output device 2 prints and prints on paper, and M
/ M Display and output to the CRT display device of the device 1. FIG. 2 shows a physical quantity predicting device having the configuration shown in FIG.
6 is a flowchart showing the overall processing procedure in the storage device. Processing 1
In step 2, the target space is defined, and in process 2, this target space is determined.
Boundary conditions are defined, and the target space is divided into n regions in process 3.
Divide into zones. The position in each divided area is a complex coordinate.
z_i (I: area number i = 1,2, ..., n) = x + jy
You. Extend the boundary of each area divided in process 4
Simplify each area itself by simplifying outside the area
I do. Simplified in the area simplified in process 5
Solution of physical quantity at each position using conformal mapping method with different boundary conditions
_i = G_i (Z_i ). Where the simplified boundaries
The condition usually means that the electrode surface in z space is r_i Real number in space
Peculiar except that it is mapped to an axis and is caused by the electrode shape in a finite area
The condition is that it has no points. Then, the solution is set to the reference position (the seat) of the target space.
Conversion function r indicated by the position from the standard origin)_i Expressed by What
It is also possible to use a known solution in process 9 in advance.
Noh. Then, in the process 6, each simplified
As a solution of the correct physical quantity at each position in the area
The complex potential w (w = u + jv) of each
Conversion function r with (a, b, ...) as unknown_i Polynomial P of
_i (R_i ) (W = P_i (R_i )). In process 7, the connection line between the areas is connected.
Conditions, multiple equations to meet the boundary conditions (simultaneous linear equations
Formula) is created. In process 8, this simultaneous primary process
The equation is solved to determine each coefficient (a, b, ...). After a while
Then, by actually calculating each polynomial, the correct
A complex potential w as a physical quantity is obtained. EXAMPLE Next, a concrete example using this physical quantity predicting apparatus will be described.
The physical quantity prediction process will be described. (First Example) The most typical partial differential equation of the physical law and
Then, in the two-dimensional space using the two-dimensional Laplace equation
A method for obtaining an electrostatic field characteristic at an arbitrary position will be described. This place
In this case, the solution of the Laplace equation in the divided domain as the conversion function r
Used. Then, for the sake of convenience, this method is used for CCM (compos
ite conformal-mapping method; synthetic conformal mapping method)
Name it. As described in the conventional method, a normal conformal image is taken.
According to the convention of the image method, the plane is represented by xy coordinates, and (x,
y) points are represented by complex numbers (complex coordinates) z = x + jy, and the solution
If the complex potential that is expressed by w = u + jv is
The part u is the electric flux and the imaginary part v is the electric potential. As a first embodiment, an electric field in a rectangular coaxial tube
Example of predicting electric field characteristics by solving Laplace's equation for distribution
Is shown. FIG. 3 (a) shows an outer tube 21 having a square cross section.
FIG. 6 is a cross-sectional view of a double pipe including an inner pipe 22. Specifics of each processing 1 to 8 in the flowchart shown in FIG.
The specific processing contents will be described in order. (Process 1, 2) of physical quantity
The target space showing the prediction range is the outer tube 21 with a side of 64 mm.
It is a range surrounded by the inner tube 22 having a side of 32 mm. Soshi
The boundary condition is that the outer tube 21 has a potential of 0 volt and the inner tube has a potential of 0 volt.
The potential of 22 is 10 volts. Specifically, the shape of this double tube is symmetrical.
Has P1, P2, P3, P4, P in the lower left.
The distribution within the trapezoid indicated by the number 5 may be calculated.
Therefore, in this first embodiment, trapezoids P1 to P5
Is the target space. (Processing 3) FIG. 3B is a trapezoid of FIG.
It is the figure which expanded the target space of. A1, B1, C1, D
1 and 2 lines indicated by E1 and F1
The five points P1, P2, P3, P shown by ○ are divided into regions.
4 and P5 are the centers of the polynomial expansion of each area. And P
The position of 2 is the origin of the coordinates of the target space. Therefore,
For example, the coordinate of the position of P1 is (16,16), which is a complex number.
Is 16 + 16j. Then, each area is designated as P1 to P5.
Name it. (Processing 4) Each of the areas P1 to P5 is shown in FIG.
To simplify. The dotted line is the axis of symmetry when there is line symmetry
Show. For example, the area P1 is at the lower left of the point P1 as shown in the figure.
In the area to calculate the electric field sandwiched between the electrodes inside and outside the
is there. In the case of region P1, the surface shape of the internal electrode is
In 3 (a), at the point 32mm to the right from the corner of P1,
There is one horn. However, in Figure 4, a straight line to infinity
It is assumed that it is growing normally. Also, in FIG. 4, the outer electrode is
Assuming that they do not exist, this area P1 is
Suppose the space extends infinitely. By simplifying the area as described above,
The problem of finding the electric potential at each position can be easily derived. other
Areas P2 to P5 are also outside the areas P2 to P5.
Assuming that there is no change in the electrode shape,
Set. (Process 5) The process 5 is simplified.
The solution of each position in each area is obtained. That is, the region P1
Has a right-angled electrode and finds the electric field in the space outside it
Things. To do this, set the point on the electrode surface to z (= x + j
y), the corresponding complex potential w (=
Find a function such that u + jv) is a real number. Normal
By convention, the imaginary part v of the complex potential w represents the potential.
However, the real part u of the complex potential w represents an electric flux.
I do. The imaginary part of the complex potential w on the electrode surface
That is, a solution with a potential of 0 will be found. This is complex
This is a well-known fact in functional theory. Boundary of electrodes in region P1
The line of the field is assumed to be a half line extending to infinity. Origin at the corner
Then, the potential expressed by a complex function (complex potential)
Is w = (-jz)^2/3 It is expressed as If the point of symbol P1 is the origin,
Since the electrode surface on the right side of xmm is z = x, w = −x^2/3 And the electrode surface ymm above the position P1 is z = jy.
So w = y^2/3 And if x and y are real numbers, then w is also a real number
Can be Position P1 is coordinate when position P2 is the origin
Since (16, 16), the conversion function r of this region P1
₁ Is r₁ = [-J {z- (16 + 16j)}]^2/3 Becomes Here, the numerical value (16 + 16j) is the actual point
A value to move to a position. As shown in FIG. 4, the area P2 has a concave right angle.
There is an electrode, and the electric field in the space inside is determined. This place
The electrode boundaries extend from the corner to the right and upwards to infinity
Assuming a half line. If the angle is the origin, it is represented by a complex function
The potential is w = z^Two It is expressed as If z = x, w = x^Two And z = jy
Then, w = -y^Two In both cases, w becomes a real number. did
Therefore, the conversion function r of the region P2_Two Is r_Two = Z^Two Becomes The area P3 has a flat surface as shown in FIG.
There is a pole, and the electric field in the space above it is determined. Electrode border
If the line is a line that extends to infinity, the voltage expressed by a complex function
The position is expressed as w = z. Therefore, the conversion function r of the region P3_Three Is r_Three = Z-8. In addition. Numerical value (-8) aligns the origin with the deployment center
It is for. Similarly, the conversion function r of the area P4_Four Is r_Four = Z-32. Furthermore, the conversion function r of the region P5_Five Is r_Five =-[Z- (32 + 16j)]. (Process 6) In the process 6, each area P1
~ Create a polynomial to find the correct solution within P5
I do. That is, as a solution in each of the regions P1 to P5
Complex potential w is conversion function r_i Polynomial P of_i (R
_i ). This polynomial originally assumes an infinite series.
However, it is cut off at an appropriate order. In this example, 5
Calculate up to next time. The potential of the outer electrode is the above-mentioned boundary
It is 0V depending on the condition, and also contacts the outer electrode
Polynomial coefficients a, b. ... are all real numbers. This is
Convert when the prime number z (= x + jy) is on the surface of the outer electrode
Function r_i Is a real number, and the complex potential w is also a real number
It is. Since the regions P1 and P5 are in contact with the inner electrode,
Set the constant term to 10j to fix the position to 10 volts.
You. In the region P1, w = 10j + a₁ r₁ + B₁ r₁ ^Three + C₁ r₁ ^Five = P₁ (R₁ ) Is assumed. a₁ , B₁ , C₁ Is an undetermined coefficient
You. The even-order terms are unnecessary because of symmetry. It is
It depends on the reason. If there is line symmetry passing through the origin, the pair
The nominal axis is one line of electric force and the electric potential of the complex potential w.
The real part u representing the bundle is 0. At that time, the conversion function in the area
Since the numbers also have the same symmetry, the conversion function r_i The real part of
0. That is, the conversion function r_i Is a pure imaginary number. So
Here, the conversion function r_i Is a pure imaginary number, the complex potential w
= P_i (R_i ) Is a pure imaginary number, the polynomial P_i 
(R_i ) Must be of odd order only. if,
When the coefficient of an even-order term is not 0, the term is r_iIs pure
This is because when it is a number, it becomes a real number. The axis of symmetry does not pass through the origin
If not, add one real number as an unknown. The constant 10j corresponds to the potential of the inner tube electrode.
In the region P2, w = a_Two r_Two + B_Two r_Two ^Three + C_Two r_Two ^Five = P_Two (R_Two ) Is assumed. a_Two , B_Two , C_Two Is an undetermined coefficient
You. It can be understood from the symmetry that the even-order terms are unnecessary.
The fact that the undetermined coefficient is a real number is the same as above. In the region P3, w = a_Three + B_Three r_Three + C_Three r_Three ^Two + D_Three r_Three ^Three + E_Three r_Three ^Four + F_Three r_Three ^Five = P_Three (R_Three ) Is assumed. a_Three , ..., f_Three Is an undetermined coefficient
You. Since it is asymmetric, even-order terms are also necessary. In the region P4, w = a_Four + B_Four r_Four + C_Four r_Four ^Three + D_Four r_Four ^Five = P_Four (R_Four ) Is assumed. a_Four , ..., d_Four Is an undetermined coefficient
You. In this region P4, the axis of symmetry is off the origin.
Therefore, the constant a_Four It is necessary to add. Other even
The order term is unnecessary. In the area P5, w = 10j + a_Five + B_Five r_Five + C_Five r_Five ^Three + D_Five r_Five ^Five = P_Five (R_Five ) Is assumed. a_Five , ..., d_Five Is an undetermined coefficient
You. In this area P5, the axis of symmetry is off the origin,
As in the area P4, the constant a_Five Need to be adjusted. other
No even-order terms are needed. Constant 10j is the internal electrode potential
It corresponds to. Further, all coefficients a₁ , ..., d_Five Is real
Suppose it is a number. Based on this assumption, the boundary line of the electrode potential
The matter is satisfied. (Process 7) In Process 7, the boundary conditions between the regions
Create a system of linear equations consisting of multiple equations that satisfy
You. The five polynomials P created in the process 6₁ 
(R₁ ) ~ P_Five (R_Five ) Has a total of 20 unknowns
Therefore, make an equation that determines this by the boundary condition. each
The boundaries of the regions P1 to P5 are line segments A1B1C1D1 and E1F.
It is one. Select an appropriate number of points on this line and
In the equation, the condition that the electric potentials on both sides of the boundary line are continuously connected
I do. The continuous condition is that the electric potential is continuous and the electric flux is continuous.
It is necessary to continue, and the continuity of the potential is a complex potential w on the boundary.
The imaginary part v indicating the electric potential of is continuous, and the electric flux is continuous on the boundary.
The real part u indicating the electric flux of the elementary potential w must be continuous.
And are required. Therefore, from one boundary point to two
You can make a linear equation. For example, the boundary line A1B1 is the area P1 and the area
At the boundary of P2, P on this₁ (R₁) And P_Two (R_Two ) Is
Must match. Therefore, the following equation holds
You. P₁ (R₁ ) = P_Two (R_Two ) This equation becomes the boundary condition equation. Suitable on A1B1 line
Select as many points (coordinates z = x + iy) as to
Coordinate z to r₁ And r_Two When is calculated and substituted into the above equation,
a₁ , B₁ , C₁ And a_Two , B_Two , C_Two Those whose is unknown
You can formulate. Since the above equation is a complex number, the real part on the left side
Since the imaginary parts are each 0, two linear equations
can get. Of these, the imaginary part shows the continuity of the potential and
Some parts show continuity of electric flux. The potential v is normally used as a reference point to ground the outer electrode.
The potential is set to 0 volt. Similarly for the electric flux u
It is necessary to set a standard, and here, in the area P1,
The target axis was used as a reference. 10 Volts on inner electrode, 0 on outer electrode
The boundary condition of the potential of the bolt has already been woven in the process 6.
You. (Process 8) In Process 8, the simultaneous one created in Process 7
Solve the following equation to determine the undetermined coefficient. The primary method
When the number of expressions is greater than the unknown, the least squares method is used.
This is done automatically by the known modified Gram-Schmidt method.
Can calculate. The modified Gram-Schmidt method is described in
It is described in detail. Toru Nakagawa and Yoshio Koyanagi “Least Square Method
Analyzing experimental data "University of Tokyo Press (1982)
When formulas are created, 102 equations are created. Solve this
Number a₁ ~ F_Five Get. Then, each obtained coefficient a₁ ~ F
_Five Is a polynomial P of the regions P1 to P5₁ (R₁ ) ~ P_Five (R
_Five ), Calculate each polynomial for each position,
The electric potential v and the electric flux u at each position are calculated. This calculation
Graphic of electric potential v and electric flux u which are the solutions of physical quantity
It is processed and printed out to the print output device 2, and M
/ M Display and output to the CRT display device of the device 1. The calculation processing time until this display output
Is a BASIC program with an old PC (CPU286LS).
It takes about 4 minutes. FIG. 5A shows this first embodiment.
Equipotential line showing potential distribution as physical quantity predicted in the example
Fig. 5 (b) is an electric flux diagram showing the electric flux distribution.
You. From the complex potential w,
The electric capacity C per unit length is obtained, and the characteristic impedance
Dance Z₀ Is also easily obtained. Square coaxial of the first embodiment
The prediction result of the electric capacity of the tube is 90.55 pF / m. In the shape of the electrode in the first embodiment,
Indicates that the point P1 is a singular point at a right angle and electric field concentration occurs.
However, the conversion function r in process 5₁ = Z^3/2 To use
Therefore, the peculiarity in the region P1 is removed, and the physical quantity
The solution complex potential w is the conversion function r₁ Holomorphic function of
Can be approximated by a fifth-order polynomial. As a result, the physical quantity in the first embodiment
The prediction accuracy of the potential distribution and the electric flux distribution is extremely high.
Yes. In order to further improve the prediction accuracy, the conventional finite element
There is no need to increase the number of divisions of the target space like the prime method,
You can increase the degree of the polynomial. Therefore significantly calculated
There is no increase in throughput. (Second Embodiment) As a second embodiment, FIG.
The conductor board 23 shown in FIG.
Potential distribution between the strip line 24 and the strip line
Predict the electric capacity per unit length of the cable. (Processing 1) A2, B2, C2, D2, E2
Is a conductor substrate 23 having an infinitely wide ground potential. F2, G
2, H2 placed on it 2mm apart, width F2H2 =
It is a strip line 24 formed of a 4 mm conductor. Strike
The thickness of the lip line 24 is zero. Space other than conductor is dielectric
The rate is constant. The structure of the strip line 24 is symmetrical
The left half is the target space. And point 1 in the figure
Is the reference position (origin) of the entire target space. (Processing 2) As a boundary condition of this object space
And the dielectric constant is the value of ordinary vacuum (air) 8.854 pF /
m is used. In addition, the electric potential of the conductor substrate 23 is 0 volt,
The potential of the trip line 24 is set to 1 volt. (Processing 3) As shown in FIG.
The target space is divided into six areas Q1 to Q6. (Processing 4) Area Q1 is an infinite space excluding areas Q2 to Q6
It is. Therefore, in this area Q1, as shown in FIG.
As shown in (c), the F2G2H2 strip electric wire 24
Simplify by removing the electrode. In the areas Q2 to Q6, as shown in FIG.
The strip line 24 infinitely without cutting at position H2.
Lengthened and simplified, and the electrode 2E2 on the conductor substrate 23 is removed.
Remove and simplify. (Processing 5) Simplified physical quantity of the area Q1
The complex potential w as the solution of
Then, w = 1 / z. Therefore, the conversion function r of the region Q1₁ Is the C2 point
Since the mark is (5,0), it is as follows. R₁ = 1 / (z-5) Further, in other regions Q2 to Q6, the position F2 in FIG.
Assuming the origin, the complex potential w as a solution is
Become. W = z^1/2 Therefore, by incorporating the coordinates (z = 3 + 2j) of the position F2,
Conversion function r_Two ~ R₆ Are all equal and can be shown below
it can. R_Two = R_Three = R_Four = R_Five = R₆ = [Z- (3 + 2i)]^1/2  (Process 6) In the area Q1, the following polynomial is set.
You. W = a₁ r₁ + B₁ r₁ ^Three + C₁ r₁ ^Five = P₁ (R₁ ) A₁ , B₁ , C₁ Is an undetermined coefficient. No even-order terms required
What can be understood from symmetry. Undetermined coefficient a₁ , B₁ ,
c₁ Is a real number, the electric potential of the conductor substrate 23 is 0 volt.
Satisfies the boundary conditions. In the area Q2, w = j + a_Two + B_Two r_Two + C_Two r_Two ^Two + D_Two r_Two ^Three + E_Two r_Two ^Four + F_Two r_Two ^Five = P_Two (R_Two ) The constant term j corresponds to the potential of the strip line 24. this
In this case, since it is asymmetric, even-order terms are necessary. a_Two ,
..., f_Two Is an undetermined coefficient. a_Two , ..., f_Two Is a real number
At this time, the stripline 24 potential meets the boundary condition of 1 volt.
Add. In the area Q3, w = j + a_Three + B_Three r_Three + C_Three r_Three ^Two + D_Three r_Three ^Three + E_Three r_Three ^Four + F_Three r_Three ^Five = P_Three (R_Three ) The constant term j corresponds to the potential of the strip line 24. a
_Three , ..., f_Three Is a real undetermined coefficient. Asymmetric so even
We also need terms of several orders. In the area Q4, the following polynomial is set.
I do. w = a_Four + B_Four r_Four + C_Four r_Four ^Two + D_Four r_Four ^Three + E_Four r_Four ^Four + F_Four r_Four ^Five = P_Four (R_Four ) Is assumed. a_Four , ..., f_Four Is a complex number
Since it is a number, this unknown number is 12. In the area Q5, the following polynomial is set.
I do. w = j + a_Five + B_Five r_Five + C_Five r_Five ^Two + D_Five r_Five ^Three + E_Five r_Five ^Four + F_Five r_Five ^Five = P_Five (R_Five ) A_Five Is a real number, b_Five , ..., f_Five Is an undetermined coefficient of complex number
You. In the area Q6, the following polynomial is set.
I do. w = j + a₆ + B₆ r₆ + C₆ r₆ ^Two + D₆ r₆ ^Three + E₆ r₆ ^Four + F₆ r₆ ^Five = P₆ (R₆ ) A₆ Is a real number, b₆ , ..., f₆ Is an undetermined coefficient of complex number
You. (Processing 7) In processing 7, connection conditions and boundaries are set.
Create a linear equation that satisfies the boundary conditions. That is,
On the boundary line as a connection condition between all adjacent regions
Create an equation for the continuous condition of electric potential and electric flux. As a boundary condition, the regions Q2 and Q5 are conductors.
Conditions for setting the potential on the substrate 23 to 0 V, regions Q2 and Q
No.3 and Q4 are the line symmetry axes at the center and there is no electric flux
Conditions are required. The coordinates on the boundary line are shown in Fig. 6 (b).
To satisfy these conditions in all points 1 to 25
Create a linear equation for. Furthermore, in FIG.
As shown by 26 to 28, it is above and below the singular point F2.
And added 1/2 point to the left. The right side of F2 is on the electrode
So it is unnecessary. (Processing 8) In the processing 8, it was created first.
Undetermined coefficient a by solving simultaneous linear equations₁ ~ F₆ Determine
You. When there are more equations than unknowns, in the first embodiment
As described, it is determined by the method of least squares. Then, similarly to the first embodiment, each obtained
Coefficient a₁ ~ F₆ Is a polynomial P of the areas Q1 to Q6₁ (R₁ )
~ P₆ (R₆ ) And assign each polynomial for each position
By calculation, the electric potential v and the electric flux u at each position are calculated.
You. The electric potential v and the electric flux u, which are the solutions of the calculated physical quantities, are
Graphically process and print out to printout device 2.
Together with the display output to the CRT display device of the M / M device 1.
You. FIG. 7A is predicted in this first embodiment.
FIG. 8 is an equipotential diagram showing a potential distribution as a physical quantity.
(B) is an electric force diagram showing an electric flux distribution. Note that this
Since the calculation can be solved analytically by the conformal mapping method, stripline
The correct value of the electric capacity of 24 can be calculated, and the value is 37.
467 pF / m. On the other hand, this second embodiment
The electric capacity predicted by the method is 37.47 pF / m
You. In this way, it is possible to obtain extremely high prediction accuracy.
It is. (Third Embodiment) As a third embodiment, the target space is
Two-dimensional for predicting electrostatic characteristics at each position in the space
An example of solving the Poisson formula will be described. Two-dimensional Poisson method
As is well known, the equation is:It is. The right-hand side f (x, y) is the Fourier series or
If it is expanded to terms, the special solution v₁ Is easily requested
And the solution that satisfies the boundary condition is v₁ + V_Two Then v_Two Is
The solution of the Laplace equation should be found. As an example, a square of 0 ≦ x ≦ 1.0 ≦ y ≦ 1
Solve for f (x, y) = sinπx cosπy with v = 0 boundary conditions around the shape. The solution is easily obtained and is as follows. v₁ =-(1 / 2π^Two ) Sinπx cosπy v = v₁ + V_Two To satisfy the boundary condition, v₁ V
_{twenty one}And v_{twenty two}Divided into₁₂Satisfies the condition of x = 0 and y = 0.
Decide to help. V on the x-axis_{twenty one}= (1 / 2π^Two ) Sin π x v_{twenty two}= 0 on the y-axis v_{twenty one}= 0 v_{twenty two}= 0. v_{twenty one}Uses the Schwarz-Christofel transformation
However, the explanation is omitted because the method is classical.
And the result is v_{twenty one}= (1 / 2π^Two ) Sinπxcoshπy = I [(j / 2
π^Two ) Sin πz]. I [] is a symbol representing an imaginary part. v_{twenty two}Is
It becomes a polynomial of the conversion function r, which is a feature of the present invention. θ = 3
Using the angle conversion function of π / 2, r = z^Two v_{twenty two}= I [ar + br^Two + Cr^Three + ……] v = v₁ + V₁₂+ V_{22 =-(1 / 2π} 2_{) Sinπ x cosπy + (1 / 2π}Two
_{) Sinπxcoshπy + I [ar + br}Two_{+ Cr}Three
_{+ ……] The undetermined coefficients a, b, c ...
, [0112] (Equation 2)  And it is sufficient. Create this condition at multiple boundary points
And simultaneous linear equations. Therefore, this simultaneous primary
Solving the equation, the coefficient a. b. c. ... is obtained. Static electricity characteristics at each position in this target space
The boundary condition for predicting Two} Direct conversion function r
Singular term, the singularity that the first derivative of the solution at the origin is discontinuous
Dots occur. V is responsible for this₁₂The role of
Therefore v_{twenty two}Has no specificity. (Fourth Embodiment) The lapla described so far.
The space equations and the Poisson equation example divide the space
The solution of the region simplified by doing the analytical calculation method
(Theory of conformal mapping method, S-C conversion method, Fourier transform method, etc.
Method). However, the numerical solution
It is also possible to use a solution by an arithmetic method. Numerical methods include the substitute charge method and the boundary method.
Element method, finite element method, etc. These methods are CCM
It is better to use the CCM of the present invention because it is less efficient
Yes. However, using the result already calculated by other methods
Use this numerical method when you want to. In this method, the solution of the region is the substitute charge method, the boundary
If a specific solution and a general solution are obtained by the element method,
The procedure is exactly the same as in the third embodiment. Domain specific solution is v₁ 
And v_{twenty one}, And the general solution is obtained as the conversion function r
General solution v_{twenty two}Is a polynomial of the conversion function r,
The undetermined coefficient may be determined from the boundary condition. That is, v
₁ , V_{twenty one}, R were obtained by analytical method Numerical calculation method
However, the processing procedure and contents are different.
There is no. However, since only the specific solution is obtained, the general solution is obtained.
If is not obtained, the calculation of the general solution must be supplemented.
Must. The calculation of this supplement is higher than the calculation of the special solution.
It ’s easy. It can be done in the same way as a solution calculation,
The only difference is that the boundary condition is 0. Simultaneous equations to solve
Since the coefficient of is also the same, the number of unknowns is N
It is known that the interval becomes 1 / N, and N = 100
If so, the calculation amount is only increased by 1/100. When the solution of the region is obtained by the finite element method
Are a little different. Use the general solution as the conversion function r
To use, the real part u is required in addition to the imaginary part v
However, in the finite element method, unlike the substitute charge method, u is usually
unknown. However, u can be easily calculated from v
Therefore, the increase in calculation amount is even smaller. The required u is at the boundary of the two regions
Value. The differentiation of u along this value makes v perpendicular to this line
Equal to the differentiated direction. In electromagnetics terms,
The density of energy lines, or the rate of increase in electric flux, is the rate of increase in potential.
That is, it is equal to the electric field (when the dielectric constant is 1). [Equation 3]However, s is the coordinate in the length direction along the boundary line, and n is
Coordinates perpendicular to the border. And this is the border
Should be integrated along. Next, the above-mentioned first embodiment, second embodiment,
The prediction results shown in the third and fourth examples are obtained.
Each feature of the physical quantity prediction device of the embodiment of the present application shown in FIG.
explain. (1) Convergence of Prediction In the embodiment, the solution in each area is defined by a polynomial.
I do. Therefore, even if the "power series" that represents the unknown function
Is an infinite series, but if the convergence is good, the result with less error
Is obtained. As is known in the theory of complex functions, power classes
Number is the distance from the center of expansion in the region to the nearest singular point
r_a Converge in a circle with radius as. r_a Is the convergence radius
U. The conversion function is r_i Then the power series up to the nth term
When there is only one singular point, the error when censoring is (r
_i / r_a )^{n + 1} / [1- (r_i / r_a )] did
And then r_a Is large, the convergence is good. In the first embodiment, the conversion function r is set in the area P1.
₁ The singularity of the right-angled vertex is eliminated by using No.
In the second embodiment, r is within the area Q1.₁ Infinity by using
The peculiarity of the area is removed, and in other areas Q2-Q6
Conversion function r_Two Singularity of F2 point is removed by using
ing. Therefore, the convergence radius r_a Is the next singular
The distance to the point (H2 point and the mirror image point of F2) becomes
Bundle radius r_a Is larger, the convergence is better, and the prediction error is
Become smaller. The convergence radius r_a Is a complex space (z =
x + jy) distance, not conversion function r_i Must be measured at the distance
It is necessary. The linear equation of the connection condition in the second embodiment is
The point to make is to supplement each position of 26, 27, 28
Conversion function r_i This is because the change in is large. Ie
w = z^1/2 Is z = 0,1, and w = 0,1,
Since it becomes 1.41 and the intervals are not equal, z = 1/2
Replenished. (2) Conventional conformal mapping method in process 9
Use of results Obtain solution for each region simplified in process 5 of FIG.
It is necessary to use the accumulated technology of the past.
You. In the first embodiment, the complex potato
Initial w = z^3/2 Was quoted from past technology accumulation,
For example, a solution has already been obtained for the following condition. 2-1 Angle θ Electrode For an angle θ electrode, w = z^a And However, a = π
/ (2π−θ) θ = 0 and w = z^1/2 θ = π / 2 and w = z^3/2 θ = 3π / 2 w = z^Two 2-2 Infinite Area The area Q1 in the second embodiment is the upper half of the conductor substrate 23.
It was an infinite space, but there was no space under the conductor board 23.
In the limit space, the logarithmic function is used. In this case, the solution is
Indicated by W = log z 2-3 pipe The distance between the strip line 24 and the conductor substrate 23 in the second embodiment.
When the height is only 0.1 mm, a very elongated pipe-shaped area
it can. Under these conditions, the solution is exponential
Is used. If the thickness of the pipe is d, the convergence can be further improved by adopting w = exp [−πz / d] as the solution w.
Note that even if the length of the pipe is infinite, the singularity is removed.
It is possible to 2-4 Disc The disc electrode has a radius r_b The center of the circle is z₀ = -Jr_b In
The solution w at that time is given as follows. W = jlog (1-z / r_b ) R is inside the circle_b You can use expressions with different signs of
You. Circle around the circle by using this solution
It is possible to effectively represent streamlines or vortices in such a path.
Wear. If the disc electrode does not touch the boundary of the area,
The area becomes a double connected area. That is, the center of the vortex is at z = a and z = b.
The solution w at this time is as follows. w = [log (1-jz / a) -log (1-jz / b)] The potential distribution of the two disc electrodes is expressed by a function obtained by multiplying this by i.
The region becomes a triple junction region. Further, the radius r_b With disc electrode plate
The solution w when there is no electric line of force terminating at the distance is as follows
become. w = jz / (z + 2jr_b ) (3) The number of domain divisions, the expansion order, and the size of the equation
If the target space is divided into multiple regions, singular points can be removed.
It is advantageous to divide it into
If the area is further divided, the calculation accuracy is improved. But,
At that time, the number of unknowns increases and the amount of calculation processing increases.
There is a title. In addition, the polynomial P without changing the number of divisions_i (R_i )
Prediction accuracy is improved by increasing the expansion order (power) of
You. Therefore, the degree is increased and the division is increased.
And consider which is more advantageous. The convergence radius is r_a , Exhibition
Open expression degree = n, half of divided area (hereinafter referred to as mesh)
Let R be the diameter. The number of meshes is 1 / R^Two Proportional to
If the number of unknowns in the cache is 2 (n + 1), the total number of unknowns
Is N = 2k (n + 1) / R^Two … (1). k is a proportional constant. The censoring error of expansion is E = (R / r_a )^{n + 1} … (2) If N is constant and N is small, the mesh becomes
Become smaller. How to set R or n to minimize the error
Find out if you should decide. Solving n from equation (1) (2)
In the formula, logE = (n + 1) (logR−logr_a ) = (NR^Two / 2k) (logR-logr_a ) (3) Differentiate this by R to find the condition that the error E is minimized.
And (logE) '= NR / k · log (R / r_a ) + (NR^Two /2k)/R=0...(4) Therefore log (R / r_a ) =-1/2 R = r_a exp (-0.5) = 0.6r_a (5) The mesh has a convergence radius r_a Half to 60% of
It is better to increase the prediction accuracy by increasing n.
It can be understood. Note that the number of variables changes slightly due to symmetry.
However, the result of Eq. (5) does not change much. (4) Comparison with conventional method 4-1 Comparison with conformal mapping method In the conformal mapping method shown in FIG. 8, as described above,
If the target space has a complicated shape,
I can't cut it. However, in the device of the embodiment of the present application,
In R5, if a solution of conformal mapping is obtained, leave it as it is.
If it is not available, the target space is divided into multiple areas.
Split and simplify. Therefore, even if the target space is
Even if it has a complicated shape like
You can solve the problem by dividing it into regions. 4-2 Comparison with the substitute charge method In this substitute charge method, as described above, the conformal mapping is performed.
Unlike the law, it can be used for any problem, but it requires a lot of calculation
There is a problem that the accuracy of the result is bad. On the other hand, in the device of the embodiment of the present application,
For example, using simultaneous equations of unknown
In this case, the error is about 1/100 compared to the conventional substitute charge method
It is reduced gradually. This is the case in the device of the embodiment of the present application.
Solution g according to the shape of the polar boundary_i (z_i ) While using
The surrogate charge method can be applied to infinite space regardless of the shape of the boundary.
Solution g_i (z_i ) Is used. for that reason
The surrogate charge method is accurate near the singularity (for example, the corner of the electrode)
Is bad. Another difference is that the charge function used in the surrogate charge method is different.
The number has a singular point at the origin, becomes infinite at that point, and is calculated
It is easy to give an error to the result. (Place the charge far from the boundary
The solution becomes unstable and the error increases. Charge near the border
When placed, the solution is disturbed by the influence of the singularity. In any case
However, it is not highly accurate. However, in the apparatus of the present embodiment, the polynomial is calculated.
Is easy and does not generate a singularity like the charge function. Sand
However, in the device of the present application, depending on the shape of the boundary, the optimum
Solution g_i (z_i ) Is used. Then the shape of the boundary is singular
If you have a right angled corner, you can remove it
There is a characteristic. However, in the surrogate charge method, the boundary singularity is
The error becomes extremely large near. 4-2 Comparison with the finite element method In the finite element method, the prediction accuracy of the physical quantity was improved.
In order to achieve this, it is necessary to subdivide the mesh,
Becomes huge. On the other hand, in the device of the embodiment of the present application,
If you increase the expansion order without subdividing the mesh,
I mentioned earlier that this method is more efficient and more efficient.
As expected. In the previous example of the SC conversion theory, the singularity is 1
If the number of singularities is 2 or more, then the region can be solved.
The singularity can be removed by dividing the area into two to three times. This
Here, treat infinity (infinity point) as a kind of singularity.
You. Therefore, the device of the present application has an infinitely wide range of problems.
Can be easily solved with high precision. Furthermore, the device of the present application
Then, easily capture the conventionally known solution in process 5.
Is possible. As described above, the physical quantity prediction of the present invention
In the method and the apparatus, the target space is divided into a plurality of areas.
Divide and simplify each area, and the polynomial of the conversion function for each area
The formula is used. Therefore, conformal mapping method, surrogate charge method
Compared with conventional methods such as
There is no need to subdivide into
A physical quantity with high accuracy can be predicted.

[Brief description of drawings]

FIG. 1 is a block diagram showing a schematic configuration of a physical quantity prediction device adopting a physical quantity prediction method according to an embodiment of the present invention.

FIG. 2 is a flow chart showing an overall processing procedure in the device.

FIG. 3 is a diagram showing a target space and each divided area of the first embodiment in the same apparatus.

FIG. 4 is a diagram showing each simplified region in the second embodiment.

FIG. 5 is a diagram showing a prediction result of the first embodiment.

FIG. 6 is a diagram showing a target space and divided areas of the second embodiment in the same apparatus.

FIG. 7 is a diagram showing a prediction result of the second embodiment.

FIG. 8 is a flowchart showing a processing procedure of a conventional conformal mapping method.

FIG. 9 is a flowchart showing a processing procedure of a conventional substitute charge method.

FIG. 10 is a flowchart showing a processing procedure of a conventional finite element method.

[Explanation of symbols]

1 ... M / M device, 2 ... print output device, 3 ... target space setting / space dividing unit, 4 ... conversion function setting unit, 5 ... region boundary condition setting unit, 6 ... polynomial generating unit, 7 ... equation calculation processing Department,
8 ... Calculation result graphic processing unit, 21 ... Outer tube, 22
… Inner tube, 23… conductor substrate, 24… strip line

Claims (5)

[Claims] 1. A boundary condition of an applicable target space and an initial condition of the physical quantity for a physical law that defines a relationship between each position in a target space including a position space and a time space and a physical quantity at each position, In a physical quantity prediction method for predicting a physical quantity at each position in the target space from the applicable physical law by setting an end condition, a space division step of dividing the target space into a plurality of areas, and each of the divided areas Region simplification step of extending the boundary of the region and simplifying the region outside, and the transformation to obtain the solution of the physical quantity of each position in the region to the simplified boundary condition of this simplified region and to make it the transformation function. A function setting step, a polynomial setting step of approximating the solution of the physical quantity at each position of each area by a polynomial of the conversion function whose coefficient is an unknown number, and a boundary condition between each area are used. Physical quantity prediction comprising: a simultaneous linear equation creating step of generating simultaneous linear equations from the polynomials; and a physical quantity calculating step of obtaining the unknowns from the simultaneous linear equations and calculating a physical quantity at each position. Method. 2. The physical law is a partial differential equation consisting of a two-dimensional Laplace equation or a two-dimensional Poisson equation,
The physical quantity predicting method according to claim 1, wherein the simplified conversion function of each region is a solution of a two-dimensional Laplace equation. 3. The physical quantity prediction method according to claim 2, wherein the conversion function setting step includes obtaining a solution of an equation having all or a part of terms of the partial differential equation as the physical law. . 4. A boundary condition of the target space and an initial condition of the physical quantity for a physical law that defines a relationship between each position in the target space including a position space, a time space, and the like and a physical quantity at each position. In a physical quantity predicting device that predicts a physical quantity at each position in the target space from a corresponding physical law by setting an end condition, a space dividing unit that divides the target space into a plurality of areas, and each of the divided areas. Area simplification means for extending the boundary of the area to simplify the outside of the area, and conversion to obtain a solution of the physical quantity at each position in the area for the simplified boundary condition of each area, and to obtain a conversion function. Using function setting means, polynomial setting means for approximating the solution of the physical quantity at each position of each area with a polynomial of the conversion function whose coefficient is an unknown number, and boundary conditions between each area Physical quantity prediction comprising means for generating simultaneous linear equations from the respective polynomials, and physical quantity calculating means for obtaining the unknowns from the simultaneous linear equations and calculating the physical quantity at each position. apparatus. 5. The physical quantity predicting device according to claim 4, further comprising graphic output means for graphing and outputting the physical quantity of each position obtained by the physical quantity calculating means.