JP2002366537A - Method for analyzing potential problem to satisfy laplace's equation - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a calculating method to find a coefficient of terms for analysis of a mathematical expression without solving simultaneous equations in a mixed boundary value problem in which Dirichlet conditions and Neumann conditions coexist. SOLUTION: An approximate solution V1 (x, y, z) to satisfy the Dirichlet conditions is first found at an early stage. Since the solution satisfies no Neumann conditions, it is made to establish the Neumann conditions by adding ρV'1 (x, y, z) to it. Since the solution satisfies no Dirichlet conditions, it is made to satisfy the Dirichlet conditions by adding ρV2 (x, y, z) to it. These processes are repeated afterwards and at a place where fluctuation of the solutions becomes equal to or less than a prescribed value, it is defined as the approximate solution.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for analyzing a potential problem that satisfies the Laplace equation such as an electromagnetic field and has a mixed boundary value in which a Dirichlet condition and a Neumann condition are mixed in one boundary region. is there.

[0002]

2. Description of the Related Art Conventionally, as a method of calculating an electromagnetic field, a numerical calculation method such as a finite element method or a surface charge method has been used.
In order to guarantee the degree of accuracy of the numerical analysis method, calculation by mathematical analysis is indispensable.

[0003]

Such a method based on mathematical analysis is simple in a two-dimensional field, but rapidly becomes complicated in a three-dimensional field. In particular, to solve Poisson's equation for potential problems such as electromagnetic fields, if the boundary condition at one boundary surface (one boundary line in a two-dimensional field) is either the Dirichlet condition or the Neumann condition, it can be simply expressed mathematically. Although it can be obtained, it is not easily obtained when the Dirichlet condition and the Neumann condition coexist.

There is a method for solving such a mixed boundary value problem of the Poisson equation by solving a simultaneous equation. However, even if this mixed boundary value problem is solved by a method based on the simultaneous equation, the calculation accuracy of the electromagnetic field distribution can be solved. In order to guarantee
The number of unknowns in the simultaneous equations needs to be considerably increased. In other words, if the number of terms in the mathematical analysis is N in one direction, the number of terms in the three-dimensional field is N ×
N. Therefore, the number of unknowns in the system of equations is N square, and to solve the system of equations, the memory of the computer is required to be N 4 powers. For example, if N = 100, 100 million memories are required, but a calculation that requires more memory is not only impossible as the capacity of the current computer, but also cannot guarantee the calculation accuracy.

There is an example in which the three-dimensional mixed boundary value problem is solved by sophisticated mathematical analysis. In this example, the disk is zero at infinity, the surface of the disk is equipotential, and satisfies the Dirichlet condition. It is limited to a special example that satisfies the condition.

The present invention has been made in view of such circumstances, and a mixed boundary in which a Dirichlet condition and a Neumann condition coexist in a finite rectangular parallelepiped boundary surface as in an electron beam scanning device or the like. It is an object of the present invention to provide a calculation method for calculating a coefficient of a term of a mathematical expression analysis without solving a simultaneous equation in a value problem.

[0007]

A first means for solving the above problem is to solve a potential problem which satisfies the Laplace equation and has a mixed boundary value in which a Dirichlet condition and a Neumann condition are mixed in one boundary region. The way to ask
A method for analyzing a potential problem that satisfies the Laplace equation characterized by solving a richlet condition and a Neumann condition alternately and repeatedly (claim 1).

As will be described later in detail in the embodiments, when the Dirichlet condition is considered alone, and when the Neumann condition is considered alone, the Laplace equation can be solved relatively easily. In this means, focusing on this, first, a solution is obtained for the case in which the Dirichlet condition is considered alone, and a correction term is added to the solution to obtain the Neum.
Obtain a solution that satisfies the ann condition. Since the solution does not satisfy the Dirichlet condition,
Meet Dirichlet conditions. The solution is Neumann
Since the condition is not satisfied, the Neumann condition is satisfied by adding a correction term. By repeatedly performing convergence, a solution satisfying both the Dirichlet condition and the Neumann condition can be obtained within a practical range without solving the simultaneous equations.

[0009] A second means for solving the above-mentioned problems is as follows.
A solution is obtained by the first means, and when the iterative calculation is terminated, a Laplace equation characterized by the fact that the final solution is the arithmetic mean of the solution satisfying the Dirichlet condition and the solution satisfying the Neumann condition is satisfied. (Claim 2).

In the first means, the solution that satisfies the Dirichlet condition and the solution that satisfies the Neumann condition are asymptotic to each other but do not match. In this means, the arithmetic mean of these solutions at the time of terminating the iterative calculation is obtained as the final solution, so that a solution closer to the true solution than in the first means is obtained.

A third means for solving the above-mentioned problem is as follows.
The solution is obtained by the first means, and the value of the coefficient when the maximum number of terms of the series constituting the solution is infinite is estimated from the coefficient when the maximum number of terms is finite by the least square method. A method for analyzing a potential problem that satisfies the Laplace equation, characterized in that a final solution is obtained using the obtained coefficients.

As described above, in the first means, when the maximum number of terms of the series constituting the solution is increased, the obtained solution asymptotically approaches the true solution but does not coincide with the true solution. To get a true solution, the maximum number of terms in the series must be infinite. Therefore, in this means, the coefficient of the solution obtained when the maximum number of terms of the series is set to infinity is estimated by the least squares method from the coefficient of the solution obtained as a result of calculation using a finite number of series. ing. By doing so, it is possible to obtain a solution close to the case where calculation is performed using an infinite number of series, without performing calculation using an infinite number of series.

A fourth means for solving the above-mentioned problem is as follows.
A method for solving a potential problem that satisfies the Laplace equation in a two-dimensional space and has a mixed boundary value in which the Dirichlet condition and the Neumann condition coexist in one boundary region. The method according to claim 1, wherein in the second and third steps, the analytical solution is superimposed on the line of the boundary condition including only the Dirichlet condition so as to satisfy the Laplace equation, and a correction term is added thereto. A method for analyzing a potential problem that satisfies the Laplace equation, characterized in that a solution is obtained by the method described in any one of (3) and (4).

In a two-dimensional space, an analytical solution of zero potential at infinity is easily obtained, as will be described later in detail in the embodiment section.
In the second and third steps, a correction term is added so as to satisfy the Laplace equation on the line of the boundary condition of only the Dirichlet condition, and the analytical solution is superimposed, so that an approximate solution with some accuracy can be obtained. Using the approximate solution as an initial condition and a correction term added thereto, the first condition
By applying the third means to the third means to obtain a solution, the convergence of the solution can be made faster than by applying the third means from the first means from the beginning.

[0015] A fifth means for solving the above problems is as follows.
In a two-dimensional space, a method for finding a solution to a potential problem that satisfies the Laplace equation and has a mixed boundary value in which a Dirichlet condition and a Neumann condition coexist in one boundary region, wherein the first or second means The obtained solution is linearly combined with the solution obtained by the fourth means,
The coefficient of the linear combination is determined by the least squares method so that the variation of the constant term of the series is minimized when the maximum number of terms of the series forming the solution changes, and the determined linear combination A potential that satisfies the Laplace equation, characterized in that the final solution is a combination of the solution obtained by the first means or the second means and the solution obtained by the fourth means, which is linearly combined by coefficients. This is a problem analysis method (claim 5).

As will be described later in detail in the embodiment section, the solution obtained by the first means or the second means and the solution obtained by the fourth means are linearly combined,
When the coefficient of the linear combination is determined by the least-squares method so that the variation of the constant term of the series is minimized when the maximum number of terms of the series forming the solution changes, using the coupling coefficient The linearly combined solution changes only slightly even if the maximum number of terms constituting the solution changes. This means that the solution is very close to the true solution.
Therefore, according to this means, a solution close to the true solution can be easily obtained at a stage where the maximum number of terms in the series is small.

A sixth means for solving the above-mentioned problem is:
In any one of the first to fifth means, the potential problem is a potential problem in an electromagnetic field (claim 6).

By applying any one of the first to fifth means to a potential problem in an electromagnetic field, the degree of accuracy of a numerical analysis method such as a finite element method or a surface charge method in an electromagnetic field analysis is guaranteed. Can be.

A seventh means for solving the above problem is as follows.
In any one of the first to fifth means, the potential problem is a potential problem in a thermal energy field (claim 7).

By applying any one of the first to fifth means to a potential problem in an electromagnetic field, the degree of accuracy of the finite element method or the like in heat flow analysis can be guaranteed.

[0021]

Embodiments of the present invention will be described below with reference to the drawings.

First, the three-dimensional mixed boundary value problem to be solved will be described. In FIG. 1, a rectangular parallelepiped ABCDEFG is shown.
H, each vertex of which is A (a, b, c), in the xyz coordinate system.
B (a, -b, c), C (-a, -b, c), D (-a, b, c), E (a, b, 0), F
(a, -b, 0), G (-a, -b, 0), and H (-a, b, 0).

Of the six faces of this rectangular parallelepiped, only the face ABCD has a non-zero boundary potential distribution, an electrode having a constant voltage of 1000 V at the center of the face, which satisfies the Dirichlet condition, and has a peripheral part of the face. At

[0024]

(Equation 1)

It is assumed that the Neumann condition is satisfied and the other five surfaces satisfy the Dirichlet condition of 0 volt on the entire surface.

The analytic solution in the case where only the plane of z = c among the planes of -a≤x≤a, -b≤y≤b, 0≤z≤c has a non-zero potential distribution is

[0027]

(Equation 2)

Is as follows. Here, a _i = (2i−1) π / (2a), b _j = (2j−1) π / (2b) (2) where i and j are integers. Although the maximum value N of the number of terms in the equation (1) is ideally infinite, it must be a certain finite value due to the restriction of the calculation time. The problem is how to solve the coefficient A _ij in the equation (1).

Before describing the embodiment of the present invention, a method of solving the coefficients _Aij by simultaneous equations will be described.
There is a rectangular electrode at the center of the surface ABCD, and the coordinates of the vertices of the rectangle are (x ₁ , y ₁ , c), (x ₁ , y ₂ , c), (x ₂ , y ₁ ,
_{c), (x 2, y} 2, c) and then, when the constant voltage value of the electrode and V _M, that is _{x 1 ≦ x ≦ x 2,} y 1 ≦ y in the range of ≦ y ₂ V (x,
y, c) = satisfied satisfies Dirichlet condition V _M, within the range of the other surface ABCD

[0030]

(Equation 3)

When the Neumann condition is satisfied, the voltage V (x, y, z) in the expression (1) satisfies the following expression.

[0032]

(Equation 4)

The first term on the right-hand side of the equation (3) is Diri on the electrode.
satisfies the chlet condition, and Neumann is the sum of the second and third terms
Satisfies the conditions. By substituting the equation (1) into the equation (3), a simultaneous equation relating to the coefficient A _ij is obtained as follows.

[0034]

(Equation 5)

However, on the right side of this equation, i ≠ k, j
Since the term of ≠ l does not become zero, that is, it cannot be orthogonalized, it is necessary to solve a simultaneous equation in order to obtain the coefficient A _ij . As described above, calculating in this manner requires a large amount of computer memory.

Therefore, the following calculation method was developed in order to obtain the coefficients of the terms of the mathematical analysis without solving the simultaneous equations. That is, first, an initial voltage distribution on the surface ABCD is assumed. This voltage distribution is expressed as x ₁ ≦ x ≦ x ₂ , y ₁ ≦ y ≦ y ₂
If within the range of V (x, y, c) = V M Otherwise V (x, y, c) = 0 of, (1) a first Dirichlet approximate solution expressed in the form, such as expression,

[0037]

(Equation 6)

Which satisfies the Dirichlet condition

[0039]

(Equation 7)

Substituting into

[0041]

(Equation 8)

Where the integral part on the right side is i = k, j =
Terms other than l are zero and can be orthogonalized, so

[0043]

(Equation 9)

Thus, the coefficient A can be obtained without solving the simultaneous equations.
¹ _kl can be determined. This first order Dirichlet approximate solution is out of the range of x ₁ ≦ x ≦ x ₂ , y ₁ ≦ y ≦ y ₂

[0045]

(Equation 10)

Since the Neumann condition is not satisfied,

[0047]

[Equation 11]

By adding Neuman in this range.
Satisfy n conditions. For that purpose, the following expression may be satisfied.

[0049]

(Equation 12)

However, the integral range of the first term of the equation (10)

[0051]

(Equation 13)

Is

[0053]

[Equation 14]

Represents By substituting equation (9) into equation (10), as in the case of equation (7), the integral part of the second term becomes zero except for i = k and j = l and can be orthogonalized. So

[0055]

(Equation 15)

Thus, the coefficient B can be obtained without solving the simultaneous equations.
¹ _kl can be determined.

If the coefficients obtained by the equation (12) are directly added, the solution may diverge. Therefore, the coefficient B ¹ _kl
And a coefficient A ¹ _kl + ρB ¹ _{kl obtained} by adding ρB ¹ _kl multiplied by a factor ρ to a coefficient A ¹ _kl is obtained as a coefficient A in the equation (1).
_ij to obtain a first-order Neumann approximate solution. That is, V ₁ (x, y, z) + ρV ′ ₁ (x, y, z) is defined as a first-order Neumann approximate solution. An appropriate value of the factor ρ is within a range of 0 <ρ <1, and ρ = 0.5 is appropriate. The solution is x ₁ ≦ x ≦ x ₂ , y ₁
≦ y ≦ y V in the range of ₂ (x, y, c) because it does not satisfy the Dirichlet conditions = V _M, voltage distribution

[0058]

(Equation 16)

By adding, the following expression is satisfied so as to satisfy the Dirichlet condition again.

[0060]

[Equation 17]

By substituting equation (13) into equation (14),
As in the case of equation (7), the integral part on the right side is i = k, j =
Terms other than l are zero and can be orthogonalized, so

[0062]

(Equation 18)

Thus, the coefficient A can be obtained without solving the simultaneous equations.
² _kl can be determined.

As described above, the solution under the Dirichlet condition and Neu
If the solution based on the mann condition is satisfied repeatedly and alternately, the coefficients A ¹ _kl and B ¹ can be obtained without solving the simultaneous equations.
_kl, ^A _{2 kl,} ... it can be obtained. By superimposing the obtained solutions one after another as V (x, y, z) = V ₁ + ρV ′ ₁ + ρV ₂ +... (16), the solution can be converged. . This calculation method will be referred to as a DN iteration method. FIG. 2 shows an algorithm diagram of the DN repetition method described above.

A ¹ _kl , B ¹ _kl , A ² _kl ,... If the maximum value of the number of terms k and l is infinite, if the superposition is sufficiently large, the solution under the Dirichlet condition and the solution under the Neumann condition completely agree. Although it is conceivable, in practice, the maximum value of the number of terms k and l must be cut off at a certain finite value due to the limitation of calculation time. In that case, the solution under the Dirichlet condition and the solution under the Neumann condition do not completely match, and the solution will oscillate with a certain small width. So Dirich
Let the arithmetic mean of the solution under the let condition and the solution under the Neumann condition be a reasonable solution that balances both conditions.

As a first example of calculation, a rectangular parallelepiped ABCD
Each vertex of the EFGH is represented by A (a, b, c), B (a,
-b, c), C (-a, -b, c), D (-a, b, c), E (a, b, 0), F (a,-
b, 0), G (-a, -b, 0), H (-a, b, 0), where a = b = 50m
m, c = 100 mm, the coordinates of the vertices of the rectangular electrode are (x ₁ , y ₁ ,
_{c), (x 1, y} 2, c), (x 2, y 1, c), a _{(x 2, y 2, c} ), x 1 = y 1 = -25mm, x 2 = y 2 = 25mm and the electrode voltage is 10
It is a constant voltage of 00 V, and in a range inside the surface ABCD other than the electrode surface.

[0067]

[Equation 19]

The voltage distribution inside the rectangular parallelepiped ABCDEFGH when the Neumann condition of the above was satisfied was calculated. Of the calculation results, the equipotential lines on the surface ABCD are shown in FIG.
FIG. 4 shows equipotential lines at a cross section of zero.

The curve a in FIG. 5 shows the first coefficient A ₁₁ when the above embodiment is solved by the DN iterative method and sufficiently converged.
Is expressed as a function of the maximum number of terms N. Then, also all coefficients other than the coefficient A ₁₁ also varies similarly. If the coefficient is obtained by orthogonalization instead of the mixed boundary value problem, the coefficient does not vary depending on the maximum number of terms N. However, when the mixed boundary value problem is solved by the DN iteration method or the simultaneous equations, at least one boundary surface,
Since the Dirichlet condition and the Neumann condition must be balanced, the coefficient is considered to vary depending on the maximum number N of terms. Then, the value of the converged coefficient when the maximum number of terms N is infinite is considered to be a true solution.

As is apparent from FIG. 5, the maximum number of terms is 1000
But, since the coefficient A ₁₁ is not sufficient convergence, the calculation result can not be said to be sufficient convergence. So,
A method to more reliably calculate the absolute value was developed for the two-dimensional mixed boundary value problem.

As a two-dimensional mixed boundary value problem, a rectangle A
Considering the case of finding the potential distribution inside the boundary of the BCD, each vertex of the rectangle is represented by A (a, 0), B (-a, 0), C in the xz coordinate system.
(a, c) and D (-a, c). Only within the central -x ₂ ≦ x ≦ x ₂ sides CD V (x, c) satisfies the Dirichlet conditions = V _M,
It is assumed that the Neumann condition is satisfied in other ranges on the side CD. All three sides other than side CD are Dirichlet with V = 0
Assume that the conditions are satisfied.

As a first step, the center of the side CD -x ₂ ≦ x ≦ x
Only ₂ of the range V (x, c) = satisfied Dirichlet condition V _M, a range on the side CD of the otherwise satisfies the Neumann condition, determining the analytical solution as is zero at infinity. Such a solution is obtained from a mapping of a complex function. That is, the mapping x + i (cz) = x ₂ cosh (u + iv) to the complex numbers x + iz and u + iv
X = x ₂ cosh (u) cos (v), cz = x ₂ s
From inh (u) sin (v),

[0073]

(Equation 20)

Then, using u as a parameter, an equipotential line
Because you can draw, (x, c), -x ₂≦ x ≦ x₂ Range
And V₁(x, c) = V_M , C (a, c), D (-a, c) and V₁(a,
c) = V₁(-a, c) = 0.

As a second step,

[0076]

(Equation 21)

However, considering a potential distribution where c _i = (2i−1) π / (2c), V ₁ (x, z) + V ₂ (x, z) is defined by the side AC
0V at both boundaries of the edge and the side BD. That is,
V ₁ (± a, z) + V ₂ (± a,
z) = 0.

[0078]

(Equation 22)

When the equation (18) is substituted into the second term of the equation (19), terms other than i = j become zero due to the orthogonal relation, and

[0080]

(Equation 23)

The coefficient _Aj is obtained from this.

As a third step,

[0083]

(Equation 24)

However, considering a potential distribution where a _i = (2i−1) π / (2a), V ₁ (x, z) + V ₂ (x, z) + V ₃ (x,
z) is set to 0 V at the boundary line of the side AB. That is,
V ₁ (x, 0) + V ₂ (x, 0) + V ₃ so that the voltage becomes 0 V at z = 0.
(x, 0) = 0. V ₁₂ (x, z) = V ₁ (x, z) + V ₂ (x, z) (22)

[0085]

(Equation 25)

When the equation (21) is substituted into the second term of the equation (23), the terms other than i = j become zero due to the orthogonal relation, and

[0087]

(Equation 26)

Thus, the coefficient B _j is obtained from this.

As the fourth stage,

[0090]

[Equation 27]

However, considering a potential distribution where ai = (2i-1) π / (2a), V ₁ (x, z) + V ₂ (x, z) + V ₃ (x, z) + V ₄ (x, z) is the center of the boundary of side CD and V is only within the range of -x ₂ ≦ x ≦ x ₂
(x, c) = satisfied Dirichlet condition V _M, the other side C
In the range on D, DN is satisfied so as to satisfy the Neumann condition.
A solution is obtained by iterative calculation using an iterative method. That is, V ₁₃ = V ₁ (x, z) + V ₂ (x, z) + V ₃ (x, z) (26), and the first order Neumann approximate solution is obtained.

[0092]

[Equation 28]

And an expression satisfying the Neumann condition

[0094]

(Equation 29)

Then, the integral part on the right side becomes i = j
The other terms are zero and can be orthogonalized, so

[0096]

[Equation 30]

Thus, the coefficient D can be obtained without solving the simultaneous equations.
¹ _j can be determined. Next, V (x, z) _14,1 = V ₁₃ (x, z) + ρV _4,1 (x, z) (30) is set. The factor ρ is for preventing the divergence of the solution as in the case of the three-dimensional case, and its value is suitably 0.5 in the range of 0 <ρ <1. First approximation Dirichlet solution

[0098]

(Equation 31)

And an expression satisfying the Dirichlet condition

[0100]

(Equation 32)

Substituting into, the integral part on the right side is i = j
The other terms are zero and can be orthogonalized, so

[0102]

[Equation 33]

Thus, the coefficient D ′ ¹ _j can be obtained without solving the simultaneous equations. Similarly, one after another,
The coefficients D ² _j , D ′ ² _j ,...

[0104]

(Equation 34)

The solution can be converged by superimposing one after another as described above. Such a calculation method will be referred to as a “multi-stage method + DN iteration method”. FIG. 6 shows an algorithm diagram of the “multi-stage method + DN repetition method” described above.

As a second embodiment, calculations were performed by this method. That is, it is assumed that each vertex of the rectangle ABCD is located at A (a, 0), B (-a, 0), C (a, c), D (-a, c) in the xz coordinate system, and a = 50 mm, c = 100mm, the coordinates of the vertices of the linear electrode are
(x _{1, c),} assumed to be _{(x 2, c), x} 1 = -25mm, x 2 = 25m
m, the voltage of the electrode is a constant voltage of 1000 V, and within a range inside the surface ABCD other than the electrode surface.

[0107]

(Equation 35)

The voltage distribution inside the rectangular ABCD when the Neumann condition of the above was satisfied was calculated. FIG. 7 shows equipotential lines by an analytical solution that is zero at infinity in the first stage of the “multi-stage method + DN iteration”, and FIG. FIG. 8 shows the total of the solutions from the first stage to the third stage, and FIG. 9 shows the total of the solutions from the first stage to the fourth stage. It is.

In the two cases of the above-described “multi-stage method + DN iteration method” and the calculation using only the DN iteration method, the variation of the first coefficient A ₁ when sufficiently converging is expressed by FIG. 11 shows the function as a function of the maximum number N of terms. The curve a represents the variation of the coefficient in the case of only the DN iteration method, and the curve b represents the variation of the coefficient by the “multi-stage method + DN iteration method”. Factor A ₁ is than with only D-N iterations, it can be seen that converges much faster towards the case of "multi-stage process + D-N iterative method".

Further, as a method of effectively utilizing the above two methods, the following calculation method has been devised. That is,
Potential distribution by only DN repetition method

[0111]

[Equation 36]

The potential distribution obtained by developing the potential distribution by the “multi-stage method + DN repetition method” is

[0113]

(37)

Let F _i be a linear combination of the coefficients of the potential distribution. Potential distribution obtained by substituting the coefficient obtained by F _i = (1-k) E _i + kD _i (37)

[0115]

(38)

Is a linear combination of the two. In the equation (37), k is a constant number not depending on the maximum number of terms N, and 0 <k
Desirably <1. k is determined by the least-squares method so that the variation of F _i is minimized even when the maximum number of terms N changes. In the case of this embodiment, k determined by this method is k = 0.89131633. The coefficients F _i which is determined by the method is estimated to be very close to the true value of the coefficient. The variation of the first coefficient of the coefficient F _i by this combination method is represented by a curve c in FIG.

In a two-dimensional field, an analytical solution to such a mixed boundary value problem can be obtained by complex polygon conversion. That is, the complex number domain Z (t) = x +
iy and W (t) = u + iv are calculated by the parameter t and elliptic integral

[0118]

[Equation 39]

Assume that they are related as follows. Above
Where P and Q are complex constants, t₁, T₂, T₃, T
₄Is a real number and 0 <t₁<T₂, 0 <t₃<T₄ 
And Equation (39) means that Z in FIG.
Like the (t) plane, Z (t₁), Z (-t ₁), Z (t₂), Z (-
t₂If the points above are A, A ', B, and B', respectively,
Real number t is -t₁<T <t₁Z (t) is a real number within the range
But the point t = ± t₁Rotate at right angles with t₁≦ t ≦ t₂ Also
Is -t₂≦ t ≦ -t₁ In the range of
Becomes a complex number and the point t = ± t₂And rotate at right angles
Each t₂≤t or t≤-t₂Within the range of x
Less or more.

Z (t), t₁, T₂To W (t) and t
₃, T₄ , The same is true for W in FIG.
This can be said in the (t) plane. Here, W (t₃), W (-t₃),
W (t ₄), W (-t₄) Are points C, C ', D,
D ′. t₂= T₃, The points B and B ′ on Z (t)
And points C and C 'on W (t) are in a mapping relationship, and points on Z (t)
A, A 'is a point W (t) between points C and C' on W (t).₁), W
(-t₁), And points D and D ′ on W (t) are Z
(t) a point Z (t) between points B and B '₄), Z (-t₄) And the mapping function
In charge. At a point W (t) = u + iv inside the rectangular CDD'C '
Then, when the parameter t is obtained from the equation (40), the elliptic product
By the elliptic function which is the inverse function of minute,

[0121]

(Equation 40)

Are represented as follows. The parameter t is a real number on the boundary of the rectangle CDD'C ',
It is a complex number inside DD'C '. By substituting the parameter t in equation (41) into equation (39), it is possible to determine which position on point W (t) = U + iV on W (t) corresponds to Z (t). Understand.

By the way, when the voltage V (x, y) is on the Z (t) plane, the Laplace equation

[0124]

[Equation 41]

When the above condition is satisfied, the voltage V (u, v) becomes W
Laplace equation even in the (t) plane

[0126]

(Equation 42)

It is known that the above is satisfied.

Therefore, in the area Z (t), the rectangle AB
Uniformly V _M volts' side AA of 'B'A, when taking a boundary voltage uniformly 0 volts sides BB', the point Z
The voltage on (t) represents the imaginary part of Z (t) as Im (Z (t)), and T = Im (Z (t ₂ )).

[0129]

[Equation 43]

Can be expressed as follows. When the boundaries AA 'and BB' on the area Z (t) satisfy the Dirichlet condition in this way, the boundaries AB and A'B '
As is clear from the Z (t) plane, the Neumann condition is satisfied. The boundary line AA ′ is on W (t), and W (t ₁ ) and W (−t ₁ )
, Satisfying the Dirichlet condition, and the boundaries AB and A′B ′ are represented by W (t) on W (t), respectively.
The boundary between (t ₁ ) and C and the boundary between W (−t ₁ ) and C ′ are mapped to satisfy the Neumann condition. Therefore, the Dirichlet condition and Neumann condition on one boundary line CC ′ Can be solved with high accuracy analytically. That is, by using the equations (41), (39), and (44) in this order, the potential amount at W (t) = u + iv can be obtained.

The square of the difference between the rectangular internal potential distribution obtained by the complex number and the internal potential distribution obtained by the equation (38) when k is varied from zero to 1 in the equation (37) FIG. 13 shows the variation of the value of the square root of the average (this will be referred to as _VRMS ) when the maximum number of terms N is from 100 to 500. As is clear from FIG. 13, the value of the root mean square is the smallest around k = 0.9, and at this value, the coefficient in equation (37) can be said to be closest to the true value.

[0132] Further, FIG. 14, when k = 0 (i.e., when only D-N iteration) and _{V R MS} of, when k = 1 (i.e., "multi-step process + D-N iteration" and V _RMS only when) the V _RMS when the k = 0.9, is shown how the change in the time of varying the maximum number of terms N from 100 to 2000. In k = 0 and k = 1, _{V RMS} as the maximum number of terms N increases becomes smaller but, _{V R MS} in k = 0.9 is regardless of a change in the maximum number of terms N, stable at lower amounts You can see that it is doing. That is, when k = 0.9, it can be seen that even if the maximum number of terms N is small, it is very close to the true solution.

However, in the case of a three-dimensional problem, it is almost impossible to find an analytical solution that is zero at infinity in the first stage of the “multi-stage method + DN iteration”. The value of the coefficient A _ij cannot be determined by the method. Further, in the case of three dimensions, it is impossible to analytically obtain a complex number. Therefore, a method of finding a true value by fitting the coefficient A _ij as a function of the maximum number of terms N and finding a convergence value in the case of N → ∞ has been developed. The method will be described below.

First, the one having the coefficient A _ij is _converted to the maximum number of terms N
C (N) = C ₀ + C ₁ N ^α (45) C ₀ and C ₁ are constants independent of the maximum number of terms N. Then

[0135]

[Equation 44]

And put

[0137]

[Equation 45]

In the above embodiment of the three-dimensional problem,
When d ₁ and d ₂ are approximated by difference and the index α is obtained by the equation (49), α ≒ −0.8 is obtained by numerical calculation.

Another method for obtaining the index α will be described. When the maximum number of terms is N ₁ , N ₂ , N ₃ , the values of the coefficients A _ij calculated by the D-N iterative method are D ₁ , D ₂ ,
When a D _3, when the fitting (45) below,

[0140]

[Equation 46]

R = (D ₃ −D ₁ ) / (D ₂ −D ₁ ) (53) and N ₂ = N ₁ r ₂ , N ₃ = N ₁ r ₃

[0142]

[Equation 47]

[0143] next, since R is unknown is only the exponent α, f (α) = R (r 2 α -1) - putting the ^{_{(r 3 α -1) ... (}} 55), f (α) = 0 (56), the expression (54) is satisfied. Equation (56) is a non-linear equation related to the index α, and is obtained by solving it by the Newton method. That is, if the i-th approximate value of the index α is α _i ,
(i + 1) The approximate value α _{i + 1} is given by α _{i + 1} = α _i −f (α _i ) / f ′ (α _i ) (57) The index α is obtained by successive approximation using Expression (57). here,

[0144]

[Equation 48]

Is as follows. Also in this case, when numerical calculation was performed in the embodiment of the three-dimensional problem, α ≒ −0.8 was obtained.

As a result, as shown in equation (45), the approximate function can be roughly estimated. However, since the index α slightly varies depending on the maximum number N of terms, more strict fitting is required. Therefore, instead of equation (45), a power series

[0147]

[Equation 49]

Fitting is performed. Constant Ci (i = 0 to M)
Is determined by the least squares method. From α ≦ 0, the limit of C (N) converges,

[0149]

[Equation 50]

Is obtained. C _{0 in} equation (60) is the maximum number of terms N
Is the value of the asymptote of the function C (N), which is the parameter, and it is estimated that this is the true value of the coefficient.

FIG. 15 shows an embodiment of the two-dimensional problem described above, where M = 5 in equation (59) and the exponent α is -0.5 to -1.5.
In the case of varied up, shows a variation of the V _RMS when modifying the coefficients E _i in (35) of the D-N iteration. As is clear from FIG.
The V _RMS becomes minimum around = −0.8. Therefore, (5
9) When fitting according to the formula, if α = -0.8,
It turns out that it is very close to the true solution. Similarly to the three-dimensional problem, in the two-dimensional problem, the exponent α calculated by the expression (49) or the expression (54) by the expression (58) is
α ≒ −0.8, which is almost the same as the result of FIG.

In FIG. 15, all the coefficients E _i
The index α is uniformly determined and calculated, but the coefficients E _i are individually calculated by the method of equation (49) or the equations (54) to (58).
When calculating the value of the index α, there is a slight difference. However, after individually obtaining the value of the index α by the method, (59)
V _RM in the case of fitting in the power series expansion, such as the formula
It was found that the minimum value of V _RMS calculated by uniformly determining the index α was smaller than _S. This is the coefficient E _i (i
> If the absolute value of 1) is smaller than the coefficient E ₁ etc., (4
It is considered that when the calculation is performed by the expression (58) from the expression 9) or the expression (54), the calculated value of the index α includes a considerable error.

Further, regarding the embodiment of the above three-dimensional problem,
Assuming that α = −0.8, the coefficient A ₁₁ is expanded by a power series as shown in Expression (59) and is obtained by the least squares method. A ₁₁ = 1117.5288
4, which is indicated by the asymptote b in FIG. As is clear from the figure, even when the maximum number of terms N is 1000, there is a considerable difference from the asymptote. Higher accuracy than the restricted accuracy can be obtained, and a true solution can be reached.

In the above-described embodiments and examples, an example in which the Laplace equation is satisfied in the electrostatic potential field has been described.
Lapla in thermal energy field
Even for a problem that satisfies the ce equation, such a mixed boundary value problem can be solved by a similar method, and analysis of a magnetic field and analysis of a heat conduction problem can be performed.

[0155]

As described above, according to the present invention,
A solution to a potential problem that satisfies the Laplace equation, which has been difficult in the past, and has a mixed boundary value in which a Dirichlet condition and a Neumann condition coexist in one boundary region can be easily obtained without solving a simultaneous equation. In particular, the method of estimating the coefficient at infinity at the maximum number of terms makes it possible to solve this problem with an accuracy higher than that restricted by the maximum number of terms.

[Brief description of the drawings]

FIG. 1 is a diagram showing boundary conditions of a three-dimensional mixed boundary value problem to be solved.

FIG. 2 is a diagram illustrating an algorithm of a DN iteration method.

FIG. 3 is a diagram showing equipotential lines on a surface ABCD in the example of the present invention.

FIG. 4 is a diagram showing equipotential lines in a cross section at y = 0 according to the embodiment of the present invention.

In the embodiment of the present invention; FIG is a graph showing the relationship between the D-N iteration at the calculated coefficients A ₁₁ and the maximum number of terms N.

FIG. 6 is a diagram showing an algorithm of a “multi-stage method + DN iteration method”.

FIG. 7 is a diagram showing a potential distribution calculated in a first step of the “multi-step method + DN repetition method”.

FIG. 8 is a diagram illustrating the sum of the potential distribution calculated in the first step and the potential distribution calculated in the second step of the “multi-step method + DN repetition method”.

FIG. 9 is a diagram showing a potential distribution calculated in the first step, a potential distribution calculated in the second step, and a third step of the “multi-step method + DN repetition method”.
It is a figure showing the sum of the potential distribution calculated at the stage.

FIG. 10 is a diagram showing the sum of potential distributions calculated from the first stage to the fourth stage of the “multi-stage method + DN repetition method”.

In [11] two-dimensional problem, the coefficient A ₁₁ calculated in D-N iteration, the coefficient A ₁₁ calculated in "multistage process + D-N iteration", is calculated by linear combination of both a coefficient a _11, is a graph representing the relationship between the maximum number of terms N.

FIG. 12 is an explanatory diagram of a method of solving a two-dimensional mixed boundary value problem by a complex number mapping transformation.

FIG. 13 shows an analytical solution obtained by mapping conversion of a complex number, and D
The variation of the root mean square of the difference between the solution obtained by the linear combination of the -N iteration method and the "multi-stage method + DN iteration method" is
FIG. 6 is a diagram showing a function of a parameter k.

FIG. 14 shows an analytical solution obtained by mapping conversion of a complex number and D
FIG. 9 is a diagram showing a root mean square of a difference between a solution obtained by a linear combination of a −N iteration method and a “multi-stage method + DN iteration method”, where parameters are k = 0 and k = 1. FIG. 9 is a diagram when the maximum number of terms is varied in the case of k = 0.9.

FIG. 15 shows an analytical solution obtained by a complex number mapping transformation and D
It is a figure at the time of changing an index (alpha) about the square root of the root mean square of the difference with the -N iterative method.

Claims (7)

[Claims] 1. Satisfies the Laplace equation, and in one boundary region
A method for solving potential problems with mixed boundary values in which Dirichlet conditions and Neumann conditions coexist.
A method of analyzing a potential problem that satisfies the Laplace equation, characterized by alternately and repeatedly solving the chlet condition and the Neumann condition. 2. A solution is obtained using the method for analyzing a potential problem satisfying the Laplace equation according to claim 1,
A method of analyzing a potential problem that satisfies the Laplace equation, characterized in that an arithmetic mean of a solution that satisfies the Dirichlet condition and a solution that satisfies the Neumann condition is the final solution when the iterative calculation is terminated. 3. A solution is obtained using the method for analyzing a potential problem satisfying the Laplace equation according to claim 1,
The coefficient value when the maximum number of terms of the series constituting the solution is infinite is estimated by the least square method from the coefficient when the maximum number of terms is finite, and the final solution is calculated using the estimated coefficients. A method for analyzing a potential problem that satisfies the Laplace equation, characterized by finding. 4. A method for finding a solution to a potential problem that satisfies the Laplace equation in a two-dimensional space and has a mixed boundary value in which a Dirichlet condition and a Neumann condition are mixed in one boundary region. Analytical solution as the first step, and in the second and third steps Dirichlet
4. The method according to claim 1, wherein the analytical solution is superimposed on the line of the boundary condition including only the condition so as to satisfy the Laplace equation, and a correction term is added thereto.
A method for analyzing a potential problem that satisfies the Laplace equation, characterized in that a solution is obtained by the method described in the section. 5. A method for solving a potential problem that satisfies the Laplace equation in a two-dimensional space and has a mixed boundary value in which a Dirichlet condition and a Neumann condition are mixed in one boundary region. 2 and the solution obtained by the method of claim 4 are linearly combined, and the coefficient of the linear combination is changed when the maximum number of terms of the series constituting the solution changes. 3. A solution determined by the method according to claim 1 or 2, which is determined by a least square method so that a variation of a constant term of the series is minimized, and linearly combined by the determined coefficient of linear combination. A method for analyzing a potential problem that satisfies the Laplace equation, wherein a combination of solutions obtained by the method according to claim 4 is a final solution. 6. The method for analyzing a potential problem satisfying the Laplace equation according to claim 1, wherein the potential problem is a potential problem in an electromagnetic field. 7. The method according to claim 1, wherein the potential problem is a potential problem in a thermal energy field.