JPH09160903A - Method for estimating physical quantity distribution - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain a physical quantity distribution estimating method by which the calculating time for a matrix equation can be remarkably shortened and the physical quantity distribution can be estimated at a high speed by converting a close matrix into a non-dense matrix in which many zero elements are included by a wavelet conversion processing and solving the matrix equation. SOLUTION: An imparted governing equation is converted into a boundary integral equation (ST2), the boundary integral equation is converted into a digitizing equation (ST3), the digitizing equation is converted into a matrix equation A.x=b (ST4), a wavelet conversion is performed for the obtained matrix equation A.x=b (ST4), the converted matrix equation At .xt =bt is approximately solved by substituting zero for the matrix element whose absolute value is threshold or below in the matrix At , (ST6) approximate physical quantity distribution x* is determined by performing a wavelet inverse conversion for the approximate solution xt (ST7) and the correction of the approximate physical quantity distribution x* is performed by the repeated calculation (ST8).

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for rapidly estimating various physical quantity distributions such as potential distribution, electromagnetic field distribution, stress distribution, displacement distribution, temperature distribution and the like. The present invention relates to a physical quantity distribution estimation method in which the physical quantity distribution estimation method called the element method is significantly speeded up.

[0002]

2. Description of the Related Art FIG. 9 shows a modification of the conventional physical quantity distribution estimation method by the boundary element method shown on page 19 of "Boundary element analysis" (Masato Enokizono, published by Baifukan in 1986). FIG. In the figure, 1 is a governing equation setting unit for giving a governing equation representing a physical phenomenon of a target system, and 2 is a boundary integral equation converting unit for converting the given governing equation into a boundary integral equation.
Is the integral theorem used to convert this boundary integral equation. Reference numeral 4 is a discretization equation conversion unit that converts the boundary integral equation into a discretization equation, and reference numeral 5 is a boundary element obtained by dividing the boundary into a plurality of elements at the time of conversion into the discretization equation. 6 is a simultaneous linear equation conversion unit that derives a simultaneous linear equation (actually, matrix equation A · x = b) from the discretized equation, and 7 solves this matrix equation A · x = b by the Gaussian elimination method. And a matrix equation calculation section for obtaining the unknown vector x.

A conventional physical quantity distribution estimating method will be described below with reference to FIG. First, in the governing equation setting unit 1, a governing equation representing a physical phenomenon of the target system is given by a differential equation and a boundary condition expression. Next, in the boundary integral equation conversion unit 2, the governing equation given by the governing equation setting unit 1 is converted into a boundary integral equation in the boundary region of the target system using the integration theorem 3. Next, the boundary is divided into many boundary elements 5, and the discretization equation conversion unit 4 creates a discretized equation from the boundary integral equation obtained by the boundary integral equation conversion unit 2. Next, the simultaneous linear equation conversion unit 6 derives a simultaneous linear equation (actually matrix equation A · x = b) from the discretized equation obtained by the discretized equation conversion unit 4. Then, the matrix equation calculation unit 7 solves the unknown vector x from the matrix equation A · x = b obtained by the simultaneous linear equation conversion unit 6 by using the Gaussian elimination method or the like. Here, the discrete equation conversion unit 4, the simultaneous linear equation conversion unit 6, and the matrix equation calculation unit 7 are realized by computer blog programming.

In the matrix equation A · x = b, A
Is a system matrix depending on the target system, x is an unknown vector representing an unknown physical quantity distribution on the boundary, and b is a constant vector.

Here, in the actual calculation, the time for solving the matrix equation A · x = b becomes much longer than the time for deriving the system matrix A depending on the target system. This requires solving a matrix equation for a matrix with few zero elements (usually called a dense matrix), and because this matrix is asymmetric, it is a computationally inefficient Gaussian elimination or LU factorization. This is because it is necessary to use this method.

[0006]

Since the conventional physical quantity distribution estimation method is configured as described above, it is necessary to solve a matrix equation for a matrix called a dense matrix with almost zero elements, and this matrix is asymmetric. Since it is a matrix, the solution has to be obtained by a method such as Gaussian elimination method or LU decomposition, which has low calculation efficiency, and there is a problem that the calculation time becomes very long.

The present invention has been made to solve the above-mentioned problems, and by the wavelet transform processing,
By converting a dense matrix into a sparse matrix containing many zero elements and solving the matrix equation, the calculation time is greatly shortened, and a physical quantity distribution estimation method that can estimate the physical quantity distribution at high speed is obtained. And

In the physical quantity distribution estimating method according to the present invention, a given governing equation is a boundary integral equation, the boundary integral equation is a discretizing equation, and the discretizing equation is a matrix equation A · x = b. And the resulting matrix equation A · x = b is wavelet-transformed A _t · x _t =
The b _t, the matrix A absolute value in _t approximately solved by replaces the following matrix elements threshold to zero, the amount of approximate physical The resulting approximate solution x _t by the inverse wavelet transform distribution x ^* And the approximate physical quantity distribution x ^*
The accuracy of the solution is improved by correcting by using the iterative calculation.

In the physical quantity distribution estimating method according to the second aspect of the present invention, the wavelet transform is performed using the four coefficient type wavelet of Dobby.

A physical quantity distribution estimating method according to a third aspect of the present invention is a wavelet-transformed matrix equation A _t · x.
_t = the threshold value in solving approximately the b _t, to 0.01% of the maximum value of the absolute values of the matrix elements of the matrix A _t 0.0000
It is selected in the range of 1%.

In the physical quantity distribution estimating method according to the present invention, the matrix equation A is obtained by using the preconditioned bi-conjugate gradient method.
_This is an approximate solution of _t · x _t = b _t .

In the physical quantity distribution estimating method according to the fifth aspect of the present invention, the number of boundary elements when discretized is a power of 2 when the boundary integral equation is converted into a discretized equation.

In the physical quantity distribution estimating method according to the sixth aspect of the present invention, the correction of the approximate physical quantity distribution x ^* by iterative calculation is calculated from the approximate physical quantity distribution x ^* , the system matrix A and the constant vector b. than the error d _b, obtains an error d _x based on the error matrix equation a · d _x = d _b, approximate the true solution x ₁ obtained by subtracting the error d _x from the approximate physical quantity distribution x ^* Substituting the physical quantity distribution x ^* ,
Computes the solution x ₂ by executing the above process again, until the relative error between the solution x ₂ and true solution x ₁ is smaller than the predetermined set value, the solution x ₂ in true solution x ₁ This is done by substituting and repeating the calculation of the solution x ₂ .

In the physical quantity distribution estimating method according to the present invention, an error matrix equation A · d is obtained according to a procedure in which a matrix equation is wavelet-transformed and approximately solved, and the obtained approximate solution is further wavelet-inverse-transformed. _x = d _b
The error d _x is obtained from

In the physical quantity distribution estimating method according to the present invention, a predetermined set value for testing the relative error between the true solution x ₁ and the solution x ₂ is 0.01% to 0.0001%. The selection is made within the range.

[0016]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS One embodiment of the present invention will be described below. Embodiment 1 FIG. 1 is a flowchart showing a physical quantity distribution estimation method according to Embodiment 1 of the present invention. In the figure, ST1 is the first that gives the governing equation of the target system.
, ST2 is a second step for converting the governing equation into a boundary integral equation, ST3 is a third step for converting the boundary integral equation into a discretized equation, and ST4 is the matrix equation A · x. This is the fourth step of converting to = b. Incidentally, these first steps ST
The processing from obtaining the matrix equation A · x = b from the governing equation in the first to fourth steps ST4 is performed by the governing equation setting unit 1 and the boundary integral equation converting unit in the conventional physical quantity distribution estimating method shown in FIG. 2. This is equivalent to the processing by the discretized equation conversion unit 4 and the simultaneous linear equation conversion unit 6.

In ST5, a new matrix equation A _t · x is obtained by performing wavelet transform on both sides of the matrix equation A · x = b.
This is the fifth step of obtaining x _t = b _t , and ST6 replaces, in the matrix elements of the matrix A _t , the matrix elements whose absolute value is less than or equal to a predetermined threshold value with zero, and approximately this matrix equation A _t
The sixth step is to solve x _t = b _t to obtain an approximate solution x _t . ST7 is a seventh step of performing an inverse wavelet transform on the obtained approximate solution x _t to obtain an approximate physical quantity distribution x ^* , and ST8 is to repeatedly calculate the approximately obtained physical quantity distribution x ^*. Eighth correction
Is the step.

Next, the operation will be described. Here, FIG.
Is an explanatory diagram showing an example of a target system to be considered this time,
The analysis area surrounded by the four sides 10, 11, 12, and 13 is the inside of a square. Further, in the target system shown in FIG. 2, the boundary area is the four sides 10, 11, 1 of the square.
2 and 13, and there are boundary elements with element numbers 1 to 32 on the boundary area (in this case, the boundary line). Also,
It is assumed that the side 10 has a potential u = 0V and the side 12 has a potential u = 300V. The other two sides 11, 1
3 assumes that the electric field vector is parallel to each side and the component du / dn perpendicular to the side of the electric field is zero.

First, in the first step ST1, differential equations and boundary condition equations that govern physical phenomena of the target system are given as governing equations. As the governing equation, Laplace equation, Poisson equation, Helmholtz equation, or wave equation can be considered. For example,
In the target system shown in FIG. 2, the Laplace equation shown below is selected as the governing equation.

[0020]

[Equation 1]

Here, the boundary condition is that the side 10 has a potential u = 0.
V, the side 12 has a potential u = 300 V, and the other two sides have zero component du / dn perpendicular to each side of the electric field. In this specification, partial differentiation of j with respect to k is expressed as dj / dk.

Next, in the second step ST2, the governing equation given in the first step ST1 is converted into a boundary integral equation of the following equation for the boundary region by using the integral theorem (Green's theorem).

[0023]

(Equation 2)

Here, u _i represents the potential u at the point i on the boundary area. In the target system shown in FIG. 2, the boundary area has four sides 10, 11, 1 of a square.
2 and 13. Further, q represents the inclination du / dn in the direction perpendicular to the boundary of the potential u. It should be noted that the integral is an integral on the boundary region, and is a line integral along the four sides 10, 11, 12, and 13 of the square in FIG. Further, u ^* represents a solution (basic solution) when there is a point energy source inside the region, and q ^* is defined by du ^* / dn.

Next, in a third step ST3, the boundary integral equation obtained in the second step ST2 is converted into a discretized equation. Therefore, the boundary area is divided into many boundary elements (in the example shown in FIG. 2, the boundary area is 4
The side 10 to the side 13 are divided into 32 boundary elements to which the element numbers 1 to 32 are added, and the integration operation is approximated by an addition operation. This result can be described by the following matrix expression. H · u = G · q where H and G are matrices that describe the target system.
Note that the number of boundary elements is preferably a power of 2 in order to perform the wavelet transform.

Next, in a fourth step ST4, the discretization equation obtained in the third step ST3 is converted into a matrix equation. Here, the above form of H · u = G · q is inconvenient to solve. This is because some of the variables u and q are given as boundary conditions, and the others are unknown variables. Therefore, only the unknown variables are regrouped to consider an unknown vector x representing an unknown physical quantity distribution.
As a result, it can be converted into the following matrix equation. A · x = b Here, A is a system matrix that depends on the target system, and b is a constant vector. The processing from the first step ST1 to the fourth step ST4 is the same as the processing in the conventional physical quantity distribution estimation method shown in "Boundary Element Analysis" (Masato Enokizono, published by Baifukan in 1986). It is the same.

Next, in the fifth step ST5, the matrix equation A.x = b obtained in the fourth step ST4 is obtained.
In, the wavelet transform is performed on the matrix A and the vectors x and b. Here, the wavelet transform is a transform that separates the information of the average value of the data and the information of the differential coefficient and efficiently stores them, and is often used for image compression, etc., and is published by Cambridge University Press in 1992. Published by William Press (Wi
lliam H. Press) et al., "Numerical Recipes," Second Edition, pp. 584 to 599, and the like, and therefore, detailed description thereof is omitted here. In addition, specifically, for the wavelet transformation matrix W and its transposed matrix W ^T ,
Execute with the following conversion formula. A _t = W · A · W ^T x _t = W · x b _t = W · b

In order to perform the wavelet transform, it is preferable that the size of the matrix and the vector is a power of 2 in terms of calculation efficiency. That is, 32, 64, 128, 25
6, 512, 1024, 2048, ... Are preferred sizes. In addition, in the first embodiment, a well-known Dobby's 4-coefficient wavelet is used as the wavelet transform. When using this four coefficient wavelet of Dobby,
A comparison was made for the case of using a two-coefficient wavelet, but the results were the same. There are various other wavelets, but any one will give results similar to those described here. Therefore, by using the Davidie 4-coefficient wavelet in this way, the wavelet transform can be performed by a simpler calculation.

As a result of the wavelet transformation, the matrix equation A · x = b is transformed into the matrix equation shown by the following equation. A _t · x _t = b _t

Next, in the sixth step ST6, the matrix equation A _t · x _t obtained in the fifth step ST5 is obtained.
= B _t , a matrix element whose absolute value in the matrix element of the matrix A _t is equal to or smaller than a predetermined threshold value is replaced with zero to approximately solve it to obtain an approximate solution x _t . Here, FIGS. 3 and 4 are explanatory views in which the system matrix A and the matrix A _t after the wavelet transformation thereof are displayed in pseudo three-dimensional form, respectively. It can be seen from FIG. 3 that the system matrix A has a complicated structure. Also, it can be seen that there is almost no zero element and the matrix is asymmetric. On the other hand, in the matrix A _t after the wavelet transform in FIG. 4, it is understood that the diagonal components are the main components and the number of elements that are close to zero is very large overall. For example, in FIG. 4, there are only 120 matrix elements with absolute values of 0.5 or more (corresponding to about 10% of data), so if the absolute values of 0.5 or less are replaced with 0, a sparse matrix with many zero elements Can be approximated by. Here, if the threshold value for replacing with zero is set too large, the matrix equation A _t · x
It turns out that _t = _bt cannot be solved. Empirically,
When the absolute value of the threshold approximately 0.00001 to 0.01% of the maximum value of each matrix element of the matrix A _t, it was found that the obtained the fastest solution.

By performing such an approximation, the matrix equation A _t · x wavelet transformed in the fifth step ST5.
As a method for solving _t = b _t, it can be selected from a fast calculation method for known sparse matrix. In the first embodiment, one of them is the preconditioned bi-conjugate gradient method (preconditioned gradient method).
The matrix equation A _t · x _t = b _t is solved using a bi-conjugate gradient method). Here, the pre-processed biconjugate gradient method is to iteratively calculate the value of the unknown vector x so that A · x−b becomes as small as possible when solving the matrix equation A · x = b. By
This matrix equation A · x = b is simply solved,
For example, the above-mentioned "Numerical Recipes" second
Edition (Outside William Press Cambridge University Press 1
Since it is a popular one described in (1992), etc., its detailed description is omitted here.

As a high-speed calculation method for this sparse matrix, there are many methods other than the above-described bi-conjugated gradient method, and any method may be used. By using the biconjugate gradient method, the matrix equation A _t · x _t = b _t can be solved faster.

Next, in the seventh step ST7, the approximated solution x _t obtained by solving the matrix equation A _t · x _t = b _t in the sixth step ST6 is subjected to the wavelet inverse transform by the following equation. Is performed to obtain an approximate physical quantity distribution x ^* . x ^* = W _^T · x _t

Here, in FIG. 5, the matrix equation A _t · x _t = b _t is solved as described above, the obtained approximate solution x _t is subjected to inverse wavelet transform, and an approximate physical quantity distribution (potential in this example) is obtained. FIG. 6 is an explanatory diagram showing a result of obtaining a distribution of electric field component) x ^* . Further, FIG. 6 is an explanatory diagram showing the result calculated by the conventional method for comparison. 5 and 6, the horizontal axis coordinates 1 to 32 correspond to the element numbers 1 to 32 in FIG. 2, and the horizontal axis coordinates 1 to 8 and 17 to 24 indicate the potentials at the center of each element. It represents. On the other hand, the horizontal axis coordinates 9 to 16 and 25
Reference numerals 32 to 32 represent electric field components perpendicular to the sides at the center of each element. Comparing FIG. 5 and FIG. 6, it can be seen that the solution is approximately obtained, but the accuracy is low. This is because the matrix elements whose absolute value is 0.5 or less are replaced with 0 after the wavelet transform.

Next, in the eighth step ST8, the physical quantity distribution x ^* obtained approximately in the seventh step ST7 ^.
To improve its accuracy. That is, the accuracy of the approximate physical quantity distribution x ^* is improved by correcting the approximately calculated physical quantity distribution x ^* by iterative calculation.

Embodiment 2 Next, a specific method for correcting the approximately calculated physical quantity distribution x ^* in the eighth step ST8 will be described as a second embodiment. FIG. 7 is a flowchart showing the procedure of correction by iterative calculation of the approximate physical quantity distribution x ^* . In the figure, ST
9 is the ninth step of calculating the error d _b from the approximate physical quantity distribution x ^* , the system matrix A and the constant vector b,
ST10 is a tenth step of determining the error d _x based than the error d _b calculated in step ST9 of the ninth to the error matrix equation, the error d _x obtained in ST11 in step ST10 of the first 10 Approximate physical quantity distribution x ^*
Is the eleventh step of finding the true solution x ₁ . In ST12, the true solution x ₁ obtained in the 11th step ST11 is substituted into the approximate physical quantity distribution x ^* , and the ninth step to the 11th step are executed again to obtain a more accurate solution x ₂ a twelfth step of obtaining the, ST13 compares the true solution x ₁ obtained as the solution x ₂ obtained in the twelfth step ST12 in the eleventh step ST11 of their relative errors If it is smaller than a predetermined set value (for example, if it is less than the set value), a series of calculation processing is terminated, and if it is larger than the set value (for example, if it is the set value or more), the solution x ₂ is the true solution x _This is the thirteenth step in which the process is returned to the twelfth step 12 after the value is assigned to ₁ .

Next, the operation will be described. First, in the ninth step ST9, the seventh step ST shown in FIG.
From the physical quantity distribution x ^* approximately obtained in 7 and the system matrix A and the constant vector b in the matrix equation A · x = b obtained in the fourth step ST4, the error d _b is calculated using the following equation. I do. d _b = A · x ^* -b

Next, in the tenth step ST10,
Using the error d _b calculated in the ninth step ST9, the error d _x is obtained from the error matrix equation shown below. A · d _x = d _b However, more errors d _x By actual calculation solves the error matrix equation A · d _x = d _b to approximately wavelet transform and inverse wavelet transform the approximate solution obtained Calculate fast. That is, this error matrix equation A · d _x = d _b is solved by the same procedure as the fifth step ST5 to the seventh step ST7 in FIG.

Next, in the eleventh step ST11,
The error d _x obtained in the tenth step ST10 is subtracted from the approximate physical quantity distribution x ^* by the following equation,
Find the true solution x ₁ . _{x 1} = x ^* -d x

Next, in a twelfth step ST12,
True solution x ₁ obtained in the 11th step ST11
Is substituted into the approximate physical quantity distribution x ^*, and the ninth step ST9 to the eleventh step ST11 are executed again,
Find a more accurate solution x ₂ .

Next, in a thirteenth step ST13,
Comparing the solution x ₂ with improved accuracy than obtained by the eleventh true solution x ₁ and the 12 step ST12 of obtained in step ST11, the predetermined in which both relative error predetermined set value If it is less than this, this series of calculation processing is terminated. On the other hand, if the relative error between the two is greater than or equal to the set value, the solution x ₂ obtained in the 12th step ST12 is substituted for the true solution x _1, and the processing in the 12th step ST12 is repeated. As a set value used for testing the relative error between the two, 0.01% to 0.0001% is a realistic value. If 0.01% is selected as the set value, the solution The accuracy is a little rough, but the solution can be found at high speed, and when 0.0001% is selected,
Although it takes some time to obtain the solution, it is possible to obtain an accurate solution with high accuracy.

Here, FIG. 8 is an explanatory diagram showing the result of the above-mentioned iterative calculation. Comparing this FIG. 8 with FIG. 5, which shows the result before correction by the iterative calculation, it can be seen that the accuracy of the solution is greatly improved. It can also be understood from the comparison with FIG. 6 that a solution with substantially the same accuracy as that obtained by the conventional method that requires a time-consuming but accurate solution is obtained.

Here, in the above embodiment, the system matrix having 32 boundary elements, that is, 32 rows and 32 columns has been described, but in an actual estimation problem, the number of elements of the matrix is further increased, and the estimation time is increased. Increase significantly. For example,
Taking the case of 1024 elements as an example, the time required to calculate each element of the system matrix A is 34 seconds in total, and the time required to solve the matrix equation A · x = b by the Gaussian elimination method is 976 seconds. On the other hand, in the physical quantity distribution estimation method according to the first and second embodiments of the present invention, the matrix equation A.
We were able to solve x = b. The accuracy of the solution is 0.01
The convergence was judged as%. Also, the computer has a DEC A
lpha was used, and the calculation code was FORTRAN language. As a result, it was found that the physical quantity distribution can be estimated about 40 times faster than in the conventional case.

In each of the above embodiments, the governing equation of the target system is the Laplace equation relating to the potential distribution, but the present invention is not limited to this, and any applicable target of boundary element analysis can be treated similarly. For example, electromagnetic field distribution, temperature distribution,
According to the physical quantity distribution estimation method of the present invention, various physical quantity distributions such as strain distribution, stress distribution, displacement distribution, and flow velocity distribution can be estimated more than one digit faster than the conventional case. As a result, it is possible to perform dozens of calculations in the time when only one analysis can be performed in the past. For example, the parameter that represents the shape of the system is sequentially changed to generate the physical quantity distribution desired by the system designer. It becomes possible to build a shape system.

Further, the process of correcting the approximate physical quantity distribution x ^* shown in the flowchart of FIG. 7 by iterative calculation is merely the simplest example, and various methods that improve this method may be adopted. Is possible, and the same effects as those of the above-described embodiment are achieved.

[0046]

As described above, according to the invention described in claim 1, the matrix equation A · x =
Approximately solve A _t · x _t = b _t transformed from b, inversely transform the obtained approximate solution x _t by wavelet to obtain an approximate physical quantity distribution x ^*, and correct it by iterative calculation. Since it is not necessary to directly solve the matrix equation related to the dense matrix, the calculation time of the above matrix equation is significantly shortened, and the physical quantity distribution can be estimated at high speed. This has the effect of facilitating the optimum shape design of.

According to the second aspect of the present invention, since the wavelet transform is performed by using the Davidie 4-coefficient type wavelet, there is an effect that the wavelet transform can be performed by a simpler calculation.

According to the third aspect of the invention, the threshold value for approximately solving the matrix equation A _t · x _t = b _t is defined by the matrix A _t.
Of the maximum absolute value of each matrix element from 0.01% to 0.
Since the selection is made within the range of 00001%, the approximate solution x _t can be obtained at a higher speed.

According to the invention described in claim 4, since the matrix equation A _t · x _t = b _t is configured to be solved approximately by using the preconditioned bi-conjugate gradient method, calculation of the approximate solution x _t There is an effect that the speed can be improved.

According to the fifth aspect of the present invention, the boundary integral equation is converted into a discretized equation so that the number of boundary elements when discretized is a power of 2, so that the wavelet transform is performed. There is an effect that the calculation efficiency of can be improved.

According to the invention of claim 6, the error d _b =
The error d _x is obtained based on the error matrix equation A · d _x = d _b by A · x ^* −b, and the true solution x ₁ obtained from x ₁ = x ^* −d _b is approximated to the physical quantity distribution x Substitute in ^* and the solution x ₂
Until the relative error between the obtained solution x ₂ and the true solution x ₁ becomes smaller than a predetermined set value, the solution x ₂ is returned to the true solution x 1.
_Since the approximate physical quantity distribution x ^* is corrected by substituting into ₁ and repeating the calculation of the solution x ₂ , there is an effect that the accuracy of the obtained solution can be improved.

According to the invention described in claim 7, from the error matrix equation A · d _x = d _b , the matrix equation is wavelet-transformed to be approximately solved, and the obtained approximate solution is inversely wavelet-transformed to be approximated. Since the error d _x is obtained in accordance with the procedure of obtaining a physical quantity distribution, the error can be corrected at high speed.

According to the invention described in claim 8, the true solution x ₁
And a set value for the test of the relative error between the solution x ₂ and
Since it is configured to select in the range of 0.01% to 0.0001%, a solution with a relatively rough accuracy can be obtained at high speed, or a solution with a high accuracy can be obtained with some time. This has the effect of enabling different choices.

[Brief description of the drawings]

FIG. 1 is a flowchart showing a processing procedure of a physical quantity distribution estimation method according to Embodiment 1 of the present invention.

FIG. 2 is an explanatory diagram showing an example of a target system in the above embodiment.

FIG. 3 is an explanatory diagram showing the system matrix A in the above embodiment in pseudo three-dimensional display.

FIG. 4 is an explanatory diagram showing a matrix A _t after wavelet transformation in the above-described embodiment in pseudo three-dimensional display.

FIG. 5 is an approximate physical quantity distribution x ^{* in the} above embodiment.
It is explanatory drawing which shows the result of having calculated | required.

6 is an explanatory diagram showing a result calculated by a conventional method for comparison with FIG.

FIG. 7 is a flowchart showing a procedure of iterative calculation for correcting an approximate physical quantity distribution x ^{* according to} the second embodiment of the present invention.

FIG. 8 is an explanatory diagram showing a correction result by iterative calculation in the above embodiment.

FIG. 9 is an explanatory diagram showing a conventional boundary element analysis method.

[Explanation of symbols]

ST1 first step, ST2 second step, S
T3 Third step, ST4 Fourth step, ST
5 Fifth Step, ST6 Sixth Step, ST7
7th step, ST8 8th step, ST9
Ninth Step, ST10 Tenth Step, ST1
1 11th step, ST12 12th step,
ST13 Thirteenth step.

Claims (8)

[Claims] 1. A first method for giving a governing equation of a target system.
And a second step of converting the governing equation into a boundary integral equation, a third step of converting the boundary integral equation into a discretized equation, and a system in which the discretized equation depends on the target system. Matrix A, an unknown vector x representing an unknown physical quantity distribution,
And a fourth step of transforming into a matrix equation A · x = b by a constant vector b, and wavelet transforming both sides of the matrix equation A · x = b to obtain a new matrix equation A _t · x _t = b _t Fifth step of obtaining and replacing the matrix elements of the matrix A _t in the matrix equation A _t · x _t = b _t whose absolute value is equal to or less than a predetermined threshold value with zero, and the matrix equation A _t · x _t = B _t is approximately solved to obtain an approximate solution x _t, and the seventh step is an inverse wavelet transform of the approximate solution x _t to obtain an approximate physical quantity distribution x ^*. Eighth step of correcting the approximately calculated physical quantity distribution x ^* by iterative calculation. 2. The physical quantity distribution estimating method according to claim 1, wherein the wavelet transform in the fifth step uses a Davidie four-coefficient wavelet. 3. The predetermined threshold value in the sixth step,
Physical quantity distribution estimating method according to claim 1, wherein the chosen 0.01% to 0.00001 of the maximum value of the absolute values of the matrix elements of the matrix A _t. 4. The matrix equation A _t in the sixth step.
The physical quantity distribution estimation method according to claim 1, wherein a pre-processed bi-conjugate gradient method is used as a method for approximately solving x _t = b _t . 5. The physical quantity distribution estimating method according to claim 1, wherein the number of boundary elements when discretized is selected to be a power of 2 in the third step of converting the boundary integral equation into a discretized equation. . 6. The eighth step of correcting the approximately calculated physical quantity distribution x ^* by iterative calculation includes an error d _b = A · from the approximated physical quantity distribution x ^* , the system matrix A and the constant vector b. a ninth step of calculating x ^* -b, based on the error matrix equation a · d _x = d _b, a tenth step of obtaining an error d _x from the error d _b calculated, the resulting error by subtracting the d _x from the approximate physical quantity distribution x ^*, by substituting the eleventh step of obtaining a true solution x _1, a true solution x ₁ obtained in the approximate physical quantity distribution x ^* perform the ninth step to eleventh step of again, ends a twelfth step of obtaining the solution x _2, the true solution x ₁ relative error in the solution x ₂ is calculated smaller than the predetermined set value and, processing the solution x ₂ is substituted into true solution x ₁ is greater than the set value 13 physical quantity distribution estimating method according to claim 1, comprising the steps of returning to the 12th step. 7. A tenth step of obtaining an error d _x from an error matrix equation A · d _x = d _b is a process of obtaining the error d _x in the same procedure as the fifth to seventh steps. The method for estimating physical quantity distribution according to claim 6, wherein 8. The physical quantity distribution estimating method according to claim 6, wherein 0.01% to 0.0001% is selected as the predetermined set value in the thirteenth step.