JPH07191965A - Method and device for designing and estimating the objective system - Google Patents

Abstract

PURPOSE:To shorten the time required to obtain a parameter value giving the required distribution by calculating the function characteristic distribution of an object including the fluctuation parameter by the specified method and solving the equation giving the equivalence to the required distribution. CONSTITUTION:The fluctuation parameters r1-rn (n>=1) such as material values describing the coordinate value describing the form and the nature of material are inputted to a calculation means (code arithmetic type SOLVER) of the boundary element method or the limited element method. The arithmetic operation is performed on code by the machine language such as mathematica and the calculation value yk=fk (r1-rn) of the function characteristic distribution (field) is outputted. The k represents the number of the space coordinate point to calculate the function characteristic distribution and k>=1. With an unknown factor, the approximate formula by the series development formula is obtained and the variable is represented by the approximate formula. The calculated function characteristic distribution and the required function characteristic distribution are requalled to prepare the equation including the variables. By solving the equation, the parameter realizing the required function characteristic distribution is controlled or decided.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention controls or determines parameters defining an object such as shape and material physical property values in order to obtain a desired distribution of the functional characteristic distribution of the object such as magnetic flux distribution and temperature distribution. The target system is controlled or determined, and the internal state of the object such as physical properties of the material is estimated based on the measured values obtained for the object.
The present invention relates to an estimating method and an apparatus thereof.

[0002]

2. Description of the Related Art FIG.
9 is a flowchart showing a method for obtaining a desired uniform magnetic flux distribution by optimizing the shape of the electromagnet shown in P.393 of Volume 9, No. 9 (1989). Further, FIG. 28 shows a two-dimensional analysis model of a target electromagnet.

In FIG. 28, reference numeral 90 is a coil portion through which an electric current is applied to generate a magnetic field in the gap 93 of the magnetic body 91, and 92 is a pole of an electromagnet. In this case, the shape of the pole 92 determines the magnetic field distribution in the air gap 93.
Therefore, y of the evaluation area 94 surrounded by the dotted line in the void 93 is y.
Consider finding the shape of the pole 92 that makes the magnetic flux density in the direction as uniform as possible. It is assumed that this electromagnet is placed in the air region.

Next, the operation will be described. In this case, the parameter defining the object is the shape of pole 92,
The desired functional characteristic distribution is a uniform magnetic field distribution. Further, in the above literature, the boundary element method is used as an analysis method. The boundary element method is a method of analyzing a characteristic of a complicated analysis target by element division on the boundary using a boundary integral equation.

First, an initial state is set (step ST
51). The setting of the initial state is the setting of the coordinates of each node and the setting of the material physical property value in each boundary element when the analysis target is divided into a finite number of boundary elements. In this case, the initial shape of the pole 92 is flat, as shown in FIG.

Next, a finite number of evaluation points i in the evaluation area 94
A target value at (i = 1, ..., M), in this case, a desired uniform magnetic field value is set (step ST52). Next, the y component of the magnetic field at the M evaluation points is calculated by the boundary element method (step ST53). Furthermore, the sum of squares of the error between the calculated value and the target value at M evaluation points is calculated, and the error of the sum of squares is evaluated (step ST
54).

If the error is less than or equal to the predetermined value, the processing is terminated,
Otherwise, the evaluation function described in the above document is calculated (step ST56). Further, the correction amount of the shape of the pole 92 is determined using the calculation result of the evaluation function (step ST55). Then, the boundary element analysis is performed again on the corrected shape to calculate the y component of the magnetic field at the evaluation point (step ST53).

As described above, steps ST53 to ST5
By performing the processing of 5 several times, that is, by repeating the boundary element analysis, the shape of the pole 92 converges to a shape adapted to the desired uniform magnetic field distribution. FIG. 29
Shows the process of convergence of the shape of the pole 92. In the figure, (a) to (f) show the shapes of the poles 92 corresponding to the respective number of iterations of the boundary element analysis. That is, it is shown that the relative error of the generated magnetic field with respect to the target magnetic field decreases as the number of iterations increases.

[0009] FIG. 30 shows the Journal of the Institute of Electrical and Electronics Engineers, Medical Engineering, Vol. 32, No. 3 (IEEE Transaction on Biome).
FIG. 4 is a cross-sectional view showing a chest cross-section model of a human body for obtaining the distribution of conductivity inside the human body described in dical Engineering, vol. BME-32, No. 3, 1985). The method of obtaining the conductivity distribution inside the human body is an example of a method of estimating the internal state of the object from the measured value of the object. In this case, the conductivity distribution is the internal state of the object.

The chest of the human body includes the lungs (dotted portions in FIG. 30), and the electrical conductivity is greatly different spatially. Therefore, as shown in FIG. 30, when the current is applied from the outside, the electric potential distribution on the surface of the human body is measured at a number of points to estimate the conductivity distribution inside the human body. This is also called the impedance CT problem.

When processing, the inside of the human body is divided into triangular elements as shown in FIG. 30, and the conductivity of each element is σ.
_i (i = 1, ..., M). The processing procedure is similar to the procedure shown in the flowchart of FIG. 27, and first, the initial state is set. That is, the coordinates of each of the nodes 101 to 120 are set, and the node into which the current is injected is determined. For example, current is injected between the nodes 101 and 111. Also, the initial value of the conductivity σ _i of each element is set.

Next, the potential on the surface of the human body, that is, the potentials at the nodes 102 to 110 and 111 to 120 are calculated by the finite element method. The finite element method is an approximate solution method in which the inside of an analysis target having a complicated shape or physical property value distribution is divided into a finite number of elements, and the physical state assigned to each element is calculated according to the principle of minimum energy. As a result of the analysis by the finite element method, in this case, each node potential is calculated.

Therefore, the error between each calculated nodal potential and each actually measured nodal potential is evaluated. If the error exceeds a predetermined value, the conductivity σ _i (i = 1, ..., M) of each element is corrected using the evaluation result, and the corrected conductivity σ _i of each element is used. , Re-run the finite element method analysis. Further, the above processing is repeated until the error between each calculated node potential and the measured node potential becomes a predetermined value or less, for example, the sum of squares of the error becomes a predetermined value or less.

FIG. 31 shows how the average of the estimation errors, that is, the average of the errors between the calculated node potential and the measured node potential, decreases as the number of iterations of the finite element method analysis increases. It is a thing.

[0015]

Since the conventional method for controlling or determining / estimating the target system, and the apparatus therefor are configured as described above, the initial state (initial shape, initial physical property value distribution, etc.) is the final state. If there is a large deviation from the state (final shape, true physical property distribution, etc.), the number of iterations of the boundary element method or finite element method increases, and the time required to execute the boundary element method or finite element method once Since it is quite long, there is a problem that the time required to obtain the final state increases.

Further, in the error evaluation, when there are many local minimum values, there is a problem that the processing may be aborted before reaching the state that gives the true minimum error. For example, as shown in FIG. 32, when the parameter value at the point C is adopted as the initial value of the parameter defining the object, the true value that minimizes the error can be reached, but the initial value is the point A or B. When the parameter value at the point is adopted, the incorrect parameter is determined as the parameter in the final state.

The present invention has been made in order to solve the above problems, and it takes time to obtain a parameter for realizing a desired functional characteristic distribution, or to obtain an internal state of an object with a minimum error. While shortening the time of
(EN) A method for controlling or determining / estimating a target system that can prevent an incorrect parameter from being an optimum parameter or an incorrect internal state as an estimated value of a true internal state, and an apparatus therefor. And

[0018]

According to a first aspect of the present invention, there is provided a method for optimally designing a target system, wherein at least one of the following steps, (1) each parameter defining the target system is a variable. (2) Calculate the functional characteristic distribution of the target system as a function of those variables, (3) Equivalently calculate the calculated functional characteristic distribution and the desired functional characteristic distribution, and obtain an equation including those variables. A method comprising: (4) solving the equation to control or determining a parameter that realizes a desired functional characteristic distribution, in the step (2) using a series expansion equation using unknown coefficients. An approximate expression is obtained, and the variable is expressed by this approximate expression.

A method for optimally designing a target system according to a second aspect of the present invention includes the following steps, (1) at least one of the parameters defining the target system is used as a variable (2) Calculate the functional characteristic distribution of the target system as a function of those variables, (3)
A residual equation including these variables is created by the sum of squares of residuals between the calculated functional characteristic distribution and the desired functional characteristic distribution. (4) The residual equation is minimized to obtain the desired functional characteristic distribution. A method comprising controlling or deciding a parameter to be realized, wherein in step (2), the variable is expressed by this approximation formula by obtaining an approximation formula by a series expansion formula using unknown coefficients. It is characterized by that.

The method of estimating a target system according to the third aspect of the present invention comprises the following steps, (1) using the internal state of the target system as a variable, and (2) determining a physical quantity distribution that can be measured for the target system. Calculate as a function of those variables, (3) Create an equation including those variables with the calculated physical quantity distribution and the actually measured physical quantity distribution as equal values, (4) Solve the equation and target system Estimating the internal state of
In the step (2), the variable is expressed by this approximation formula by obtaining an approximation formula by a series expansion formula using unknown coefficients.

The method of estimating the target system according to the invention of claim 4 is as follows: (1) The variable of the internal state of the target system; (2) The physical quantity distribution measurable for the target system. (3) A residual equation including these variables is created by the sum of squares of residuals between the calculated physical quantity distribution and the actually measured physical quantity distribution, which is calculated as a function of those variables. (4) A method provided with minimizing the residual equation to estimate the internal state of the object, wherein in step (2), an approximate expression by a series expansion equation is obtained using unknown coefficients, The feature is that the variables are expressed by this approximate expression.

The method according to the invention of claim 5 is claim 1
In the method of the present invention described in any one of (1) to (4), the variable is used from a low-order term to a high-order term until the number of unknown coefficients agrees with the number of simultaneous equations for obtaining the unknown coefficient. Is.

According to a sixth aspect of the present invention, in the method according to the first to fifth aspects of the present invention, a polynomial is obtained by performing series expansion on a portion of an irrational expression relating to variables of a matrix equation.

The method according to the invention of claim 7 is characterized in that, in the method of the inventions of claims 1 to 6, both sides are multiplied by the least common multiple of the denominator function of each element of the matrix equation to convert into a polynomial expression. It is a thing.

The method according to the invention of claim 8 is characterized in that in the method of the invention of claims 1 to 7, Taylor expansion or McLaughlin expansion is used as the series expansion formula.

The method according to the inventions of claims 9, 10 and 11 is the method of the inventions of claims 1 to 8 wherein the finite element method, the boundary element method and the moment method are respectively used to derive the equations describing the system. I am using.

According to a twelfth aspect of the present invention, there is provided an apparatus for optimally designing a target system, wherein: (1) at least one of the parameters defining the target system is used as a variable, and the function of the target system is obtained. Means for calculating the characteristic distribution as a function of those variables, (2) means for equalizing the calculated functional characteristic distribution and the desired functional characteristic distribution, and creating an equation including those variables, (3)
An apparatus including means for controlling or determining parameters for realizing the desired functional characteristic distribution by solving the equation, wherein the means (2) obtains an approximate expression by a series expansion equation using unknown coefficients. Then, the variable is expressed by this approximate expression.

According to a thirteenth aspect of the present invention, there is provided an apparatus for optimally designing a target system, wherein: (1) at least one of the parameters defining the target system is used as a variable, and the function of the target system is obtained. Means for calculating the characteristic distribution as a function of those variables, (2) means for creating a residual equation including those variables by the sum of squares of residuals between the calculated functional characteristic distribution and the desired functional characteristic distribution, (3) An apparatus including means for controlling or determining a parameter that realizes a desired functional characteristic distribution by minimizing the residual equation, wherein the means (2) is a series expansion using unknown coefficients. It is characterized in that an approximate expression is obtained by an equation and the variable is expressed by this approximate expression.

An apparatus for estimating a target system according to a fourteenth aspect of the present invention comprises the following means, (1) using an internal state of the target system as a variable, and measuring a physical quantity distribution that can be measured for the target system as a function of those variables. (2) means for equalizing the calculated physical quantity distribution and the actually measured physical quantity distribution to create an equation including those variables, (3) solving the equation and the internal state of the object The apparatus of (2) is characterized in that the means of (2) obtains an approximate expression by a series expansion equation using unknown coefficients, and expresses the variable by this approximate expression. It is what

An apparatus for estimating a target system according to a fifteenth aspect of the present invention comprises the following means: (1) Using an internal state of the target system as a variable, a physical quantity distribution measurable for the target system as a function of those variables. And (2) means for creating a residual equation including those variables by the sum of squares of residuals of the calculated physical quantity distribution and the actually measured physical quantity distribution, (3) the residual equation Is a device for estimating the internal state of an object by minimizing the above, wherein the means (2) obtains an approximate expression by a series expansion equation using unknown coefficients and sets the variable to this approximate expression. It is characterized by being expressed in.

[0031]

In the step of calculating the function of each variable in the inventions according to claims 1 to 11, the boundary element method or the finite element method is executed by executing the boundary element method or the finite element method while leaving the parameters defining the target system as unknown variables. The functional characteristic distribution (defined by at least one point in space) or the physical quantity distribution is output as a function of the unknown variable. Further, since the functional characteristic distribution of the target system or the physical quantity distribution of the target system is expressed by an approximate expression using a series expansion expression, the calculation speed can be increased.

In the step of estimating the internal state of the target system by minimizing the residual equation in the invention of claim 2 or claim 4 or determining the parameter that realizes the desired functional characteristic distribution, By handling the error function containing, it is possible to determine the true minimum that exists among multiple error minima.

Further, the means for calculating the function of each variable in the twelfth to fifteenth aspects of the present invention executes the boundary element method or the finite element method while leaving the parameters defining the object as unknown variables. The functional characteristic distribution or physical quantity distribution of is output as a function of unknown variables. Also,
Since the functional characteristic distribution of the target system or the physical quantity distribution of the target system is represented by an approximate expression using a series expansion expression, the calculation speed can be increased.

Further, the means for minimizing the residual equation in the invention according to claim 13 or claim 15 handles an error function including an unknown variable, so that a true error exists in a plurality of error minimum values. It is possible to determine the minimum value.

[0035]

Embodiments of the present invention will be described below. First, the basic concept of each embodiment will be described. FIG. 1 shows the basic concept of the present invention. The boundary element method or the finite element method calculation means (SOLVER)
When one or more variable parameters r _1, ..., R _n (n ≧ 1) are input, the calculated value y _k = f of the functional characteristic distribution (field)
_{It is shown that k} (r ₁ , ..., R _n ) (k is the number of spatial coordinate points for calculating the functional characteristic distribution, k ≧ 1) is output.

In the past, SOLVER was a numerical operation type, but here it is a symbolic operation type. In addition, as a computer language that can perform symbolic operations, mathematica (mathemati
ca) etc. are known.

It should be noted that possible variable parameters include, for example, coordinate values that define the shape and material physical property values that specify the properties of the object. Also, as an example of the functional characteristic distribution as the calculation output, for electromagnetic equipment, the electric field distribution,
There are magnetic field distribution, electromagnetic field distribution, potential distribution and natural frequency. Mechanical structures are considered to be stress distribution, strain energy, displacement distribution, and natural frequency. Also, for heat utilization equipment, temperature distribution and heat flux distribution may be considered. For fluid application equipment, flow velocity distribution,
Streamline distribution and pressure distribution are possible.

As shown in FIG. 1, the calculated value y _k is a function of the variable parameter r _k . Conventionally, SOLVE
The input of R was a specific numerical value, and the calculated value was also a specific numerical value. Then, error evaluation is performed on the calculated value, and the value corrected according to the evaluation result is input again to SOLVER.

However, in this case, the calculated value y _k of the functional characteristic distribution including the unknown function is obtained, and the calculated value y _k is used to determine the parameter value for optimizing the shape or physical property value of the object. can do. In other words, it is possible to perform the optimized design of the shape and the physical property value of the target object without the need to repeatedly execute the boundary element method and the finite element method.

Therefore, when the computer is used, the calculation time is shortened as compared with the conventional method. Further, according to the conventional method, there is a drawback that the precision of calculation is deteriorated due to a digit cancellation and a rounding error that occur during calculation. In particular, when the handled numerical value ranges from a very small value to a very large value, a very small value may become 0.
However, according to the present invention, there is no possibility that the symbol will be treated in the calculation in the middle, and the accuracy is determined only by the numerical calculation performed last.

The method and apparatus according to the present invention will be described below in more detail with reference to FIGS. 2 to 5 and 6 to 9. As described above, that is, in steps ST1 and ST2 of FIG. 2, the variable parameter r is calculated by the means for calculating the function indicated by reference numeral 201 in FIG.
₁ , ..., R _n are selected, and the calculated value y _k of the functional distribution is obtained by the boundary element method or the finite element method.

Then, the designer who carries out the optimization design of the object has a desired distribution y _1a , ..., Y _na (n ≧ n points) at n points.
Give 1). If n variable parameters have been selected, in step ST2 of FIG.
The simultaneous equations represented by the following equation (1) are obtained by the means for creating the equations with.

[0043]

[Equation 1]

This simultaneous equation is composed of n equations, and since there are n unknown variables, it can be solved. That is, in step ST4 of FIG.
If the equation (1) is solved by a means for solving the equation given by, the optimum values of the variable parameters r ₁ , ..., R _n can be obtained.

The method described above is a method of determining the value of the variable parameter by equating the calculated value y _k with each value indicating the desired functional characteristic distribution. As shown in FIG. 3, the method of least squares is used. Accordingly, each value that minimizes the sum of squares of the residuals between the calculated value y _k and the desired functional characteristic distribution may be determined as the value of the unknown variable.

That is, in steps ST11 and ST12 of FIG. 3, the variable parameters r ₁ , ..., R are calculated by the means for calculating the function denoted by reference numeral 211 in FIG.
_{After n} is selected and the calculated value y _k is obtained, in step ST13 of FIG. 3, the means for creating the residual equation denoted by reference numeral 212 in FIG. Establish an equation showing the sum of squares.

[0047]

[Equation 2]

Then, in step ST14 of FIG. 3, unknown variables r ₁ , ..., R that minimize ε in the equation (2) are obtained by means of minimizing the residual equation denoted by reference numeral 213 in FIG. _Find the value of _n . The following two methods are conceivable as methods for minimizing ε. One method is an optimization method such as the known steepest descent method. Other methods, unknown variables r ₁ to epsilon, ..., and 0 each expression obtained by partially differentiating the respective r _n, is a way to solve it. That is,
This is a method of solving the simultaneous equations shown by the following equation (3).

[0049]

[Equation 3]

This simultaneous equation also has unknown variables r ₁ , ..., R
_{Since n} is n and consists of n expressions, it can be solved. Such a method produces unknown variables r ₁ , ..., R
_This is possible because _n remains in equation (2) in the form of a variable. In the conventional case, the sum of squares is obtained as a numerical value, a predetermined error evaluation is performed on the numerical value, and the evaluation result is reflected in the next execution of the boundary element method or the finite element method.

Further, as described above, according to the conventional method, when there are many local minimum values, there is a possibility that the processing will be terminated before the state that gives the true minimum error is reached. . However, according to the method using the equation (3),
Since we can find all maxima and minima,
From them, the true minimum can be obtained.

For example, after _obtaining an approximate solution (r ₁₁ , r ₁₂ , ... R _1n ) using a certain initial value, each equation in the equation (3) is multiplied by a product (r _1- r ₁₁ ) ・
Divided by _{(r 2 -r 12) · (} r n -r 1n), it makes each expression was lowered the order. Then, a second approximate solution is obtained by using new initial values for these expressions. By repeating this process successively, all solutions (maximum value and minimum value) can be obtained.

This processing is a known processing as described in, for example, p.376 of Information Processing Handbook (Ohmsha, 1980), and is called a deflation processing. It is also a feature of the present invention that such reduction processing is possible. Further, it is easy to find the true minimum value from all the obtained maximum values and minimum values.
For example, an area or point of r _i that satisfies the following expression (4) is searched using random numbers. In addition, α in the equation (4)
Is a small number, for example, about 0.001.

[0054]

[Equation 4]

The minimum value is searched for by the steepest descent method using these r _i as an initial value group. Then, all the local minimum values are obtained, and the true minimum value can be found from them.

The methods and apparatuses described above are used for designing equipment and the like, and a method for controlling or determining a target system for determining parameters suitable for desired functional characteristics of the apparatus, that is, an object. , And its apparatus, the method and apparatus for estimating the target system for estimating the internal state of an object can be constructed in the same manner. 4 and 5 are flowcharts showing the method, respectively, and FIGS. 8 and 9 are block diagrams showing the configuration of the apparatus.

According to the method shown in FIG. 4, the variable parameters corresponding to the internal state of the object are first selected by the means for calculating the function denoted by reference numeral 221 in FIG. r ₁ , ..., R _n (step ST21). Next, the measurable physical quantity distribution of the object is calculated by the finite element method as a function of the variable parameters r ₁ , ..., R _n (step ST22). That is, the calculated value y _k = f _k (r ₁ , ..., R _n ) is obtained. Then
By the means 222 for creating the equation of FIG.
_Equivalent to _k and the value actually measured for the object,
The simultaneous equations shown by the equation (1) are obtained (step ST2
3). When the simultaneous equations are solved by the equation solving means 223 of FIG. 8, the value of the variable parameter corresponding to the internal state is obtained (step ST24).

Further, according to the method shown in FIG.
First, variable parameters corresponding to the internal state of the object are selected by the means for calculating the function _denoted by reference numeral 231 and are set as unknown variables r ₁ , ..., R _n (step ST31). Next, the measurable physical quantity distribution of the object is calculated by the finite element method as a function of the variable parameters r ₁ , ..., R _n (step ST32). Then, the means 232 for creating the residual equation of FIG. 9 establishes the equation (2) indicating the sum of squares of the residuals (step ST33). Then, the means 233 for minimizing the residual equation in FIG. 9 obtains the value of the variable parameter that minimizes ε (step ST34).

FIG. 10 shows an apparatus configuration for specifically realizing the apparatus for controlling or determining the target system shown in FIGS. 6 and 7 and the apparatus for estimating the target system shown in FIGS. 8 and 9. Show. In the figure, reference numeral 300 is a central processing unit (hereinafter referred to as CPU) that executes a predetermined calculation control process. Reference numerals 301a and 301b are input devices such as a keyboard and a mouse for inputting computer programs and data used by the CPU 300, and 302 is the CPU 300.
It is a display device such as a CRT display for displaying the process of the process by C. Further, 303 is a hard disk device in which a computer language for symbol calculation, numerical calculation, etc. is stored, and 304 to 306 are CPUs.
Is a main memory used for storing data in the process of various processes.

The CPU 300 operates in accordance with the computer program input using the input device 301, and in the case of realizing the device for controlling or determining the target system shown in FIG. Thirty
4 to execute the equation 202 in the main memory 305, and further to solve the equation 203
In the main memory 306. The hard disk device 30 is used as necessary to execute each of these means.
The computer language stored in 3 is used. Also, C
The PU 300 sends the operation data in the middle of calculation or the result data after the end of the calculation to the display device 302 for display as needed.

The apparatus for controlling or determining the target system shown in FIG. 7 and the apparatus for estimating the target system shown in FIGS. 8 and 9 are the same as those in the above-mentioned case. ~ 213, 221-223
231-233 are executed by the main memories 304-306, respectively.

The value thus obtained is an estimated value of the physical quantity in each element, and the internal state of the object can be estimated based on these values. Hereinafter, each embodiment of the present invention will be described with reference to the drawings.

Example 1. Fig. 10 shows Finite Elements for Electrical Engineers (Finite Element) published by Cambridge University Press.
s for Electrical Engineers, 2nd Edition, Cambridga
1 described in P.1 to P.19 of University Press, 1990)
It shows a dimensional transmission line problem. Hereinafter, this will be described as an example.

FIG. 11A shows a transmission line 1 having a length of 2 m.
A medium in which (for example, two bare copper wires arranged in parallel) has an electrical loss (for example, in the ground, the loss is 0.5 s / m)
It is shown in the state of being placed in. A voltage of 1V is applied to one end of the transmission line 1 and the other end is open. Then, it is considered to obtain material characteristics of the transmission line 1 so that the node voltage on the transmission line 1 has a desired value. In this case, as shown in FIG. 11B, the transmission line 1 is divided into three parts in the length direction, and a finite element analysis with three elements is performed.

In the above literature, parameters that define the transmission line such as the size and resistivity of the transmission line 1 and the conductivity of the medium are given as numerical values, and the node voltage V ₁ across the three elements is calculated by the finite element method. , V ₂ and V ₃ are shown, this embodiment considers obtaining material characteristics of the transmission line 1 such that the node voltages V ₁ , V ₂ and V ₃ have desired values. .

First, the resistivity of each element is set to r ₁ , r ₂ , r ₃
And symbol. Other parameters, such as the length of each element and the conductivity of the medium, are expressed numerically. Then, the node voltages V ₁ , V ₂ , and V ₃ are calculated by the finite element method. Here, the calculation result is expressed as a function of the resistivities r ₁ , r ₂ and r ₃ of each element as shown in the equation (5).

V ₁ = (9r ₁ −0.333333r ₁ ² −0.333333r ₁ r ₃ +0.0123457 r ₁ ² r ₃ ) · (3r ₃ −0.111111r ₂ r ₃ ) / (− 27r ₁ r ₃ −2r ₁ ² r ₃ -14 r ₁ r ₂ r ₃ −0.703704r ₁ ² r ₂ r ₃ −8 r ₁ r ₃ ² −0.259259r ₁ ² r ₃ ² −1.14815 r ₁ r ₂ r ₃ ² −0.0356653 r ₁ ² r ₂ r ₃ ² ) V ₂ = (9r ₁ + 0.666667r ₁ ² −0.333333r ₁ r ₃ −0.0246914 r ₁ ² r ₃ ) · (3r ₃ −0.111111r ₂ r ₃ ) / (− 27r ₁ r ₃ −2r ₁ ² _{r 3 -14r 1 r 2 r 3} -0.703704r 1 2 r 2 r 3 - 8r 1 r 3 2 -0.259259r 1 2 r 3 2 -1.14815 r 1 r 2 r 3 2 -0.0356653 r 1 2 r 2 r 3 ^{_{2) V 3 = (9r 1}} + 0.666667r 1 2 -2.66667 r 1 r 3 +0.0864198 r 1 2 r 3) · (3r 3 -0.111111r 2 r 3) / (- 27r 1 r 3 - 2r 1 2 r ₃ -14 r ₁ r ₂ r ₃ -0.703704r ₁ ² r ₂ r ₃ -8 r ₁ r ₃ ² -0.259259r ₁ ² r ₃ ² -1.14815 r ₁ r ₂ r ₃ ² −0.0356653 r ₁ ² r ₂ r ₃ ² ) ··· (5)

When obtaining the equation (5), specifically, coding was performed by using the Mamatica language capable of executing symbol operation, and symbol calculation was performed by a computer. Also, because there is no symbolic operation function such as Fortran,
You can't use it.

For example, as a desired value, V ₁ = 0.2
When V, V ₂ = 0.4V and V ₃ = 0.9V are selected, the ternary simultaneous equations obtained by substituting those values into the equation (5) may be solved. Therefore, the present embodiment is an example of the process according to the flowchart of FIG. In this case, this equation is non-linear and can be solved using the well-known Newton method or the like. Solving using the Newton's method provided in the Mathematica language gives the following solution in about 2 seconds (using Apple Computer's Macintosh).

R ₁ = 6.75 Ω / m r ₂ = 0.317647 Ω / m r ₃ = 3.85714 Ω / m

In this way, the parameter that realizes the desired voltage value is easily determined. In other words, if the design target value is given, the material characteristic distribution can be obtained immediately.

Although the number of elements is set to 3 by using a one-dimensional model for ease of explanation, the number of elements can be increased to improve accuracy. Also, two-dimensional and three
The method according to the present embodiment can be applied to the dimensional model (the same applies to the following embodiments). That is, the present invention can provide an object design means for determining a complicated material characteristic distribution for realizing an arbitrary function distribution characteristic for an arbitrary object.

Example 2. In the first embodiment, the node voltage V _{1 in the} equation (5) obtained by the finite element method,
Let V ₂ and V ₃ be equivalent to the desired values, and the resistivity r ₁ ,
Although r ₂ and r ₃ are determined, in this embodiment, the values of the resistivities r ₁ , r ₂ and r ₃ that minimize the sum of squared residuals are obtained. Therefore, this embodiment is an example of the process according to the flowchart of FIG. The sum of squares of the residuals of the node voltages V ₁ , V ₂ , V ₃ and the desired value is expressed by the following equation (6).

Ε = (V ₁ −0.2) ² + (V ₂ −0.4) ² + (V ₃ −0.9) ² (6)

By substituting the node voltages V ₁ , V ₂ and V ₃ represented by the equation (5) into the equation (6), the values of the resistivities r ₁ , r ₂ and r ₃ which minimize ε are obtained. . For example, the well-known steepest descent method (to find the valley bottom of a function that has the minimum value, always descend in the direction in which the function has the steepest gradient) is the minimum value of ε. Can be determined. By the steepest descent method, the following values were obtained as the values of the resistivities r ₁ , r ₂ and r ₃ .

R ₁ = 6.78157 Ω / m r ₂ = 0.317731 Ω / m r ₃ = 3.85387 Ω / m

Further, as already described, in order to obtain the minimum value of ε, the method of partially differentiating the equation (6) by r ₁ , r ₂ and r ₃ and setting the obtained value as 0 to formulate the equation There is. In this case, the obtained formula is the following formula (7).

[0078]

[Equation 5]

That is, by substituting the node voltages V ₁ , V ₂ and V ₃ represented by the equation (5) into the equation (6), and applying the equation (7) to the resulting equation, the three-dimensional simultaneous equations are expressed. Is obtained. The equation is non-linear and can be solved by Newton's method. V ₁ = 0.255594V, V ₂ =
When the Newton method prepared in the Mashimatika language was used to solve with 0.31742V and V ₃ = 0.531V as desired values, the following solution was obtained.

R ₁ = 2.0 Ω / m r ₂ = 2.00127 Ω / m r ₃ = 1.99811 Ω / m

Example 3. FIG. 12 shows an example of a boundary value problem in a rectangular area in which a phenomenon is expressed by the Laplace equation. That is, it is considered to determine the dimension c in order to set the potential at the point P to a desired value.
The length of the other side is 1 m. As a boundary condition,
The potentials of the two opposite sides are given as u = 1 and u = 0, and the rate of change of the potential in the direction normal to the boundary line is 0 (the potential contour line is perpendicular to the boundary line) for the other two sides. . It should be noted that 11 to 14 indicate nodes, and the numerical values in parentheses near the nodes are coordinate values.

As a specific example corresponding to the problem shown in FIG. 12, there is a heat conduction problem shown in FIG. This is, for example, boundary element analysis by microcomputer (Baifukan, 1986) p. 6
3 are described. This heat conduction problem considers two parallelly arranged walls 2a and 2b having different temperatures (for example, the outer surface temperature of the left wall is 500 ° C. and the outer surface temperature of the right wall is 0 ° C.), and the gap space (region This is a problem of considering the temperature potential of (3).

As another example, as shown in FIG. 14, there is a problem of obtaining the internal potential when a voltage is applied between the parallel plate electrodes 4a and 4b. Near the centers of the parallel plate electrodes 4a and 4b, the equipotential lines are parallel to the parallel plate electrodes 4a and 4b. Therefore, if a virtual boundary 6 is provided, this problem will occur.
The model is modeled as shown in.

It is considered that the dimension c such that the potential of the point P in FIG. 12 becomes a desired value is determined by the processing according to the flowchart of FIG. In this case,
The variable parameter is the dimension c. Then, the potential at the point P is obtained using the boundary element method. By the analysis by the boundary element method, the potential of the point P is given as a function of the dimension c.

For example, when the potential at the point P is set to 0.5, the function of the dimension c and 0.5 are set equal to each other to establish an equation. Then, when the equation is solved, c = 0.99
8469 was obtained.

Further, since the potential of the point P is given as a function of the dimension c, the characteristic diagram of the potential (u (0.5,0.5)) of the point P versus the dimension c as shown in FIG. 15 can be obtained. Is also easy. In the conventional numerical calculation method,
The potential at the point P was calculated by the boundary element method after the dimension c was set to a specific numerical value. Then, the numerical value of the dimension c is changed and the potential of the point P is obtained again by the boundary element method. By repeating the processing, the characteristics shown in FIG. 15 were obtained.

In particular, when this characteristic changes in a complicated manner, it is difficult to properly select the step size of the numerical value when determining various numerical values of the dimension c. However, according to the present embodiment, the potential at the point P is represented by a function of the dimension c, so that the characteristic can be easily obtained. Also, the differential coefficient and the like can be obtained by analytically differentiating the function, and the state of the potential change can be accurately grasped.

Example 4. FIG. 16 shows a disk-shaped conductor model. This model is described in, for example, Hyundai Electric Finite Element Method (Ohm) P.5. FIG.
As shown in (a), + 100V,-
A voltage of 100V is applied. Note that FIG.
Since the one shown in (1) has symmetry, only the portion of the first quadrant can be targeted in the analysis. Therefore, as shown in FIG. 16B, the part in the first quadrant is set as an analysis target, and the part is divided into seven elements.

The coordinates of the nodes 21 to 28 are shown in FIG.
As shown in. That is, here, node 2
Let the x coordinate of 2 be the variable parameter d. And node 2
Consider determining d such that the voltage of 3 becomes a desired value by using the method according to the flowchart of FIG. 2 (that is, how much the disc shape is distorted will be determined).

By the finite element method, the voltage at the node 23 is
It is represented as a function of the variable parameter d. Also, node 2
The desired voltage of 3 is 40.767V. Then, an equation was obtained with the function and 40.767 V as equal values, and when the equation was solved, d = 4.5 cm was obtained. Also, since the voltage at the node 23 is represented as a function of the variable parameter d,
The characteristic diagram shown in FIG. 17 can be easily obtained.

Example 5. In the model shown in FIG. 16B, it is considered that the value of the variable parameter d at which the voltage at the node 23 and the voltage at the node 24 have desired values is obtained using the method according to the flowchart of FIG. According to the finite element method, the voltage V _{3 at the} node 23 and the voltage V _{4 at the} node 24 are
It is given as a function of the variable parameter d. The desired voltage at node 23 is 40.7V and the desired voltage at node 24 is 27.3V. Then, the sum of squares of the residuals is given by the following equation (8).

Ε = (V ₃ −40.7) ² + (V ₄ −27.3) ² (8)

When d that minimizes ε is obtained, d =
Obtaining 4.49345. Further, the same result was obtained by solving the equation obtained by setting 0 as the value obtained by differentiating the equation (8) by d. Note that FIG. 18 shows the characteristics of V ₃ , V ₄ and ε with respect to the variable parameter d. V ₃ ,
Since V ₄ and ε are both expressed as a function of d, each characteristic can be easily obtained.

Although each of the above embodiments relates to the electrical field, the present invention is also applicable to the field of fluid engineering. Figure 19 What's published by Marcel Decker
Every Engineer Sud Know About Finenight Element Analysis (What Every Engin
eer Should Know About Finite Element Analysis, Ma
rcel Dekker, 1988) P.192 shows an example of analysis of air flow around a vehicle.

In the figure, 31 is an automobile and 32 is the ambient air. The inside of each network in FIG. 19B is each divided element. Further, FIG. 19 (c) shows the flow of air, and FIG. 19 (d) is a velocity vector distribution diagram. According to the present invention, element division is performed as shown in FIG. 19 (b), the shape of the automobile 31 is used as a variable parameter, an equation is established using the finite element method, and the equation is analyzed. By doing so, it is possible to design the outer shape of the automobile 31 that does not most hinder the flow of air.

The present invention is also applicable in the field of mechanical engineering. Figure 20 shows The Finite Element Method published by McGraw-Hill
Element Method. 4th edition, McGraw-Hill, 1989) P.
This is an analysis example of the stress distribution inside the metal that occurs when the hollow metal shaft described in 269 is threaded.

In the figure, 41a and 41b are respectively different metals. Further, FIG. 20 (a) is a perspective view showing the hollow metal shaft, and FIG. 20 (b) shows each element set inside the hollow metal shaft. In addition,
The stress value at each node is also shown in the above literature. According to the present invention, the element is divided as shown in FIG. 20 (b), and the shape of the hollow metal shaft and the metal 41a, 4a are formed.
1b is used as a variable parameter, an equation is set up using the finite element method, and the equation is analyzed. It is possible to design.

Example 6. Next, an example of a method of estimating the internal state of the object from the measurement value obtained for the object will be described. Here, the impedance CT problem will be described using the square conductor model shown in FIG. Consider the case where the inside of the conductor is divided into eight elements as shown in FIG. In the figure, reference numerals 51 to 59 indicate nodes, and the numerical values in parentheses near the nodes are coordinate values.

In this case, a voltage is externally applied to a square conductor whose conductivity distribution is unknown, and the voltage distribution on the conductor surface is measured. Then, the conductor distribution is estimated from the measured values by using the method according to the flowchart of FIG. The voltages at the nodes 51 to 59 are V ₁ , ... V _9, and the conductivity of each element is σ ₁ , ... σ ₈ . The length of each side of the square is 2 m.

Eight nodes 51 to 54,5 on the surface of the conductor
When a voltage is applied to two points of 6 to 59, the voltages of the other 6 nodes can be measured. Then, using the finite element method, an equation expressing those six nodes as a function of the conductivity σ _{1 to} σ ₈ is obtained. That is, the conductivity σ _{1 to} σ ₈ is a variable parameter. Then, using these equations and the measured values of the voltages at the six nodes, the residual sum of squares is created, and the values of the conductivity σ _{1 to} σ ₈ are calculated so that the sum of squares is minimized. ,
Those values give an estimate of the conductivity.

For example, 1 V is applied to the node 52, and the node 58
Consider the case where is set to 0V. In order to simplify the calculation, it is assumed that σ _{4 to} σ ₈ of the conductivity σ _{1 to} σ ₈ are known values. Moreover, the measured value of voltage was created by simulation calculation. That is, σ ₁ = 0.1 s in advance
/ M and σ _{2 to} σ ₈ are set to 1.0 s / m, the respective node voltages are determined by a normal finite element method (conventional method), and the node voltages are used as voltage measurement values.

In this case, the three conductivity σ ₁
An equation showing the node voltage with σ ₃ as a variable is obtained. Then, the sum of squares of the residuals between the node voltages indicated by these equations and the above-mentioned voltage measurement values is obtained. This sum of squares is
Because it contains a σ ₁ ~σ ₃ as unknown variables, determine the value of such σ ₁ ~σ ₃ Part square sum is minimized. In that way the following values were determined:

Σ ₁ = 0.0927542 S / m σ ₂ = 1.03182 S / m σ ₃ = 0.971097 S / m

In the sixth embodiment, the method according to the flowchart of FIG. 5 is used, that is, the method of minimizing the sum of squares of the residuals is used. However, the method according to the flowchart of FIG. 4 is used. You can also However, since the measured value (actual measured value) of the object is used, the measured value may include a measurement error (noise or the like). Therefore, when the possibility that a measurement error is included cannot be ignored, it is preferable to use the method according to the flowchart of FIG. 5 rather than the method according to the flowchart of FIG. 4 in which the obtained function and the measured value are made equal. This is also understood from the fact that the method of investigating the characteristics by using the least square method is generally adopted when processing the measurement data.

The above description with reference to FIG. 21 has been made on the assumption that the internal state is estimated using the measurement data, but more generally, the internal state of a tomographic image represented by x-ray CT, electrical impedance CT, etc. It can also be used for obtaining images of the body and for structural diagnosis. It can also be used for deterioration diagnosis and non-destructive inspection of reactor cooling pipes and various structures. Further, it can be similarly used for controlling the internal state (that is, the conductivity distribution in this case) in order to bring the surface potential of FIG. 21 close to a certain control target.

In summary, the present invention relates to a method and device for controlling or determining / estimating a target system. A supplementary explanation of the control of the target system is, for example, control of a plant using a chemical reaction or the like. For example, in this case, the temperature and internal pressure of the reactor wall may be measured to estimate and control the internal temperature distribution. FIG. 22 is a conceptual diagram showing such an application example.
In the figure, 61 is a reaction furnace, 61 is its inner wall, and 6
3 is a temperature sensor arranged on the inner wall 62, and 64 is a pressure sensor arranged in the reaction furnace 61. In particular,
The reactor system may be calculated using the finite element method, the boundary element method, etc., which have already been described.

Next, a supplementary explanation will be given regarding the determination of the target system. For example, the optimum design of various electromagnetic devices such as an electromagnet system can be considered. In this case, the magnetic field distribution generated by the electromagnet is given to optimally design the shape of the magnet.
Further, it can be applied to a method of shielding electromagnetic noise and a design of a shield, which have recently become a problem. FIG. 23 is a conceptual diagram showing such an application example, in which 65 is an electromagnetic noise source, 66 is a shield arranged so as to surround the electromagnetic noise source 65, and 67. Is a cooling hole formed in the shield 66. In this case, for example, the intensity distribution of the generated noise is measured, the current distribution of the electromagnetic noise generation source 65 is estimated, and the structure of the shield is determined based on this. For example, as shown in FIG. 23, where should the hole 67 for cooling the inside of the shield 66 be located?
The structure of the shield 66 can be determined.

The application examples shown in FIGS. 22 and 23 can be generally regarded as control or determination of the process device shown in FIG. In the figure, 71 is the process device, 72 is a sensor arranged at a predetermined point of the process device 71, and 73 is the sensor 72.
Is a data processing unit that processes the output data of Reference numeral 74 is an internal state quantity estimation unit that estimates the internal state quantity of the process device 71 based on the processing result of the data processing unit 73, and 75 is a process quantity that optimizes the process quantity based on the estimated internal state quantity. The optimization unit 76 is a process amount control unit that controls the process amount of the process device 71 according to the output of the process amount optimization unit 75.

Further, as an apparatus for implementing the method of the present invention, for example, a plurality of engineering workstations (EWSs) are connected to a computer network, and a program stored in the memory is used to store the programs shown in FIG.
It can be obtained by implementing the respective means shown in FIG.
Further, as shown in FIG. 26, by using a plurality of EWS 81 and a large-scale computer in a hierarchical structure to form a network, it is possible to efficiently use the computer.

The superiority of each of the above embodiments can be explained as follows. For example, the overall size of the target is represented by a variable x, and the size of each element is also represented by x. FIG. 26 (a)
Shows an example including three dimension elements each of which has a dimension of x / 3. As in the conventional case, a method for obtaining the overall dimension y of the target including the dimensionally fixed portion (see FIG. 2).
6 (b)), y> 1 must be satisfied, and
As y approaches 1, collapse occurs in the shaded elements. However, according to the method described with reference to FIG. 26A, x> 0 and the size can be changed more greatly. That is, each element can be expanded and contracted at the same rate. That is, when the present invention is applied to the design, the dimensional range that can be designed can be expanded.

Next, speeding up by approximation calculation when solving a matrix equation will be described. The embodiment described above has already been described in the application of Japanese Patent Application No. 3-329699, and the matrix equation including the symbol elements can be solved by using the symbol operation, but when the size of the matrix becomes large, The calculation time will increase significantly.

The characteristic feature of the present invention that the calculation time can be greatly reduced by finding an approximate solution using series expansion when solving a matrix equation will be described in detail below. For example, describe the target system 2
Consider the row-by-two matrix equation (9).

[0113]

[Equation 6]

In this matrix equation, x and y are variables giving the functional characteristic distribution, and a is a parameter variable that defines the target system such as the size of the target object. At this time, first, the case of solving this matrix equation without using series expansion will be described. Using the Mathmatica (Mathematica) language, which is capable of symbolic operations, this equation can be solved by directly inputting the command, and the calculation time and the solution to solve are given as follows.

[0115]

[Equation 7]

This expression (10) displays the output of the execution result in the Masmatica language, where "->" means the equal sign "=" in mathematics, and the solutions x and y are expressed by a function of a. ing. As shown in this equation (10), the calculation time of the matrix equation was about 1.47 seconds. The computer used was an Apple Macintosh IIci model with an acceleration board added.

Next, a case will be described in which the approximate solution method by series expansion, which is a feature of the present invention, is applied to the equation (9). As can be seen from the equation (10), the solutions x and y are functional expressions of the unknown parameter a. So the solution x, y
Is approximated by a series expansion of the symbol parameter a, and this expansion equation is substituted into the equation (9). That is, an approximate solution of Expression (9) is obtained when a is near the constant a ₀ . In the following description, the case of a ₀ = 0 will be considered. When a ₀ = 0 is not satisfied, similar processing can be performed by obtaining an approximate solution using aa ₀ as a new variable. The order of the approximation formula is set according to the required accuracy. In the following description, for example, a case where x and y are approximated by a first-order polynomial of a will be described. Further, in order for this approximation to hold, it is necessary that a be a value close to zero. f ₁₁ , f
_{With 12} , f ₂₁ and f ₂₂ as constant coefficients, x and y are as follows.
Express x = f ₁₁ + f ₁₂ · a, y = f ₂₁ + f ₂₂ · a (11)

Next, by substituting the equation (11) into the matrix equation (9), the following equations (12) and (13) are obtained. _{(F 12 + f 22) a} 2 + (f 11 + f 21 + f 22) a + f 21 -1 = 0 ······ (12) (f 12 + 3f 22) a 2 + (f 11 + 2f 12 + 3f 21) a + f ₁₁ = 0 ... (13)

Next, the highest-order term of a in equations (12) and (13) is ignored (in this case, the second-order term), and the coefficient is used as the identity polynomial of a (hereinafter referred to as the identity). Is zero. This is the condition that the equation holds regardless of the value of a.
In this example, the following four expressions are obtained. f ₁₁ + f ₂₁ + f ₂₂ = 0 .................. (14) f ₂₁ -1 = 0 ........ (15) f ₁₁ + 2f ₁₂ + 3f ₂₁ = 0 ........ (16) f ₁₁ = 0 ... (17)

As a result, the undetermined coefficient is the simultaneous equation (14).
(17) is solved. Numerical calculation may be performed in the process of solving the undetermined coefficient. It is generally known that numerical calculation is significantly faster than calculation including symbols. Therefore, the calculation for obtaining these undetermined coefficients can be performed in a short time. The reason for ignoring the highest-order term of a is to match the number of undetermined coefficients with the number of equations. In the above example, the unknowns are f ₁₁ , f
There are four numbers of ₁₂ , f ₂₁ , and f ₂₂ , and only four equations are required. Although the approximation is performed here as well, since a is considered to be a value close to zero, the approximation error is small even if the highest-order term of a is ignored. The above is considered as the point a ₀ = 0 in the series expansion, but if not, set a−a ₀ to b,
A similar argument can be made by considering the identity of the new variable. The calculation results for obtaining the undetermined coefficient are shown below.

[0121]

[Equation 8]

The expression (18) represents the execution result output in the Mathematica language, and "->" means the mathematical equal sign "=". Further, as shown in this equation, the values of f ₁₁ , f ₁₂ , f ₂₁ , and f ₂₂ are calculated. The time taken for this calculation is also displayed. Therefore, by substituting equation (18) into equation (11), x
= -1.5a, y = 1-a was obtained as an approximate solution. The calculation time was about 0.083 seconds.

When the exact solution formula (10) is expanded into a series around a = 0 and the first-order term of a is considered, x = −
1.5a, y = 1-a was obtained. Therefore, the validity of the above-mentioned approximate solution method was confirmed. Further, the calculation time of the approximate solution method can be shortened to about 1/20 that of the conventional method. In the above example, the highest-order term of a is ignored, and the identity of a is derived and each coefficient is set to zero. However, in general, until the number of undetermined coefficients and the number of equations match, It is necessary to disregard the order from the highest order to the lowest order. In other words, until the number of undetermined coefficients and the number of equations match, it is necessary to create an equation in which the coefficients of those terms are zero in order from the terms of low order of a. For example, if the number of undetermined coefficients and the number of equations are the same, ignoring the highest-order term of a and the (highest-order-1) order term, the identity of a is derived in that state, and each coefficient is set to zero. I will leave it. Note that each element of the matrix is an n-th order term (n = 1, 2,
3. ． ． ) Is included, if the inequality is obtained by ignoring the terms from the highest order of a to the (highest order-n + 1) order, the number of undetermined coefficients and the number of equations match.

In the equation (9), the case where each element of the matrix is given by the polynomial expression of the symbol a is considered. However, if each element is given by a rational fractional function (each of the denominator and the numerator represented by a polynomial), the function corresponding to the denominator of the fractional function may be multiplied on both sides of the matrix equation.
If the denominators of the fractional functions of the elements of the matrix are different from each other, multiplying these least common multiples on both sides of the matrix equation gives each element of the matrix a polynomial expression. These processes are processes generally required by the finite element method. Furthermore, when each element of the matrix is an irrational expression instead of a polynomial, the function of each element may be expanded by Taylor series or Maclaurin series. This is the processing required by the boundary element method and the moment method. The process of converting into a polynomial in this way may be necessary for the vector on the right side of the matrix equation. For example, when the design parameter is included in the denominator part of the vector element. Also in this case, if the polynomial of the denominator part is applied to both sides, the whole expression becomes a polynomial expression. If the elements of the vector are not rational fractional functions, series expansion is sufficient. For the above reasons, it can be seen that generality is not lost even if the above constraint conditions are applied.

FIG. 33 is a flow chart showing an outline of a method for approximating and solving a matrix equation describing the target system using a series expansion formula. As shown in this flowchart, first, in step ST3301, a variable giving a functional characteristic distribution is expressed by a series expansion formula regarding a parameter variable defining an object. The order of this series expansion formula is set according to the precision. Furthermore, the coefficient of the series expansion formula is set as an undetermined coefficient to be obtained. This series expansion formula of undetermined coefficients is substituted into the matrix equation in step ST3302. Next, the substituted expression is calculated in step ST33.
At 03, the identity polynomial for the parameter variable is obtained. Then, in step ST3304, the undetermined coefficient of the series expansion formula is obtained by solving a simultaneous equation obtained from the identity polynomial. In this way, the matrix equation can be solved by the approximate expression.

Next, description will be made on the case where the determinant solution method based on the above-mentioned approximate expression is applied to a finite element model. As a result of applying the above-mentioned approximate solution method of the matrix equation to the symbolic operation type finite element inverse analysis, the calculation time of the matrix equation can be reduced to about 1/80 as compared with the case where the series expansion is not used. The point is that the solution is approximated by a series expansion formula with respect to the design parameter, and the numerical coefficient is replaced.

FIG. 34 shows a thin film resistor 2 having an elliptical shape.
It is a dimensional model. The resistor has a uniform conductivity, and the length of the major axis radius is given by an unknown parameter a. The minor axis radius is 2. Further, both ends of the long axis are connected to a direct current source.

FIG. 35 is a finite element model of the quarter region of the thin film low antibody shown in FIG. 34, and the number of element divisions is 11 for simplicity of explanation. The DC current source is a finite element model connected to an electrode with an infinitely small size, but in order to avoid the resistance value of the connection part becoming infinite, it was formulated using a delta function. There is. When the major axis radius a is changed, the nodes on the elliptic circumference move correspondingly. We have written a finite element calculation program by the variational method, using the Mathematica language capable of symbolic operations. As a result, each node voltage is given as a non-linear function including the input variable a. Therefore, the shape parameter a that gives various design target values such as a desired voltage distribution can be obtained directly by the Newton method or the like. For example, assuming that the applied current in FIG. 34 is 1 [A], coordinates (a, 0)
The resistance value of this low antibody can be determined by determining the potential of.

First, the result of the calculation method in the case where the series expansion is not performed will be described by presenting a calculation code in the Mathematica language. Expression (19) is used to represent the final simultaneous equations using the Gaussian elimination method and the command Solve in the Mathematica language. Where matrix.va == con
stant is a matrix equation to be solved, and in the usual writing, matrix.va = constant, but in the Mathematica language, == is used in the sense of = in such a case. Also, matrix is a matrix depending on the target system, and va is shown in FIG.
A vector representing the potential at each node of is expressed in terms of components as follows. va = {v [1], v [2], v [3], v [4], v [5], v [6], v [7], v [8]} In addition, constant is the target system It is a vector that depends on boundary conditions. Solve [matrix.va == constant, {v [1], v [2], v [3], v [4], v [5], v [6], v [7], v [8]} ]; (19) This calculation took about 30 seconds. The potential v [1] at the coordinates (a, 0) is expressed by the following expression (20) by simplifying the expression. In Expression (20), the potential v [1] is the design parameter a
It is displayed by the fractional function of. That is, the denominator and the numerator are each expressed by the polynomial of a.

It should be noted that the expression (20) also shows that it took about 18 seconds in the final step of simplifying the expression. In addition, since it takes about 35 seconds in the intermediate step not shown here, it takes about 80 seconds in total. (18.05 Second, (19.2 a (9.7695 10 ²² + 1.44044 10 ²³ a ² + 6.5415 10 ²² a ⁴ + 1.16432 10 ²² a ⁶ + 7.89177 10 ²⁰ a ⁸ + 1.41635 10 ¹⁹ a ¹⁰ + 1. a ¹² )) / ( ^{9.77857 10 23 + 1.94639 10 24 a} 2 + 1.14856 10 24 a 4 + 2.41449 10 23 a 6 + 1.84653 10 22 a 8 + 3.72923 10 20 a 10 - 12.8 a 12 + 1. a 14)} ······ (20) Since the inflow current was set to 1 [A] when v [1] expressed by equation (20) was calculated, v [1] is the resistance value of this model,
It is a rational fractional function expression of the design parameter a. FIG. 36 is a graph in which the resistance value of the thin film low antibody is plotted against the design parameter a. The calculation for plotting took about 1 second.

Next, the desired design parameter a is obtained by the Newton method or the secant method (hereinafter abbreviated as the Newton method). In the Mathmatica language, if you use the FindRoot command to execute equation (21), the solution will be
It is required as in 2). Here, the Timing command is intended to display the calculation time. Also, the value is 2.8.
And 2.98 are initial values required for calculation. Timing [FindRoot [Resistance == 2.8, {a, {2.9,2.98}}]] (21) {0.2 Second, {a-> 2.94911}} ・ ・ ・ ・ ・ ・ (22)

For example, it can be seen from the equation (22) that the parameter a for designing to 2.8 ohms should be about 2.9 meters. Also, it can be seen that the calculation took 0.2 seconds. Subsequently, it is understood from the equations (23) and (24) that the parameter a for designing to 2.85 ohms should be about 3.0 meters. In this way, even if there are a plurality of design goals, it is possible to easily cope with them. Timing [FindRoot [Resistance == 2.85, {a, {3., 3.05}}]] ... (23) {0.133333 Second, {a-> 3.0231}} ......... (24)

Next, the case of using series expansion will be described. In this example, the approximate solution is found near a = 3, so
We introduce a new variable d = a-3 and decide to find an approximate solution around d = 0. That is, the solution v [1], ..., v [8]
Is given by a complex function of a, and is approximated to a = 3 neighborhood (that is, d = 0 neighborhood) by the following equation (25):
Given the following equation, replace it with the problem of finding the coefficient. Increasing the degree of the polynomial improves the degree of approximation. v [1] = f ₁₁ + f ₁₂・ d + f ₁₃・ d ² v [2] = f ₂₁ + f ₂₂・ d + f ₂₃・ d ² v [3] = f ₃₁ + f ₃₂・ d + f ₃₃・ d ²・ ・ ・ v [8] = f ₈₁ + f ₈₂・ d + f ₈₃・ d ² (25) v [1], ..., v [8] of the equation (25) is described above. Matrix equation ma
Substituting in trix.va == constant, we get 8 polynomials on d. For example, one of the polynomials is represented by the following equation. _{1.99751 f 11 + 0.226134 df 11 +} 0.0376889 d 2 f 11 + 1.99751 df 12 + 0.226134 d 2 f 12 + 1.99751 d 2 f 13 - 0.678401 f 21 - 0.452267 df 21 - 0.0753778 d 2 f 21 - 0.678401 df 22 - 0.452267 d ^{_{2 f 22 - 0.678401 d 2 f}} 23 - 1.31911 f 31 + 0.226134 df 31 + 0.0376889 d 2 f 31 - 1.31911 df 32 + 0.226134 d 2 f 32 - 1.31911 d 2 f 33 = 3 + d (26)

Next, regarding the above eight polynomials, these are considered as identities with respect to d, and the undetermined coefficients f ₁₁ ,
Find f _{12 ,,} ..., f ₈₃ . Solve using the Solve command in the Mathematica language as shown below. Where c [1,1], c
[1,2], ..., c [8,3], etc. give the left side of each equation derived from the identities such as equation (26). For example, c [1,1] is a collection of only zero-order terms (terms not including d) of d on the left side of Expression (26), and specifically, 1.97551 f
₁₁ - becomes 1.31911 f ₃₁ - 0.678401 f _21. Formula (2
The zero-order term (the term that does not include d) on the right side of 6) is 3, which is considered to be equal. This is the following formula (27)
Corresponds to c [1,1] == 3. The same thing should be executed after c [1,2]. That c [1, 2] is a collection of only the coefficient of the primary term of the left side of d of formula (26), _{specifically, 0.226134 f 11 + 1.99751 f 12} + 0.452267 f 21 -
A _{1.31911 f 32 - 0.678401 f 22 +} 0.226134 f 31.
The coefficient of the first-order term of d on the right-hand side of the equation (26) is 1, and these are said to be equal. This is c in equation (27)
Corresponds to [1,3] == 1. The same applies hereinafter.

Solve [{c [1,1] == 3., c [1,2] == 1., c [1,3] == 0., C [2,1] == 0., c [2,2] == 0., c [2,3] == 0., c [3,1] == 0., c [3,2] == 0., c [3,3] == 0., C [4,1] == 0., c [4,2] == 0., c [4,3] == 0., C [5,1] == 0., c [5,2] == 0., c [5,3] == 0., C [6,1] == 0., c [6,2] == 0., c [6,3] = = 0., C [7,1] == 0., c [7,2] == 0., c [7,3] == 0., C [8,1] == 0., c [ 8,2] == 0., c [8,3] == 0.}, {F ₁₁ , f ₁₂ , f ₁₃ , f ₂₁ , f ₂₂ , f ₂₃ , f ₃₁ , f ₃₂ , f ₃₃ , f ₄₁ , f ₄₂ , f ₄₃ , f ₅₁ , f ₅₂ , f ₅₃ , f ₆₁ , f ₆₂ , f ₆₃ , f ₇₁ , f ₇₂ , f ₇₃ , f ₈₁ , f ₈₂ , f ₈₃ }] ・ ・ ・ ・ ・・ (27)

The solution was given by: _{{{f 11 -> 2.83439,} f 12 -> 0.675867, f 13 -> 0.00303785, f 21 -> 1.28006, f 22 -> 0.5525539, f 23 -> 0.003331, f 31 -> 1.3595, f 32 -> 0.26128, _{f 33 -> 0.0207778, f 41} -> 1.06114, f 42 -> 0.401662, f 43 -> -0.00789728, f 51 -> 0.951082, f 52 -> 0.418465, f 53 -> -0.00645431, f 61 -> 0.424813, _{f 62 -> 0.201634, f 63} -> -0.00645838, f 71 -> 0.517817, f 72 -> 0.178864, f 73 -> -0.00472476, f 81 -> 0.598782, f 82 -> 0.127763, f 83 -> 0.0153675} } (28) The calculation time of equation (28) is about 1 second, which is about 80 times faster than the case without series expansion.

Next, the validity of using the above approximate polynomial will be verified. v [1] = f ₁₁ + f ₁₂ d + f ₁₃ d ² , and by substituting the above numerical values, we obtain v [1] = 2.83439 + 0.675867d + 0.00303785d ² . The result of solving the equation accurately by symbolic operation is d
It was found that when they were expanded in the vicinity of = 0, they were completely in agreement. As described above, v [1] corresponds to the resistance value, and the graph for the design parameter a is given as shown in FIG. FIG. 38 shows the result of comparison with the case without series expansion. As shown in the figure, the calculated error of the resistance value is 6 × 1 at most even if the value is changed by 10% around a = 3.
It was found to be about 0 ^-6 . In the optimization of design parameters, usually, the optimum value is often searched in the vicinity of the initial value, for example, in the range of 10 to 30%, so this result is very effective. That is, the calculation time can be significantly shortened by using the approximate polynomial.

Further, the design parameter a = d + 3 that gives a desired resistance value was obtained by the Newton method as follows. 1 when compared with the case where the series expansion of equations (21) to (24) is not performed.
It can be seen that the speed is about 0 times faster. Timing [FindRoot [Resistance == 2.8, {a, {2.9,3.}}]] (29) {0.0166667 Second, {a-> 2.94911}} ・ ・ ・ ( 30) Timing [FindRoot [Resistance == 2.85, {a, {3.0,3.05}}]] (31) {0.0166667 Second, {a-> 3.0231}} ・ ・ ・ ( 32) Timing [FindRoot [Resistance == 2.9, {a, {3.0,3.1}}]] ・ ・ ・ ・ ・ ・ (33) {0.0166667 Second, {a-> 3.09704}} ・ ・ ・ ・ ・ ・ ・ ・(34)

For example, from the equation (30), the resistance value (Resist
The design parameter for designing the ance) to be 2.8 ohms was found to be 2.9491 meters.

Next, an embodiment when applied to the boundary element method will be described. The calculation model is shown in FIG. 12, which will be briefly described again. Considering a rectangular parallelepiped with two sides of potential u = 0 [V] and 1 [V], the problem is to analyze the potential at the inner point. As an inverse problem, the length c of one side of a rectangular parallelepiped when the potential of the inner point P is given as 0.5 [V] will be obtained.

In the calculation without series expansion, the result of plotting the electric potential u (0.5,0.5) of the interior point P against the design parameter c has already been shown in FIG. The calculation time required to plot this graph was about 420 seconds. This is because u (0.5,0.5) is a very long and complicated function of c. In addition, the potential u (0.5,0.5) of the inner point P is set to 0.
Equations (35) and (36) show the results obtained by Newton's method for c to reach 5 [V]. It took about 420 seconds to get the solution. Timing [FindRoot [u (0.5,0.5) == 0.5, {c, 1., 1.1}]] ・ ・ ・ ・ ・ ・ (35) {415.367 second, {c-> 0.998469}} ・ ・ ・ ・ ・ ・(36)

As described above, in the method that does not use the series expansion, even if the matrix size is as small as 4 rows and 4 columns, the function that represents the interior point potential is a long expression, so that the graph drawing and the Newton method calculation are performed. It takes too long. Also,
This is difficult to apply because a memory of 32 MB or more is required for matrix calculation with a matrix size of about 10 rows and 10 columns.

Since the procedure has already been described for the case of using the series expansion, only the result is shown. However, each element of the matrix of the matrix equation that describes the target system in the boundary element method is an irrational expression regarding the design parameters. Therefore, it is necessary to expand that part by Taylor series. This is because unless this is done, the polynomial expression will not be obtained eventually. FIG. 39 is a graph in which the potential at the point P is plotted against the design parameter c. The potential at the point P was expressed by the function of the design parameter c in the following equation. 0.49931-0.450138 (c-1) + 0.530316 (c-1) ^{2 The} calculation time was 0.7 seconds, which was about 600 times faster. FIG. 40 shows the result of the comparison when the series expansion is not performed.
As shown in the figure, the calculated error of the obtained potential is
Even if it is changed by 20% around c = 1, at most 6 × 10 ^-3
It turned out to be about.

Next, equations (37) to (40) show the results obtained by the Newton method for c for the potential u (0.5,0.5) of the interior point P to become 0.5 [V] and 0.4 [V]. . To get the solution
Each took about 0.1 seconds. Also, the calculation time is about 1
/ 4000. Timing [FindRoot [u (0.5,0.5) == 0.5, {c, 0.9,1.}]] ・ ・ ・ ・ ・ ・ (37) {0.116667 Second, {c-> 0.998469}} ・ ・ ・ ・ ・ ・・ ・ (38) Timing [FindRoot [u (0.5,0.5) == 0.4, {c, 0.9,1.}]] ・ ・ ・ ・ ・ ・ (39) {0.133333 Second, {c-> 1.12921}} ・・ ・ ・ ・ (40)

It should be noted that in this approximate solution method, the problem is how much the order should be taken when the series expansion is performed. For this, it is possible to perform the calculation twice for the n-th order and the n + 1-th order to check the convergence state, to expand the series around different values and compare the results. Larger matrix calculations and multiple design parameters can be accommodated. For example, when the calculation results obtained by the second-order approximation and the third-order approximation are compared and a greatly different result is obtained, the calculation of the fourth-order approximation may be further performed to check whether the calculation has converged. This is a well-known method that is usually performed when calculating something by series approximation.

Next, a description will be given of an embodiment applied to the moment method inverse analysis. FIG. 41 is a one-dimensional model of a dipole antenna having a circular cross section with a radius a, and the length is L. The input impedance is Z _in . Consider deciding the design parameter L for setting the real part of the input impedance, that is, the input resistance to a predetermined value.
In addition, in order to simplify the description, a is a constant. First,
The case of solving a matrix equation without depending on the series expansion is described. This is Fujimoto, M. Fuji.
Analoui) and Y.Kagawa, "Symbolic approach to inverse problem in metho
d of moments by mathematica --one dimensional cas
e "), The 14th Symposium on Computational Electrical and Electronic Engineering, Japan Society for Simulation, 261 pages (1
The constants are partially changed using the calculation method by the method of moments described in (993). The method of moments is a method similar to the boundary element method, in which the characteristics of an antenna or the like are analyzed by dividing elements on the boundary using a boundary integral equation. FIG. 42 is a graph showing the L / λ dependence of the input resistance (real part of input impedance) R _in of the antenna shown in FIG. λ is the wavelength of the electromagnetic wave used. The calculation time was about 40 seconds.

Next, equations (41) to (46) show the results obtained by the Newton method for L / λ for making the input resistance R _in a predetermined value. Here, L / λ is set to b. It took about 23 seconds to get the solution. FindRoot [R _in == 2.5, {b, 0.1,0.11}] // Timing ・ ・ ・ ・ ・ (41) {23.08 Second, {b-> 0.099190}} ・ ・ ・ ・ ・ ・ (42) FindRoot [ R _in == 2.54399, {b, 0.1,0.11}] // Timing ・ ・ ・ (43) {22.40 Second, {b-> 0.099999}} ・ ・ ・ ・ ・ ・ ・ (44) FindRoot [R _in == 4.0, {b, 0.14,0.15}] // Timing ・ ・ ・ ・ ・ (45) {23.35 Second, {b-> 0.1234}} ・ ・ ・ ・ ・ ・ ・ (46)

Next, the case where the matrix equation is solved not by the Gaussian elimination method but by the method of obtaining an approximate solution by series expansion will be described. Since the method has already been explained, only the results are shown. However, as with the boundary element method, each element of the matrix of the matrix equation that describes the target system is an irrational expression related to the design parameters, so that part must be expanded by Taylor. It was found that the input impedance is given by the following equation. [1./{0.0000182479 + 0.00114837 I + (-0.000462939-
0.000937144 I) b + (0.00348229 + 0.0577471 I) b ² }] Here, b represents L / λ, and I is an imaginary unit. The input resistance is given by the real part of this equation.

FIG. 43 is a graph plotting the input resistance R _in with respect to 0.05 <L / λ <0.15. The calculation time was about 1.3 seconds. The speed was about 30 times faster than the method described in the above-mentioned paper collection of electrical and electronic engineering symposium. FIG. 44 shows the case where the series expansion is not performed and the case where the series expansion is performed in an overlapping manner. The thick line is the case where the series expansion is not performed, and the thin line is the case where the series expansion is performed. When 0.08 <L / λ <0.12, the error is sufficiently small. In order to further reduce the error over a wider range, increase both or at least one of the order of the series expansion for each matrix element and the expansion order of the variables corresponding to x and y in equation (11). Just go.

Finally, by the Newton method, a predetermined input resistance R
Find the antenna length b = L / λ that gives _in . Expression (4
As shown in 7) to (52), it was found that the calculation can be performed in about 1/100 time as compared with the case without series expansion. FindRoot [R _in == 3., {B, 0.1,0.11}] // Timing ・ ・ ・ ・ (47) {0.18 Second, {b-> 0.1080}} ・ ・ ・ ・ ・ ・ ・ ・ (48 ) FindRoot [R _in == 4., {B, 0.1,0.11}] // Timing ・ ・ ・ ・ ・ ・ (49) {0.22 Second, {b-> 0.1258}} ・ ・ ・ ・ ・ ・ ( 50) FindRoot [R _in == 1.8, {b, 0.1,0.11}] // Timing ・ ・ ・ ・ (51) {0.25 Second, {b-> 0.0839}} ・ ・ ・ ( 52)

To reiterate, in this approximate solution method, the problem is how much the order should be taken when the series expansion is performed. For this, it is possible to perform two calculations of the nth order and the n + 1th order, check the convergence state, expand the series around different values, and compare the results.

In each of the above embodiments, there is one design parameter, but the same applies to two or more design parameters. In this case, multivariable series expansion may be used. Further, it is also applicable to other analysis methods including the finite difference method. In other words, as long as the physical behavior or behavior of the target system is described by the matrix equation, this method can be applied.

[0153]

As described above, according to the first aspect of the present invention, the method for controlling or determining the target system is calculated by calculating the functional characteristic distribution of the object including variable parameters, and calculating the functional characteristic distribution and the desired characteristic distribution. Create an equation by equalizing the distributions and
Since it is configured to solve the equation to determine the parameter value that gives the desired distribution, it is not necessary to repeatedly execute the boundary element method or the finite element method, and the time to obtain the parameter value can be shortened, and Since the calculation in the middle can be performed by using the symbol, there is an effect that the deterioration of the calculation accuracy during the processing can be prevented and an accurate result can be obtained. Further, since the functional characteristic distribution of the object including the variable parameter can be obtained, there is an effect that the characteristic of the distribution with respect to the parameter can be easily grasped.

Further, since the functional characteristic distribution to be evaluated can be expressed directly by an unknown parameter such as the physical property value of the shape,
There is also an effect that it is possible to simultaneously calculate shapes and physical property values that give a plurality of desired functional characteristic distributions. According to the conventional method, when the design work or the like for one functional characteristic distribution is completed, it is necessary to calculate again for the other distributions from the beginning. In this way, even if there are multiple design goals,
The calculation time is greatly reduced as compared with the conventional method.
Further, the variate of the functional characteristic distribution can be expressed by an approximate expression using a series expansion expression, which has the effect of speeding up the calculation.

According to the second aspect of the present invention, the method for controlling or determining the target system is calculated by calculating the functional characteristic distribution of the object including variable parameters, and calculating the residual difference between the functional characteristic distribution and the desired distribution by 2 Since it is configured to determine the value that minimizes the sum of multiplications as the parameter value, it is not necessary to repeatedly execute the boundary element method and the finite element method, the time to obtain the parameter value can be shortened, and Since the calculation can be performed by using the symbol, there is an effect that the calculation accuracy is prevented from being lowered during the processing and an accurate result can be obtained. Further, since the functional characteristic distribution of the object including the variable parameter is obtained, there is an effect that the characteristic of the distribution with respect to the parameter can be easily grasped. Furthermore, when minimizing the sum of squares of the residuals, it is possible to prevent a value that is not the minimum value from being mistaken as the minimum value, and there is an effect that the optimum parameter value can be correctly determined.

Further, for example, by weighting a certain coordinate value and calculating the residual, the deviation between the design value and the calculated value at a certain place in space can be made particularly small. In this case, evaluation of the sum of squares of a plurality of residuals (various evaluations such as evaluation when weights are uniform) can be performed at the same time. According to the conventional method, if the types of evaluation are different, it is necessary to perform each of them sequentially. Further, the variate of the functional characteristic distribution can be expressed by an approximate expression using a series expansion expression, which has the effect of speeding up the calculation.

According to the third aspect of the present invention, the method of estimating the target system is calculated by calculating the physical quantity distribution of the object including variable parameters and equating the physical quantity distribution with the actually measured distribution. Since it is configured to solve the equation and obtain the estimated value of the internal state of the object, it is not necessary to repeatedly execute the boundary element method and the finite element method, and the time to obtain the estimated value can be shortened. In addition, since the calculation in the middle can be performed using the symbol, there is an effect that the calculation accuracy is prevented from being lowered during the processing, and the accurate result can be obtained. In addition, the variable of the physical quantity distribution can be expressed by an approximate expression using a series expansion expression, which has the effect of speeding up the calculation.

According to the fourth aspect of the present invention, the method of estimating the target system is such that the physical quantity distribution of an object including variable parameters is calculated, and the square of the residual between the physical quantity distribution and the actually measured distribution is squared. Since it is configured to obtain the value that minimizes the sum as the estimated value, it is not necessary to repeatedly execute the boundary element method and the finite element method, the time to obtain the estimated value can be shortened, and the calculation in the middle can be reduced. Since it is possible to use symbols, it is possible to prevent a decrease in calculation accuracy during processing and obtain an accurate result. Furthermore, in minimizing the residual sum of squares, it is possible to prevent a local minimum value that is not the minimum value from being mistaken as the minimum value, and it is possible to correctly determine the optimum estimated value. In addition, the variable of the physical quantity distribution can be expressed by an approximate expression using a series expansion expression, which has the effect of speeding up the calculation.

According to the fifth to eleventh aspects of the present invention, the symbol calculation is replaced with the numerical calculation, so that the calculation time can be significantly reduced.

According to the twelfth aspect of the present invention, the apparatus for optimally designing the target system includes means for calculating the function characteristic distribution of the target system as a function of variables, and the function characteristic distribution and the desired function specific distribution. Since it is composed of means for creating equations by equalization and means for solving the equations to realize the desired functional characteristic distribution, it is possible to shorten the time until the parameter values are obtained, resulting in accurate results. And the characteristics of the distribution with respect to the parameters can be easily grasped, and the shape and the physical property value that give a plurality of desired functional characteristic distributions can be calculated all at once. In addition, since the variate of the functional characteristic distribution is represented by an approximate expression using series expansion, there is also an effect that the calculation becomes faster.

According to the thirteenth aspect of the present invention, the apparatus for optimally designing the target system calculates the functional characteristic distribution of the target system as a function of the variable, and the remaining of the functional characteristic distribution and the desired functional characteristic distribution. Since it is configured by means for creating a residual equation from the sum of squared differences and means for minimizing the residual equation to realize a desired functional characteristic distribution,
It is possible to shorten the time to obtain the parameter values, obtain accurate results, and easily understand the characteristics of the distribution with respect to the parameters. There is an effect that multiple sums of squares can be evaluated simultaneously. In addition, since the variate of the functional characteristic distribution is represented by an approximate expression using series expansion, there is also an effect that the calculation becomes faster.

According to the fourteenth aspect of the present invention, the apparatus for estimating the target system comprises means for calculating the physical quantity distribution of the target system including variable parameters as a function of variables.
Since the physical quantity distribution and the measured physical quantity distribution are equalized to create an equation and the equation is solved to estimate the internal state of the object, the time to obtain the estimated value is shortened. Therefore, there is an effect that an accurate result can be obtained. In addition, since the variate of the physical quantity distribution is expressed by an approximate expression using series expansion, there is also an effect of speeding up the calculation.

According to the fifteenth aspect of the present invention, the apparatus for estimating the target system comprises means for calculating the physical quantity distribution of the target system including variable parameters as a function of variables.
The estimation is made by means of means for creating a residual equation from the sum of squares of residuals between the physical quantity distribution and the measured physical quantity distribution, and means for estimating the internal state of the object by minimizing the residual equation. It is possible to shorten the time until the value is obtained, obtain an accurate result, and further, it is possible to correctly determine the optimum estimated value. In addition, since the variate of the physical quantity distribution is expressed by an approximate expression using series expansion, there is also an effect of speeding up the calculation.

[Brief description of drawings]

FIG. 1 is an explanatory diagram for explaining a basic concept of the present invention.

FIG. 2 is a flowchart showing a method for controlling or determining a target system according to an embodiment of the present invention.

FIG. 3 is a flowchart showing a method for controlling or determining a target system according to an embodiment of the invention as set forth in claim 2;

FIG. 4 is a flowchart showing a method for estimating a target system according to an embodiment of the invention as set forth in claim 3;

FIG. 5 is a flowchart showing a method for estimating a target system according to an embodiment of the present invention.

FIG. 6 is a block diagram showing an apparatus for controlling or determining a target system according to an exemplary embodiment of the present invention.

FIG. 7 is a block diagram showing an apparatus for controlling or determining a target system according to an embodiment of the present invention.

FIG. 8 is a block diagram showing an apparatus for estimating a target system according to an embodiment of the present invention.

FIG. 9 is a block diagram showing an apparatus for estimating a target system according to an embodiment of the present invention.

FIG. 10 is a block diagram showing a specific configuration example of the device shown in FIGS. 6 to 9;

FIG. 11 is an explanatory diagram showing a one-dimensional transmission line problem.

FIG. 12 is an explanatory diagram showing a boundary value problem in a rectangular area in which a phenomenon is expressed by the Laplace equation.

FIG. 13 is an explanatory diagram showing a model of a heat conduction problem.

FIG. 14 is an explanatory diagram showing a model of a problem of obtaining an internal potential between parallel plate electrodes.

FIG. 15 is a characteristic diagram showing characteristics with respect to a potential size.

FIG. 16 is an explanatory diagram showing a disc-shaped conductor model.

FIG. 17 is a characteristic diagram showing characteristics of node voltages with respect to dimensions.

FIG. 18 is a characteristic diagram showing characteristics with respect to a voltage of a node and a dimension of a residual sum of squares.

FIG. 19 is an explanatory diagram showing an example of analysis of the flow of air around a vehicle.

FIG. 20 is an explanatory diagram showing an example of analysis of stress distribution inside a hollow metal shaft.

FIG. 21 is an explanatory diagram for explaining a method of estimating the internal conductivity of a square conductor model.

FIG. 22 is a conceptual diagram showing an application example in which the method for controlling or determining / estimating a target system and the apparatus thereof according to the present invention is applied to temperature distribution estimation / control in a reactor.

FIG. 23 is a conceptual diagram showing an application example in which the method for controlling or determining / estimating the target system and the apparatus according to the present invention are applied to the design of a shield for electromagnetic noise.

FIG. 24 is a block diagram showing a configuration of a process device that generalizes each of the application examples.

FIG. 25 is a block diagram showing a computer system that realizes an apparatus for controlling or determining / estimating a target system according to the present invention.

FIG. 26 is an explanatory diagram for explaining the superiority of the present invention.

FIG. 27 is a flowchart showing a process of a conventional object parameter determining method.

FIG. 28 is an explanatory diagram showing a two-dimensional analytical model of an electromagnet.

FIG. 29 is an explanatory diagram showing how the shape of the electromagnet converges.

FIG. 30 is an explanatory diagram showing a chest cross-section model of a human body.

FIG. 31 is an explanatory diagram showing a relationship between an estimated error average and the number of iterations of analysis by the finite element method.

FIG. 32 is an explanatory diagram showing a state in which a local minimum value exists.

FIG. 33 is a flowchart showing an outline of a method for approximating and solving a matrix equation describing a target system using a series expansion formula.

FIG. 34 is a diagram showing a two-dimensional model of a thin film resistor having an elliptical shape.

35 is a diagram showing a finite element model of a quarter region of the thin film resistor shown in FIG. 34.

FIG. 36 is a graph chart in which the resistance value of the thin film resistor is plotted against the design parameter a in the case where the series expansion is not performed.

FIG. 37: Design parameter a in case of series expansion
It is the graph figure which plotted the resistance value of the thin film resistor with respect to.

FIG. 38 is a graph showing an error when a series expansion is performed.

39 is a graph diagram in which the potential at the point P in FIG. 12 is plotted against the design parameter c.

FIG. 40 is a graph showing an error when the potential of the point P in FIG. 12 is obtained by series expansion.

FIG. 41 is a diagram showing a one-dimensional model of a dipole antenna having a circular cross section with a radius a.

42 is an L / of the input resistance R _in of the antenna shown in FIG. 41;
It is a graph which shows lambda dependence.

43] 0.05 <L / λ <of the antenna shown in FIG. 41
Is a graph plotting the input resistance R _in for 0.15.

FIG. 44 is a graph diagram in which, when obtaining the input impedance of the antenna shown in FIG. 41, a case where series expansion is performed and a case where series expansion is not performed are overlapped and shown.

[Explanation of symbols]

 201 means for calculating a function 202 means for creating an equation 203 means for solving an equation 211 means for calculating a function 212 means for creating a residual equation 213 means for minimizing a residual equation 221 means for calculating a function 222 creating an equation Means 223 means to solve equations 231 means to calculate functions 232 means to create residual equations 233 means to minimize residual equations

Claims (15)

[Claims] 1. A step of using at least one parameter among parameters defining a target system as a variable, a step of calculating a functional characteristic distribution of the system as a function of the variable, and the calculated functional characteristic distribution. And a desired functional characteristic distribution are equalized to create an equation including the variable, and a step of solving the equation to control or determine a parameter for realizing the desired functional characteristic distribution In the method for optimally designing a system, the step of calculating the functional characteristic distribution of the system as a function of the variable includes a first step of expressing a variable giving the functional characteristic distribution by a series expansion formula of undetermined coefficients relating to the variable. , The functional characteristic distribution expressed by the series expansion equation of the undetermined coefficient in the matrix equation describing the system The second step of substituting a given variable, the third step of obtaining an identity polynomial for the variable from the equation obtained by the substitution in the second step, and the value of the undetermined coefficient in the series expansion equation And a fourth step of obtaining it by using simultaneous equations obtained from the identity polynomial. 2. A step of using at least one parameter among parameters defining a target system as a variable, a step of calculating a functional characteristic distribution of the system as a function of the variable, and the calculated functional characteristic distribution. And a step of creating a residual equation including the variable by the sum of squares of residuals of the desired functional characteristic distribution, and a step of minimizing the residual equation to realize the desired functional characteristic distribution. In the method of optimally designing a target system, the step of calculating the functional characteristic distribution of the system as a function of the variable is a first step of expressing a variable giving the functional characteristic distribution by a series expansion equation of an undetermined coefficient related to the variable. And a transformation giving the functional characteristic distribution expressed by the series expansion equation of the undetermined coefficient to the matrix equation describing the system. A second step of substituting a quantity, and a third step of obtaining an identity polynomial for the variable from the expression obtained by the substitution in the second step,
A fourth step of obtaining the value of the undetermined coefficient in the series expansion equation by using a simultaneous equation obtained from the identity polynomial, the method for optimally designing a target system. 3. A step of using an internal state of a target system as a variable, a step of calculating a physical quantity distribution measurable with respect to the system as a function of the variable, the calculated physical quantity distribution and an actually measured physical quantity distribution. And a method of estimating an object system for estimating the internal state of the system by solving the equation, the physical quantity distribution measurable for the system being the variable. The step of calculating as a function of is the first step of expressing the variable giving the physical quantity distribution by a series expansion formula of the undetermined coefficient relating to the variable, and the matrix expansion describing the system by the series expansion formula of the undetermined coefficient. The second step of substituting the expressed variable for giving the physical quantity distribution and the substituting in the second step Therefore, the third step of obtaining the identity polynomial for the variable from the equation obtained, and the fourth step of obtaining the value of the undetermined coefficient in the series expansion equation using the simultaneous equation obtained from the identity polynomial equation. A method for estimating a target system characterized by having. 4. A step of using an internal state of a target system as a variable, a step of calculating a physical quantity distribution measurable for the target system as a function of the variable, the calculated physical quantity distribution and an actually measured physical quantity. A step of creating a residual equation including the variable by a sum of squares of residuals with a distribution; and a method of estimating a target system, which estimates the internal state of the target system by minimizing the residual equation, The step of calculating a physical quantity distribution measurable for the system as a function of the variable comprises a first step of expressing a variable giving the physical quantity distribution by a series expansion equation of an undetermined coefficient relating to the variable, and a matrix describing the system. A second step of substituting into the equation a variable giving the physical quantity distribution expressed by the series expansion of the undetermined coefficient; , A third step of obtaining an identity polynomial for the variable from the equation obtained by the substitution in the second step, and a simultaneous equation obtained by obtaining the value of the undetermined coefficient in the series expansion equation from the identity polynomial. And a fourth step of obtaining the target system. 5. The method according to any one of claims 1 to 4, wherein the third step includes a step of adopting a low-order term to a high-order term of the variable until the number of undetermined coefficients and the number of equations of the simultaneous equations match. the method of. 6. The method according to claim 1, wherein in the third step, series expansion is performed on the variables to obtain a polynomial for the matrix equation. 7. The method according to claim 1, wherein the least common multiple of the denominator function of each element in the matrix equation is multiplied to convert both sides into a polynomial expression. 8. The method according to claim 1, wherein a Taylor expansion formula or a Maclaurin expansion formula is used as the series expansion formula. 9. A method according to claim 1, wherein a finite element method is used in deriving an equation describing the system. 10. A method according to claim 1, wherein the boundary element method is used in the derivation of the equations describing the system. 11. A method according to claim 1, wherein the method of moments is used in the derivation of the equations describing the system. 12. Means for calculating a functional characteristic distribution of the target system as a function of the variable by using at least one parameter among parameters defining the target system as a variable, and the calculated functional characteristic distribution and a desired value. And a means for creating an equation that includes the variable by equalizing the functional characteristic distribution of, and a means for controlling or determining a parameter that solves the equation to realize the desired functional characteristic distribution. In the apparatus for optimal design, the means for calculating the functional characteristic distribution of the target system as a function of the variable expresses the variable giving the functional characteristic distribution by a series expansion equation of undetermined coefficients relating to the variable, and describes the system. Substituting a variable that gives the functional characteristic distribution expressed by the series expansion equation of the undetermined coefficient into the matrix equation, and by this substitution, The identity polynomial for the variable is obtained from the obtained equation, and the value of the undetermined coefficient in the series expansion equation is calculated as a function of the variable by using a simultaneous equation obtained from the identity polynomial. A device that optimally designs the target system. 13. A means for calculating a function distribution of the system as a function of the variable by using at least one parameter among the parameters defining the target system as a variable, and the calculated function characteristic distribution and a desired value. Means for creating a residual equation including the variable by the sum of squares of residuals with the functional characteristic distribution, and means for controlling or determining parameters for minimizing the residual equation to realize the desired functional characteristic distribution. In a device for optimally designing a target system including and, means for calculating the functional characteristic distribution of the target system as a function of the variable expresses a variable giving the functional characteristic distribution by a series expansion formula of an undetermined coefficient related to the variable. Then, substitute a variable that gives the functional characteristic distribution expressed by the series expansion equation of the undetermined coefficient into the matrix equation that describes the system, The identity polynomial for the variable is obtained from the equation obtained by substituting, and the value of the undetermined coefficient in the series expansion equation is calculated as a function of the variable by using the simultaneous equation obtained from the identity polynomial. A device that optimally designs the target system. 14. The internal state of the target system is used as a variable,
Means for calculating a physical quantity distribution measurable for the target system as a function of the variable; means for creating an equation including the variable by equalizing the calculated physical quantity distribution and the actually measured physical quantity distribution; In the apparatus for estimating the target system, which comprises means for solving the equation and estimating the internal state of the system, the means for calculating a physical quantity distribution measurable for the system as a function of the variable is the physical quantity distribution. Is expressed by a series expansion formula of an undetermined coefficient relating to the variable, and the variable giving the physical quantity distribution expressed by the series expansion formula of the undetermined coefficient is substituted into a matrix equation that describes the system, and obtained by this substitution. The identity polynomial for the variable is obtained from the given equation, and the value of the undetermined coefficient in the series expansion equation is obtained from the identity polynomial. Apparatus for estimating the target system and calculates as a function of said variable by determining using the simultaneous equations were. 15. The internal state of the target system is used as a variable,
Means for calculating a physical quantity distribution that can be measured for the target system as a function of the variable, and a residual equation including the variable by a sum of squares of residuals between the calculated physical quantity distribution and the actually measured physical quantity distribution. Means to create
In an apparatus for estimating a target system, which comprises means for estimating the internal state of the system by minimizing the residual equation, means for calculating a physical quantity distribution that can be measured for the system as a function of the variable, A variable giving a physical quantity distribution is expressed by a series expansion formula of an undetermined coefficient relating to the variable, and the variable giving the physical quantity distribution expressed by the series expansion formula of the unknown coefficient is substituted into a matrix equation describing the system, and this substitution is performed. The identity polynomial for the variable is obtained from the equation obtained by, and the value of the undetermined coefficient in the series expansion equation is obtained by using the simultaneous equation obtained from the identity polynomial to calculate as a function of the variable. An apparatus for estimating a target system characterized by the above.