JP4891810B2 - Error estimation apparatus and error estimation method - Google Patents

Description

  The present invention relates to an error estimation device and an error estimation method for estimating the magnitude of error of a numerical solution when performing numerical analysis.

  In recent years, computer science has greatly improved along with the improvement of computer capabilities, and numerical analysis (numerical simulation) has been attempted, such as reducing the number of model tests that require large costs by using many numerical calculations at the development site of equipment. Are also becoming indispensable in industry.

  However, when designing with the safety factor and safety margin based on the numerical analysis results, the value that is focused on in the numerical solution (for example, the force that the structure receives from the flow in the case of thermal flow analysis or cooling If the heat removal performance of the element is evaluated by the flow of the medium, how much error is in the temperature rise of the element), that is, when the test measurement is performed with the same system, how much the measured value is based on the result of numerical analysis There is an increasing demand to evaluate whether there is a possibility of a difference without conducting tests as much as possible.

  Especially for large, high-temperature, high-pressure equipment and devices such as nuclear reactors, it is very expensive to make mockups (models) and perform test measurements. There is a great merit in evaluating only with numerical analysis and small-scale tests or simplified tests. As a numerical analysis method, for example, there is a method of discretizing and solving a space such as a difference method, a finite volume method, a finite element method and the like.

  Generally, there are three types of errors, that is, a modeling error, a discretization error, and a calculation error, between the numerical analysis result and the original physical phenomenon targeted by the analysis. Here, these three types of errors are collectively referred to as numerical analysis errors. The modeling error includes, for example, a turbulent flow model, and the discretization error includes an error due to a difference scheme. In some cases, the residual when solving the equation by an iterative method is defined as an error (see Patent Document 1).

  However, when a product is actually manufactured and tested, or when a product is actually manufactured and operated, there is a geometrical gap between the phenomenon occurring there and the phenomenon calculated by numerical analysis, in addition to the numerical analysis error described above. Some are caused by geometrical tolerances, and some are caused by errors in input values such as boundary conditions given in numerical analysis. The error evaluation method for numerical solutions required for equipment / product development in the industry will be extremely useful if these are included.

  In addition, a method has been proposed in which a verification problem is specified, and a numerical analysis program is a program in which test results are well reproduced in numerical analysis targeting the verification problem (see Patent Document 2). Turbulence models that are almost always used for solving industrial flows are known to vary greatly in accuracy depending on the analysis target.If a certain value is compared with the measurement result in this verification problem, the error percentage is calculated. Even if it is obtained, the error will not necessarily be the same percentage when analyzing other systems, so this validation problem is used to determine the percentage of the numerical solution error in the analysis object that an analyst is facing. It is impossible to guess.

  On the other hand, there is an evaluation method called CSAU (Code Scaling, Applicability, and Uncertainty). It describes the conceptual framework for assessing the uncertainty of analysis required by the US Nuclear Regulatory Commission (NRC) when performing safety analysis of nuclear power plants using optimal evaluation models. . The optimum evaluation model is not an analysis model that always outputs a value on the maintenance side, but a numerical analysis model that is created with the intention of reproducing the phenomenon as faithfully as possible.

  However, although the CSAU method has a track record in the safety analysis of the reactor intended when the method was developed, for example, there is no track record in multidimensional heat flow analysis, and there is no specific method for realizing CSAU. .

  For example, in the numerical analysis of multi-dimensional thermal flow fields such as inside equipment, the flow to be analyzed with good accuracy and the flow that can only be analyzed with poor accuracy are mixed in one analysis target area. It is normal. In this respect, the multidimensional heat flow analysis is greatly different from the so-called design code consisting of empirical formulas, etc., specifying equipment and phenomena, making it difficult to apply the CSAU method to error estimation in multidimensional heat flow analysis. Yes.

As mentioned above, at present, in numerical analysis, especially multidimensional heat flow analysis, unless the test measurement is performed in the same system, the error that the analyst himself faces is accurately estimated. I can't.
JP-A-4-232568 JP 2002-7947 A

  In the technique of estimating the error of the numerical analysis result, the method of the conventional patent document described above can handle only a part of the error type, or the verification problem for grasping the accuracy is specified. The problem is that it is impossible to know the error of a numerical solution in an arbitrary analysis target. Further, the CSAU method showing a conceptual framework for evaluating the uncertainty of numerical analysis does not have a specific procedure for realizing it by, for example, multidimensional heat flow analysis.

The present invention has been made to solve the above-described problems, and can be applied to numerical analysis of an arbitrary analysis target, and has input values such as errors and boundary conditions caused by geometric dimensional tolerances. An object of the present invention is to provide an error estimation device and an error estimation method for estimating an error width of a value focused by an analyst in a numerical solution in consideration of errors caused by errors and numerical analysis errors. To do.

In order to solve the above problem, the error estimation apparatus according to the embodiment of the present invention calculates an error of a predetermined attention value in a numerical solution obtained when performing a numerical analysis on a model equation for a predetermined phenomenon. In the error estimation apparatus for estimation, the geometric dimension of the region targeted by the numerical analysis is used as an error factor, and the influence of the error value on the error in the numerical solution of the predetermined phenomenon can be clearly ignored. The geometric sensitivity calculation means for calculating the sensitivity of the error width and the increase / decrease in the value of the error factor to the attention value in the numerical solution of the predetermined phenomenon, and the numerical analysis and noise factor input value provided from the input device when performing, in effect clearly ignored for the error with a target value in the numerical solution of the given phenomenon of this error factor The error factor considered that it is assumed that no error widths, and an input value sensitivity calculating means increases or decreases the value of the error factor to calculate the sensitivity on target values in the numerical solution of the given phenomenon, the predetermined phenomenon The numerical analysis error width acquisition means for acquiring the error width of the numerical analysis error of the value of interest in the numerical solution as an error factor, the error width calculated by the geometric sensitivity calculation means and the error factor of sensitivity, and the input with an error factor of error range and sensitivity calculated by the value sensitivity calculating means, and a noise factor of the acquired error range by the numerical analysis errors width obtaining means, error estimating means for estimating the error of the target value And.

The error estimation method according to the embodiment of the present invention is a method for estimating an error of a predetermined attention value in a numerical solution obtained by a computer when performing a numerical analysis on a model equation for a predetermined phenomenon. Yes, the geometric sensitivity calculation means uses the geometric dimension of the region targeted by the numerical analysis as an error factor, and the influence of the error factor on the error of the attention value in the numerical solution of the predetermined phenomenon is obvious It is assumed that there is no error factor considered to be negligible , and the geometric sensitivity calculation step for calculating the sensitivity of the error width and the increase / decrease in the value of the error factor to the attention value in the numerical solution of the predetermined phenomenon; the input value sensitivity calculating means, the input value provided from the input device in performing the numerical analysis and error factors, interest in the numerical solution of the given phenomenon of this error factor Input error factors influence on the error is considered to be clearly negligible with the regarded as not present, error range, and the increase or decrease in the value of the error factor to calculate the sensitivity on target values in the numerical solution of the given phenomenon A value sensitivity calculation step, a numerical analysis error width acquisition means, a numerical analysis error width acquisition step of acquiring, as an error factor, an error width of a numerical analysis error of a target value in the numerical solution of the predetermined phenomenon, and an error estimation means, The error width and sensitivity error factor calculated in the geometric sensitivity calculation step, the error width and sensitivity error factor calculated in the input value sensitivity calculation step, and the numerical analysis error width acquisition an error factor of the acquired error range in step, using, characterized in that it comprises a and error estimation step of estimating an error of the target value.

According to the error estimation device and the error estimation method of the present invention, it can be applied to numerical analysis of an arbitrary analysis target, and is caused by an error of an input value such as an error caused by a geometric dimensional tolerance or a boundary condition. It is possible to estimate the error width of the value that the analyst pays attention to in the numerical solution in consideration of the error to be performed and the numerical analysis error.

  Embodiments of an error estimation device and an error estimation method for estimating the magnitude of an error of a numerical solution according to the present invention will be described with reference to the accompanying drawings.

  FIG. 1 is a configuration diagram of an error estimation apparatus 1 according to the present invention. As shown in FIG. 1, the error estimation device 1 includes a CPU (Central Processing Unit) 2 that performs error estimation processing for estimating an error, and various other arithmetic processing and control processing, and a CPU 2 performs processing. Controls RAM (Random Access Memory) 3 that temporarily stores data when performing, ROM (Read Only Memory) 4 that stores data and programs necessary for processing by CPU 2, and hard disk 5 A hard disk control unit 6 that performs control, a display control unit 8 that controls a display device 7 such as a display, and an input control unit 10 that controls an input device 9 such as a keyboard and a mouse. Is done.

  First, a first embodiment of the error estimation apparatus 1 according to the present invention will be described.

  The error estimation device 1 performs an error estimation process for estimating an error of an analysis result when performing numerical analysis of a “certain phenomenon”. The procedure of this error estimation process will be described based on the flowchart shown in FIG. Note that an object on which the error estimation apparatus 1 performs numerical analysis is referred to as an “original system” or an “overall system”. Hereinafter, “step S101” will be described as “S101”, and “step” will be omitted.

  The CPU 2 performs numerical analysis of “a certain phenomenon” in the entire system, and obtains a value for which an error is to be estimated from the obtained numerical solution. Here, the value to be estimated for error is set as “attention value” y (S101).

Next, the CPU 2 causes errors to be caused by errors caused by geometric dimensional tolerance (procedure A), errors caused by errors of input values such as boundary conditions (procedure B), and numerical analysis errors (procedures). C). The process of procedure A is a geometric sensitivity calculation unit, the process of procedure B is an input value sensitivity calculation unit, and the process of procedure C is a numerical analysis error width acquisition unit or a numerical analysis sensitivity calculation unit.

  In the procedure A (S103 to S107), a factor related to a geometric dimension is handled as a factor (error factor) that causes an error of the attention value y.

First, CPU 2 extracts the error factors _{x i} about geometric tolerances and errors, to grasp the error width ε _xi (S103). The error width is determined from the manufacturing tolerance, the range of manufacturing error, the range of dimensional change due to temperature change, the measurement error range if the dimension is a measured value, or the standard deviation σ of these errors and changes. The value (for example, 1.5σ, 2σ, 3σ, etc.) may be taken. Note that it may be considered that there is no error factor that is considered to be negligible for the influence of the attention value y on the error. This is called error factor screening. S103 includes this screening work.

In the numerical analysis of “a certain phenomenon” in the entire system in S101, the CPU 2 performs an analysis in which each error factor x _i extracted in S103 is independently changed, and grasps the value of the attention value y at that time. (S105). Here, in order to require only a primary sensitivity in a later step, at least once in the analysis at different x _i value is required at this step. However, the model equation to be solved and the numerical analysis method to be used (in the example of multidimensional heat flow analysis, the numerical analysis program, the turbulent model to be used, the difference scheme, the time integration method, etc. correspond to this) are those used in S101. Same as The range in which the value of each error factor x _i is changed from the value used in S101 is basically the above-described error range ε _xi .

CPU2, based on the result of S105, for each error factor, increase or decrease the value of the error factors _{x i} is determined the sensitivity ∂y / ∂x _i on target value y in the numerical solution of "an Symptoms" (S107) . As an example, for an error factor x _i , in the analysis in S101, a value of x _i0 is used to obtain a value of y ₀ as an attention value, and in the previous step, this value is numerically analyzed as x _i0 + ε _xi to obtain the attention value. Assume that the value y ₁ is obtained. At this time, the first order sensitivity with respect to the error factor x _i can be obtained by the following equation (1).

In addition, asking for ∂y / ∂x _i by the least squares method from 3 or more cases of the analysis result, by taking the derivative by creating an approximate curve may be obtained ∂y / ∂x _i. Alternatively, a well-known combination of experimental design and response surface method may be used to create a response surface function of the attention value y with respect to x _i , and obtain the differential.

  In the procedure B (S109 to S113), a factor (error factor) that causes an error of the attention value y is handled as a factor (error factor) due to an error of the input value such as a boundary condition.

CPU2 extracts the error factors _{x i} about errors with the input values such as boundary conditions, to grasp the error width ε _xi (S109). As the error width, the range of change of this value, the range of measurement error if this value is a measured value, the control range and the range of control error if this value is controlled by equipment, etc. Alternatively, values determined from these errors and standard deviation σ of change (for example, 1.5σ, 2σ, 3σ, etc.) may be taken. As in S103, it is considered that there is no one that the influence of the attention value y on the error can be clearly ignored (error factor screening).

Processing of S111 and S113, to the error factors _{x i} extracted in S109, performs processing respectively similar to the processing of S105 and S107. This description is omitted.

  In the procedure C (S115), a factor relating to a numerical analysis error is handled as a factor (error factor) that causes an error of the attention value y.

The CPU 2 gives the value of the numerical analysis error ε _anal (S115). For example, if a similar test and analysis are compared in the past, or if the error percentage α for a phenomenon relatively close to “a certain phenomenon” is described in the literature, this percentage α and S101 can be obtained. What is necessary is just to obtain | require as (epsilon) _anal = (alpha) y etc. with the obtained attention value y.

CPU2 is Procedure A, Procedure B, and the procedure When C is completed, the following ∂y / ∂x _i of the right side of the equation _{(2), ε xi, ε} anal assigns a value obtained in Step A~ Procedure C to By doing this, an error width ε _y of the value of the attention value y in the numerical solution of the “certain phenomenon” is obtained (S117). At this time, for example, if all ε _xi substituted in the equation is 3σ (σ: standard deviation), the obtained ε _y is 3σ of the attention value y.

By S117, the error width of the attention value y is estimated as “± ε _y ”.

  According to the first embodiment of the error estimation device 1, the error width of the target value obtained from the numerical solution has all three effects of geometrical tolerance, input error such as boundary condition, and numerical analysis error. The included value can be estimated only by numerical analysis.

  Next, a second embodiment of the error estimation apparatus 1 according to the present invention will be described with reference to FIG.

  A procedure when the error estimation apparatus 1 performs the error estimation process will be described based on the flowchart shown in FIG.

  First, the CPU 2 performs numerical analysis of “a certain phenomenon” in the entire system, and obtains the attention value y from the numerical solution (S201).

Next, the CPU 2 causes errors to be caused by errors caused by geometric dimensional tolerance (procedure A), errors caused by errors of input values such as boundary conditions (procedure B), and numerical analysis errors (procedures). C).

  In the procedure A (S203 to S207), a factor relating to a geometric dimension is handled as a factor (error factor) that causes an error of the attention value y.

First, CPU 2 extracts the error factors _{x i} about geometric tolerances and errors, to grasp the error width ε _xi (S203). The error width is determined from the manufacturing tolerance, the range of manufacturing error, the range of dimensional change due to temperature change, the measurement error range if the dimension is a measured value, or the standard deviation σ of these errors and changes. The value (for example, 1.5σ, 2σ, 3σ, etc.) may be taken. Note that it may be considered that there is no error factor that is considered to be negligible for the influence of the attention value y on the error. This is called error factor screening. S203 includes this screening work.

CPU2 is performs the same processing as S105, only that it requires an analysis of more than once you change the value of the error factors _{x i} are different (S205).

Based on the result in the previous step, the CPU 2 determines, for each error factor, the linear (primary) sensitivity as the sensitivity that the increase / decrease in the error factor x _i has on the attention value y in the numerical solution of “a certain phenomenon”. and ∂y / ∂x _i is, obtains the second-order sensitivity ^{_{∂ 2 y / ∂x i 2 (}} S207). As an example, for an error factor x _i , in the analysis in S101, the value x _i0 is used to obtain a value of y ₀ as the attention value, and in the previous step, this value is set to x _{i1 and} a value of y ₁ is obtained. and to obtain a value of _{y 2} as _i2. At this time, three values of (x _i0 , y ₀ ), (x _i1 , y ₁ ), and (x _i2 , y ₂ ) are substituted into an expression of the form “y = ax _i ² + bx _i + c”. To obtain the values of a, b, and c.

a, b, the value of c (3) wherein, by substituting the equation (4), the sensitivity ∂y / ∂x _i and secondary sensitivity ∂ ² y / ∂x _i ² of linear (primary) Can be sought. Further, an analysis of a larger number of cases may be performed to calculate an approximate curve formula and the same processing may be performed. It also creates a second or higher order function of the response surface of the target value y on the error factor x _i using a combination of the response surface method and well-known experimental design, these primary and secondary by taking the derivative You may ask for sensitivity.

  In the procedure B (S209 to S213), a factor (error factor) that causes an error of the attention value y is handled as a factor (error factor) caused by an error of the input value such as a boundary condition.

CPU2 extracts the error factors _{x i} about errors with the input values such as boundary conditions, to grasp the error width ε _xi (S209). As the error width, the range of change of this value, the range of measurement error if this value is a measured value, the control range and the range of control error if this value is controlled by equipment, etc. Alternatively, values determined from these errors and standard deviation σ of change (for example, 1.5σ, 2σ, 3σ, etc.) may be taken. As in S103, it is considered that there is no one that the influence of the attention value y on the error can be clearly ignored (error factor screening).

Processing of S211 and S213, to the error factors _{x i} extracted in S209, performs processing respectively similar to the processing of S205 and S207. This description is omitted.

  In the procedure C (S215), a numerical analysis error is handled as a factor (error factor) that causes an error of the attention value y.

  The CPU 2 performs the same process as S115 (S215).

CPU2 is (5) of the right side of _{∂y / ∂x i, ε xi,} ∂ 2 y / ∂x i 2, ε xi 2, ε substituting the values obtained in Step A~ Step C in _anal Thus, the error width ε _y of the value of the attention value y in the numerical solution of “a certain phenomenon” is obtained.

Further, for example, if all ε _xi assigned to the equation is 3σ (σ: standard deviation), the obtained ε _y is the value of 3σ of the attention value y, as in the first embodiment. is there.

According to the second embodiment of the error estimator 1, since the second-order term of Taylor expansion is taken, ε _y obtained thereby has an estimation accuracy in principle higher than the value obtained in the first embodiment. improves.

  Next, a third embodiment of the error estimation apparatus 1 according to the present invention will be described with reference to FIG. In the third embodiment, the process of S117 in the first embodiment is improved.

  First, the CPU 2 performs numerical analysis of “a certain phenomenon” in the entire system, and obtains the attention value y from the numerical solution (S301).

  The CPU 2 performs the same processing as the procedure A to the procedure C (S103 to S115) of the first embodiment as the procedure A to the procedure C (S303 to S315).

When all of the procedure A to the procedure C are completed, the error width ε _y of the value of the attention value y in the numerical solution of “a certain phenomenon” is obtained by substituting the value obtained in the procedure A to the procedure C into the equation (6). Ask.

Further, for example, if all ε _xi assigned to the equation is 3σ (σ: standard deviation), the obtained ε _y is the value of 3σ of the attention value y. Same as 2.

As an alternative to the first embodiment, ε _y can also be evaluated by this equation.

  Next, a fourth embodiment of the error estimation apparatus 1 according to the present invention will be described with reference to FIG. In the fourth embodiment, the process of S217 in the second embodiment is improved.

  First, the CPU 2 performs numerical analysis of “a certain phenomenon” in the entire system, and obtains the attention value y from the numerical solution (S401).

  The CPU 2 performs the same processing as the procedure A to the procedure C (S203 to S215) of the second embodiment as the procedure A to the procedure C (S403 to S415).

When all of the procedure A to the procedure C are completed, the error width ε _y of the value of the attention value y in the numerical solution of “a certain phenomenon” is obtained by substituting the value obtained in the procedure A to the procedure C into the equation (7). Ask.

Further, for example, if all ε _xi substituted in the equation is 3σ (σ: standard deviation), the obtained ε _y is the value of 3σ of the attention value y. It is the same.

As an alternative to the second embodiment, it is possible to evaluate ε _y also by equation (7).

Next, a fifth embodiment of the error estimation apparatus 1 according to the present invention will be described with reference to FIG. In the fifth embodiment, the processes of the procedure C and S117 in the first embodiment are improved.

  First, the CPU 2 performs numerical analysis of a “certain phenomenon” in the entire system, and obtains the attention value y from the numerical solution (S501).

  The CPU 2 performs the same processing as the procedure A and the procedure B (S103 to S113) of the first embodiment as the procedure A and the procedure B (S503 to S513).

  In the procedure C (S515 to S523), a factor relating to a numerical analysis error is handled as a factor (error factor) that causes an error of the attention value y.

  The CPU 2 decomposes the “certain phenomenon” into one or more “simple phenomena” included in the phenomenon (S515). Note that it may be considered that there is no simple phenomenon in which the influence of the attention value y on the error is clearly negligible (screening for error factors), and S515 includes this screening operation.

The CPU 2 prepares to perform numerical analysis for each “simple phenomenon”, and the amount (error factor x _i ) for evaluating the numerical analysis error of the simple phenomenon (there may be a plurality for one simple phenomenon). ) Is determined (S517). This error factor x _i is treated as an error factor of the attention value y of “a certain phenomenon” later.

As to what is defined as the evaluation amount (error factor x _i ) of the numerical analysis error of a simple phenomenon, the influence of the change range due to the error of the simple phenomenon on the attention value y in “a certain phenomenon” What is considered to be the most prominent should be selected. If you think there are multiple factors that will be most noticeable, choose one of them. To apply seeking sensitivity ∂ to target values y on the error factor x _i selected in step after Anyway _{^{(j) y / ∂x i (}} j), by selecting which in principle Also good.

CPU2 performs numerical analysis to solve the same model equation as that used in S501 for each simple phenomena in the same numerical method, x _i value is the amount for evaluating the numerical analysis errors simple phenomena numerical solutions Is obtained (S519). Here, the same numerical analysis method is, in the example of multi-dimensional heat flow analysis, in addition to a numerical analysis program, a turbulent model to be used, a difference scheme, a time integration method, and grid division conditions, for example, y + (Waiplus) value of wall lattice points. Necessary values such as the enlargement ratio of the lattice spacing and the maximum lattice width are also matched with the calculation in the entire system in S101.

The CPU 2 uses z _i , which is a value obtained separately from a literature value, an experimental correlation equation, a small scale test of a simple phenomenon, and the like for each _xi value obtained in S519, and the numerical analysis error of the simple phenomenon the epsilon _xi, determined by _{ε xi = x i -z i (} S521).

The CPU 2 performs one or more numerical analyzes of “a certain phenomenon” by changing the analysis conditions (hereinafter referred to as parameters) such as boundary conditions and dimensions so that the respective _xi values change in the “a certain phenomenon”. performed, decrease of _{x i} values determine the sensitivity ∂y / ∂x _i on target value y in the numerical solution of "an Symptoms" (S523). However, the model equation to be solved and the numerical analysis method used are the same as those used in S101. The method for calculating the sensitivity conforms to S107.

CPU2, the procedure A (S503~S507), procedure B (S509~S513), error range determined for each error factor _{x i} obtained from Step C (S515~S523) ε _xi and sensitivity ∂y / Using ∂x _i , ε _y , which is a value determined from the error width of the value of the target value y or the standard deviation of the error in the numerical solution of “a certain phenomenon”, is obtained using equation (8) (S525).

  Here, the concept and feasibility of a method for estimating an error of a numerical solution will be described for Example 5 of the error estimation device 1.

There attention value y extracted from the numerical solutions, it is determined by the n different factors x _i affecting its value can be expressed by the following expression f as a function. However error factor x _i is the n pieces there.

  It should be noted that all factors that cannot ignore the influence on the attention value y are taken up as factors.

The difference when the attention value y taken out from the numerical solution is compared with the test measurement value, that is, the error is written as ε _y . This time can be expressed by the following equation using the "error of the target value y is the value to influence determined by the error epsilon _xi with the various error factors x _i" that is the Taylor expansion of g as a function.

The error factor x _{i to which} this equation can be applied must be a continuous quantity and ε _y must be a minor quantity. ∂ ^(j) y / ∂x _i ^(j) is the j-th order sensitivity of the error factor x _i on the target value y.

Here, the categories of error factors that must be considered in this analysis are the following (1) to (3) as described above.
(1) Geometric dimensional errors and tolerances (2) Errors of input values such as boundary conditions (3) Numerical analysis errors

The geometric dimensional tolerance (1) is a continuous amount, and the dimensional tolerance is usually considered to be very small and often satisfies the condition. In many cases, the error of the input value of (2) may be considered to be very small. These In the two months of the category, errors and tolerance corresponds to ^{_{ε xi, ∂ (j) y}} / ∂x i (j) is the magnitude of the impact on the attention value of y when the value of those factors has changed ( Sensitivity).

However, regarding the numerical analysis error of (3), the definition of the error factor x _i and the error ε _xi is not simple unlike the above two error factors. In the method described in Example 5 of the present invention, the numerical analysis error is handled as follows. As a result, for example, even in multi-dimensional heat flow analysis, the number of events that can be handled as a minute value with a continuous amount as in the other two factor categories increases.
・ Whole phenomenon (for example, multi-dimensional flow field) to be analyzed is a complicated phenomenon, but here it is broken down into one or more “simple phenomena” (the influence on the attention value y cannot be ignored) all).
・ Determine (unify) the analysis method to be used (turbulent flow model, difference scheme, grid division width, etc.).
When only each simple phenomenon is analyzed alone, a numerical analysis error is obtained for a certain evaluation amount (error factor x _i ). The error in the simple phenomenon analysis corresponds to ε _xi .

What is defined as the evaluation amount (error factor x _i ) at the time of simple phenomenon analysis may be appropriately selected from those that are considered to have the greatest influence on the value of interest y. If there are multiple factors that are most likely to be affected, choose one of them. To apply seeking sensitivity ∂ to target values y for the selected factor _{^{(j) y / ∂x i (}} j), it may be selected which is in principle.

The error of the evaluation amount (error factor x _i ) when each simple phenomenon is numerically analyzed alone is the value obtained from the literature value or empirical formula, or the simple phenomenon of the evaluation amount (error factor x _i ) extracted from the numerical solution. It is only necessary to perform a test on and compare with test values. If there is a literature value, it can usually be evaluated much cheaper than the cost of the test, and even if a simple phenomenon is tested, it may be a simple system or may not require a large area. Breaking down into phenomena is often much less expensive than testing the original complex system. If the change in error in the reduced scale model is known separately, using it can eliminate the need for large-scale tests and further reduce costs.

  If errors in numerical analysis are broken down into simple phenomena in this way and each simple phenomenon is numerically analyzed, the error may be considered as an error factor in the numerical analysis of the overall phenomenon. Many. In this case, it can be handled in the same manner as a geometric dimensional error or an error of an input value, and it can be seen that the following properties are obtained.

If the error ε _xi of various error factors can be considered to be a minute amount, the error of the attention value y can be expressed as follows by ignoring the high-order minute term in the equation (10). That is, (10) of the formula, the sensitivity ∂ ^(j) y / ∂x _i It can be seen that no significant loss of accuracy even only consider linear term ∂y / ∂x _i in ^(j).

Further, the function g can be expressed as the following equation (12) when, for example, “n = 3”.

Here, in the equation (12), the interaction term becomes a second-order or higher-order minute amount of the minute amount ε _xi and can be ignored, and can be expressed only by the linear coupling equation of the first-order term. This is the same whether n is “2” or “4” or more.

That is, the error epsilon _y attention value y, when an error with the various error factors x _i affecting attention value y is good consider minute error epsilon _xi in continuous quantity ignores higher small amount only It was found that the linear expression can be expressed as

When it is possible to represent the error epsilon _y attention value y (14) shaped like a formula, the error propagates as a statistical estimation method of error epsilon _y If the error with the error factor can be approximated to exhibit a normal distribution Can be used. The law of error propagation is as follows. Standard deviation sigma _y of the error of the target value y is the sensitivity ∂y / ∂x _i standard deviation sigma _xi and error factors x _i of the error with the various error factors x _i affecting attention value y is applied to y It can be evaluated by the following formula.

Here, the “standard deviation” may be read as “n times the standard deviation (n is a positive real number)”, in which case it can be expressed as the following equation.

When only errors and tolerances are given for each error factor x _i , as a common idea, the maximum value ε _xi (ε _xi > 0) of the error or tolerance is 3σ of variation (σ: standard deviation) ) Is used. That is, it is a method of thinking that the absolute value of the error falls within ε _xi with a probability of about 99.7%. As described above, when Expression (16) is expressed using the error and the tolerance range instead of the standard deviation, the following form is obtained.

Ε _y obtained by this is a statistical value. For example, according to the idea that the evaluation on the right side represents 3σ of variation as described above, “the variation range of the error of the attention value y is 99.7%. It is in the range of ± ε _y evaluated by the equation (17) with the probability of “.

(17) 1-order sensitivity ∂y / ∂x _i appearing in the right-hand side of create a function response surface error factors x _i for the target value y by DOE can be obtained as a gradient. However, according to the above consideration, if the error width of each error factor x _i can be regarded as being minute, linear approximation can be performed and the interaction can be ignored. Therefore, the error per error factor x _i Sensitivity can be obtained by performing two numerical analyzes in which the value of the factor x _i is changed independently.

  Here, the procedure of the fifth embodiment of the error estimation apparatus 1 will be described in detail with reference to FIGS. 7 to 11 by applying it to the error evaluation of the heat flow analysis.

The flow channel shape shown in FIG. 7 is a system in which a fluid flows in from the left side, there is a step when going to the right side, the flow channel expands downward, and flows out from the right end, and has a certain length as depth 3 Suppose that it is a dimensional system. Fluid flow 20 flowing flows from the inlet portion 21 having a flow channel height B _0, and flows out from the flow passage outlet portion 22 of height B _1. The outflow portion 22 is provided with a tube group 23 composed of a plurality of tubes. In FIG. 7, the flow of the fluid is indicated by the flow line 24, but as can be seen from the flow line 24, a part of the fluid that has flowed back flows at the reattachment point 25. A flow path in which the flow path expands downward in this manner is called a back step flow.

  Usually, the back step flow often has a flow field as shown by the streamline 24 in FIG. A region squeezed by the hatch portion in this flow field is referred to as a backflow region 26.

  FIG. 9 shows a view of the tube group 23 shown in FIG. 7 as viewed from above. As shown in FIG. 9, the tube group 23 is arranged in a staggered arrangement, for example.

  Here, it is assumed that the analyst wants to know the reattachment point position s in the system of FIG. 7, and in S501, the flow field is analyzed by numerical analysis for the system of FIG.

In the analysis of S501, the flow path height B ₀ at the inflow portion is calculated as Amm, the flow rate through the system is Qm ³ / sec , the diameter of one tube is Dmm, and the tube group pitch is Pmm. Further, it is assumed that a value s ₁ is obtained as the position s of the reattachment point 25.

  This analyst wants to know how much error the position s of the reattachment point 25 obtained from the numerical solution has. That is, here, the position s of the reattachment point 25 is taken as an attention value. The error estimation process at this time will be described according to the procedure of the fifth embodiment.

(1) Step S503 (S103)
The error factors related to the geometric dimensions are: the flow path height B ₀ of the inflow part 21, the flow path height B ₁ of the outflow part 22 after the flow path is enlarged, the diameter D of one tube of the tube group 23, and the tube group There is a pitch P or the like. Each dimension shall have a value indicated in the drawing with a certain manufacturing tolerance.

Here, screening is performed. The subsequent work can be simplified by discarding elements that are considered to have negligible influence on the position s of the reattachment point 25 by order evaluation on the desk. If you don't understand the impact, consider everything. Here, in order to simplify the explanation, it is assumed as an example that the influence on the position s of the reattachment point 25 can be ignored except for the flow path height B ₀ of the inflow portion 21 whose dimensional drawing instruction is A ± 1 mm. Here, the flow path height B ₀ of the inflow portion 21 is defined as an error factor x _i . The error width is ε _xi = 1 [mm].

(2) Step S505 (S105)
In this step, it is assumed that a numerical analysis is performed by increasing the flow channel height B ₀ of the inflow portion 21 calculated in S501 to A + 1 mm, and a value of s ₂ is obtained as the attention value y from the numerical solution.

(3) Step S507 (S107)
The sensitivity ∂s / ∂x ₁ with respect to the position s of the reattachment point 25 at the flow path height B ₀ of the inflow portion 21 is obtained by the following equation (18).

(4) Step S509 (S109)
As an error factor related to input values such as boundary conditions, there is a flow rate per unit time of a fluid flowing through the system. The flow rate is a measured value, and the value is Q ± 1 m ³ / sec including the error range. It is assumed that the error range ± 1 m ³ / sec of the flow rate Qm ³ / sec is determined to have a negligible effect on the attention value s. Here the error factors x ₂ the flow rate. The error width is ε _x2 = 1 [m ³ / sec].

(5) Step S511 (S111)
In this step, it is assumed that a numerical analysis is performed by increasing the flow rate Q calculated in S501 by 1 m ³ / sec, and a value of s ₃ is obtained as an attention value from the numerical solution.

(6) Step S513 (S113)
The following equation (19), obtaining the sensitivity ∂y / ∂x ₂ relative to the position s of the reattachment point of the flow rate Q (2104) (2105).

(7) Step S515
The whole phenomenon that occurs in the original system of FIG. 7 is decomposed into a simple phenomenon. As simple phenomena, the flow in the tube group, the back step flow, and the flow affected by the wall friction can be cited. As a result of the screening, the influence of the wall friction on the position s of the reattachment point 25 is almost negligible. truncate. That is, here, as a simple phenomenon, “flow in a tube group” shown in FIG. 10 and “back step flow (without a tube group)” shown in FIG. 11 are taken up. The diameter D and the tube group pitch P of one tube of the tube group 23 shown in FIG. 10 are Dmm and Pmm, respectively, similarly to FIG. Moreover, the dimensional shape shown in FIG. 11 is equal to the dimensional shape obtained by removing the tube group 23 from the dimensional shape shown in FIG.

(8) Step S517
In performing the numerical analysis on the two simple phenomena taken up, the amount (error factor x _i ) for evaluating the numerical analysis error of the simple phenomenon is determined.

First, we will consider “flow in the tube group”. As an influence on the calculation accuracy of the position s of the reattachment point 25 in the system shown in FIG. 7, the prediction accuracy of the pressure loss due to the flow orthogonal to the tube group 23 in the “flow in the tube group” which is one of the simple phenomena. The impact is considered large. Therefore, as an amount for evaluating the numerical analysis error regarding the simple phenomenon “flow in the tube group”, the pressure loss per unit length of the tube group due to the flow orthogonal to the tube group is taken, and this is set as an error factor x ₃ .

Next, consider “backstep flow (no tube group)”. As an influence on the prediction accuracy of the position s of the reattachment point 25 in the system shown in FIG. 7, there is an influence of the prediction accuracy of the position s of the reattachment point 25 in the backstep flow without a tube group, which is one of simple phenomena. It is considered large. Therefore, when numerically analyzing the simple phenomenon “backstep flow without tube group”, the position s of the reattachment point 25 in FIG. 11 is taken as an amount to evaluate the numerical analysis error, and this is written as x _4. To.

(9) Step S519
Numerical analysis of “flow in the tube group” and “back step flow (without the tube group)” is performed by numerical analysis in which the same model equation as that used in S501 is solved by the same numerical analysis method. From the numerical solution of “flow in the tube group”, a pressure loss x _3A per tube group unit length due to the flow orthogonal to the tube group 23 is obtained, and the numerical solution of “back step flow (without tube group)” , It is assumed that the position s, x _4A of the reattachment point 25 is obtained.

  Here, the lattice width distribution of the lattice model used for the numerical analysis of “flow in the tube group” is assumed to match the lattice division of the tube group region in the lattice model in FIG. 7 performed in S501. Further, regarding the distribution of the lattice width of the lattice model used for the numerical analysis of “back step flow (without tube group)”, the numerical analysis of the lattice line spacing in the space, the roughness near the wall surface, etc. was performed in S501. It is assumed that it matches with the lattice model targeting 7.

(10) Step S521
For the amount to evaluate the numerical analysis error in each of “flow in tube group” and “back step flow (without tube group)”, obtain this value separately from literature values and simple phenomenon tests. This value is written as z _i .

Regarding pressure loss per unit length of “flow in tube group” (x ₃ ), literatures (eg “Hydrodynamics Handbook” (Maruzen), “Consideration on Estimation Method of Flow Resistance Coefficient of Tube Group”, Japan There are empirical formulas for obtaining the resistance coefficient in the JSME Proceedings (Part B), v. 51, n.461, pp. 356. ”, using these, the x _3A value for the system of FIG. Of course, the z _3A value may be obtained by performing a new test measurement on the system of FIG.

Similarly, with respect to the position s (x ₄ ) of the reattachment point 25 in the “back step flow (without a tube group)”, similarly, a reliable document value is referred to, or if there is no document value, FIG. The z _4A value is determined by performing a new simple test measurement.

Using the obtained z _3A value and z _4A value and the x _3A and x _4A values obtained in S519, numerical analysis errors ε _x3 and ε _x4 of a simple phenomenon are expressed by the following equations (20) and (21). Ask for.

(11) Step S523
Here, the sensitivities ∂ s / ∂x ₃ and ∂ s / ∂x ₄ with respect to the position s of the redeposition point 25 in the system shown in FIG. 7 are calculated for the amount for evaluating the numerical analysis error of each simple phenomenon.

First, Fig. 7 shows the "pressure loss per unit length of the pipe group due to the flow perpendicular to the pipe group", which is the amount to evaluate the numerical analysis error of the first elementary phenomenon "flow in the pipe group". The sensitivity ∂ s / ∂ x ₃ with respect to the position s of the reattachment point 25 of the system is calculated. There are a plurality of methods for calculating the sensitivity, but the procedure is shown below as an example.

In the system shown in FIG. 10, by solving the same model equation using the same numerical analysis method as that used in S501, the numerical analysis in which the pitch is Pmm and the tube diameter of the tube group is changed from D to D + D'mm is used. As a result, it is assumed that the pressure loss x _3A per unit length of the tube group due to the flow orthogonal to the tube group 23 obtained in S519 is changed to x _3B .

In the original system shown in FIG. 7, the same model equation as that used in S501 is the numerical analysis of the system in which the pipe diameter is changed to D + D'mm without changing the pitch of the tube group 23 in Pmm. Is solved by the same numerical analysis method, and the position of the reattachment point 25 obtained thereby is defined as s ₄ .

Using the distance s ₁ the reattachment point 25 obtained from the numerical analysis in system of FIG. 7 that the results obtained and was carried out at S501 in S515～S521, the pressure loss per tube bundle unit length, 7 The sensitivity with respect to the position s of the redeposition point 25 in the system shown in FIG.

Next, “the position s of the reattachment point 25 in the back step flow (without the tube group)” is an amount for evaluating the numerical analysis error of the second elementary phenomenon “back step flow (without the tube group)”. , The sensitivity ∂s / ∂x ₄ with respect to the position s of the redeposition point 25 in the system shown in FIG. 7 is calculated. An example of this calculation method will be described below.

In the analysis for the model of FIG. 11, an analysis is performed with some parameters changed so that the position s of the reattachment point 25 changes. Here, the inflow flow rate per unit time is changed from Q to Q + Q ′, and the numerical analysis at that time is performed by solving the same model equation as that used in S501 by the same numerical analysis method, and as a result, obtained in S519. position s of the reattachment point _{25, x 4A} is to have changed to _{x 4B.}

Similarly, in the numerical analysis in the system of FIG. 7 performed in S501, a numerical analysis in which the inflow flow rate per unit time is changed from Q to Q + Q ′ is performed to see the influence on the reattachment point distance. If only the flow rate is changed in the system of FIG. 7, the tube group pressure loss, which is another simple phenomenon, also changes. Since the calculation of the sensitivity here needs to be performed independently for each simple phenomenon, the tube diameter D of the tube group 23 is determined so that the pressure loss does not change when the flow rate is Q + Q ′, and such a tube diameter is obtained. Numerical analysis is performed on the assumption that the tube group 23 having D is provided in the outflow portion 22 of FIG. If the tube diameter D satisfying this condition is not known in advance, it may be obtained by performing a plurality of numerical analyzes by trial and error. The location s of the reattachment point 25 obtained at this time is s _5.

Using this result, the sensitivity of the position s of the reattachment point 25 in the backstep flow without the tube group to the position s of the reattachment point 25 in the system shown in FIG.

(12) Step S525
At _{S503~S523, ∂ s / ∂x 1,} ∂ s / ∂x 2, ∂ s / ∂x 3, ∂ s / ∂x 4, _{and, ε x1, ε x2, ε} x3, ε x4, is calculated It was done. Using these, the error width ε _s of the position s of the reattachment point 14 in the numerical solution of the system of FIG. 7 obtained in S501 is evaluated by the following equation.

That is, in the method of Example 5 of the present invention, the error of the position s ₁ of the reattachment point 25 obtained by the numerical analysis of S501 was estimated to be ± ε _s .

  According to the fifth embodiment of the error estimation apparatus 1, it is possible to use a document value by decomposing a numerical analysis error into a simple phenomenon, and even if a test is performed, the error is estimated only by a small-scale test. It becomes possible. For example, when evaluating errors in numerical solutions such as the flow on the lower plenum and shroud head of a boiling water reactor with a large number of tube groups, it would be very expensive to test the entire system. The simple system test can reduce the number of tubes and only a small system test, so a large cost reduction effect can be expected.

Next, a sixth embodiment of the error estimation apparatus 1 according to the present invention will be described with reference to FIG. In the sixth embodiment, the processes of the procedure C and S217 in the second embodiment are improved.

  First, the CPU 2 performs numerical analysis of a “certain phenomenon” in the entire system, and obtains an attention value y from the numerical solution (S601).

  The CPU 2 performs the same processing as the procedure A to the procedure B (S203 to S213) of the second embodiment as the procedure A to the procedure B (S603 to S613).

  In the procedure C (S615 to S625), a factor relating to a numerical analysis error is handled as a factor (error factor) that causes an error of the attention value y.

  CPU2 performs the process of S515-S523 of Example 5 as a process of S615-S623.

CPU2 performs numerical analysis of each x _i value is "there Symptoms""ThereSymptoms" analysis condition twice or more changing the (parameter), such as boundary conditions and dimensions to vary in, x _i as the sensitivity on the target value y changes in value in the numerical solution of "an Symptoms", and ∂y / ∂x _i is the sensitivity of linear (primary), secondary sensitivity ∂ ^{2 y} / ∂x _i ² Is obtained (S625). However, the model equation to be solved, the numerical analysis method used, and the sensitivity calculation method are the same as in S207.

The CPU 2 obtains ε _y , which is a value determined from the error width of the value of the target value y or the standard deviation of the error in the numerical solution of “a certain phenomenon”, according to the following equation (25) (S627).

According to the sixth embodiment of the error estimation apparatus 1, since ε _y is obtained up to the quadratic term of Taylor expansion, the error of the value of the attention value y in the numerical solution of the “certain phenomenon” obtained thereby. The estimation accuracy of the width ε _y is theoretically improved compared to the value obtained in the fifth embodiment.

In the equations of ε _y described in the first to sixth embodiments, evaluation is separately performed on the side larger than the attention value y and the smaller side, and the error ε _y− on the smaller side and the error ε _{y +} on the larger side are expressed as It may be used. For example, if the embodiment 5, the evaluation of the error range of the target value y by the equation (26) below, the nature of the error factors x _i and the error range epsilon _xi included in the right side of the error evaluation, Featured value y Are separately evaluated on the large side and the small side, and the error ε _y− on the small side and the error ε _{y +} on the large side are used. In this case, the range of the value considering the error width of the attention value y is the range from y− | ε _y− | to y + | ε _{y +} |.

For example, in the expression of ε _y in Example 3, when it is known that only the numerical analysis error ε _anal is an error that necessarily increases the attention value y,

The block diagram of the error estimation apparatus which concerns on this invention. The flowchart which shows the procedure at the time of the error estimation apparatus which concerns on this invention performing an error estimation process (Example 1). The flowchart which shows the procedure (Example 2) when the error estimation apparatus which concerns on this invention performs an error estimation process. The flowchart which shows the procedure at the time of the error estimation apparatus which concerns on this invention performing an error estimation process (Example 3). The flowchart which shows the procedure at the time of the error estimation apparatus which concerns on this invention performing an error estimation process (Example 4). The flowchart which shows the procedure at the time of the error estimation apparatus which concerns on this invention performing an error estimation process (Example 5). The figure which shows the example of the system used as the object which the error estimation apparatus which concerns on this invention performs an error estimation process. The figure explaining the typical back step flow in the system of FIG. The top view which looked at the pipe group in the system of Drawing 7 from the top. The figure which shows the system for carrying out the numerical analysis of the simple phenomenon "flow in a pipe group" in the system of FIG. The figure which shows the system for carrying out the numerical analysis of the simple phenomenon "the back step flow (there is no tube group)" in the system of FIG. The flowchart which shows the procedure (Example 6) when the error estimation apparatus which concerns on this invention performs an error estimation process.

Explanation of symbols

1 Error estimation device 2 CPU
20 Inflowing flow 21 Inflowing portion 22 Outflowing portion 23 Tube group 24 Streamline 25 of inflowing fluid Reattachment point 26 Backflow region

Claims (12)

In an error estimation device for estimating an error of a predetermined attention value in a numerical solution obtained when performing a numerical analysis on a model equation for a predetermined phenomenon,
The geometric dimension of the area targeted by the numerical analysis is used as an error factor, and there is an error factor that can be clearly ignored for the error of the attention value in the numerical solution of the predetermined phenomenon. The geometric sensitivity calculation means for calculating the sensitivity that the error width and the increase / decrease in the value of the error factor affect the attention value in the numerical solution of the predetermined phenomenon,
The input value given by the input device when performing the numerical analysis is an error factor, and there is an error factor that can be considered to be clearly negligible for the error of the attention value in the numerical solution of the predetermined phenomenon. The input value sensitivity calculation means for calculating the sensitivity of the error range and the increase / decrease in the value of the error factor to the attention value in the numerical solution of the predetermined phenomenon,
Numerical analysis error width acquisition means for acquiring, as an error factor, the error width of the numerical analysis error that the attention value has in the numerical solution of the predetermined phenomenon;
The error width and sensitivity error factor calculated by the geometric sensitivity calculation means, the error width and sensitivity error factor calculated by the input value sensitivity calculation means, and the numerical analysis error width acquisition means An error estimation means for estimating an error of the attention value using an error factor of an error width;
An error estimation device comprising: In an error estimation device for estimating an error of a predetermined attention value in a numerical solution obtained when performing a numerical analysis on a model equation for a predetermined phenomenon,
The geometric dimension of the region targeted by one or more of the numerical analyzes is defined as an error factor x _i, and the influence of the error factor on the error of the attention value in the numerical solution of the predetermined phenomenon is clearly negligible. error factor is to be regarded as not present, error range epsilon _xi, and geometric sensitivity increase or decrease of the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon A calculation means;
The input value supplied from the input device when performing one or more of the numerical analysis and error factors x _i, considered to have influence on the error with the target values in the numerical solution of the given phenomenon of this error factor can apparently negligible error factor is to be regarded as not present, error range epsilon _xi, and the input value sensitivity calculation increase or decrease of the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon Means,
Numerical analysis error width acquisition means for acquiring, as an error factor, an error width ε _anal of a numerical analysis error of a target value in the numerical solution of the predetermined phenomenon;
The calculated by the geometric sensitivity calculating means error range epsilon _xi and sensitivity ∂y / ∂x _i, the error width epsilon _xi calculated by the input value sensitivity calculating means and sensitivity ∂y / ∂x _i and the numerical Error estimation means for estimating the error ε _{y of the} attention value by the following equation (a) using the error width ε _anal acquired by the analysis error width acquisition means;

An error estimation device comprising: The geometric sensitivity calculating means may increase or decrease the value of the error factors x _i is calculated sensitivity ∂ ^{2 y} / ∂x _i ² on the target value y in the numerical solution of the given phenomenon,
The input value sensitivity calculation means calculates the sensitivity ∂ ² y / ∂x _i ² that the increase or decrease in the value of the error factor x _i has on the attention value y in the numerical solution of the predetermined phenomenon,
The error estimation unit includes an error width ε _xi , sensitivity ∂y / ∂x _i , sensitivity ∂ ² y / ∂x _i ² , and numerical analysis error calculated by the geometric sensitivity calculation unit and the input value sensitivity calculation unit. Using the error width ε _anal acquired by the width acquisition means, the error ε _{y of the} attention value is estimated by the following equation (b):

The error estimation device according to claim 2. In an error estimation device for estimating an error of a predetermined attention value in a numerical solution obtained when performing a numerical analysis on a model equation for a predetermined phenomenon,
The geometric dimension of the region targeted by one or more of the numerical analyzes is defined as an error factor x _i, and the influence of the error factor on the error of the attention value in the numerical solution of the predetermined phenomenon is clearly negligible. error factor is to be regarded as not present, error range epsilon _xi, and geometric sensitivity increase or decrease of the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon A calculation means;
The input value supplied from the input device when performing one or more of the numerical analysis and error factors x _i, considered to have influence on the error with the target values in the numerical solution of the given phenomenon of this error factor can apparently negligible error factor is to be regarded as not present, error range epsilon _xi, and the input value sensitivity calculation increase or decrease of the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon Means,
Numerical analysis error width acquisition means for acquiring, as an error factor, an error width ε _anal of a numerical analysis error of a target value in the numerical solution of the predetermined phenomenon;
Error range calculated by the geometric sensitivity calculating means epsilon _xi and sensitivity ∂y / ∂x _i, the error width epsilon _xi and sensitivity ∂y / ∂x _i and the numerical analysis calculated by the input value sensitivity calculating means Using the error width ε _anal acquired by the error width acquisition means, error estimation means for estimating the error ε _{y of the} attention value by the following equation (c);

An error estimation device comprising: The geometric sensitivity calculating means may increase or decrease the value of the error factors x _i is calculated sensitivity ∂ ^{2 y} / ∂x _i ² on the target value y in the numerical solution of the given phenomenon,
The input value sensitivity calculation means calculates the sensitivity ∂ ² y / ∂x _i ² that the increase or decrease in the value of the error factor x _i has on the attention value y in the numerical solution of the predetermined phenomenon,
The error estimation unit includes an error width ε _xi , sensitivity ∂y / ∂x _i , sensitivity ∂ ² y / ∂x _i ² , and numerical analysis error calculated by the geometric sensitivity calculation unit and the input value sensitivity calculation unit. Using the error width ε _anal acquired by the width acquisition means, the error ε _{y of the} attention value is estimated by the following equation (d):

The error estimation apparatus according to claim 4. In an error estimation device for estimating an error of a predetermined attention value in a numerical solution obtained when performing a numerical analysis on a model equation for a predetermined phenomenon,
The geometric dimension of one or more regions targeted for the numerical analysis is defined as an error factor x _i, and the influence of the error factor x _{i on} the error of the attention value y in the numerical solution of the predetermined phenomenon is clarified. error factors thought to be negligible is deemed not to exist, an error width epsilon _xi, and increase or decrease the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon Geometric sensitivity calculation means;
The input value given from the input device when performing one or more numerical analyzes is defined as an error factor x _i, and the influence of the error factor x _{i on} the error of the attention value y in the numerical solution of the predetermined phenomenon is clearly ignored. error factors thought to be able to be regarded as not present, error range epsilon _xi, and inputs changes in the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon Value sensitivity calculation means;
The predetermined phenomenon is decomposed into one or more simple phenomena, one or more evaluation quantities are defined in a numerical solution of each simple phenomenon, and a difference between values obtained by a plurality of methods for the evaluation quantities is defined as an error width ε _xi . as well as the numerical analysis sensitivity calculating means increases or decreases the evaluation of each simple phenomenon to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon,
Error range calculated by the geometric sensitivity calculating means epsilon _xi and sensitivity ∂y / ∂x _i, the error width epsilon _xi and sensitivity ∂y / ∂x _i and the numerical analysis calculated by the input value sensitivity calculating means an error estimation means using the error width epsilon _xi and sensitivity ∂y / ∂x _i calculated to estimate the error epsilon _y of the target value by the following equation (e) by the sensitivity calculating means,

An error estimation device comprising: The geometric sensitivity calculating means calculates sensitivity ∂ ^{2 y} / ∂x _i ² which increase or decrease the value of the error factors x _i is on target value y in the numerical solution of the given phenomenon,
The input value sensitivity calculation means calculates the sensitivity ∂ ² y / ∂x _i ² that the increase or decrease in the value of the error factor x _i has on the attention value y in the numerical solution of the predetermined phenomenon,
The numerical analysis sensitivity calculation means calculates a sensitivity ∂ ² y / ∂ _xi ² that an increase or decrease in an evaluation amount of a simple phenomenon has on a target value y in a numerical solution of the predetermined phenomenon,
Said error estimation means, the geometric sensitivity calculating means, the input value sensitivity calculating means, and numerical analysis sensitivity error width epsilon _xi calculated by the calculation _means, the sensitivity ∂y / ∂x _i, sensitivity ∂ ^{2 y} / ∂x _{Using i} ² , the error ε _{y of the} attention value is estimated by the following equation (f):

The error estimation apparatus according to claim 6. The geometric sensitivity calculation means and the input value sensitivity calculation means obtain the error width ε _xi separately for the side that increases the attention value y and the side that decreases the attention value y , respectively, while the numerical analysis The error width acquisition means acquires the error width ε _anal separately for a side that increases the attention value y and a side that decreases the attention value y ,
The error estimator calculates the error widths ε _xi and ε _anal obtained separately for the side that increases the attention value y and the side that decreases the attention value y, from the equations (a), (b), (c ) And (d), respectively, and calculating an error ε _y− on the smaller side of the target value y and an error ε _{y +} on the larger side of the target value y, thereby calculating the value of the target value y. The error ε _y− on the smaller side and the error ε _{y +} on the larger side of the attention value y are estimated , and at the time of the substitution , at least one error of the error widths ε _xi and ε _anal is necessarily the attention value y. If it is known that the error is on the side of increasing the target value y, in the calculation for estimating the error ε _y− on the side of the small target value y, a term including an error on the side of increasing the target value y is necessarily included. while ignoring to calculating the error range ε upon the assignment If it is known that at least one error of _i and epsilon _anal are errors on the side always reduce the attention value y, in the calculation for estimating the error epsilon _{y +} a larger side of the target value y, always The error estimation apparatus according to claim 2, wherein the calculation is performed while ignoring a term including an error on the side of reducing the attention value y . The geometric sensitivity calculation means, the input value sensitivity calculation means, and the numerical analysis sensitivity calculation means each have an error width ε _xi Are divided into a side for increasing the attention value y and a side for decreasing the attention value y,
The error estimator obtains an error width ε obtained separately for a side for increasing the attention value y and a side for decreasing the attention value y. _xi Is substituted into any one of the equations (e) and (f), and the error ε on the smaller side of the attention value y is substituted. _y- And the error ε on the larger side of the attention value y _{y +} By calculating the error ε on the smaller side of the target value y _y- And the error ε on the larger side of the attention value y _{y +} And the error width ε during the substitution _xi When it is known that at least one of the errors is an error on the side of increasing the attention value y, an error ε on the side of the smaller attention value y _y- In the calculation for estimating the error value, the calculation is performed while ignoring the term including the error on the side where the attention value y is increased. _xi If it is known that at least one of the errors is necessarily an error on the side where the attention value y is reduced, the error ε on the side where the attention value y is larger _{y +} The error estimation apparatus according to claim 6 or 7, wherein in the calculation for estimating the value, the calculation is performed while ignoring a term including an error on the side of reducing the attention value y. When performing numerical analysis on a model equation for a predetermined phenomenon, a method of estimating an error of a predetermined attention value by using a computer in a numerical solution obtained ,
The geometric sensitivity calculation means uses the geometric dimension of the area targeted by the numerical analysis as an error factor, and the influence of the error factor on the error of the attention value in the numerical solution of the predetermined phenomenon is clearly ignored. A geometric sensitivity calculation step for calculating the sensitivity that the error factor and the increase / decrease in the value of the error factor calculate the sensitivity to the attention value in the numerical solution of the predetermined phenomenon, assuming that there is no error factor that can be considered .
The input value sensitivity calculating means uses the input value given from the input device when performing the numerical analysis as an error factor, and the influence of the error factor on the error of the attention value in the numerical solution of the predetermined phenomenon can be clearly ignored. An input value sensitivity calculation step for calculating the sensitivity of the error width and the increase / decrease in the value of the error factor to the attention value in the numerical solution of the predetermined phenomenon ,
Numerical analysis error width acquisition means, a numerical analysis error width acquisition step of acquiring, as an error factor, an error width of a numerical analysis error of a target value in the numerical solution of the predetermined phenomenon;
The error estimation means includes an error width and sensitivity error factor calculated in the geometric sensitivity calculation step, an error width and sensitivity error factor calculated in the input value sensitivity calculation step, and the numerical analysis error. an error estimation step using the error factor of the acquired error range in width acquisition step, to estimate the error of the target value,
Error estimation method characterized by comprising a. When performing numerical analysis on a model equation for a predetermined phenomenon, a method of estimating an error of a predetermined attention value by using a computer in a numerical solution obtained ,
The geometric sensitivity calculation means sets one or more geometric dimensions of the region targeted by the numerical analysis as an error factor x _i, and out of the error factor x _i , the target value y in the numerical solution of the predetermined phenomenon Assuming that there is no error factor whose influence on the error it can be clearly ignored , the sensitivity 存在 y that the error width ε _xi and the increase / decrease in the value of the error factor have on the attention value y in the numerical solution of the predetermined phenomenon geometric sensitivity calculating a / ∂x _i,
The input value given from the input device when performing one or more numerical analyzes is defined as an error factor x _i, and the influence of the error factor x _{i on} the error of the attention value y in the numerical solution of the predetermined phenomenon is clearly ignored. error factors thought to be able to be regarded as not present, error range epsilon _xi, and inputs changes in the value of the error factor to calculate the sensitivity ∂y / ∂x _i on target value y in the numerical solution of the given phenomenon A value sensitivity calculation step;
A numerical analysis error width acquisition step of acquiring, as an error factor, an error width ε _anal of a numerical analysis error of a target value in the numerical solution of the predetermined phenomenon;
The calculated by the geometric sensitivity calculating step error range epsilon _xi and sensitivity ∂y / ∂x _i, the calculated in the input value sensitivity calculating step the error range epsilon _xi and sensitivity ∂y / ∂x _i and Using the error width ε _anal acquired in the numerical analysis error width acquisition step, an error estimation step of estimating the error ε _{y of the} attention value by the following equation (a) :

Error estimation method characterized by comprising a. When performing numerical analysis on a model equation for a predetermined phenomenon, a method of estimating an error of a predetermined attention value by using a computer in a numerical solution obtained,
The geometric sensitivity calculation means calculates one or more geometric dimensions as an error factor x. _i And this error factor x _i It is assumed that there is no error factor that the influence on the error of the attention value y in the numerical solution of the predetermined phenomenon can be clearly ignored, and the error width ε _xi , And the sensitivity ∂y / ∂x that the increase or decrease of the error factor value has on the attention value y in the numerical solution of the predetermined phenomenon _i A geometric sensitivity calculating step for calculating
The input value sensitivity calculating means converts an input value of 1 or more into an error factor x _i And this error factor x _i It is assumed that there is no error factor that the influence on the error of the attention value y in the numerical solution of the predetermined phenomenon can be clearly ignored, and the error width ε _xi , And the sensitivity ∂y / ∂x that the increase or decrease of the error factor value has on the attention value y in the numerical solution of the predetermined phenomenon _i An input value sensitivity calculating step for calculating
Numerical analysis sensitivity calculation means decomposes the predetermined phenomenon into one or more simple phenomena, defines one or more evaluation quantities in the numerical solution of each simple phenomenon, and obtains values obtained by a plurality of methods for the evaluation quantities. Difference to error width ε _xi And the sensitivity ∂y / ∂x that the increase or decrease in the evaluation amount of each simple phenomenon has on the attention value y in the numerical solution of the predetermined phenomenon _i Numerical analysis sensitivity calculation step for calculating
The error width ε calculated in the geometric sensitivity calculation step _xi And sensitivity ∂y / ∂x _i , Error width ε calculated in the input value sensitivity calculation step _xi And sensitivity ∂y / ∂x _i And the error width ε calculated in the numerical analysis sensitivity calculation step. _xi And sensitivity ∂y / ∂x _i Using the following equation (e), the error ε of the attention value _y An error estimation step for estimating

An error estimation method comprising:

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