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CN114969029A - Taylor expansion-based multi-dimensional interpolation scheme and inter-dimensional point value taking method thereof - Google Patents

Taylor expansion-based multi-dimensional interpolation scheme and inter-dimensional point value taking method thereof Download PDF

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CN114969029A
CN114969029A CN202210426534.5A CN202210426534A CN114969029A CN 114969029 A CN114969029 A CN 114969029A CN 202210426534 A CN202210426534 A CN 202210426534A CN 114969029 A CN114969029 A CN 114969029A
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CN114969029B (en
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罗连潭
黄向华
张天宏
蔡晓晓
伏子豪
邓晶蕾
闵思凯
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Nanjing University of Aeronautics and Astronautics
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Abstract

本发明涉及一种基于泰勒展开的多维插值方案及其维间点取值方法，属于离散数学应用领域。本发明公开的方法包括：插值方法，插值表存储法和插值表维间取值方法，所述的插值方法以多元泰勒展开为载体，所述的插值表存储法是指通过一个二维区间数组和一维值域数组进行数据存储，所述的二维区间数组只存储各维间的独立的维间点值，所述的插值表维间取值方法采用的是通过对目标函数的量纲分析来求解各维自变量的阶次，依据阶次选择最少的维间点的个数。本发明基于一阶多元泰勒展开降低了计算量，并通过二维区间数组降低了数据存储空间和数据调用算法的复杂性，还利用量纲分析出各维变量的阶次，以便在不破坏变化规律的前提下选择最经济的维间点个数。

Figure 202210426534

The invention relates to a multi-dimensional interpolation scheme based on Taylor expansion and a method for selecting values between dimensions, belonging to the application field of discrete mathematics. The method disclosed in the present invention includes: interpolation method, interpolation table storage method and interpolation table dimension value selection method, the interpolation method uses multivariate Taylor expansion as a carrier, and the interpolation table storage method refers to a two-dimensional interval array Perform data storage with a one-dimensional range array, the two-dimensional interval array only stores the independent inter-dimensional point values between each dimension, and the inter-dimensional value selection method of the interpolation table adopts the dimension of the objective function. Analyze to solve the order of independent variables in each dimension, and select the minimum number of points between dimensions according to the order. The invention reduces the amount of calculation based on the first-order multivariate Taylor expansion, reduces the data storage space and the complexity of the data calling algorithm through the two-dimensional interval array, and also uses the dimension to analyze the order of each dimension variable, so as not to destroy the change Under the premise of regularity, select the most economical number of interdimensional points.

Figure 202210426534

Description

Taylor expansion-based multi-dimensional interpolation scheme and inter-dimensional point value taking method thereof
Technical Field
The invention relates to a multi-dimensional interpolation scheme based on Taylor expansion and an inter-dimensional point value taking method thereof, belonging to the field of discrete mathematics application.
Background
When a complex multi-influence-factor physical model is described, an objective function f is often used for description, but an analytical expression which explicitly contains each factor does not exist in f, and the solution equation is complex. When the problems are researched, high-precision numerical simulation calculation time and cost are adopted, the experiment table building cost and the experiment cost are also high by adopting an experiment method, and the intrinsic physical law can be researched only by simulating or experimenting a small number of discrete data points on the premise of limited scientific research expenditure. As a more intensive research, even a control rule needs to be designed for the physical model according to the output f, and in this case, the real-time requirement on the calculation of the objective function f is high.
The interpolation is an important method for approximating a discrete function, and the approximate values of the function at other points can be estimated by utilizing the value conditions of the function at a limited number of points, and meanwhile, the calculation time is shorter because the calculation formula of the interpolation is simpler. Therefore, interpolation is the first choice for describing the above physical model in the early stage of research to obtain a low-precision real-time estimation model of the objective function f in any state.
The most interpolation used at present is one-dimensional linear interpolation, two-dimensional bilinear interpolation and the like, and an interpolation result can be obtained according to a fixed formula. Higher dimensional interpolation lacks a general efficient solution. Patent publication No. CN110674133A proposes a compressed storage and calculation method for high-dimensional interpolation, which solves the problem of solving higher-dimensional interpolation. The interpolation calculation method adopts a recursion calculation method from leaves to roots, the calculation is the calculation of layer-by-layer weight and the final average, the calculated amount is
Figure BSA0000271787260000011
The method thereof is yet to be improved; in the aspect of data storage, a deep storage tree is adopted, if the deep storage tree is storedIn the two-dimensional array, many positions are '0' due to the asymmetry of the matrix, unnecessary memory is wasted, and the calling of a nested loop algorithm is inconvenient when a plurality of one-dimensional arrays are defined. In addition, in some cases, in order to reduce the manufacturing cost of the interpolation table, the number of discrete points, that is, the number of values of the inter-dimensional points, has to be reduced, but there is no scientific guidance.
Disclosure of Invention
The invention aims to provide a multi-dimensional interpolation scheme based on Taylor expansion and an inter-dimensional point value taking method thereof, aiming at further simplifying compressed storage for a deep storage tree, reducing memory occupation and facilitating the use of a nested loop algorithm; a more general interpolation method based on a first-order multivariate Taylor expansion is provided, so that the calculated amount is further reduced; and an inter-dimensional point value taking method of the multi-dimensional interpolation table is provided, so that more economic scientific guidance is provided for making the interpolation table.
In order to achieve the purpose, the invention provides the following technical scheme:
a multi-dimensional interpolation scheme based on Taylor expansion and an inter-dimensional point value taking method thereof comprise the following steps: the interpolation method comprises a multivariate Taylor expansion and data calling algorithm, the interpolation table storage method is characterized in that data storage is carried out through a two-dimensional interval array A [ I ] [ N ] and a one-dimensional value domain array B [ K ], the interpolation table storage data are obtained through a numerical simulation or experiment means, the two-dimensional interval array only stores independent inter-dimension point values among dimensions, the independent inter-dimension points refer to the number and the numerical value of inter-dimension points of N + 1-dimensional multi-dimensional tree diagram branches corresponding to each inter-dimension point of the nth dimension, the interpolation table inter-dimension value taking method adopts the steps of solving the order of each dimension independent variable through dimensional analysis of a target function, and the number of the least inter-dimension points is selected according to the order;
the two-dimensional interval array A [ I ]][N]Representing a two-dimensional array with rows I and columns N, where A [0 ]][n]Represents the minimum value of the nth dimension, A [ I ] n -1][n]Representing the maximum value of the nth dimension, constructed to implement a full range interpolation within the limit, I n Represents the nth dimensionNumber of discrete points between dimensions of variable, and I ═ max { I ═ I 1 ，I 2 ，…，I n ，…，I N }；
The multi-dimensional interpolation scheme based on Taylor expansion is characterized in that the interpolation method takes multi-element Taylor expansion as a carrier, a data calling algorithm is needed in the process of solving the multi-element Taylor expansion, wherein f is a point
Figure BSA0000271787260000021
At any point in a field of
Figure BSA0000271787260000022
The first order expansion of the discrete form multivariate taylor of (1) is:
Figure BSA0000271787260000023
wherein,
Figure BSA0000271787260000024
the function value is to be calculated;
Figure BSA0000271787260000025
for the nth dimension, i n Coordinate value of point between dimensions, i.e. two-dimensional interval array A [ I ]][N]N th column of (1), i th n A row;
Figure BSA0000271787260000026
is f pairs
Figure BSA0000271787260000027
A partial increase function value of; eta n For the coordinate increment to be solved of the nth dimension variable, the above formula represents lower boundary value field index interpolation, or upper boundary value field index interpolation, which are equal on the premise that each dimension variable is an independent inter-dimension point, wherein the upper boundary value field index interpolation formula is (wherein, the definition of each variable is the same as that of the lower boundary value field index interpolation):
Figure BSA0000271787260000028
the data calling algorithm takes the following boundary value domain index interpolation formula as an example and comprises the following steps:
firstly, according to the coordinate value to be solved, adopting golden section method to make A [ I ]][N]Performing double nested loop search to calculate the coordinate increment eta of n dimensional variables to be solved 1 ，η 2 ，…，η n And coordinates (i) 1 ，i 2 ，…，i N )；
② according to the coordinate (i) 1 ，i 2 ，…，i N ) To calculate a value range index value h N Here, the method of double nested loop search is also used, where a parameter Δ is defined, where n > 0, and Δ ═ 0 denotes the evaluation
Figure BSA0000271787260000029
Is the function value index of (a) n, which means that the solution is the partial increase function value
Figure BSA0000271787260000031
When Δ ≠ n, it is necessary to determine the coordinate (i) 1 ，i 2 ，…，i n +1，…，i N ) Calculating a value domain index value, and embedding a judgment statement in the loop statement to judge whether the coordinate of the nth dimension is added with 1;
the value of f to be searched, which is associated with the one-dimensional value range array B [ h ] N -1]Corresponds to the value of (b), wherein h of the nth dimension n The calculation formula of (A) is as follows:
h n ＝(h n-1 -1)·I n-1 +i n .
wherein h is n Represents the nth dimension interpolation value domain index to be solved, h n-1 Denotes an n-1-dimensional interpolation value domain index, I n-1 Representing a two-dimensional interval array A [ I ]][N]The number of n-1 th columns;
solving interpolation function values according to the items;
as an option of the invention with higher interpolation precision, the method can adopt the multivariate Taylor expansion of a high-order form instead of the multivariate Taylor expansionPrice is higher calculation amount and memory resource, and because of the definition of discrete high-order derivative, the order is increased once, and an additional data point is needed to be added on each dimension, the cost of interpolation table data is increased, if the calculation amount of the first-order multivariate Taylor is N, the Q order is
Figure BSA0000271787260000032
The advantages of the present invention are lost with the use of a high order taylor expansion, where the Q order taylor expansion is:
Figure BSA0000271787260000033
wherein,
Figure BSA0000271787260000034
expressed as a variable x of the nth dimension n The higher order requires recursive solution with discrete forms of the lower order.
The Taylor expansion-based multi-dimensional interpolation scheme and the inter-dimensional point value taking method thereof are characterized in that the inter-dimensional point value taking method of the interpolation table solves the order of each dimension independent variable through dimension analysis of an objective function, and selects the minimum number of the inter-dimensional points according to the order so as to reduce the manufacturing economic cost of the interpolation table;
the dimension analysis comprises the following steps:
firstly, the main influencing factors of the researched phenomenon are selected manually according to experience
Figure BSA0000271787260000035
Selecting independent variables which can completely cover the basic physical quantity according to the dimension of the basic physical quantity of the described phenomenon and recording the variables as m variables;
writing out expression of n-m similarity criterion pi according to pi theorem, in which any one similarity criterion pi i Can be expressed as a function of the remaining n-m-1 similarity criteria, requiring the selection of a phase containing the objective functionSimilarity criterion pi i Thus, the expression of the objective function is solved as:
Figure BSA0000271787260000036
wherein the analysis can be performed by means of an existing expression of the target parameter in order to obtain f 1 With respect to intermediate variables
Figure BSA0000271787260000037
Constitutive relation of (f) 1 (x 1 ，x 2 ，…，x N )；
The constitutive relation f 1 (x 1 ，x 2 ，…，x N ) It is difficult to determine that f is x when other dimensional variables are constants n If the independent function is a linear constitutive relation, there is
Figure BSA0000271787260000038
I.e. f 1 (x n ) C, but the real case is due to the influence of other dimensional parameters, complex functions
Figure BSA0000271787260000041
The structure of (a) is much more complicated, and f (x) can be approximated when no constitutive relation is analyzed 1 ，x 2 ，…，x N ) The description is as follows:
Figure BSA0000271787260000042
wherein, g n ＞1，a n ，g 0 ，a 0 B, c are unknown coefficients that may vary with variations in other dimensional variables, a 0 B, only the curve is translated left and right and up and down, and the selection of the number of discrete points is not influenced;
order for independent variable is g n And g is n Greater than 0, the number of interdimensional points is at least g n + 1; order g for independent variable n When the number is less than 0, the number of the inter-dimensional points is at least 3;
when the nth dimension variable can be approximated to a periodic function with the period of T, the change rule of the nth dimension variable cannot be damaged at least by setting the simulation or experiment data interval of T/2 according to the fragrance concentration sampling theorem.
Compared with the prior art, the invention has the advantages that: based on the first-order multivariate Taylor expansion, the calculation amount of interpolation calculation is N +1, compared with the calculation amount of a recursive calculation method from leaves to roots
Figure BSA0000271787260000043
Obviously reducing: the complexity of a data storage space and a data calling algorithm is reduced through a two-dimensional interval array, and the storage space is from the original deep storage tree
Figure BSA0000271787260000044
To reduce to
Figure BSA0000271787260000045
In the aspect of complexity of a data calling algorithm, the nested loop algorithm only needs to set a two-dimensional interval array A [ I ]][N]The row and column circulation can realize the search calculation of the increment of each dimension; finally, a method for obtaining the order of each dimension variable and the minimum discrete point of the inter-dimension points corresponding to each order by using dimension analysis is provided, so that the most economical number of the inter-dimension points is selected on the premise of not damaging the change rule.
Drawings
FIG. 1 is a corresponding diagram of a two-dimensional interval array A [ I ] [ N ] and a one-dimensional value range array B [ K ] and a multi-dimensional tree diagram according to the present invention.
FIG. 2 is a data call algorithm of the present invention.
FIG. 3 is a graph of the present invention for g n ＝4，a n ＝3，g 0 ＝2，a 0 An example of the minimum dimension point method is 1, 0, and 1.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1, in an embodiment of the present invention, a multi-dimensional interpolation scheme based on taylor expansion and a method for evaluating points between dimensions thereof include: the interpolation method comprises a multivariate Taylor expansion and data calling algorithm, the interpolation table storage method is characterized in that data storage is carried out through a two-dimensional interval array A [ I ] [ N ] and a one-dimensional value domain array B [ K ], the interpolation table storage data are obtained through a numerical simulation or experiment means, the two-dimensional interval array only stores independent inter-dimension point values among dimensions, the independent inter-dimension points refer to the number and the numerical value of inter-dimension points of N + 1-dimensional multi-dimensional tree diagram branches corresponding to each inter-dimension point of the nth dimension, the interpolation table inter-dimension value taking method adopts the steps of solving the order of each dimension independent variable through dimensional analysis of a target function, and the number of the least inter-dimension points is selected according to the order;
the two-dimensional interval array A [ I ]][N]Representing a two-dimensional array with rows I and columns N, where A [0 ]][n]Represents the minimum value of the nth dimension, A [ I ] n -1][n]Representing the maximum value of the nth dimension, constructed to implement a full range interpolation within the limit, I n Denotes the number of discrete points between dimensions of the nth dimension variable, and I ═ max { I 1 ，I 2 ，…，I n ，…，I N }；
The multi-dimensional interpolation scheme based on Taylor expansion is characterized in that the interpolation method takes multi-element Taylor expansion as a carrier, a data calling algorithm is needed in the process of solving the multi-element Taylor expansion, wherein f is a point
Figure BSA0000271787260000051
At any point in a field of
Figure BSA0000271787260000052
The first order expansion of the discrete form multivariate taylor of (1) is:
Figure BSA0000271787260000053
wherein,
Figure BSA0000271787260000054
the function value is to be calculated;
Figure BSA0000271787260000055
for the nth dimension, i n Point coordinate value between dimensions, i.e. two-dimensional interval array A [ I ]][N]N th column of (1), i th n A row;
Figure BSA0000271787260000056
is f to
Figure BSA0000271787260000057
A partial increase function value of; eta n For the coordinate increment to be solved of the nth dimension variable, the above formula represents lower boundary value field index interpolation, or upper boundary value field index interpolation, which are equal on the premise that each dimension variable is an independent inter-dimension point, wherein the upper boundary value field index interpolation formula is (wherein, the definition of each variable is the same as that of the lower boundary value field index interpolation):
Figure BSA0000271787260000058
referring to fig. 2, in the embodiment of the present invention, the data call algorithm, for example, has the following boundary value domain index interpolation formula, and includes the following steps:
firstly, according to the coordinate value to be solved, adopting golden section method to make A [ I ]][N]Performing double nested loop search to calculate the coordinate increment eta of n dimensional variables to be solved 1 ，η 2 ，…，η n And coordinates (i) 1 ，i 2 ，…，i N )；
② according to the coordinate (i) 1 ，i 2 ，…，i N ) To calculate the value range index value hN, here againUsing a double nested loop search method, defining a parameter delta, wherein n is greater than 0, and delta is 0 to obtain
Figure BSA0000271787260000059
Is the function value index of (a) n, which means that the solution is the partial increase function value
Figure BSA00002717872600000510
When Δ ≠ n, it is necessary to determine the coordinate (i) 1 ，i 2 ，…，i n +1，…，i N ) Calculating a value domain index value, and embedding a judgment statement in the loop statement to judge whether the coordinate of the nth dimension is added with 1;
the value of f to be searched, which is associated with the one-dimensional value range array B [ hN-1 ]]Corresponds to the value of (b), wherein h of the nth dimension n The calculation formula of (A) is as follows:
h n ＝(h n-1 -1)·I n-1 +i n .
wherein h is n Representing the nth-dimensional interpolation value domain index, h, to be solved n-1 Denotes the n-1-dimensional interpolation value domain index, I n-1 Representing a two-dimensional interval array A [ I ]][N]The number of n-1 th columns;
solving interpolation function values according to the items;
as an option of the invention with higher interpolation precision, the multivariate Taylor expansion in a high-order form can be adopted, the cost is higher calculation amount and memory resource, meanwhile, because the definition of discrete high-order derivatives, each time the order is increased once, an additional data point needs to be added on each dimension, the cost of interpolation table data can be increased, if the calculation amount of the first-order multivariate Taylor is N, the Q order is N
Figure BSA0000271787260000061
The advantages of the present invention are lost with the use of a high order taylor expansion, where the Q order taylor expansion is:
Figure BSA0000271787260000062
wherein,
Figure BSA0000271787260000063
expressed as a variable x of the nth dimension n The higher order requires recursive solution with discrete forms of the lower order.
Referring to fig. 3, in the embodiment of the present invention, the taylor expansion-based multidimensional interpolation scheme and the inter-dimensional point value taking method thereof are characterized in that the inter-dimensional point value taking method of the interpolation table solves the order of each dimension independent variable through dimensional analysis of an objective function, and selects the minimum number of inter-dimensional points according to the order to reduce the economic cost for manufacturing the interpolation table;
the dimension analysis comprises the following steps:
firstly, the main influencing factors of the researched phenomenon are selected manually according to experience
Figure BSA0000271787260000064
Selecting independent variables which can all cover the basic physical quantity according to the dimension of the basic physical quantity of the described phenomenon and recording the independent variables as m variables;
writing out expression of n-m similarity criterion pi according to pi theorem, in which any one similarity criterion pi i Can be expressed as a function of the remaining n-m-1 similarity criteria, requiring the selection of a similarity criterion π comprising the objective function i Thus, the expression of the objective function is solved as:
Figure BSA0000271787260000065
wherein the analysis can be performed by means of an existing expression of the target parameter in order to obtain f 1 With respect to intermediate variables
Figure BSA0000271787260000066
Constitutive relation of (f) 1 (x 1 ，x 2 ，…，x N )；
The constitutive relation f 1 (x 1 ，x 2 ，…，x N ) Is difficult to determine atWhen other dimensional variables are constants, f is x n If the independent function is a linear constitutive relation, there is
Figure BSA0000271787260000067
I.e. f 1 (x n ) C, but the real case is due to the influence of other dimensional parameters, complex functions
Figure BSA0000271787260000068
The structure of (a) is much more complicated, and f (x) can be approximated when no constitutive relation is analyzed 1 ，x 2 ，…，x N ) The description is as follows:
Figure BSA0000271787260000071
wherein, g n ＞1，a n ，g 0 ，a 0 B, c are unknown coefficients that may vary with variations in other dimensional variables, a 0 B is only to translate the curve left and right and up and down without affecting the selection of the number of discrete points, wherein, the figure 3 is for the nth dimension variable x n G of n ＝4，a n ＝3，g 0 ＝2，a 0 The equidistant distance of 1, 0, 1 and c is the discrete interpolation effect of 5 discrete points;
order for independent variable is g n And g is n Greater than 0, the number of interdimensional points is at least g n + 1; order g for independent variable n When the number is less than 0, the number of the inter-dimensional points is at least 3;
when the nth dimension variable can be approximated to a periodic function with the period of T, the change rule of the nth dimension variable cannot be damaged at least by setting the simulation or experiment data interval of T/2 according to the fragrance concentration sampling theorem.
The dimensional analysis is performed below with specific examples:
according to observation and analysis, factors influencing the pressure loss of the flow in the pipe comprise the average flow velocity V in the pipe, the flow density rho, the diameter d of the pipeline, the length l of the pipe, the dynamic viscosity mu of the fluid and the roughness delta of the pipe wall. The pressure loss Δ p of the pressure line placed horizontally is determined experimentally.
Solution: according to the theme, the main factors influencing the pressure flow in the pipe are 7, namely the average flow velocity V in the pipe, the flow density rho, the diameter d of the pipe, the length l of the pipe, the dynamic viscosity mu of the fluid, the roughness delta of the pipe wall and the pressure loss delta p of the pipe.
V, d and rho three independent variables are selected as basic physical quantities, and the three basic dimensions comprise length [ L ], mass [ M ] and time [ t ]. Writing an expression of 7-3 ═ 4 π:
Figure BSA0000271787260000072
according to the principle that the dimensions of the numerator denominator are the same, the index (a) in each pi term can be calculated 1 ，b 1 ，c 1 ) …, of equal value, now pi 1 For example, as
Figure BSA0000271787260000073
Figure BSA0000271787260000074
Get it solved
Figure BSA0000271787260000075
Substituting the original formula to obtain a similarity criterion pi related to pressure loss 1 ＝Δp/(ρV 2 )＝Eu
Similarly, can get pi 2 ＝μ/(ρVd)＝1/Re，π 3 ＝l/d，π 4 ＝Δ/d。
Selecting a similarity criterion comprising an objective function, which is pi 1 We can relate the similarity criterion pi to pressure loss 1 Expressed as a function of the remaining 3 criteria:
π 1 ＝f(π 2 ，π 3 ，π 4 ) I.e. Eu ═ f (1/Re, l/d, Delta/d)
For the in-line flow given in this example, it was found experimentally that the pressure loss Δ p is proportional to the relative tube length l/d, and the above equation can be rewritten as
Figure BSA0000271787260000081
Namely, it is
Figure BSA0000271787260000082
Then pressure loss of pipe flow
Figure BSA0000271787260000083
Wherein, λ ═ f is defined 1 (1/Re, Δ/d), the following is an empirical formula obtained by experimental measurements:
for laminar flow λ 64/Re
For turbulent transition asperity zone λ 0.11(Δ/d +68/Re) 0.25
Lambda for turbulent completely rough tube area ═ 2lg (Δ/d) +1.72 -2
It can be seen that the dependence of the dimension analysis method on theoretical experience is high, in the case that the Reynolds number and the relative roughness are determined, the precise discrete point taking of the average flow velocity V, the flow density rho, the pipe length l and the dynamic viscosity mu of the fluid can be independently carried out, and for the parameters of other dimensions, the function can be used
Figure BSA0000271787260000084
To approximate the analysis.
The present invention is not limited to the above embodiments, and based on the technical solutions disclosed in the present invention, those skilled in the art can make some simple modifications, equivalent changes and modifications to some technical features without creative efforts based on the disclosed technical contents, and all fall into the technical solution of the present invention.

Claims (3)

1.一种基于泰勒展开的多维插值方案及其维间点取值方法，包括：插值方法，插值表存储法和插值表维间取值方法，所述的插值方法包括多元泰勒展开和数据调用算法，所述的插值表存储法是指通过一个二维区间数组A[I][N]和一维值域数组B[K]进行数据存储，插值表存储数据通过数值仿真或实验的手段获得，所述的二维区间数组只存储各维间的独立的维间点值，所述的独立的维间点是指第n维的每一个维间点对应的第n+1维的多维树状图分支的维间点的个数和数值都一样，所述的插值表维间取值方法采用的是通过对目标函数的量纲分析来求解各维自变量的阶次，依据阶次选择最少的维间点的个数；1. a multidimensional interpolation scheme based on Taylor expansion and a method for obtaining values between dimensions thereof, comprising: interpolation method, interpolation table storage method and interpolation table dimension value method, and described interpolation method comprises multivariate Taylor expansion and data calling Algorithm, the interpolation table storage method refers to data storage through a two-dimensional interval array A[I][N] and a one-dimensional range array B[K], and the data stored in the interpolation table is obtained by means of numerical simulation or experiment. , the two-dimensional interval array only stores the independent inter-dimension point values of each dimension, and the independent inter-dimension point refers to the n+1-th dimension multidimensional tree corresponding to each inter-dimension point of the n-th dimension The number and value of the inter-dimensional points of the shape graph branch are the same. The inter-dimension value selection method of the interpolation table adopts the dimensional analysis of the objective function to solve the order of the independent variables of each dimension, and selects the order according to the order. The minimum number of interdimensional points; 所述二维区间数组A[I][N]的A[0][n]表示第n维的最小值，A[In-1][n]表示第n维最大值。A[0][n] of the two-dimensional interval array A[I][N] represents the minimum value of the nth dimension, and A[I n -1][n] represents the maximum value of the nth dimension. 2.如权利要求1所述的一种基于泰勒展开的多维插值方案，其特征在于，所述的插值方法以多元泰勒展开为载体，在求解多元泰勒展开式的过程需要数据调用算法，其中f在点
Figure FSA0000271787250000011
的某一领域内的任意一点
Figure FSA0000271787250000012
的离散形式多元泰勒的一阶展开式为：2. a kind of multidimensional interpolation scheme based on Taylor expansion as claimed in claim 1, is characterized in that, described interpolation method is carrier with multivariate Taylor expansion, needs data calling algorithm in the process of solving multivariate Taylor expansion, wherein f at the point
Figure FSA0000271787250000011
any point in a field of
Figure FSA0000271787250000012
The discrete form of multivariate Taylor's first-order expansion is:
Figure FSA0000271787250000013
Figure FSA0000271787250000013
 其中，
Figure FSA0000271787250000014
为待求函数值；
Figure FSA0000271787250000015
为插值表第n维，第in个维间点坐标值，即二维区间数组A[I][N]的第n列，第in行；
Figure FSA0000271787250000016
为f对
Figure FSA0000271787250000017
的偏增函数值；ηn为第n维变量的待求坐标增量，上式表示下边界值域索引插值，也可以上边界值域索引插值，两者在各维变量都是独立的维间点的前提下是相等的，其中上边界值域索引插值公式为(其中，各变量的定义和下边界值域索引插值的定义一样)：in,
Figure FSA0000271787250000014
is the value of the function to be evaluated;
Figure FSA0000271787250000015
is the nth dimension of the interpolation table, the coordinate value of the point between the i nth dimension, that is, the nth column and the i nth row of the two-dimensional interval array A[I][N];
Figure FSA0000271787250000016
for f pair
Figure FSA0000271787250000017
The partial increase function value of ; η n is the coordinate increment of the nth dimension variable to be determined, the above formula represents the lower boundary value domain index interpolation, and the upper boundary value domain index interpolation, both of which are independent dimensions in each dimension variable On the premise that the interval points are equal, the upper boundary value range index interpolation formula is (wherein, the definition of each variable is the same as the definition of the lower boundary value range index interpolation):
Figure FSA0000271787250000018
Figure FSA0000271787250000018
 所述的数据调用算法，以下边界值域索引插值公式为例，具有如下步骤：The described data calling algorithm, taking the following boundary value range index interpolation formula as an example, has the following steps: ①根据待求坐标值，采用黄金分割法对A[I][N]进行双重嵌套循环查找，以计算n个维度变量的待求坐标增量η1，η2，…，ηn，和坐标(i1，i2，…，iN)；①According to the coordinate values to be found, use the golden section method to perform double nested loop search on A[I][N] to calculate the coordinate increments η 1 , η 2 , ..., η n , and coordinates (i 1 , i 2 , ..., i N ); ②根据坐标(i1，i2，…，iN)来计算值域索引值hN，这里同样采用双重嵌套循环查找的方法，定义参数Δ，其中n＞0，Δ＝0表示求
Figure FSA0000271787250000019
的函数值索引，Δ＝n，表示求解的是偏增函数值
Figure FSA00002717872500000110
Δ≠n时需要根据坐标(i1，i2，…，in+1，…，iN)来计算值域索引值，需要在所述的循环语句里面嵌入判断语句进行判断第n维的坐标是否加1；② Calculate the range index value h N according to the coordinates (i 1 , i 2 , ..., i N ), and here the method of double nested loop search is also used to define the parameter Δ, where n>0, Δ=0 means to find
Figure FSA0000271787250000019
The function value index of , Δ=n, indicating that the solution is a partial increase function value
Figure FSA00002717872500000110
When Δ≠n, it is necessary to calculate the index value of the range according to the coordinates (i 1 , i 2 , ..., i n +1, ..., i N ), and it is necessary to embed a judgment statement in the loop statement to judge the nth dimension. Whether the coordinates are incremented by 1; 所查找f的数值，其与一维值域数组B[hN-1]的数值相对应，其中第n维的hn的计算式为：Find the value of f, which corresponds to the value of the one-dimensional range array B[h N -1], where the nth dimension h n is calculated as: hn＝(hn-1-1)·In-1+in.h n =(h n-1 -1)·In -1 +i n . 其中，hn表示待求解的第n维插值值域索引，hn-1表示第n-1维插值值域索引，In-1表示二维区间数组A[I][N]第n-1列的个数；Among them, h n represents the index of the n-th dimensional interpolation range to be solved, h n-1 represents the index of the n-1 th dimensional interpolation range, and I n-1 represents the two-dimensional interval array A[I][N] The n-th 1 column number; ③根据各项，求解插值函数值；③ According to each item, solve the interpolation function value; 作为本发明插值精度更高的一种选择，采用高阶形式的泰勒展开，但计算量更高。As an option for higher interpolation accuracy in the present invention, a higher-order form of Taylor expansion is used, but with a higher amount of computation. 3.如权利要求1所述的一种基于泰勒展开的多维插值方案及其维间点取值方法，其特征在于，所述的插值表维间点取值方法采用的是通过对目标函数的量纲分析来求解各维自变量的阶次，依据阶次选择最少的维间点的个数，以降低插值表的制作经济成本；3. a kind of multi-dimensional interpolation scheme based on Taylor expansion as claimed in claim 1 and the method for obtaining the value between the dimensions thereof, it is characterized in that, what the method for obtaining the value between the dimensions of the described interpolation table adopts is to pass through the method to the objective function. Dimensional analysis is used to solve the order of the independent variables of each dimension, and the minimum number of points between dimensions is selected according to the order to reduce the economic cost of making the interpolation table; 所述的量纲分析的步骤为：The steps of the dimensional analysis are: ①先根据经验，人为的选择所研究现象的主要影响因素
Figure FSA0000271787250000021
①According to experience, artificially select the main influencing factors of the phenomenon under study
Figure FSA0000271787250000021
 ②依据所描述现象的“基本物理量”的量纲，选择能全部涵盖所述“基本物理量”的相互独立的变量，记为m个；②According to the dimension of the "basic physical quantity" of the described phenomenon, select mutually independent variables that can fully cover the "basic physical quantity", denoted as m; ③依据π定理，写出n-m个相似准则π的表达式，其中任意一个相似准则πi都能表达为其余n-m-1个相似准则的函数，需要选择一个包含目标函数的相似准则πi，从而求解出目标函数的表达式为：
Figure FSA0000271787250000022
其中可以借助目标参数的已有表达式进行分析，以便获得f1关于中间变量
Figure FSA0000271787250000023
的本构关系f1(x1，x2，…，xN)；③According to the π theorem, write the expressions of nm similarity criteria π, any of which can be expressed as a function of the remaining nm-1 similarity criteria π i , we need to choose a similarity criterion π i that includes the objective function, so that The expression to solve the objective function is:
Figure FSA0000271787250000022
The analysis can be carried out with the help of the existing expressions of the target parameters in order to obtain f 1 with respect to the intermediate variables
Figure FSA0000271787250000023
The constitutive relation f 1 (x 1 , x 2 , ..., x N ) of ; 所述本构关系f1(x1，x2，…，xN)很难确定，在其它维变量为常数时，f为xn的独立函数，如果是线性的本构关系，有
Figure FSA0000271787250000024
即f1(xn)＝c，但是真实情况由于其它维参数的影响，复合函数
Figure FSA0000271787250000025
的结构要复杂的多，在没有分析出本构关系时，可以近似地将f(x1，x2，…，xN)描述为：The constitutive relation f 1 (x 1 , x 2 , ..., x N ) is difficult to determine. When other dimensional variables are constants, f is an independent function of x n .
Figure FSA0000271787250000024
That is, f 1 (x n )=c, but the real situation is due to the influence of other dimension parameters, the composite function
Figure FSA0000271787250000025
The structure of is much more complicated. When the constitutive relation is not analyzed, f(x 1 , x 2 , ..., x N ) can be approximately described as:
Figure FSA0000271787250000026
Figure FSA0000271787250000026
 其中，gn＞1，an，g0，a0，b，c为未知系数，可能会随着其它维变量值的变化而变化，a0，b只是对曲线的左右和上下平移，不影响离散点个数的选取；Among them, g n > 1 , a n , g 0 , a 0 , b, c are unknown coefficients, which may change with the changes of other dimensional variables. Affects the selection of the number of discrete points; 对于自变量的阶次为gn时，且gn＞0，维间点个数至少为gn+1个；对于自变量的阶次gn＜0时，维间点个数至少为3个；When the order of the independent variable is g n , and g n > 0, the number of inter-dimensional points is at least g n +1; for the order of the independent variable g n <0, the number of inter-dimensional points is at least 3 indivual; 当第n维变量可以近似为周期为T的周期性函数时，根据香浓采样定理至少需要设置T/2的仿真或实验数据间隔才能不破坏其变化规律。When the nth-dimensional variable can be approximated as a periodic function with a period of T, according to the Shannon sampling theorem, at least a simulation or experimental data interval of T/2 needs to be set so as not to destroy its variation law.
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