Interval field finite element method based on Chebyshev polynomial

Abstract

The invention discloses an interval field finite element method based on a Chebyshev polynomial, and relates to the technical field of structural uncertainty analysis, and the method comprises the steps: obtaining a fluctuation range of uncertainty parameters of a to-be-tested structural member; constructing an interval field model based on a B spline principle; determining an interval field of the uncertain parameters based on the fluctuation range of the uncertain parameters and an interval field model; determining nodes and units of the structural member by adopting a finite element method, and establishing a finite element control equation; determining a response interval of the uncertainty parameters based on the interval field of the uncertainty parameters and the finite element control equation; and performing approximation processing on the response interval of the uncertainty parameter by adopting a Chebyshev polynomial to obtain a Chebyshev polynomial approximation function, solving the upper and lower bounds of the response by adopting an SQP algorithm, and determining the response interval of the structural member, so that the uncertainty analysis of the structural member can be completed under the condition of ensuring the precision and the calculation efficiency, and the reliability of the structural member is improved. And performing security assessment.

Description

Technical Field

[0001] The present application relates to the technical field of non-probabilistic uncertainty propagation, and in particular to an interval field finite element method based on Chebyshev polynomials. Background Art

[0002] In modern engineering design, the performance of structures and materials is often affected by various uncertainties such as uncertainty in material properties, geometric fluctuations, and variations in the manufacturing process. In order to ensure the safety of structural design, the impact of these uncertainties on the performance of structural components must be considered. Therefore, uncertainty propagation analysis has become a key issue in the engineering field to evaluate the impact of uncertainty on system response.

[0003] At present, uncertainty propagation methods can be mainly divided into the following categories: sample-based methods, local expansion methods, numerical integration methods, and methods based on surrogate models. However, when faced with high nonlinearity or multiple extreme value problems, traditional methods often cannot guarantee sufficient computational accuracy. Especially in the case of large-scale uncertainty, the accuracy of traditional methods drops rapidly, resulting in unreliable analysis results; for high-dimensional uncertainty problems, the computational complexity of traditional methods is extremely high. For example, Monte Carlo simulation requires a large number of samples to obtain sufficiently accurate results, while the computational complexity of numerical integration methods in high-dimensional problems increases exponentially, which limits their application in large-scale systems due to their low computational efficiency. Summary of the invention

[0004] The purpose of this application is to provide an interval field finite element method based on Chebyshev polynomials, which can complete the uncertainty analysis of structural parts while ensuring accuracy and computational efficiency.

[0005] To achieve the above objectives, this application provides the following solutions.

[0006] In a first aspect, the present application provides an interval field finite element method based on Chebyshev polynomials, comprising the following steps.

[0007] Obtain the fluctuation range of the uncertainty parameters of the structural component to be tested.

[0008] The interval field model is constructed based on the B-spline principle.

[0009] Based on the fluctuation range of the uncertainty parameter and the interval field model, the interval field of the uncertainty parameter is determined.

[0010] The finite element method is used to determine the nodes and units of the structural parts, and the finite element control equations are established based on the nodes and units of the structural parts.

[0011] Based on the interval field of the uncertain parameters and the finite element control equations, the response interval of the uncertain parameters is determined.

[0012] The Chebyshev polynomial is used to approximate the response interval of the uncertain parameters and the Chebyshev polynomial approximation function is obtained.

[0013] The SQP algorithm is used to solve the extreme value of the Chebyshev polynomial approximation function and obtain the response interval of the structural component, thereby analyzing the uncertainty of the structural component and determining the safety of the structural component.

[0014] According to the specific embodiments provided in this application, this application has the following technical effects.

[0015] The present application provides an interval field finite element method, system and equipment based on Chebyshev polynomials. Compared with the traditional uncertainty propagation method, the response interval of the uncertainty parameters determined by the finite element control equation is approximated by Chebyshev polynomials, so that the operation of inverting the interval matrix is ​​avoided in the calculation process, thereby significantly improving the calculation efficiency while ensuring the accuracy; according to the final response interval of the structural component, the uncertainty analysis and safety assessment of the structural component can be completed. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the technical solutions in the embodiments of the present application or related technologies, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0017] Figure 1 A schematic flow chart of an interval field finite element method based on Chebyshev polynomials provided in one embodiment of the present application.

[0018] Figure 2 A schematic diagram of a refinement process of an interval field finite element method based on Chebyshev polynomials provided in one embodiment of the present application.

[0019] Figure 3 A schematic diagram of the decomposition of a steel plate using the finite element method provided in one embodiment of the present application; wherein (a) is the geometric model and constraints of the steel plate; and (b) is the finite element meshing model (the blue numbers in the figure are mesh nodes, i.e., the nodes of the structural parts, and the black numbers in the mesh in the figure are mesh numbers, i.e., the units of the structural parts).

[0020] Figure 4A schematic diagram for comparing the structural displacement response provided in an embodiment of the present application with the Interval Perturbation Analysis (IPA) method and the Monte Carlo Simulation (MCS) method; wherein the red line and the dark blue line are the upper and lower boundaries of the Monte Carlo simulation method (MCS), the brown line and the pink line are the upper and lower boundaries of the Interval Perturbation Method (IPA), and the green line and the light blue line are the upper and lower boundaries of the Chebyshev Interval Polynomials Approximation (CIPA).

[0021] Figure 5 A schematic diagram comparing the structural stress response provided in one embodiment of the present application with the interval perturbation analysis (IPA) method and the Monte Carlo simulation (MCS) method; wherein the red line and the dark blue line are the upper and lower boundaries of the Monte Carlo simulation method (MCS), the brown line and the pink line are the upper and lower boundaries of the interval perturbation method (IPA), and the green line and the light blue line are the upper and lower boundaries of the Chebyshev approximation method (CIPA). DETAILED DESCRIPTION

[0022] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0023] The traditional uncertainty propagation methods are introduced as follows.

[0024] Sample-based methods: Monte Carlo simulation (MCS) and Latin Hypercube Sampling (LHS) are the most common sample-based analysis methods. These analysis methods calculate the statistical characteristics of system responses through random sampling. Although these analysis methods can handle complex nonlinear problems, the amount of calculation is huge, especially in high-dimensional problems, the computational overhead is extremely high, which limits their application.

[0025] Local expansion methods: such as the first reliability method FORM and the second reliability method SORM. The local expansion method approximates the behavior of the system by expanding the system response near the design point. These reliability methods do not perform well in highly nonlinear systems, especially when the response function has multiple extreme values, and the accuracy is low.

[0026] Numerical integration method: Numerical integration method calculates the statistical moments of system response by selecting specific integration rules. Although it is suitable for low-dimensional problems, when faced with high-dimensional uncertainty, the computational complexity increases significantly, leading to dimensionality disaster and low computational efficiency.

[0027] Surrogate model-based methods: Surrogate model-based methods use machine learning techniques (such as artificial neural networks ANN, support vector regression SVR, etc.) to replace high-computational cost functions. Although these methods improve computational efficiency, their accuracy depends on the quality of training data and their applicability to complex uncertainty parameters is limited.

[0028] At the same time, traditional methods usually assume that the distribution of input variables is known and stable, but in reality many sources of uncertainty have spatial variation characteristics, such as the spatial distribution of materials, fluctuations in geometric dimensions, etc. These characteristics are often difficult to effectively model and propagate through traditional methods.

[0029] Therefore, although the above methods have been widely used in engineering problems, when dealing with large-scale uncertainty and high-dimensional problems, the above methods still have the problems of low precision, low computational efficiency, and difficulty in dealing with complex uncertainty parameters. The method of the present application can analyze the fluctuation range of the uncertainty parameters of the structural parts, and complete the uncertainty analysis of the structural parts and the safety assessment of the structural parts while ensuring accuracy and computational efficiency.

[0030] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0031] In an exemplary embodiment, Figure 1 and Figure 2 As shown, an interval field finite element method based on Chebyshev polynomials is provided, and the method includes the following steps 1 to 8.

[0032] Step 1: Obtain the fluctuation range of the uncertainty parameters of the structure to be tested. The uncertainty parameters include: any one of the applied loads, geometric dimensions and material properties.

[0033] Step 2: Construct an interval field model based on the B-spline principle.

[0034] Specifically, the expression of the interval field model is as follows.

[0035] (1) in, is the uncertainty parameter in The interval field at each position; is the central value of the uncertainty parameter; is the radius value of the uncertainty parameter; is the number of basis functions; For the The basis function is The weight at each position; For the interval field The basis function is The coordinate factor at the position is [-1, 1], that is, .

[0036] Step 3: Determine the interval field of the uncertainty parameter based on the fluctuation range of the uncertainty parameter and the interval field model.

[0037] When the uncertainty parameter is an external load, the function expression of the interval field of the uncertainty parameter is as follows.

[0038] (2) in, The external load is The interval field function at each position; is the center value of the applied load; is the radius value of the applied load.

[0039] When the uncertainty parameter is a geometric dimension or a material property, the function expression of the interval field of the uncertainty parameter is as follows.

[0040] (3) in, The stiffness matrix of the structural member affected by geometric dimensions or material properties is The interval field function at each position; is the center value of the structural member stiffness matrix; is the radius value of the structural member stiffness matrix; For the The basis function is The weight at each position; For the The interval is The coordinate factor at the position.

[0041] Step 4: Use the finite element method to determine the nodes and units of the structural parts, and establish the finite element control equations based on the nodes and units of the structural parts.

[0042] Specifically, the expressions of the finite element control equations are as follows.

[0043] (4) in, is the structural component stiffness matrix; is the displacement vector of the node; is the external load vector.

[0044] Step 5: Based on the interval field of the uncertainty parameter and the finite element control equation, determine the response interval of the uncertainty parameter. The response interval includes: displacement response interval and stress response interval.

[0045] According to the finite element governing equation (4), the displacement calculation expression is obtained as follows.

[0046] (5) The stress calculation expression is as follows.

[0047] (6) in, is the stress interval vector of the element; is the elastic matrix of the structural member; is the stress-to-displacement conversion matrix; is the unit transposed matrix.

[0048] The external load is Interval field function at positions Substituting it into the displacement calculation expression, we can obtain the function expression of the displacement response range of the uncertainty parameter when the uncertainty parameter is an external load as follows.

[0049] (7) in, For the node The displacement interval vector at each position; is the structural member stiffness matrix.

[0050] The functional expression of the stress response interval of the uncertainty parameter is as follows.

[0051] (8) in, For the unit The stress interval vector at each position; is the elastic matrix of the structural member; is the stress-to-displacement conversion matrix; is the unit transposed matrix.

[0052] The stiffness matrix of the structural member affected by the geometric dimensions or material properties is Interval field function at positions Substituting it into the stress calculation expression, we can obtain the following function expression of the displacement response range of the uncertainty parameter when the uncertainty parameter is the geometric dimension or material property.

[0053] (9) in, For the node The displacement interval vector at each position; is the external load vector.

[0054] The functional expression of the stress response interval of the uncertainty parameter is as follows.

[0055] (10) in, For the unit The stress interval vector at each position; is the elastic matrix of the structural member; is the stress-to-displacement conversion matrix; is the unit transposed matrix.

[0056] Step 6: Use Chebyshev polynomials to approximate the response interval of the uncertainty parameter to obtain a Chebyshev polynomial approximation function. The Chebyshev polynomial approximation function includes: a Chebyshev polynomial approximation function of displacement and a Chebyshev polynomial approximation function of stress.

[0057] Specifically, for the expressions of displacement response and stress response finally obtained in step 5, the Chebyshev polynomial order and interpolation points are selected, and the first kind of Chebyshev polynomial is selected by approximating the structural response calculation equation with Chebyshev polynomials, and its basis function expression is as follows.

[0058] (11) in, is the 0th order Chebyshev polynomial basis function; is the first-order Chebyshev polynomial basis function; For the Chebyshev polynomial basis functions of order; For the Chebyshev polynomial basis functions of order; For the Chebyshev polynomial basis functions of order; is the coordinate factor.

[0059] set up , since in step 2 ,so .

[0060] Substitute the formulas (7) and (9) in step 5 into Replace with , and we get the following formulas (12) and (13).

[0061] (12) (13) Using Chebyshev to approximate equations (12) and (13), through the projection of equations (12) and (13) to the Chebyshev basis function and the Mailer integral formula, interpolation points are selected for approximation, then each The value range of as follows.

[0062] (14) in, is the number of interpolation points, its value must be higher than the order of the Chebyshev polynomial ,Right now By approximating the initial displacement interval formula at the sampling point, namely, equation (12) and equation (13), the expression of the Chebyshev polynomial approximation function of the displacement can be obtained as follows.

[0063] (15) in, is the Chebyshev polynomial approximation function of the displacement; are the coefficients of the Chebyshev polynomial shift; For the The order of the dimension Chebyshev polynomial, where the dimension is the result of processing the basis function using the Chebyshev polynomial; For the The influencing factors of each dimension.

[0064] Among them, the coefficients of Chebyshev polynomial stress are The expression of is as follows.

[0065] (16) in, for is the number of 0s.

[0066] Substitute the formulas (8) and (10) in step 5 into Replace with , and we get the following formulas (17) and (18).

[0067] (17) (18) By using Chebyshev to approximate equations (17) and (18), and using the same interpolation points as equations (12) and (13) for approximation, the expression of the Chebyshev polynomial approximation function of stress can be obtained as follows.

[0068] (19) in, is the Chebyshev polynomial approximation function of stress; are the coefficients of Chebyshev polynomial stress; For the The order of the dimension Chebyshev polynomial, where the dimension is the result of processing the basis function using the Chebyshev polynomial; For the The influencing factors of each dimension.

[0069] Among them, the coefficients of Chebyshev polynomial stress are The expression of is as follows.

[0070] (20) Step 7: Use the SQP algorithm to solve the extreme value of the Chebyshev polynomial approximation function and obtain the response interval of the structural member, so as to analyze the uncertainty of the structural member and determine the safety of the structural member.

[0071] Specifically, the Chebyshev polynomial approximation function is solved by the SQP optimization algorithm. For the Chebyshev polynomial approximation function of displacement and the Chebyshev polynomial approximation function of stress obtained in step 6, that is, formula (15) and formula (18), considering that SQP is an algorithm suitable for optimization problems with constraints, we use an optimization method to solve the result boundary of the Chebyshev polynomial approximation function. The original nonlinear optimization problem is solved by converting it into a series of quadratic programming (QP) subproblems. Each QP subproblem is a quadratic approximation of the objective function and constraints near the current point, so it is optimized and solved step by step. Finally, by solving the extreme value of the polynomial, the displacement or stress response of the structure is obtained.

[0072] The basic framework of the SQP algorithm is as follows.

[0073] (twenty one) According to formula (20), the response range of the structural component can be obtained, so as to analyze the uncertainty of the structural component and determine the safety of the structural component according to the response range of the structural component.

[0074] In an exemplary embodiment, a Q235 steel plate with an uncertain thickness parameter is used as an example to illustrate the technical solution of the present application. The bottom edge of the square steel plate is fixed, the top edge is uniformly pulled, the pulling force is P=100MPa, the side length is 1m, the material properties: Young's model is E=2.1×105MPa, Poisson's ratio The calculation of the response interval of a structural member using the interval field finite element method based on Chebyshev polynomials of the present application includes the following steps.

[0075] Step 1: Determine the uncertainty parameter. In this example, the thickness of the steel plate is determined as the uncertainty parameter, and its fluctuation range is [0.7, 1.3] mm.

[0076] Step 2: Construct an interval field model for the steel plate thickness and obtain the following formula.

[0077] (twenty two) in, The thickness of the steel plate is The interval field at each position; is the center value of the steel plate thickness; is the radius value of the steel plate thickness; is the design domain; for dimension; Indicates The basis functions are The weights at each position are used to simulate the correlation between the internal components of the interval field; For the interval field The basis function is The coordinate factor at the position, .

[0078] Step 3: The geometric model and constraints of the steel plate, as well as the finite element meshing model Figure 3 The uncertainty thickness interval field in the second step is input into the finite element unit control equation as follows.

[0079] (twenty three) in, is the displacement vector of the node; is the external load vector. According to the finite element principle, the stiffness matrix of the steel plate element is It is obtained by integrating the following equation.

[0080] (twenty four) in, is the unit area.

[0081] Through finite element calculation, the calculation formula of the steel plate stiffness matrix is ​​as follows.

[0082] (25) in, is the stiffness matrix of the steel plate; is the number of units; is the transposed matrix. The interval field finite element formula of the displacement response obtained through finite element operation is as follows.

[0083] (26) (27) Step 4: Using step 6 of the present application, the response interval of the uncertainty parameter is approximated using Chebyshev polynomials to obtain a Chebyshev polynomial approximation function.

[0084] Step 5: Solve the Chebyshev polynomial approximation function through the SQP optimization algorithm. For the Chebyshev polynomial approximation function obtained in the fourth step, the original nonlinear optimization problem is converted into a series of quadratic programming (QP) sub-problems to solve it. Each QP sub-problem is a quadratic approximation of the objective function and constraints near the current point, so as to optimize and solve it step by step. Finally, by solving the extreme value of the polynomial, the displacement or stress response of the steel plate is obtained, and the results are compared with the interval perturbation analysis (IPA) method and the Monte Carlo simulation (MCS) method. The comparison results are shown as follows: Figure 4 and Figure 5 As shown, the proposed CIPA method is more computationally efficient than MCS, with a 100-fold reduction in computational cost while maintaining 93%-99% accuracy. This efficiency stems from the elimination of repeated sampling through Chebyshev spectral decomposition. In addition, unlike the traditional IPA method, which suffers from extreme accuracy degradation when the uncertainty is large (i.e., when the perturbation radius is large), the CIPA framework maintains a stable relative error over all tested perturbation ranges. This accuracy robustness stems from the method's inherent ability to capture nonlinear dependencies through Chebyshev orthogonal polynomials. Notably, theoretical analysis further determines that the optimal expansion center selection in IPA is a difficult problem to solve when spatial correlation exists, which introduces systematic errors, while the CIPA formulation fundamentally avoids these errors.

[0085] The beneficial effects of the interval field finite element method based on Chebyshev polynomials proposed in this application are mainly manifested in: compared with the traditional uncertainty propagation method, Chebyshev polynomials are used to approximate the response interval of the uncertainty parameters determined by the finite element control equation, thereby avoiding the matrix inversion operation in the calculation process, improving the calculation efficiency while ensuring the accuracy, and finally analyzing the uncertainty of the structural parts through the response interval of the structural parts, while evaluating the safety of the structural parts.

[0086] The present application also provides an application scenario, which applies the above-mentioned interval field finite element method based on Chebyshev polynomials. Specifically: The interval field finite element method based on Chebyshev polynomials provided in this embodiment can be applied in engineering design scenarios. The engineering design scenario includes a structural component design link, a structural component response interval determination link, a structural component safety determination link, and a structural component redesign link; the structural component response interval determination link determines the response interval according to the currently designed structural component, and determines the safety of the structural component according to the structural component response interval, so as to determine whether to design the structural component again. The interval field finite element method based on Chebyshev polynomials provided in this embodiment belongs to the structural component response interval determination link and the structural component safety determination link. Specifically, in the structural component response interval determination link and the structural component safety determination link, by establishing an interval field model and a finite element control equation, the Chebyshev polynomial is used to approximate the response interval of the uncertainty parameter, and finally the SQP algorithm is used to determine the structural component response interval, thereby determining the safety of the structural component.

[0087] The database involved in each embodiment provided in this application may include at least one of a relational database and a non-relational database. The non-relational database may include a distributed database based on blockchain, etc., but is not limited thereto. The processor involved in each embodiment provided in this application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but is not limited thereto.

[0088] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0089] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. An interval field finite element method based on Chebyshev polynomials, characterized in that: The interval field finite element method based on Chebyshev polynomials includes: Obtain the fluctuation range of the uncertainty parameters of the structural component to be tested; Construct interval field model based on B-spline principle; Based on the fluctuation range of the uncertainty parameter and the interval field model, the interval field of the uncertainty parameter is determined; The finite element method is used to determine the nodes and units of the structural parts, and the finite element control equations are established based on the nodes and units of the structural parts; Based on the interval field of the uncertain parameters and the finite element control equation, the response interval of the uncertain parameters is determined; The Chebyshev polynomial is used to approximate the response interval of the uncertainty parameter, and the Chebyshev polynomial approximation function is obtained; The SQP algorithm is used to solve the extreme value of the Chebyshev polynomial approximation function and obtain the response interval of the structural component, thereby analyzing the uncertainty of the structural component and determining the safety of the structural component.

2. The interval field finite element method based on Chebyshev polynomials according to claim 1 is characterized in that: The uncertainty parameter includes any one of an applied load, a geometric dimension and a material property.

3. The interval field finite element method based on Chebyshev polynomials according to claim 2 is characterized in that: When the uncertainty parameter is an external load, the function expression of the interval field of the uncertainty parameter is: ； in, The external load is The interval field at each position; is the center value of the applied load; is the radius value of the applied load; is the number of basis functions; For the The basis function is The weight at each position; For the interval field The basis function is The coordinate factor at the position.

4. The interval field finite element method based on Chebyshev polynomials according to claim 2 is characterized in that: When the uncertainty parameter is a geometric dimension or material property, the function expression of the interval field of the uncertainty parameter is: ； in, The stiffness matrix of the structural member affected by geometric dimensions or material properties is The interval field at each position; is the center value of the structural member stiffness matrix; is the radius value of the structural member stiffness matrix; is the number of basis functions; For the The basis function is The weight at each position; For the interval field The basis function is The field coordinate factor at each position.

5. The interval field finite element method based on Chebyshev polynomials according to claim 3 is characterized in that: The response interval includes: a displacement response interval and a stress response interval; When the uncertainty parameter is an external load, the function expression of the displacement response interval of the uncertainty parameter is: ； in, For the node The displacement interval vector at the position; is the structural component stiffness matrix; The functional expression of the stress response interval of the uncertainty parameter is: ； in, For the unit The stress interval vector at each position; is the elastic matrix of the structural member; is the stress-to-displacement conversion matrix; is the unit transposed matrix.

6. The interval field finite element method based on Chebyshev polynomials according to claim 4 is characterized in that: The response interval includes: a displacement response interval and a stress response interval; When the uncertainty parameter is a geometric dimension or material property, the function expression of the displacement response interval of the uncertainty parameter is: ； in, For the node The displacement interval vector at the position; is the external load; The functional expression of the stress response interval of the uncertainty parameter is: ； in, For the unit The stress interval vector at each position; is the elastic matrix of the structural member; is the stress-to-displacement conversion matrix; is the unit transposed matrix.

7. The interval field finite element method based on Chebyshev polynomials according to claim 1 is characterized in that: The Chebyshev polynomial approximation function includes: a Chebyshev polynomial approximation function of displacement and a Chebyshev polynomial approximation function of stress; The expression of the Chebyshev polynomial approximation function of the displacement is: ； in, is the Chebyshev polynomial approximation function of the displacement; are the coefficients of the Chebyshev polynomial shift; is the order of the Chebyshev polynomial; For the The order of the dimension Chebyshev polynomial, where the dimension is the result of processing the basis function using the Chebyshev polynomial; is the number of basis functions, that is, the number of dimensions; For the The influencing factors of each dimension.

8. The interval field finite element method based on Chebyshev polynomials according to claim 7 is characterized in that: The expression of the Chebyshev polynomial approximation function of stress is: ； in, is the Chebyshev polynomial approximation function of stress; are the coefficients of Chebyshev polynomial stress; For the The order of the dimension Chebyshev polynomial, where the dimension is the result of processing the basis function using the Chebyshev polynomial; For the The influencing factors of each dimension.

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