Prediction method of tunnel excavation deformation characteristic curve based on surrogate model

Abstract

The present invention discloses a method for predicting tunnel excavation deformation characteristic curves based on a proxy model, comprising the steps of: 1. determining input variables and their value ranges; 2. parameterizing and characterizing the tunnel excavation deformation characteristic curve; 3. establishing and training a proxy model; 4. predicting the tunnel excavation deformation characteristic curve parameters; and 5. drawing the tunnel excavation deformation characteristic curve. The present invention parameterizes and characterizes the tunnel excavation deformation characteristic curve and establishes a proxy model for predicting tunnel excavation deformation characteristic curve parameters. Without the need for finite element modeling and analysis of the tunnel excavation construction process, the corresponding tunnel excavation deformation characteristic curve parameter values ​​are directly obtained, thereby rapidly predicting the tunnel excavation deformation characteristic curve, reducing the prediction workload and significantly improving prediction efficiency. This method overcomes the bottleneck of traditional simulation analysis and prediction methods, which are unable to rapidly predict tunnel excavation deformation characteristic curves, and has broad application prospects in tunnel construction safety control and support optimization design.

Description

Technical Field

[0001] The present invention relates to the field of tunnel structure morphology prediction and analysis, and in particular to a method for predicting tunnel excavation deformation characteristic curves based on an agent model. Background Art

[0002] Hydraulic tunnels are a crucial component of water conservancy and hydropower infrastructure systems and must be excavated and constructed within mountains or underground. Due to the disturbances caused by tunnel excavation, the surrounding rock continues to deform during construction and may experience deformation instability. Therefore, rationally predicting tunnel excavation deformation characteristics is a prerequisite and foundation for implementing tunnel construction safety control and optimizing the design of excavation support schemes. Tunnel excavation deformation characteristics are influenced by multiple factors, including burial depth, tunnel diameter, cyclic footage, and surrounding rock mechanical parameters. They also involve variable structural nonlinearity (nonlinearity caused by changes in the surrounding rock structure due to tunnel excavation) and material nonlinearity (nonlinearity caused by the nonlinear stress-strain relationship in the surrounding rock), making them complex and variable. Because tunnel surrounding rock deformation gradually decreases from the excavation boundary to the depth of the surrounding rock, tunnel design and construction focus on how the radial displacement of the tunnel vertex changes with the distance between the predicted profile and the excavation surface, described using the tunnel excavation deformation characteristic curve.

[0003] Currently, tunnel excavation deformation characteristic curves are primarily predicted using simulation analysis methods. This involves using numerical analysis methods such as finite element methods to simulate the tunnel excavation process to predict the distribution, magnitude, and evolution of surrounding rock deformation during the excavation face. This results in a curve showing how the tunnel vertex at the predicted section changes with the distance between the predicted section and the excavation face. The aforementioned existing technologies suffer from the following deficiencies:

[0004] 1. In the existing technology, predicting the deformation characteristic curve of tunnel excavation requires the establishment of a complex finite element simulation model and the implementation of corresponding calculation and analysis, which is labor-intensive, time-consuming, and has low prediction efficiency.

[0005] 2. In the existing technology, when the tunnel design and construction plan or the surrounding rock mechanical parameters are adjusted, the corresponding finite element simulation modeling analysis needs to be re-performed, which makes it impossible to quickly predict the tunnel excavation deformation characteristic curve under different design and construction plans or parameter conditions. Summary of the Invention

[0006] Purpose of the invention: In order to overcome the deficiencies of the above-mentioned prior art, the present invention provides a method for predicting the tunnel excavation deformation characteristic curve based on a proxy model. This method parameterizes the tunnel excavation deformation characteristic curve and establishes a proxy model for predicting the tunnel excavation deformation characteristic curve parameters. It breaks through the bottleneck that traditional simulation analysis and prediction methods cannot quickly predict the tunnel excavation deformation characteristic curve, greatly improves the prediction efficiency, and has broad application prospects in tunnel construction safety control and support optimization design.

[0007] To solve the technical problem raised in the background technology: the present invention provides a method for predicting tunnel excavation deformation characteristic curve based on a proxy model, comprising the following steps:

[0008] Step 1: Determine the input variables: Select the factors directly related to the tunnel stress as the main influencing factors among the factors affecting the deformation characteristics of tunnel excavation. The main influencing factors include tunnel depth H, tunnel diameter D, excavation cycle footage f p , deformation modulus E of surrounding rock, Poisson's ratio μ, internal friction angle φ and cohesion c;

[0009] Among the main factors affecting the deformation characteristics of tunnel excavation, N p Input variables P i , and determine the value range of each input variable [l i ,m i ];

[0010] Step 2: Parametrically characterize the tunnel excavation deformation characteristic curve: A three-parameter exponential function is used to characterize the tunnel excavation deformation characteristic curve. The function expression is as follows:

[0011]

[0012] Where u is the radial displacement of the tunnel vertex at the predicted section; L is the distance between the predicted section and the excavation surface; u max is the parameter representing the maximum value of excavation deformation; k is the parameter representing the rate of change of deformation rate of tunnel excavation deformation characteristic curve; a is the parameter representing the excavation deformation when L=0 and u max Parameters of the proportional relationship between

[0013] Step 3: Establish and train the proxy model: Construct a training sample set S, and train the proxy model through the model input and model output of each sample in the sample set S to obtain the proxy model for predicting the tunnel excavation deformation characteristic curve parameters;

[0014] Step 4: Parameter prediction: Using the agent model trained in step 3, the given input variable value P i * (i=1,2,…,N p ) is the model input to predict the parameters of tunnel excavation deformation characteristic curve;

[0015] Step 5. Result drawing: Draw the tunnel excavation deformation characteristic curve based on the parameters predicted in step 4.

[0016] Furthermore, the P i In the equation, 1≤i≤N p .

[0017] Furthermore, in step 3, the steps of establishing and training the proxy model include:

[0018] Step 31: Extract N in the tunnel excavation deformation characteristic curve parameter prediction agent model input space s Sample S j ,j∈[1,N s ], and obtain the model input of each sample in the proxy model training sample set S, where the jth sample S j The model input is denoted as P i j ;

[0019] Step 32, with N s Sample S j (j=1,2,…,N s ) is used as the modeling parameter, and the corresponding tunnel excavation finite element modeling calculation is carried out to establish a finite element simulation model of the tunnel excavation process and simulate the tunnel excavation process. The established finite element model is the same as the j-th sample S j The model input P i j The corresponding tunnel excavation finite element model is denoted as M j ;

[0020] Step 33: Substitute the finite element modeling calculation results of each tunnel excavation into the three-parameter exponential function for fitting, and obtain the tunnel excavation deformation characteristic curve characterization parameter u max A set of fitting values ​​of , k and a, based on the tunnel excavation finite element model M j The characterization parameter u obtained by fitting the calculation results max The fitted values ​​of , k and a are recorded as u maxj 、k j and a j , with u maxj 、k j and a j As the basis for the construction and the j-th sample S j Model output in are equal to u maxj 、k j 、a j ;

[0021] Step 34: Use the sample set S to train the selected proxy model to obtain a tunnel excavation deformation characteristic curve parameter prediction proxy model.

[0022] Furthermore, in step 31, the input space of the tunnel excavation deformation characteristic curve parameter prediction agent model is composed of P i N p dimensional bounded space, and P i ∈[l i ,mi ].

[0023] Furthermore, in step 32, the finite element simulation model of the tunnel excavation process uses the predicted section as the middle cross-section of the model, the axial length is ≥15 times the tunnel diameter, the center of the cross-section is located at the center of the tunnel, and the vertical distance between the center of the cross-section and the surrounding boundaries is ≥5.5 times the tunnel diameter.

[0024] Furthermore, in step 4, the input variable value P i * where i∈[1,N p ], and input variable value P i * With input variable P i The multiple influencing factors correspond one to one.

[0025] Furthermore, the process of predicting the parameters of tunnel excavation deformation characteristic curve is as follows:

[0026] Enter the given input variable value The established tunnel excavation deformation characteristic curve parameter prediction proxy model is used to obtain the corresponding model output With P i * The predicted value u of the corresponding tunnel excavation deformation characteristic curve parameter max * 、k * 、a * are equal to

[0027] Furthermore, in step 5, the specific process of drawing the tunnel excavation deformation characteristic curve is as follows: let u in the three-parameter exponential function be max , k and a take the corresponding predicted value u max * 、k * and a * , draw the function curve according to the function expression after determining the parameters.

[0028] The present invention has the following beneficial effects:

[0029] The present invention parameterizes the tunnel excavation deformation characteristic curve and establishes a tunnel excavation deformation characteristic curve parameter prediction proxy model. After the tunnel excavation deformation characteristic curve parameter prediction proxy model is established, for any given tunnel excavation deformation characteristic curve prediction input variable, the corresponding tunnel excavation deformation characteristic curve parameter value is directly obtained without the need to carry out finite element modeling analysis of the tunnel excavation construction process, thereby quickly predicting the tunnel excavation deformation characteristic curve, reducing the prediction workload, reducing work time, and significantly improving the prediction efficiency; in the actual work process, even if the tunnel design and construction plan or the surrounding rock mechanical parameters change, the proxy model in the method is still applicable, and there is no need to re-carry out the corresponding finite element simulation modeling analysis. This solution breaks through the bottleneck of the existing technology that is unable to quickly predict the tunnel excavation deformation characteristic curve, and provides an effective solution for efficiently carrying out tunnel construction safety control and support optimization design. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 This is the workflow diagram of the tunnel excavation deformation characteristic curve prediction method based on the surrogate model.

[0031] Figure 2 This is a finite element simulation model diagram of the tunnel excavation process in the embodiment provided by the present invention.

[0032] Figure 3 An example diagram of parameter fitting of the tunnel excavation deformation characteristic curve in the embodiment provided by the present invention.

[0033] Figure 4 The tunnel excavation deformation characteristic curve predicted in the embodiment provided by the present invention. DETAILED DESCRIPTION

[0034] The technical solution of the present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0035] like Figure 1 As shown in FIG, the method for predicting the deformation characteristic curve of tunnel excavation based on the agent model includes the following steps:

[0036] Step 1: Determine the input variables and their value ranges for predicting the tunnel excavation deformation characteristic curve. The specific process is as follows:

[0037] Among the factors affecting the deformation characteristics of tunnel excavation, the factors directly related to the tunnel stress are selected as the main influencing factors. The main influencing factors include the tunnel burial depth H, the tunnel diameter D, the excavation cycle footage f p , surrounding rock deformation modulus E, Poisson's ratio μ, internal friction angle φ and cohesion c; collect N from the main influencing factors of tunnel excavation deformation characteristics p factors, as N for predicting tunnel excavation deformation characteristic curve p Input variables Pi (i=1,2,…,N p ), and determine the value range of each input variable, the i-th input variable P i The value range of [l i ,m i ].

[0038] The tunnel excavation deformation characteristic curve refers to the relationship curve between the radial displacement u of the tunnel vertex at the predicted section and the distance L between the predicted section and the excavation surface.

[0039] In this embodiment, a water diversion tunnel is approximately 3.5 km long and has a burial depth range of approximately 260-460 m. The rock strata along the tunnel are steeply inclined and the lithology is complex and diverse. Different tunnel sections have different excavation deformation characteristics. The tunnel cross-section is circular, the excavation diameter is 6.8 m, and the excavation cycle footage is 2 m.

[0040] According to the actual situation of the project, taking into account the tunnel diameter D and the excavation cycle footage f p Keeping the surrounding rock deformation modulus E, Poisson's ratio μ, internal friction angle φ, cohesion c and burial depth H unchanged, the five factors of surrounding rock deformation modulus E, Poisson's ratio μ, internal friction angle φ, cohesion c and burial depth H are selected as the five input variables for predicting the tunnel excavation deformation characteristic curve. The surrounding rock deformation modulus E, Poisson's ratio μ, internal friction angle φ, cohesion c and burial depth H are denoted as P1, P2, P3, P4 and P5 respectively. According to the engineering geological survey data, the reasonable value range of each input variable is determined, as shown in Table 1.

[0041] Table 1 Input variables and their value ranges for prediction of tunnel excavation deformation characteristic curve

[0042] <![CDATA[P1(GPa)]]> <![CDATA[P2]]> <![CDATA[P3(°)]]> <![CDATA[P4(MPa)]]> <![CDATA[P5(m)]]> [6,10] [0.24,0.28] [35,45] [0.5,1] [260,460]

[0043] Step 2: Parametrically characterize the tunnel excavation deformation characteristic curve: A three-parameter exponential function is used to characterize the tunnel excavation deformation characteristic curve. The function expression is as follows:

[0044]

[0045] Where L is the distance between the predicted section and the excavation surface; u is the radial displacement of the tunnel vertex at the predicted section; u max is the parameter representing the maximum value of excavation deformation; k is the parameter representing the rate of change of deformation rate of tunnel excavation deformation characteristic curve; a is the parameter representing the excavation deformation when L=0 and u max Parameters of the proportional relationship between them.

[0046] Step 3: Establish and train the proxy model: Construct a training sample set S, train the proxy model through the model input and model output of each sample in the sample set S, and obtain the proxy model for predicting the tunnel excavation deformation characteristic curve parameters. The steps include:

[0047] Step 31: Sampling in the tunnel excavation deformation characteristic curve parameter prediction proxy model input space to obtain N in the proxy model training sample set S. s Sample S j (j=1,2,…,N s ) model input, where the j-th sample S j The model input is denoted as P i j .

[0048] The input space of the tunnel excavation deformation characteristic curve parameter prediction agent model is composed of P i (i=1,2,…,N p ) constitutes N p dimensional bounded space, and P i ∈[l i ,m i ].

[0049] In this embodiment, the input space of the tunnel excavation deformation characteristic curve parameter prediction agent model is composed of P i (i=1,2,…,5), P1∈[6,10], P2∈[0.24,0.28], P3∈[35,45], P4∈[0.5,1], P5∈[260,460]; Latin hypercube sampling technique is used to sample in the input space of the proxy model for predicting tunnel excavation deformation characteristic curve parameters. The number of samples included in the proxy model training sample set S is taken as 30, and the 30 samples S obtained by sampling are j (j=1,2,…,30) model input, as shown in Table 2, the jth sample S j The model input is denoted as P i j (i=1,2,…,5; j=1,2,…,30).

[0050] Table 2 Model input of each sample in the proxy model training sample set S

[0051]

[0052]

[0053] Step 32: For each sample model input in the training sample set S, the corresponding tunnel excavation finite element modeling calculation is carried out, that is, N s Sample S j (j=1,2,…,N s ) is used as the modeling parameter, and the corresponding tunnel excavation finite element modeling calculation is carried out. The established j The model input P i jThe corresponding tunnel excavation finite element model is denoted as M j .

[0054] Finite element modeling calculation for tunnel excavation refers to establishing a finite element simulation model of the tunnel excavation process using the model input of each sample in the training sample set S as the modeling parameter, and simulating the tunnel excavation process.

[0055] The finite element simulation model of the tunnel excavation process uses the predicted section as the middle cross-section of the model, with an axial length of not less than 15 times the tunnel diameter, the center of the cross-section located in the center of the tunnel, and the vertical distance between the center of the cross-section and the surrounding boundaries not less than 5.5 times the tunnel diameter.

[0056] In this embodiment, the axial length of the finite element simulation model of the tunnel excavation process is taken as 15 times the tunnel diameter (102m), and the vertical distance between the cross-section center and the surrounding boundaries is taken as 5.5 times the tunnel diameter (37.4m). Figure 2 shown.

[0057] Step 33: Based on the finite element modeling calculation results of each tunnel excavation, multiple calculation results are sequentially substituted into the three-parameter exponential function for fitting to obtain the tunnel excavation deformation characteristic curve characterization parameter u max A set of fitting values ​​of , k and a, based on the tunnel excavation finite element model M j The characterization parameter u obtained by fitting the calculation results max The fitted values ​​of , k and a are recorded as u maxj 、k j and a j , with u maxj 、k j and a j As the basis for the construction and the j-th sample S j Model output (m=1,2,3), where Q1 j 、 are equal to u maxj 、k j 、a j .

[0058] The calculation results of tunnel excavation finite element modeling refer to the change process of the radial displacement u of the tunnel vertex at the predicted section obtained by simulating the tunnel excavation process as the distance L between the predicted section and the excavation surface changes.

[0059] In this embodiment, based on the finite element modeling calculation results of 30 tunnel excavations, a three-parameter exponential function is used as the fitting function to fit the change process of the radial displacement u of the tunnel vertex at the predicted section as the distance L between the predicted section and the excavation surface changes ( Figure 3 ), and 30 sets of tunnel excavation deformation characteristic curve characterization parameters u are obtained max , k and the fitted value u of a maxj 、kj and a j , j=1,2,…,30, let Q1 j =u maxj 、 The model output of 30 samples in the proxy model training sample set S can be constructed (m=1,2,3;j=1,2,…,30), as shown in Table 3.

[0060] Table 3 Model output of each sample in the proxy model training sample set S

[0061]

[0062]

[0063] Step 34: Use the sample set S to train the selected proxy model to obtain a tunnel excavation deformation characteristic curve parameter prediction proxy model.

[0064] In this embodiment, the Kriging proxy model is selected as the proxy model for predicting the tunnel excavation deformation characteristic curve parameters, and 30 samples S in the training sample set S are used. j (j=1,2,…,30) model input (Table 2) and model output (Table 3) are used to train it. enter, As the model output of the prediction agent model, the tunnel excavation deformation characteristic curve parameter prediction agent model can be obtained. The specific expression is:

[0065]

[0066] Among them, r is the 30th order correlation vector, which is the input variable P of the prediction i * The model input P of the training sample i j The Gaussian correlation function values ​​between them are composed; R is the 30th order correlation matrix, which is composed of the model input P of the training sample i j The Gaussian correlation function values ​​between them are composed; F is a 30-order unit array; β0 is a scalar, which represents the constant term output by the trained model, and β0=(F T R -1 F) -1 F T R - 1 Y.

[0067] Step 4: Parameter prediction: using the proxy model trained in step 3, with the given input variable value P i * (i=1,2,…,Np ) is the model input to predict the tunnel excavation deformation characteristic curve parameters. The specific process is as follows:

[0068] Enter the given input variable value P i * (i=1,2,…,N p ),Right now The established tunnel excavation deformation characteristic curve parameter prediction proxy model is used to obtain the corresponding model output With P i * The predicted value u of the corresponding tunnel excavation deformation characteristic curve parameter max * 、k * 、a * Equal to Q1 * 、Q2 * 、Q3 * .

[0069] In this embodiment, three typical tunnel sections (1 # , 2 # and 3 # tunnel section), of which 1 # , 2 # and 3 # Different tunnel sections have different design and construction plans (including burial depth H) and surrounding rock mechanical parameters (including deformation modulus E, Poisson's ratio μ, internal friction angle φ, and cohesion c). The established tunnel excavation deformation characteristic curve parameter prediction proxy model is used to predict the excavation deformation characteristic curve parameters. The input variables for the excavation deformation characteristic prediction of each typical tunnel section, determined based on tunnel topographic and geological data, are shown in Table 4. The predicted values ​​of the excavation deformation characteristic parameters are shown in Table 5.

[0070] Table 4 Input variables for predicting deformation characteristics of typical tunnel sections

[0071]

[0072] Table 5 Predicted values ​​of parameters of deformation characteristic curve for typical tunnel sections

[0073]

[0074] Step 5. Result plotting: Draw the tunnel excavation deformation characteristic curve based on the parameters predicted in step 4. The specific process is as follows:

[0075] Let u in the three-parameter exponential function be max , k and a are equal to the predicted values ​​u of the tunnel excavation deformation characteristic curve parameters max* 、k * and a * , draw the function curve, which is the predicted tunnel excavation deformation characteristic curve, such as Figure 4 shown.

[0076] The above embodiments are only preferred embodiments of the present invention and are not limitations on the technical solutions of the present invention. Any technical solution that can be implemented on the basis of the above embodiments without creative work should be deemed to fall within the scope of protection of the patent of the present invention.

Claims

1. A method for predicting tunnel excavation deformation characteristic curve based on a proxy model, characterized in that: The following steps are involved: Step 1: Determine the input variables: Select the factors directly related to the tunnel stress as the main influencing factors among the factors affecting the deformation characteristics of tunnel excavation. The main influencing factors include tunnel depth H, tunnel diameter D, excavation cycle footage f p , deformation modulus E of surrounding rock, Poisson's ratio μ, internal friction angle φ and cohesion c; Among the main factors affecting the deformation characteristics of tunnel excavation, N p Input variables P i , and determine the value range of each input variable [l i ,m i ]; Step 2: Parametrically characterize the tunnel excavation deformation characteristic curve: A three-parameter exponential function is used to characterize the tunnel excavation deformation characteristic curve. The function expression is as follows: Where u is the radial displacement of the tunnel vertex at the predicted section; L is the distance between the predicted section and the excavation surface; u max is the parameter representing the maximum value of excavation deformation; k is the parameter representing the rate of change of deformation rate of tunnel excavation deformation characteristic curve; a is the parameter representing the excavation deformation and u when L=0 max Parameters of the proportional relationship between Step 3: Establish and train the proxy model: Construct a training sample set S, and train the proxy model through the model input and model output of each sample in the sample set S to obtain the proxy model for predicting the tunnel excavation deformation characteristic curve parameters; Step 4: Parameter prediction: Using the agent model trained in step 3, the given input variable value P i * As model input, the parameters of tunnel excavation deformation characteristic curve are predicted; Step 5. Result drawing: Draw the tunnel excavation deformation characteristic curve based on the parameters predicted in step 4.

2. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 1, characterized in that: The P i In the equation, 1≤i≤N p .

3. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 1, characterized in that: In step 3, the steps of building and training the proxy model include: Step 31: Extract N in the tunnel excavation deformation characteristic curve parameter prediction agent model input space s Sample S j ,j∈[1,N s ], and obtain the model input of each sample in the proxy model training sample set S, where the jth sample S j The model input is denoted as P i j ; Step 32, with N s Sample S j The model input is used as the modeling parameter, and the corresponding tunnel excavation finite element modeling calculation is carried out to establish a finite element simulation model of the tunnel excavation process and simulate the tunnel excavation process. The established finite element model is the same as the j-th sample S j The model input P i j The corresponding tunnel excavation finite element model is denoted as M j ; Step 33: Substitute the finite element modeling calculation results of each tunnel excavation into the three-parameter exponential function for fitting, and obtain the tunnel excavation deformation characteristic curve characterization parameter u max A set of fitting values ​​of , k and a, based on the tunnel excavation finite element model M j Characterization parameter u obtained by fitting the calculation results max The fitted values ​​of , k and a are recorded as u maxj 、k j and a j , with u maxj 、k j and a j As the basis for the construction and the j-th sample S j Model output m∈[1,3], where are equal to u maxj 、k j 、a j ; Step 34: Use the sample set S to train the selected proxy model to obtain a tunnel excavation deformation characteristic curve parameter prediction proxy model.

4. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 3, characterized in that: In step 31, the input space of the tunnel excavation deformation characteristic curve parameter prediction agent model is composed of P i N p dimensional bounded space, and P i ∈[l i ,m i ].

5. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 3, characterized in that: In step 32, the finite element simulation model of the tunnel excavation process uses the predicted section as the middle cross-section of the model, with an axial length ≥15 times the tunnel diameter, the center of the cross-section located at the center of the tunnel, and the vertical distance between the center of the cross-section and the surrounding boundaries ≥5.5 times the tunnel diameter.

6. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 3, characterized in that: In step 4, the input variable value P i * where i∈[1,N p ], and input variable value P i * With input variable P i The multiple influencing factors correspond one to one.

7. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 6, characterized in that: The process of predicting tunnel excavation deformation characteristic curve parameters is as follows: Enter the given input variable value P1 * 、 The established tunnel excavation deformation characteristic curve parameter prediction proxy model is used to obtain the corresponding model output With P i * The predicted value u of the corresponding tunnel excavation deformation characteristic curve parameter max * 、k * 、a * are equal to 8. The method for predicting tunnel excavation deformation characteristic curve based on agent model according to claim 7, characterized in that: In step 5, the specific process of drawing the tunnel excavation deformation characteristic curve is as follows: let u in the three-parameter exponential function max , k and a take the corresponding predicted value u max * 、k * and a * , draw the function curve according to the function expression after determining the parameters.

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