Method for Adjusting the Extrapolated Hardening Curve of a Material

Abstract

The present invention discloses a method for adjusting the extrapolated hardening curve of a material, comprising the steps of: 1. Starting a uniaxial tensile experiment on the material and obtaining an engineering stress-strain curve; 2. Obtaining an experimental hardening curve from the engineering stress-strain curve of the material; 3. Supplementing the true strain of the experimental hardening curve to 1, i.e., the model fitting curve; 4. Taking a tangent line at the necking point of the experimental hardening curve; 5. Generating a Bézier curve using a second-order Bézier curve, i.e., the extrapolated hardening curve; 6. Adjusting the shape and stress value of the Bézier curve by controlling the parameters x1 and y2 to make the Bézier curve coincide with the model fitting curve and determining the reference values of x1 and y2; 7. Adjusting the extrapolated hardening curve according to actual needs. The present invention can determine the extrapolated hardening curve through three points, realize the adjustment of the extrapolated hardening curve by adjusting the parameters of the Bézier curve, and smoothly connect with the experimental hardening curve, meeting the requirements for identifying the material flow curve during the material tensile experiment and simulation benchmarking process.

Description

Technical Field

[0001] The present invention relates to an experimental method for the mechanical properties of materials, and particularly to a method for adjusting the extrapolated hardening curve of materials. Background Art

[0002] Accurate computer simulation is a prerequisite for the application of materials in automobiles. Accurately characterizing the mechanical behavior of materials and then inputting it into simulation software is a basic requirement for simulation. Under the uniaxial tensile stress state, the true stress-strain curve (hardening curve) of the material within the range of 0-1 true strain is one of the key data required in current general finite element simulation software. The experimental method for obtaining the hardening curve of the material is to conduct a uniaxial tensile experiment on the material.

[0003] For the tensile experiments of most sheet materials, the necking phenomenon during the experiment is inevitable. The existence of the necking phenomenon brings great difficulties to the characterization of materials. The two main reasons are respectively: (1) It is difficult to obtain the true cross-sectional area after necking, and the stress calculated through some simple formulas is inaccurate; (2) The stress distribution is uneven after necking, and the stress state of the test section gradually deviates from uniaxial tension. The existence of the necking phenomenon makes it difficult to obtain the hardening curve of the material in the large strain range through a simple uniaxial tensile experiment, and it cannot meet the requirement of the hardening curve in the 0-1 true strain range in the simulation software. Currently, there are mainly two types of solutions to this problem: 1. Using empirical formulas to correct the experimental results. 2. The reverse method combining experiment and simulation, that is, making a reverse supplement to the hardening curve part after the material necking, comparing the simulation results with the experimental results, and obtaining a hardening curve that can make the two match. The second type of method has a relatively high practical application value because it requires fewer experimental parameters to be input during the process and the steps are relatively simple.

[0004] The existing hardening curve reverse supplement method is to fit the curve before necking using a stress-strain constitutive model, and then use the model to extrapolate and supplement the curve part after necking. The level of the extrapolated stress is adjusted by adjusting the model parameters or introducing weights between multiple models after fitting multiple models to adjust the extrapolated stress. Chinese Patent Application CN201910056296.1 discloses a method for measuring the hardening curve of a metal round bar specimen in the large strain range of uniaxial tension, and specifically discloses: using the Voce model to fit the stress-strain curve before necking, and combining and adjusting the model parameters to adjust the extrapolated curve for flow curve identification. The existing hardening curve reverse supplement method has the following problems:

[0005] (1) Adjusting the model parameters will reduce the goodness of fit of the model to the pre-necking curve, resulting in deviation of the benchmark results of the uniform deformation section of the material during application. If the model only supplements the hardening curve after necking, the post-necking curve cannot be connected with the experimental curve before necking, making it difficult to ensure the continuity and smoothness of the entire hardening curve, and the processing is more cumbersome.

[0006] (2) The existing stress-strain constitutive model is a function of a certain form, and its shape can only be adjusted within a certain range. With the application of advanced high-strength steel, the microscopic strengthening mechanism of the material has become more complex, and the hardening behavior has become more diverse. The shape of the extrapolation curve of the existing constitutive model is not flexible enough.

[0007] (3) During the process of identifying the flow curve, it is necessary to adjust the stress of the extrapolated curve multiple times. Adjusting the model parameters to change the extrapolated stress is not simple, intuitive, and efficient. Summary of the invention

[0008] The purpose of the present invention is to provide a method for adjusting the extrapolated hardening curve of a material, which can determine the extrapolated hardening curve of an experimental hardening curve through three points, adjust the extrapolated hardening curve by adjusting the Bezier curve parameters, and smoothly connect the extrapolated hardening curve with the experimental hardening curve to meet the needs of identifying the material flow curve during material tensile testing and simulation benchmarking.

[0009] The present invention is achieved in that:

[0010] A method for adjusting a material extrapolation hardening curve comprises the following steps:

[0011] Step 1: Start the uniaxial tensile test of the material to obtain the engineering stress-strain curve of the material;

[0012] Step 2: intercept the uniform deformation section of the material engineering stress-strain curve and convert it into a true stress-strain curve, remove the elastic strain, and obtain the experimental hardening curve;

[0013] Step 3: Fit the experimental hardening curve by the model, add the true strain of the experimental hardening curve to 1, and form a model fitting curve;

[0014] Step 4: The coordinates of the necking point on the experimental hardening curve are (ε, σ), and the tangent line of the experimental hardening curve is taken with the necking point as the tangent point;

[0015] Step 5: Set the necking point (ε, σ) as the first endpoint P0 (x0, y0) of the Bezier curve, set the point where the true strain becomes 1 as the second endpoint P2 (x2, y2) of the Bezier curve, that is, the parameter x2 = 1, and use the point P1 (x1, y1) as the shape control point of the Bezier curve, and the point P1 falls on the tangent of the experimental hardening curve; provisionally set the parameters x1 and y2, and use the second-order Bezier curve to generate the Bezier curve, that is, the extrapolated hardening curve;

[0016] Step 6: Adjust the shape and stress value of the Bézier curve by controlling the parameters x1 and y2 to make the Bézier curve coincide with the model fitting curve, and determine the reference values of the parameters x1 and y2.

[0017] Step 7: Adjust the extrapolated hardening curve according to actual needs.

[0018] The parametric equation of the described Bézier curve is:

[0019] x = (1 - t) 2 *x0 + 2 * t * (1 - t) * x1 + t 2 *x2

[0020] y = (1 - t) 2 *y0 + 2 * t * (1 - t) * y1 + t 2 *y2

[0021] where the parameter t ∈ [0, 1];

[0022] The parameter y2 corresponds to the magnitude of the extrapolated stress, and the parameter x1 corresponds to the shape of the extrapolated curve.

[0023] The Bézier curve contains N pairs of point data. Let t = 0, 1 * Δt, 2 * Δt,..., (N - 2) * Δt, 1, and Δt = 1 / (N - 1).

[0024] The tentative principle of the parameter x1 is: ε < x1 < 1; the tentative principle of the parameter y2 is: y2 > σ.

[0025] The number of pairs of point data N on the Bézier curve is not less than 100, i.e., N ≥ 100.

[0026] The tangent equation of the described experimental hardening curve is y3 = kx3 + b, where k is the slope and b is the intercept;

[0027] Take n points (x 3i , y 3i ) including the necking point on the experimental hardening curve, where n is a natural number and i ∈ [1, n];

[0028] The calculation formula for the slope k is:

[0029] The calculation formula for the intercept b is:

[0030] where, is the average value of the abscissas x 3i of the n points, is the average value of the ordinates y 3i of the n points.

[0031] Compared with the prior art, the present invention has the following beneficial effects:

[0032] 1. Since the present invention adopts an extrapolated hardening curve based on a Bessel curve, the extrapolated hardening curve is determined by three points. Only by adjusting the relative positions of the three points can the adjustment of the extrapolated hardening curve be realized, and it can be smoothly connected with the experimental hardening curve. The entire hardening curve after extrapolated supplementation can ensure continuity and smoothness, solving the problems of inconvenient operation, non-smooth curve, and poor shape adjustability of the extrapolated hardening curve using traditional constitutive models, and improving the efficiency and accuracy of flow curve recognition.

[0033] 2. Since the present invention adopts an extrapolated curve based on a Bessel curve, the shape of the extrapolated curve and the corresponding stress values can be adjusted through spreadsheet software. A calculation template can be quickly generated, and the hardening curve can be adjusted quickly and conveniently. Moreover, the Bessel curve can well fit the hardening curves of various materials, meeting the requirements for identifying the flow stress of materials in simulation applications.

[0034] 3. Since the present invention adjusts the extrapolated curve by using two parameter values, the parameter value y2 directly corresponds to the stress value at a true strain of 1. When it is necessary to adjust the magnitude of the extrapolated stress during the subsequent flow stress recognition process, the parameter value of y2 can be directly adjusted. When it is necessary to adjust the shape of the extrapolated curve during the flow stress recognition process, the parameter value of x1 can be adjusted. The flexibility of the shape of the extrapolated curve is better, and the adjustment of the stress level can be carried out conveniently and intuitively.

[0035] Based on a second-order Bessel curve, the present invention can determine the extrapolated hardening curve of the experimental hardening curve through three points by using spreadsheet software, and realize the adjustment of the extrapolated hardening curve by adjusting the Bessel curve parameters, and make it smoothly connected with the experimental hardening curve. The number of parameters to be adjusted is small, the operation is simple and easy, and it can assist in aligning simulation and experimental data, and then determine the accurate hardening curve of the material in the large strain range, having a wide range of engineering application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is a flowchart of the method for adjusting the extrapolated hardening curve of the material in the present invention;

[0037] Figure 2 is an engineering stress-strain curve diagram of Embodiment 1 of the method for adjusting the extrapolated hardening curve of the material in the present invention;

[0038] Figure 3 is a graph showing the fitting of the hardening curve and the model of Embodiment 1 of the method for adjusting the extrapolated hardening curve of the material in the present invention;

[0039] Figure 4is an extrapolation curve diagram of Example 1 of the method for adjusting the extrapolation hardening curve of a material according to the present invention;

[0040] Figure 5 is a fitting curve diagram of an extrapolation curve and a model fitting curve of Example 1 of the method for adjusting a material extrapolation hardening curve of the present invention;

[0041] Figure 6 It is an extrapolated stress magnitude adjustment diagram (y2=960) of Example 1 of the method for adjusting the material extrapolation hardening curve of the present invention;

[0042] Figure 7 Extrapolation curve shape adjustment diagram of Example 1 of the method for adjusting the material extrapolation hardening curve of the present invention (x1=0.4);

[0043] Figure 8 1 is a diagram showing different extrapolated stress magnitude adjustments in Example 1 of the method for adjusting the material extrapolation hardening curve of the present invention;

[0044] Figure 9 is a diagram of adjusting shapes of different extrapolated curves in Example 1 of the method for adjusting the extrapolated hardening curve of a material according to the present invention;

[0045] Figure 10 It is a force-displacement curve diagram of the experiment and simulation of Example 1 of the method for adjusting the material extrapolation hardening curve of the present invention;

[0046] Figure 11 It is a hardening curve diagram after benchmarking in Example 1 of the method for adjusting the material extrapolation hardening curve of the present invention. DETAILED DESCRIPTION

[0047] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0048] Please see attached Figure 1 , a method for adjusting a material extrapolation hardening curve, comprising the following steps:

[0049] Step 1: Start the uniaxial tensile test of the material to obtain the engineering stress-strain curve of the material.

[0050] Preferably, the stress and strain of the material can be detected by the DIC (Digital Image Correlation) method of the prior art, and the fracture position is placed in the exact center of the virtual scale length, so as to facilitate accurate calibration with the simulation.

[0051] Step 2: Intercept the uniform deformation section of the material engineering stress-strain curve and convert it into a true stress-strain curve, remove the elastic strain, and obtain the experimental hardening curve.

[0052] Step 3: Fit the experimental hardening curve through the model, supplement the true strain of the experimental hardening curve to 1, and form a model fitting curve.

[0053] Preferably, the model can adopt the stress-strain constitutive model of the prior art, such as: Swift model: σ = k * (ε0 + ε pl ) n Or MPL model:

[0054] Step 4: The coordinates of the necking point on the experimental hardening curve are (ε, σ). Take the tangent of the necking point hardening curve with the necking point as the tangent point.

[0055] The tangent equation of the experimental hardening curve is y3 = kx3 + b, where k is the slope and b is the intercept;

[0056] Take n points (x 3i , y 3i ) including the necking point on the experimental hardening curve, n is a natural number, and i ∈ [1, n]; preferably, n = 10.

[0057] The calculation formula for the slope k is:

[0058] The calculation formula for the intercept b is:

[0059] Among them, is the average value of the abscissas x 3i of n points, is the average value of the ordinates y 3i of n points.

[0060] Step 5: Set the necking point (ε, σ) as the first endpoint P0(x0, y0) of the Bezier curve, set the point with a true strain of 1 as the second endpoint P2(x2, y2) of the Bezier curve, that is, the parameter x2 = 1, set the point P1(x1, y1) as the shape control point of the Bezier curve, and the point P1 falls on the tangent of the experimental hardening curve. Tentatively set the parameter x1 and the parameter y2, and with the help of spreadsheet software (such as excel, etc.), use the second-order Bezier curve to generate a Bezier curve with N pairs of evenly distributed points, that is, the extrapolated hardening curve.

[0061] The parametric equation of the Bezier curve is:

[0062] x = (1 - t) 2 *x0 + 2 * t * (1 - t) * x1 + t 2 *x2

[0063] y = (1 - t) 2 *y0 + 2 * t * (1 - t) * y1 + t 2*y2 formula (3)

[0064] Among them, the parameter t ∈ [0, 1].

[0065] The number N of point - pair data can be adjusted according to actual needs. To ensure that the number of point - pair data can basically meet the usage requirements, the number N of point - pair data on the B - spline curve is not less than 100, that is, N ≥ 100.

[0066] The tentative principle of the parameter x1 is: ε < x1 < 1; the tentative principle of the parameter y2 is: y2 > σ.

[0067] Step 6: N point - pair data are evenly distributed on the B - spline curve. Let t = 0, 1*△t, 2*△t, …, (N - 2)*△t, 1, △t = 1 / (N - 1); adjust the shape and stress value of the B - spline curve by controlling the parameters x1 and y2 to make the B - spline curve coincide with the model fitting curve, and determine the reference values of the parameters x1 and y2.

[0068] Step 7: Adjust the extrapolated hardening curve according to actual needs.

[0069] Example 1:

[0070] Process the JIS 5# tensile specimen of DP590 high - strength steel and conduct a quasi - static tensile experiment to obtain the engineering stress - strain curve of the DP590 material, as shown in the appendix Figure 2 as follows.

[0071] Intercept the uniform deformation section (the curve part before the peak stress) on the engineering stress - strain curve of the DP590 material, convert it into a true stress - strain curve, and remove the elastic strain to obtain the experimental hardening curve. The conversion formula is:

[0072]

[0073] σ T = σ E *(1 + ε E )

[0074] Among them, is the true plastic strain, ε E is the engineering strain, σ T is the true stress, E is the elastic modulus, σ E is the engineering stress, ε E is the engineering strain.

[0075] Use the existing MPL model to fit the experimental hardening curve of the DP590 material, and extrapolate and supplement the experimental hardening curve to a true strain of 1 to form a model fitting curve. The experimental hardening curve and the model fitting curve of DP590 are shown in the appendix Figure 3As shown, where "―" represents the model fitting curve and "●" represents the experimental hardening curve.

[0076] From the appendix Figure 3 it can be seen that the necking point is the last point on the experimental hardening curve. The necking point coordinates of DP590 material are (0.1525, 741.28). Take the tangent of the experimental hardening curve with the necking point as the tangent point. Take n = 10. According to formula (2) and formula (3), calculate the slope k of the tangent = 769.15, and the intercept b of the tangent = 623.92. Then the tangent equation is y3 = 769.15 * x3 + 623.92.

[0077] Define the necking point (0.1525, 741.28) as the first endpoint P0 of the Bezier curve, that is, x0 = 0.1525, y0 = 741.28. The true strain of the second endpoint P2(x2, y2) of the Bezier curve is 1, that is, the parameter x2 = 1. Take the point P1(x1, y1) as the shape control point of the Bezier curve, and the point P1 falls on the tangent of the experimental hardening curve. Tentatively set x1 = 0.2, y2 = 800, N = 501, △t = 0.002, that is, t = 0, 0.002, 0.004,..., 1. So P1(0.2, 769.15 * 0.2 + 623.92), P2(1, 800). According to the point P0, point P1 and point P2, using spreadsheet software, generate a Bezier curve containing 501 pairs of point data based on the second-order Bezier curve algorithm, that is, the extrapolation curve segment of DP590 material, as shown in the appendix Figure 4 shown. In the appendix Figure 4 "■" represents the second endpoint P2; "·" represents the shape control point P1 of the Bezier curve; "―" represents the experimental hardening curve; "-------" represents the model fitting curve; "--" represents the Bezier curve; "-·-·" represents a control line between the point P1 and the necking point, used to adjust the shape of the Bezier curve; "··········" represents a control line between the point P1 and the second endpoint P2, used to adjust the shape of the Bezier curve.

[0078] x = (1 - t) 2 *x0 + 2 * t * (1 - t) * x1 + t 2 *x2

[0079] y = (1 - t) 2 *y0 + 2 * t * (1 - t) * y1 + t 2 *y2

[0080] Adjust the parameter values of x1 and y2 so that the Bezier curve coincides with the model fitting curve, as shown in the appendix Figure 5 shown. At this time, the parameter reference values are: x2 = 1, y2 = 881.98, x1 = 0.315, y1 = 866.2. In the appendixFigure 5 In it, "■" represents the second endpoint P2; "·" represents the shape control point P1 of the Bezier curve; "―" represents the experimental hardening curve; "-------" represents the model fitting curve; "--" represents the Bezier curve; "-·-·" represents a control line between point P1 and the necking point, used to adjust the shape of the Bezier curve; "··········" represents a control line between point P1 and the second endpoint P2, used to adjust the shape of the Bezier curve.

[0081] If it is necessary to adjust the extrapolation stress magnitude during the alignment process, it can be achieved by adjusting the parameter value of y2. For example, when the parameter value of y2 is adjusted to 960 and other parameter values remain unchanged, i.e., x2 = 1, x1 = 0.315, y1 = 866.2, the extrapolation curve is as shown in the appendix Figure 6 shown. In the appendix Figure 6 In it, "■" represents the second endpoint P2; "·" represents the shape control point P1 of the Bezier curve; "―" represents the experimental hardening curve; "------" represents the model fitting curve; "--" represents the Bezier curve; "-·-·" represents a control line between point P1 and the necking point, used to adjust the shape of the Bezier curve; "··········"" represents a control line between point P1 and the second endpoint P2, used to adjust the shape of the Bezier curve.

[0082] The hardening curves with different extrapolation stress magnitudes are as shown in the appendix Figure 8 shown. In the appendix Figure 8 In it, "―" represents the experimental hardening curve; "----" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 881.98; "---" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 981.98; "-·-" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 1081.98; "--" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 1181.98; "-·-·" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 1281.98; "-··-··" represents the Bezier curve with parameter x1 = 0.315 and parameter y2 = 1381.98.

[0083] If it is necessary to adjust the shape of the extrapolation curve during the alignment process, it can be achieved by adjusting the parameter value of x1. For example, when the parameter value of x1 is adjusted to 0.4, then according to the tangent equation y1 = 931.58 and other parameter values remain unchanged, i.e., x2 = 1, y2 = 960, the extrapolation curve is as shown in the appendix Figure 7 shown. In the appendix Figure 6In it, "■" represents the second end point P2; "·" represents the shape control point P1 of the Bezier curve; "―" represents the experimental hardening curve; "-------" represents the model fitting curve; "--" represents the Bezier curve; "-·-·" represents a control line between the point P1 and the necking point, used to adjust the shape of the Bezier curve; "··········" represents a control line between the point P1 and the second end point P2, used to adjust the shape of the Bezier curve.

[0084] Hardening curves with different extrapolation curve shapes are shown in the appendix Figure 9 as follows. In the appendix Figure 9 "―" represents the experimental hardening curve; "-------" represents the Bezier curve with parameter x1 = 0.23 and parameter y2 = 960; "---" represents the Bezier curve with parameter x1 = 0.33 and parameter y2 = 960; "-·-·" represents the Bezier curve with parameter x1 = 0.43 and parameter y2 = 960.

[0085] From appendix Figure 8 and appendix Figure 9 it can be seen that the entire hardening curve composed of the experimental hardening curve and the extrapolated hardening curve is a continuous and smooth curve, which can meet the requirements of adjusting the magnitude of the extrapolated stress and the curve shape after adjusting the hardening curve during the experimental and simulation comparison of DP590 materials. When the parameter x1 is set to 0.37 and y2 is set to 920, after the entire hardening curve is input into the simulation software, the coincidence degree of the force-displacement curves of the experiment and the simulation is relatively high, and a good comparison effect is obtained, as shown in appendix Figure 10 where "---" represents the force-displacement curve of the simulation, and "—" represents the force-displacement curve of the experiment. The hardening curve after the comparison of the force-displacement curves of the experiment and the simulation of DP590 is shown in appendix Figure 11 as follows.

[0086] The above is only a preferred embodiment of the present invention, and is not used to limit the protection scope of the present invention. Therefore, any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for adjusting the extrapolated hardening curve of a material, characterized in that: The following steps are involved: Step 1: Start the uniaxial tensile test of the material to obtain the engineering stress-strain curve of the material; Step 2: intercept the uniform deformation section of the material engineering stress-strain curve and convert it into a true stress-strain curve, remove the elastic strain, and obtain the experimental hardening curve; Step 3: Fit the experimental hardening curve by model, and add the true strain of the experimental hardening curve to 1, that is, the model fitting curve; Step 4: The coordinates of the necking point on the experimental hardening curve are (ε, σ), and the tangent line of the experimental hardening curve is taken with the necking point as the tangent point; Step 5: Set the necking point (ε, σ) as the first endpoint P0 (x0, y0) of the Bezier curve, set the point where the true strain becomes 1 as the second endpoint P2 (x2, y2) of the Bezier curve, that is, the parameter x2 = 1, and use the point P1 (x1, y1) as the shape control point of the Bezier curve, and the point P1 falls on the tangent of the experimental hardening curve; provisionally set the parameters x1 and y2, and use the second-order Bezier curve to generate the Bezier curve, that is, the extrapolated hardening curve; Step 6: Adjust the shape and stress value of the Bezier curve by controlling the parameters x1 and y2, so that the Bezier curve coincides with the model fitting curve, and determine the reference values ​​of the parameters x1 and y2; Step 7: Adjust the extrapolated hardening curve according to actual needs; The parametric equation of the Bezier curve is: x = (1 - t) 2 *x0 + 2**(1 - t)*x1 + t 2 *x2 y = (1 - t) 2 *y0 + 2 * t * (1 - t) * y1 + t 2 *y2 Among them, the parameter t∈[0,1]; The parameter y2 corresponds to the magnitude of the extrapolated stress, and the parameter x1 corresponds to the shape of the extrapolated curve.

2. The method for adjusting the extrapolated hardening curve of the material according to claim 1, wherein: The Bezier curve contains N point pair data, let t=0, 1*△t, 2*△t, ..., (N-2)*△t, 1, △t=1 / (N-1).

3. The method for adjusting the extrapolated hardening curve of the material according to claim 2, wherein: The number N of point pair data on the Bezier curve is not less than 100, that is, N≥100.

4. The method for adjusting the extrapolated hardening curve of the material according to claim 1, characterized in that: The provisional principle for the parameter x1 is: ε<x1<1; the provisional principle for the parameter y2 is: y2>σ.

5. The method for adjusting the extrapolated hardening curve of the material according to claim 1, characterized in that: The tangent equation of the experimental hardening curve is y3=kx3+b, where k is the slope and b is the intercept; Take n points (x 3i , y 3i ) including the necking point on the experimental hardening curve, where n is a natural number and i ∈ [1, n]; The calculation formula for the slope k is as follows: The calculation formula for the intercept b is as follows: Among them, is the average value of the abscissas x of n points 3i , is the average value of the ordinates y of n points 3i .

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