Sigmoidal Regression of Double-Hit Compression Fractional Softening Data

Marcus Ritosa

Non-linear Programming - EBGN 552 - Semester Project

Problem Description

The following problem arose during the execution of “Determination of the Non-Recrystallization Temperature (TNR) In Multiple Microalloyed Steels” thesis project [1].  This thesis was executed by Caryn N. Homsher of the Advanced Steel Processing and Products Research Center (ASPPRC) at the Colorado School of Mines, advised by Dr. Chester Van Tyne.  

Rolling is a process of plastically deforming a material passed through two or more rotating rolls.  Hot rolling, as with all hot-working processes, requires elevated temperature control, generally in the range 850 °C-1320 °C (1562 °F‑2408 °F) for steel [2].  Multiple critical temperatures exist during hot rolling, one is known as the non-recrystallization temperature, defined as the temperature below which complete static recrystallization cannot occur in the given time.  The lower flow stress of the material at high temperature requires lower tool forces and power to deform the plate [3].  The workpiece is heated to a uniform elevated temperature in the austenite region, typically above the TNR.  The TNR for the steel is alloy dependent [2], and also dependent on the deformation parameters [2], [4].

To determine TNR of microalloyed steels, various testing methods are used in both industry and academia [5].  One method is double-hit compression, commonly executed on a thermomechanical simulator, such as Gleeble®  3500.  In this method, small cylindrical samples are heated to a specific conditioning temperature then cooled at a controlled rate to various temperatures under controlled conditions and then compressed twice within a small time interval.  At half-second intervals, the Gleeble®  3500 measures pound-force applied and piston stroke location (mm), among other metrics, but only these two are needed for the present analysis.

These data are filtered to include only the measurements taken during each of the two “hits.”  Manipulation of these data is then performed to chart stress-strain curves experienced by each sample, as demonstrated in Figure 1.  Following the 5 % true-strain method [5], a power function regression is performed, within MS Excel, on the last 50 points of the first hit to determine its parameters.  Using these parameters, the stress at 5 % strain is found for the first and second hits and is predicted for the power function at the 5 % strain of the second hit, given by σo, σr, and σm, respectively.  Using the found stresses, fractional softening (FS) can be calculated, as in Equation(1).  FS is a gauge on how much recrystallization took place in the 5 seconds between hits.  TNR can be found at the temperature where FS is equal to 20 % [6].  

Figure 1: Generated stress-strain curves for double-hit compression at 1000 °C [1]

                                                (1)

The initial approach to determining TNR from the dataset calculated the 20 % FS point as a linear interpolation.  The average of each set of data points with the same temperature parameter was calculated.  In the region around which the 20 % FS point was then suspected to be, additional tests were performed using temperature inputs with a 25 degree interval.  After these data points were added, a linear interpolation of the two closest averages yielded the 20 % FS point.  This method relies on the assumption that the function around these points is approximately linear.  This approach is inherently flawed given that the 20 % FS point was chosen because it signals the beginning of a major slope change in the function.  Thus, a new effort to determine 20 % FS using a sigmoidal regression was initiated.

Regression Formulation

A plot of these data on FS-temperature axes reveals that the data points generally adhere to a sigmoidal curve.  A sigmoidal function has four parameters: a, b, c and x0, as in Equation (2):

                                (2)

• Parameter a marks the range of the function f(x).  An approximation for this parameter value would be to take the difference between the largest observed f(x) value and the lowest observed f(x) value.  In this particular study, since the dependent variable f(x) is a percentage, an initial start point of a = 100 is appropriate.
• Parameter b characterizes the curvature of the s-shaped curve.  A positive value for b yields a function with a generally positive slope.  A negative value for b yields a function with a generally negative slope.  The higher the absolute value of b, the smaller the range of the curve gets overall.  Given that the dataset has a generally positive slope, b can be restricted to positive values for the purposes of this regression.
• Parameter c marks the lower bound asymptote of the curve.  An approximation for this parameter is the lowest observed f(x) value.  In this particular study, since the dependent variable f(x) is a percentage, an initial start point of c = 0 is appropriate.
• Parameter x0 marks the dependent variable value marking the center of the curve, where the slope is the highest, and the curve is in the middle of its range.  An approximation for this parameter value can be obtained by visually approximating the midpoint of the curve after having plotted the data.  Using Figure 2 below, a good approximation would be x0 = 1050.  If a common start point is to be used for each alloy dataset, the curves should overlap greatly.  This happens to be the case in this project.  The actual value of this parameter will depend greatly on the test input temperatures.

Figure 2: Example Sigmoidal Curve

Given a dataset of about 40 points, a regression must be done to determine the optimal values of the four parameters for each alloy in the research study.  This model is an unconstrained non-linear program with the form that follows.  Since this analysis seeks to determine parameter values, the parameters for the analysis will be represented in a way typical of variables, which the variables are represented in a way typical of parameters.  x0, the variable in this analysis, and a parameter value being determined should not be confused with xk, the parameter in this analysis, and a variable in the end product model, as k will never equal 0.

Indices

i alloys tested throughout the research project

j tested samples of each alloy-temperature

k temperature levels in the research project

Parameters

xk = Peak temperature of each sample processed at temperature k

yi,j,k = FS % of test sample j = {1..3}, alloy i = {1..10}, temp k = {1..10}

Variables

a = variable to determine asymptote range parameter a >= 0

b = variable to determine function curvature parameter b

c = variable to determine lower asymptote location parameter c >= 0

x0 = asymptote midpoint parameter >= 0

Objective Function

min 

Convexity Analysis

For its successful solution to be considered global, we must show that it is the minimization of a convex function.  In doing so, we will compute the Hessian matrix and find the eigenvalues.  Since this is a sum of expressions, we will test the convexity of only one term, and it will hold for the entire expression.  Thus, to be tested, Equation (3):

f(a, b, c, x0) =         (3)

Using Wolfram|Alpha, the following partial derivatives were computed:

The presence of the variables in these second derivatives indicate that the eigenvalues will depend upon their signs and values.  The shear complexity of these Hessian elements prevents the actual eigenvalues from being computed.  Without further details from this analysis, we cannot make a judgement on the convexity of the objective function.

We will now turn to empirical evidence for non-convexity.  If we run the model as defined above using AMPL, one solution is given, as shown in Table 1, below.  However, if the initial value of a parameter is changed, for instance if the starting point of x0 = 700 and b = 1, then a much different solution is given.  For Alloy A, the following values are output as the solution:

Table 1: Disparate outcome for Alloy A

Additional empirical examples could be shown, but they are not necessary.  Given this empirical evidence, we will accept that the function is non-convex.  Thus, the end solutions found can only be confidently said to be locally optimal.  However, approximations of “good” parameters, those described above, have yielded an acceptable starting point such that the results come reasonably close to comparable studies.

Regression Output

For each alloy i, an independent objective function was optimized was run to determine the following values for a, b, c, and x0.  Using these optimized variables as parameters, the temperature, TNR, predicting 20% FS is calculated in the final column of Table 2 below.

Table 2: Regression Output - All Alloys

The prediction of 20%-FS TNR given in the last column of the table was computed in the same AMPL model using the formula derived from the newly-determined values for the four parameters, as in Equation (4):

                (4)

This is the same equation as the sigmoidal curve, except it is solved for the value of the original dependent variable, temperature.  The value of the original independent variable, FS %, is set at 20.  This predictor becomes the input data for the next set of analysis in the larger material properties study, which must determine the causes and interactions of the micro-alloy content.

Comparison of different nonlinear solvers

The above analysis was performed using the solver MINOS.  Based on the model run output, 82 iterations were performed, including 172 evaluations of the objective and 171 of the gradient all within 0.01 seconds.  The optimal objective function value is 6929.08.

Running the same model using the solver MINLP, the model run output indicates that the model executed 1 subproblem with 43 evaluations of the objective, 24 of the gradient, and 24 of the Hessian all within 0.01 seconds.  The optimal objective function value is 6932.44.  The overall difference in run time was negligible between these two solvers.  There is one significant difference in model output between these two solvers.  This is manifest when determining the parameter values for Alloy F.  The difference is illustrated in Table 3, below:

Table 3: Alloy F model outputs

A difference of 13 degrees in the predicted TNR is significant, given that all other Alloys’ function parameters were identical within hundredths of a degree.  Curiously, when the initial start point for c is set to 5 instead of 0, the following results arise for Alloy G, in Table 4:

Table 4: Alloy G model outputs

This difference of 35 degrees is very troubling, given that in the prior iteration, running MINLP with a start parameter of c = 0, there was virtually no difference in outputs for Alloy G between the two solvers.  It would seem, at this point, that the parallel objective function terms for each alloy runs into some sort of timeout or degeneracy that prevents the full optimization of each alloy’s parameters.

References

[1]           C. N. Homsher, “Determination of the Non-Recrystallization Temperature in Multiple Microalloyed Steels,” Colorado School of Mines, 2013.

[2]           T. Gladman, The Physical Metallurgy of Microalloyed Steels, 2nd ed. London: Maney Publishing, 2002, p. 363.

[3]           W. F. Hosford and R. M. Caddell, Metal Forming: Mechanics and Metallurgy, 3rd ed. New York: Cambridge University Press, 2007, p. 312.

[4]           D. Q. Bai, S. Yue, W. P. Sun, and J. J. Jonas, “Effect of deformation parameters on the no-recrystallization temperature in Nb-bearing steels,” Metallurgical Transactions A, vol. 24, pp. 2151–2159, 1993.

[5]           S. Vervynckt, K. Verbeken, B. Lopez, and J. J. Jonas, “Modern HSLA steels and role of non-recrystallisation temperature,” International Materials Reviews, vol. 57, pp. 187–207, 2012.

[6]        E. J. Palmiere, C. I. Garcia, and A. J. DeArdo, “The influence of niobium supersaturation in austenite on the static recrystallization behavior of low carbon microalloyed steels,” Metallurgical and Materials Transactions A, vol. 27A, pp. 951–960, 1996.

Appendix

NLPP13.mod

#Sets

set Alloy;

set Test;

set Temp;

set Sample within {Alloy,Test,Temp};

#Parameters

param y {Sample};

param x {Temp};

#Variables

var a {i in Alloy}  := 100;

var b {i in Alloy}  := 20;

var c {i in Alloy}  := 0;

var x0 {i in Alloy} := 1050;

var tw {i in Alloy} = x0[i] - b[i] * log(a[i]/(20 - c[i])-1);

#Objective Function

minimize SSQ: sum{(i,j,k) in Sample} (c[i] + a[i] / (1+exp((x0[i] - x[k])/b[i])) - y[i,j,k])^2;

NLPP13.run

model NLPP13.mod;

data NLPP13.dat;

option solver minos;

solve;

display a, b, c, x0, tw;

quit;

NLPP13.dat

data;

set Alloy := A B C D E F G H I J;

set Test := 1 2 3;

param: Temp: x :=

1 1200

2 1150

3 1100

4 1050

5 1000

6  950

7  900

8  850

9  800

10 750;

param: Sample: y :=

A 1 1 105.8

A 3 1  81.6

A 1 2 100.4

A 2 2  82.5

A 3 2  95.9

A 1 3  84.4

A 2 3  63.5

A 3 3  83.7

A 1 4  46.6

A 2 4  50.0

A 3 4  49.8

A 1 5  19.3

A 2 5  23.0

A 3 5  19.5

A 1 6  14.9

A 2 6  14.7

A 3 6  13.4

A 1 7  13.2

A 2 7  13.4

A 3 7  12.5

A 1 8   9.9

A 2 8   9.3

A 3 8  11.1

A 1 9   8.3

A 2 9   6.8

A 3 9   6.3

A 1 10  9.8

A 2 10  7.4

A 3 10 10.0

B 1 1  98.5

B 2 1  91.0

B 3 1  94.3

B 1 2  92.5

B 2 2  87.5

B 1 3  72.5

B 2 3  69.8

B 3 3  73.5

B 1 4  39.9

B 2 4  45.0

B 3 4  33.8

B 1 5  13.9

B 2 5  15.9

B 3 5   3.6

B 1 6  13.7

B 2 6   7.6

B 3 6  13.0

B 1 7  14.0

B 2 7  13.3

B 3 7  -3.5

B 1 8   8.3

B 2 8   6.4

B 3 8  10.8

B 1 9   7.2

B 2 9  -3.0

B 3 9   6.8

B 1 10 -6.7

B 2 10  8.2

B 3 10  5.2

C        1       1       89.7

C        1       2       89.0

C        1       3       76.2

C        1       4       66.8

C        1       5       33.7

C        1       7       10.5

C        1       8       13.2

C        1       9       11.6

C        1       10      12.3

C        2       1       86.0

C        2       2       87.5

C        2       3       80.2

C        2       4       58.4

C        2       5       33.9

C        2       6       14.1

C        2       7       18.1

C        2       8       10.7

C        2       9       8.9

C        2       10      9.2

C        3       1       94.1

C        3       2       90.1

C        3       3       76.3

C        3       5       34.5

C        3       6       16.7

C        3       7       15.9

C        3       8       8.8

C        3       9       12.7

D       1       2       87.6

D       1       3       78.6

D       1       4       49.3

D       1       5       24.5

D       1       6       13.1

D       1       7       9.2

D       1       8       4.9

D       1       9       7.1

D       1       10      6.5

D       2       1       95.4

D       2       2       83.7

D       2       3       70.2

D       2       4       33.4

D       2       5       19.2

D       2       6       10.9

D       2       7       10.0

D       2       8       6.4

D       2       9       5.0

D       2       10      3.9

D       3       1       85.0

D       3       2       88.7

D       3       3       78.3

D       3       4       52.9

D       3       5       17.0

D       3       6       10.2

D       3       7       9.6

D       3       8       5.1

D       3       9       6.1

D       3       10      7.8

E       1       1       107.8

E       1       2       85.9

E       1       3       70.1

E       1       4       56.8

E       1       5       25.7

E       1       6       12.4

E       1       7       14.5

E       1       8       9.6

E       1       9       5.9

E       1       10      7.8

E       2       2       82.0

E       2       3       78.4

E       2       4       51.5

E       2       5       22.6

E       2       6       11.0

E       2       7       13.7

E       2       8       9.2

E       2       9       8.1

E       2       10      3.7

E       3       1       93.5

E       3       2       88.4

E       3       3       70.1

E       3       4       56.6

E       3       5       23.8

E       3       6       10.9

E       3       7       13.6

E       3       8       10.2

E       3       9       4.6

E       3       10      3.7

F       1       1       81.8

F       1       2       86.0

F       1       3       77.8

F       1       4       28.3

F       1       5       16.9

F       1       7       13.7

F       1       8       7.9

F       1       10      2.5

F       2       1       86.2

F       2       2       76.6

F       2       3       81.5

F       2       4       35.7

F       2       5       16.6

F       2       6       9.0

F       2       8       5.4

F       2       10      0.6

F       3       1       75.5

F       3       2       76.4

F       3       3       88.2

F       3       4       44.1

F       3       5       13.1

F       3       6       9.5

F       3       7       14.8

F       3       8       6.0

F       3       9       8.6

F       3       10      4.1

G       1       1       83.5

G       1       2       85.0

G       1       3       89.5

G       1       4       36.1

G       1       5       15.0

G       1       6       13.6

G       1       7       15.2

G       1       8       10.4

G       1       9       6.2

G       1       10      11.5

G       2       1       71.7

G       2       2       88.1

G       2       3       79.2

G       2       4       52.2

G       2       5       14.9

G       2       6       10.0

G       2       7       14.6

G       2       8       7.1

G       2       10      4.8

G       3       1       86.6

G       3       2       86.6

G       3       3       74.7

G       3       4       64.5

G       3       5       15.8

G       3       6       16.3

G       3       7       15.0

G       3       8       11.7

G       3       9       7.4

G       3       10      8.9

H       1       1       94.0

H       1       3       79.6

H       1       5       14.8

H       1       6       11.6

H       1       7       10.2

H       1       8       2.5

H       1       9       2.5

H       1       10      2.4

H       2       1       91.5

H       2       2       79.4

H       2       3       76.8

H       2       4       58.8

H       2       5       9.7

H       2       6       13.4

H       2       7       10.9

H       2       8       3.0

H       2       9       2.6

H       2       10      -0.6

H       3       1       92.9

H       3       2       98.5

H       3       3       72.8

H       3       4       51.2

H       3       5       10.2

H       3       6       11.7

H       3       7       10.3

H       3       8       2.8

H       3       9       2.7

H       3       10      -1.4

I       1       1       87.3

I       1       2       72.7

I       1       3       69.0

I       1       4       53.2

I       1       5       17.4

I       1       6       11.4

I       1       7       13.5

I       1       8       6.9

I       1       9       2.6

I       1       10      12.4

I       2       1       73.9

I       2       2       71.8

I       2       3       66.7

I       2       4       49.0

I       2       5       18.3

I       2       6       12.3

I       2       7       12.6

I       2       8       8.1

I       2       9       7.2

I       2       10      17.3

I       3       1       72.4

I       3       2       67.4

I       3       3       65.2

I       3       4       54.1

I       3       5       15.4

I       3       6       11.9

I       3       7       13.0

I       3       8       12.8

I       3       9       7.5

I       3       10      7.0

J       1       1       94.1

J       1       2       97.3

J       1       3       76.1

J       1       4       60.1

J       1       5       10.3

J       1       6       12.1

J       1       7       9.2

J       1       8       9.7

J       1       9       1.6

J       1       10      3.3

J       2       1       78.4

J       2       2       86.8

J       2       3       87.1

J       2       4       24.3

J       2       5       14.2

J       2       6       14.5

J       2       7       13.4

J       2       8       3.5

J       2       9       1.2

J       2       10      1.6

J       3       1       94.7

J       3       3       78.2

J       3       4       38.5

J       3       5       20.4

J       3       6       13.6

J       3       7       12.0

J       3       8       4.3

J       3       9       2.2

J       3       10      0.6;

NLPP13.out

MINOS 5.51: optimal solution found.

100 iterations, objective 6929.080815

Nonlin evals: obj = 220, grad = 219.

:               a                 b                 c                 x0                 tw       :=

A   86.2614   32.0357    9.77654   1057.3     993.019

B   89.6796   30.3085    5.82463   1067.51   1016.81

C   79.6924   36.0696   10.862     1032.17    958.443

D   84.6415   32.7072    6.80098   1054.63    999.397

E   92.8748   44.9906    6.71546   1056.1     975.551

F   72.4652   11.1115    9.04294   1055.57   1036.4

G   72.8932   16.7604   10.8661    1046.55   1013.98

H   87.1241   27.8553    4.2889    1048.16   1005.99

I   63.1831   20.6185   10.0213    1037.88   1003.37

J   84.9275   24.2728    6.67897   1057.67   1016.84 ;