Prediction of flow stress by Artificial Neural Network modeling

Prediction of peak flow stress

4.7 Prediction of flow stress by Artificial Neural Network modeling

rates, 5 deformation temperatures and 6 true strain values considered) were obtained from the compression tests. It was decided to use 48 data sets for the network training, 36 data sets for the testing and 18 data sets from the remaining, for validation purpose. The validation data sets were not used earlier for the testing or training purposes. Both training and testing of the network was carried our independently. A number of numerical trials were carried out with single hidden layer neural network. The tansig, logsig and purelin transfer functions were tried with while simultaneously varying the number of neurons in the hidden layer, in order to arrive at the best network architecture and processing function for each of the investigated alloys. The RMS functional error used as a measure of performance, can be expressed as

2 f

err 2

( *)

RMS n

σ σ σ

−

∑

(4.14) where σ is the experimental value and σ* is the predicted value of the flow stress. Both training and testing errors were calculated separately. Effective error for training and testing data is given by:

Effective error = max. of [RMS of training data, ^f_err RMS of testing data] (4.15) ^f_err The sum squared training error goal for σ was fixed at 0.00001. After a number of trials with various initial weights and biases, the best neural network architecture was frozen for which (i) functional RMS error was minimum, (ii) minimum number of data sets has a deviation error of 10 %, (iii) maximum deviation during testing and training is within 20 %, and (iv) minimum variation in RMS functional error during training and testing.

Table 4.33 shows the details of the ANN architecture for all the investigated alloys.

Once the network was frozen, it was required to establish the confidence in the network architecture for the prediction of σ. Therefore, the trained network was used for prediction of σ using the 18 validation data sets. Table 4.34 gives the values of maximum absolute error, maximum percentage error and RMS error calculated for all the alloys during training, testing and validation stages separately. Table 4.35 to Table 4.40 give the detailed experimental and predicted σ values along with the associated errors for validation data sets of all the alloys studied.

Figures 4.54, 4.55 and 4.56 show the plots of experimental vs. predicted values of σ for all the alloys during training, testing and validation, respectively. Dashed lines representing the boundaries of ±10 % deviations are also shown in the figures. Figures reveal that most of the points lie very close to the line of prefect prediction. During validation, it was observed that 18, 17, 17, 16, 17 and 15 out of 18 data points fall within

±10 % deviation line for Alloy-A, Alloy-B, Alloy-C, Alloy-D, Alloy-E, and Alloy-F, respectively. Though the percentage errors in prediction were comparatively high for the few data with deviations above ±10 %, their absolute errors were low, especially in the cases of low σ.

The observation that 0, 1, 1, 2, 1 and 3 out of 18 data points fall outside ±10 % deviation line for Alloy-A, Alloy-B, Alloy-C, Alloy-D, Alloy-E, and Alloy-F, respectively shows very good prediction of σ by ANN modeling. This observation together with the low RMS errors registered, highlight the superior prediction capability of this technique.

Table 4.33. Best fit network architectures for the investigated alloys.

Sample ID Hidden neurons

1st transfer function

2nd transfer function

Alloy-A 9 tansig purelin

Alloy-B 8 logsig purelin

Alloy-C 3 logsig purelin

Alloy-D 5 tansig purelin

Alloy-E 9 tansig purelin

Alloy-F 8 logsig purelin

Table 4.34. Maximum absolute error, maximum percentage error and RMS error obtained for the alloys during training, testing and validation

Training Testing Validation

Alloy ID

Max.

Error (MPa)

Max.

% Error

RMS Error (MPa)

Max.

Error (MPa)

Max.

% Error

RMS Error (MPa)

Max.

Error (MPa)

Max.

% Error

RMS Error (MPa) Alloy-A 1.63 2.00 0.45 -6.00 -9.61 2.78 4.69 7.61 2.37 Alloy-B 2.24 4.13 0.95 -14.91 14.90 6.11 -8.96 12.86 3.41 Alloy-C 2.81 -5.84 1.22 9.58 11.82 4.26 13.54 10.25 6.70 Alloy-D -3.69 3.65 1.27 8.89 -4.59 2.50 13.98 11.19 5.40 Alloy-E 2.69 -6.26 0.83 -9.01 -19.59 4.68 -5.39 -13.75 2.11 Alloy-F -4.45 -4.67 1.27 -11.48 -15.18 4.98 17.05 23.17 8.32

Table 4.35. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-A

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 61.27

120.07 49.31 36.22 121.47

66.03 61.61 121.71

49.66 36.43 121.87

65.86 61.25 121.71

49.13 36.52 122.42

65.09

57.17 120.93

46.84 36.88 119.59

68.33 56.92 123.47

50.39 36.91 122.88

67.81 57.96 124.76

49.21 35.86 118.38

64.44

4.10 -0.86

2.47 -0.66

1.88 -2.30

4.69 -1.76 -0.73 -0.48 -1.01 -1.95 3.29 -3.05 -0.08 0.66 4.04 0.65

6.70 -0.71

5.01 -1.83

1.55 -3.48

7.61 -1.44 -1.46 -1.32 -0.83 -2.97 5.37 -2.51 -0.16 1.80 3.30 1.00

Table 4.36. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-B

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 56.09

64.5 111.82

24.96 102.19

83.61 76.57 55.84 65.91 36.91 102.58

49.07 77.1 54.82 65.08 21.82 101.9 83.02

56.01 59.71 120.78

26.69 105.86

89.37 78.90 55.54 67.90 36.95 101.04

42.76 77.80 55.33 64.73 20.55 100.48

82.44

0.08 4.79 -8.96 -1.73 -3.67 -5.76 -2.33 0.30 -1.99 -0.04 1.54 6.31 -0.70 -0.51 0.35 1.27 1.42 0.58

0.14 7.43 -8.02 -6.94 -3.59 -6.89 -3.05 0.53 -3.02 -0.12 1.50 12.86 -0.91 -0.93 0.54 5.80 1.39 0.70

Table 4.37. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-C

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 52.74

139.47 51.76 39.84 141.14

67.98 52.62 140.76

52.59 38.76 140.95

66.69 51.8 141.06

51.35 37.76 141.65

65.83

48.91 131.74

54.79 39.45 131.10

69.76 47.67 130.41

53.29 38.69 129.68

67.90 46.49 128.91

51.85 37.90 128.11

66.06

3.83 7.73 -3.03

0.39 10.04

-1.78 4.95 10.35

-0.70 0.07 11.27

-1.21 5.31 12.15

-0.50 -0.14 13.54 -0.23

7.26 5.55 -5.85

0.99 7.12 -2.63

9.40 7.35 -1.33

0.19 7.99 -1.81 10.26 8.61 -0.97 -0.38 9.56 -0.35

Table 4.38. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-D

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 67.49

46.43 124.38

34.28 125.49

82.11 66.37 45.48 118.87

30.87 124.89

78.67 64.66 44.17 112.92

27.66 122.44

76.62

68.53 45.40 122.10

32.10 116.18

80.81 64.59 44.25 122.21

28.46 110.91

79.58 63.67 42.76 118.69

26.81 109.42

75.70

-1.04 1.03 2.28 2.18 9.31 1.30 1.78 1.23 -3.34

2.41 13.98

-0.91 0.99 1.41 -5.77

0.85 13.02

0.92

-1.55 2.23 1.83 6.35 7.42 1.59 2.69 2.70 -2.81

7.80 11.20 -1.15 1.53 3.19 -5.11

3.06 10.64

1.20

Table 4.39. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-E

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 105.82

84.9 143.72

60.24 109.31

86.67 100.85

77.27 80.36 33.38 106.24

83.5 97.93 78.35 136.71

56.59 106.13

82.18

106.40 84.99 142.78

60.14 109.29

87.07 101.61

82.66 80.60 37.97 107.12

85.61 97.63 77.99 137.18

56.38 104.95

86.73

-0.58 -0.09 0.94 0.10 0.02 -0.40 -0.76 -5.39 -0.24 -4.59 -0.88 -2.11 0.30 0.36 -0.47

0.21 1.18 -4.55

-0.55 -0.10 0.65 0.17 0.02 -0.46 -0.76 -6.97 -0.29 -13.75

-0.83 -2.53 0.30 0.46 -0.34

0.38 1.11 -5.54

Table 4.40. Comparison of experimental and predicted values of flow stress, σ for the validation test of Alloy-F

Experimental flow stress(MPa)

Predicted flow stress (MPa)

Error (MPa)

Percentage error (%) 107.07

118.61 49.23 118.61

103.7 118.1 48.19 117.25 100.58 116.26 88.03 47.59 45.45 135.5 93.5 30.5 89.52

24.5

102.16 130.34 50.34 128.46 103.60 129.07 48.99 128.43 102.25 125.30 79.39 48.94 34.92 136.57

76.45 29.59 76.29 25.15

4.91 -11.73

-1.11 -9.85 0.10 -10.97

-0.80 -11.18

-1.67 -9.04 8.64 -1.35 10.53 -1.07 17.05 0.91 13.23 -0.65

4.58 -9.89 -2.26 -8.31 0.09 -9.29 -1.66 -9.53 -1.66 -7.78 9.81 -2.83 23.17 -0.79 18.23 2.98 14.77 -2.66

0 25 50 75 100 125 150