PSI - Issue 23

Petr Opěla et al. / Procedia Structural Integrity 23 (2019) 221 – 226

224

Petr O pěla et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 3. Block diagram of the utilized genetic algorithm optimization.

Table 1. Material constants of the Eq. 3 given by the genetic algorithm optimization. Material constant ε p ( - ) σ p (MPa) σ ss (MPa) c (-)

s (-)

3.10 ·10 1.69 ·10 1.25 ·10

4.74 ·10 +2 0.54 ·10 +0 2.99 ·10 +2 3.3 0·10 − 3

5.93 ·10 +2 0.53 ·10 +0 3.05 ·10 +2 3.7 0·10 − 3

− 0.12 ·10 +0 − 0.11 ·10 +0 − 1.68 ·10 +2 − 1.4 0·10 − 3

1 .33·10 +6 3.07 ·10 +0 2 .31·10 +3 1.8 0·10 − 2

p 1 p 2 p 3 p 4

(various)

+2

(-)

+0

(K)

+3

1 .00·10 − 2

− 1 )

(K

3. Graphical and statistical evaluation Experimentally achieved and calculated curves are compared in Fig. 4 – see boxes vs. lines. The full lines correspond to the Cingara & McQueen's model in combination with the ANN approach, whereas the dashed lines represent the same model in cooperation with Eq. 3 and GA optimization. In order to validate the prediction performance, the grey lines embody the additionally predicted curves at three non-experimentally studied temperature levels (698 K, 748 K and 798 K). It can be seen, both approximation approaches are practically comparable and providing a high-accuracy description. However, a bigger discrepancy can be observed at higher strain rates, especially at 10 s − 1 . The ANN-predicted ε p -values of the peak point are shifted to the greater strains (see e.g. 698 K and 723 K / 10 s − 1 ). Nevertheless, the prediction capability seems to be sufficient – almost each additionally predicted curve is in the presumed flow stress level. Small exception takes place at 698 K / 5 s − 1 when both predicted curves are too close to the curve of 723 K / 5 s − 1 – at least in the first phase (flow stress growing). In order to statistically evaluate the approximation capability, the relative percentage error, η (%), is introduced as follows Quan et al. (2017):   100 i i i i y x y y     (4) The η i (%) corresponds with the i -th error (note: i = [1, n ] ⊂ ℕ , where n is the number of flow curve datapoints). The y i (MPa) and y ( x i ) (MPa) relate to the i -th target and computed flow stress value, respectively. In Fig. 5, the histograms provide the graphical view on the η -error distribution. As can be seen, the η -error of the Cingara & McQueen / ANN methodology is ranging in the narrower magnitude (between − 3 % and 2 %) in comparison to the Cingara & McQueen / GA attitude (between − 5 % and 4 %), which is, of course, reflected also by the standard deviation, σ (%). Howbeit, both methods are of good approximation accuracy – verified also by the zero-near mean η -error, μ (%).