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materials Article

A Modified Back Propagation Artificial Neural Network Model Based on Genetic Algorithm to Predict the Flow Behavior of 5754 Aluminum Alloy Changqing Huang 1,2,3, *, Xiaodong Jia 1,2 and Zhiwu Zhang 1,2 1 2 3

*

Light Alloy Research Institute, Central South University, Changsha 410083, China; [email protected] (X.J.); [email protected] (Z.Z.) State Key Laboratory of High-performance Complicated Manufacturing, Central South University, Changsha 410083, China School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China Correspondence: [email protected]; Tel.: +86-731-8887-6183

Received: 13 April 2018; Accepted: 17 May 2018; Published: 21 May 2018







Abstract: In order to predict flow behavior and find the optimum hot working processing parameters for 5754 aluminum alloy, the experimental flow stress data obtained from the isothermal hot compression tests on a Gleeble-3500 thermo-simulation apparatus, with different strain rates (0.1–10 s–1 ) and temperatures (300–500 ◦ C), were used to construct the constitutive models of the strain-compensation Arrhenius (SA) and back propagation (BP) artificial neural network (ANN). In addition, an optimized BP–ANN model based on the genetic algorithm (GA) was established. Furthermore, the predictability of the three models was evaluated by the statistical indicators, including the correlation coefficient (R) and average absolute relative error (AARE). The results showed that the R of the SA model, BP–ANN model, and ANN–GA model were 0.9918, 0.9929, and 0.9999, respectively, while the AARE of these models was found to be 3.2499–5.6774%, 0.0567–5.4436% and 0.0232–1.0485%, respectively. The prediction error of the SA model was high at 400 ◦ C. It was more accurate to use the BP–ANN model to determine the flow behavior compared to the SA model. However, the BP–ANN model had more instability at 300 ◦ C and a true strain in the range of 0.4–0.6. When compared with the SA model and BP–ANN model, the ANN–GA model had a more efficient and more accurate prediction ability during the whole deformation process. Furthermore, the dynamic softening characteristic was analyzed by the flow curves. All curves showed that 5754 aluminum alloy showed the typical rheological characteristics. The flow stress rose rapidly with increasing strain until it reached a peak. After this, the flow stress remained constant, which demonstrates a steady flow softening phenomenon. Besides, the flow stress and the required variables to reach the steady state deformation increased with increasing strain rate and decreasing temperature. Keywords: 5754 aluminum alloy; hot compressive; constitutive model; artificial neural network; genetic algorithm; dynamic softening mechanism

1. Introduction Hot working is a significant process which directly affects the formability of materials and the mechanical properties of the product [1–4]. However, in the actual production process of hot compression, the internal structure and properties of the metal will obviously change, resulting in different mechanical behavior in the subsequent deformation. The flow behavior of materials is an important factor that needs to be considered in actual production as it is closely related to the microstructure and properties of the products [5,6]. It reflects the non-linear relationship between the Materials 2018, 11, 855; doi:10.3390/ma11050855

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flow stress and control parameters, including strain, strain rate, and temperature of deformation, which is often determined by the constitutive model [7–9]. The constitutive model can be employed to uncover the deformation mechanism of materials and to develop the dynamic model of recrystallization. Therefore, the establishment of a high-precision constitutive model for materials has become a focus of researchers. In the current study, many constitutive models including physical analysis, artificial neural network (ANN), and phenomenological models for different materials have been developed on the basis of hot deformation tests [10–13]. The physical analysis models have to regress many material coefficients, which inevitably imposes serious constraints on their practical applications [11]. Due to the effective avoidance of contact with the physical mechanism and increased accuracy, the phenomenological constitutive models, such as the strain-compensation Arrhenius (SA) model, require less material constants, and thus, are widely adopted in practice [14–17]. In order to improve the predictive accuracy, many researchers have modified the phenomenological constitutive models [18,19]. However, when it comes to different experimental data and materials with different processing states, the regression constants of physical analysis and phenomenological models using the regression method have to be recalculated [20,21]. The ANN modeling method, without the specific mathematical model, provides a new idea for the study of the constitutive behavior of materials. This does not require repeated regression calculations, and high-precision predictions can be obtained by the ANN model [22–24]. It has a great advantage of being able to deal with complicated and non-linear relationships. Hence, ANN has been widely used to predict the flow behavior of metal materials [25–28]. Nevertheless, the uncertainty and instability of ANN in predicting flow behavior limits its use. The introduction of genetic algorithm (GA) can remove this limitation, improve its stability and accuracy, and search for the optimal weight and threshold for ANN at the same time. The 5754 aluminum alloy is a typical Al–Mg alloy, and has moderate strength, favorable corrosion resistance, good weldability, and easy processing. Therefore, it is often used in the automotive and aerospace industries. Consequently, it is necessary to study the performance and indicators of the target materials, especially the dynamic softening mechanism and flow behavior. Furthermore, setting up a perfect constitutive prediction model of flow behavior for 5754 aluminum alloy would have far-reaching implications. In this work, the hot compression tests were carried out on Gleeble-3500 thermo-simulation machine at different strain rates and different deformation temperatures. In order to provide accurate data and establish the model with high accuracy and stability, the temperature rise effect was considered to optimize the stress–strain data obtained. Furthermore, the experimental data and modified flow curves were applied to construct the SA model and the back propagation artificial neural network (BP–ANN) model. The use of GA led to the determination of optimal weight and threshold for ANN simultaneously. After this, the performance of the three models was compared according to the correlation coefficient (R) and the average absolute relative error (AARE). Finally, the dynamic softening mechanism of 5754 aluminum alloy was analyzed by the stress–strain curves of a single pass. 2. Experiment and Materials The hot compression tests were carried out on a Gleeble-3500 thermo-simulation machine (Dynamic System Institution, New York, NY, USA). The nominal chemical composition (wt %) of the 5754 aluminum alloy is shown in Table 1. The geometric shapes and size of the specimens for testing are shown in Figure 1.

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Figure 1. The initial shape and size of the specimens. Figure 1. The initial shape and size of the specimens. Table 1. The nominal chemical composition of the 5754 aluminum alloy. Table 1. The nominal chemical composition of the 5754 aluminum alloy. Composition Mg Mn Si Fe Composition Mg Mn Content (wt %) 2.6–3.6 0.5 Si0.4 Fe 0.4 Content (wt %) 2.6–3.6 0.5 0.4 0.4

Cr Cr 0.3 0.3

Zn Ti Cu Impurity Al Zn Ti Cu Impurity Al 0.2 0.15 0.1 0.15 Balance Balance 0.2 0.15 0.1 0.15 Balance

First, the specimens were heated to the holding temperature (500 °C) by resistance with a heating speed 10 °C/s and insulation forthe 4 min. Aftertemperature this, we cooled at different ◦ C)specimens First, the of specimens were heated to holding (500 the by resistance with a deformation temperatures at a rate of 5 °C/s to eliminate the thermal gradients and ensure that ◦ heating speed of 10 C/s and insulation for 4 min. After this, we cooled the specimens at different heating was temperatures complete before variation the range of thegradients test parameters werethat as deformation at acompression. rate of 5 ◦ C/sThe to eliminate thermal and ensure −1 −1. The whole follows: deformation temperatures were 300–500 °C and strain rates were 0.1–10 s heating was complete before compression. The variation range of the test parameters were as follows: process of hot compression tests ended◦ C when the compression deformation 70%. The deformation temperatures were 300–500 and strain rates were 0.1–10 s−1 . Thereached whole process of specimens were quenched immediately after the experiment. The specific experimental procedure is hot compression tests ended when the compression deformation reached 70%. The specimens were shown in Figure 2. quenched immediately after the experiment. The specific experimental procedure is shown in Figure 2.

Figure 2. 2. The flowsheet of of hot hot compression compression tests. tests.

3. Results and Discussion 3.1. 3.1. The The SA SA Model Model According model, thethe constitutive relationship of the at lowatstress According to tothe theArrhenius Arrhenius model, constitutive relationship ofmaterial the material low levels stress can becan expressed as follows: levels be expressed as follows: . ε = A1 σnn1 exp(− Q/RT ) (1)   A11 n11 exp(Q / RT ) (1) For high stress levels, this equation becomes: For high stress levels, this equation becomes: . ε = A2 exp( βσ ) exp(− Q/RT ) (2)   A22 exp( ) exp(Q / RT ) (2)

For all stress levels, the following equation can be applied:

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  A3[sinh( )]n exp(Q / RT )

(3) For all stress levels, the following equation can be applied: where  represents the true stress (MPa); T and  represent the absolute temperature (K) and . n ε = A ασ)]the exp (− Q/RT )energy (J mol−1) and gas constant, (3) strain rate (s−1), respectively; and3 [sinh(are activation

Q R . respectively; and the , A3 ,T nand , and canand be strain derived  ,true n are the  (K) A1 , A(MPa);  , stress where σ represents ε represent thematerial absoluteconstants. temperature 2 1

−1 −1 rate by:(s ), respectively; Q and R are the activation energy (J mol ) and gas constant, respectively; and α, β, A1 , A2 , A3 , n1 , and n are the material constants. α can be derived by:    / n1 (4) α = β/n1 (4) The influence of the deformation parameter on the flow stress can be defined as Z (Zener–Holloman The influence parameter): of the deformation parameter on the flow stress can be defined as Z

Z .  exp(Q / RT ) (5) Z = ε exp( Q/RT ) (5) The material constants were calculated at the true strain of 0.2. As shown in Figures 3a,b, there material constantsbetween were calculatedand at the true, strain of 0.2. As shown in Figure 3a,b, there is The a linear relationship ln  . ln  where.  and ln  are the same. Linear is a linear relationship between ln σ and ln ε, where σ and ln ε are the same. Linear regression was regression was performed using the least squares method. The reciprocals of the slopes in Figures 3a performed using the least squares method. The reciprocals of the slopes in Figure 3a,b were used to and 3b were used to calculate n1 and  . After this,  can be calculated according to Equation (4). calculate n1 and β. After this, α can be calculated according to Equation (4). The calculated values The calculated values of , , and 8.1376, 0.1508, 0.0185, respectively. Asthe shown  arerespectively. n  of n1 , β, and α are 8.1376, 0.1508, and 0.0185, As and shown in Figure 3c, using samein 1 method, the using valuethe of nsame can be calculated from of then mean thethe slope for reciprocals the lines inof the Figure 3c, method, the value can bereciprocals calculated of from mean the figure = 5.3029). slope(nfor the lines in the figure (n = 5.3029).

(Zener–Holloman parameter):

. FigureFigure 3. Plots3.between (a) ln (a) vs.ln ln (c) ln[sinh( ln)]ε.. vs.  lnvs.σ vs. ln ln;ε;. (b)  ; and Plots between (b) σ vs. ε; and (c) ln [sinh (ασ)] vs.

ln  .

Q had We assumed that no correlation with temperature. Equation can converted into We assumed that Q had no correlation with temperature. Equation (3)(3) can bebe converted into Equation (6): Equation (6): . ln ε − ln A Q 1 A Q 1 ln[sinh(ασ)] = ln   ln + (6) nR ln[sinh( )]  n  T (6) n

Differentiating Equation (6) gives:

nR T

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Differentiating Equation (6) gives:

Q = Rn

Q

 ln[sinh( )] (1/ ∂ ln[sinh (ασT)]) 

Q  Rn

(7) (7)

. ∂(1/T ) ε (206,669 J mol−1) can be calculated from the mean slope of the lines in Figure 4a. After this,

−1 can be calculated from the mean slope of the lines in Figure 4a. After this, the (206,669 J mol theQZ-parameter can )be determined according to Equation (8): Z-parameter can be determined according to Equation (8): Z   exp(206669 / RT ) (8) .

Z = ε exp(206669/RT ) (8) Equation (9) can be derived by combining Equations (3) and (5). After this, A can be determined by the intercept of the line in Figure 4b (A = 3.93 × 1017), where: Equation (9) can be derived by combining Equations (3) and (5). After this, A can be determined by the intercept of the line in Figure 4b (A =Z3.93 1017), where:  A×[sinh( )]n (9) 3

Z = A3under [sinh(ασ )]n conditions by coupling the Z-parameter (9) The constitutive model can be obtained various as follows: The constitutive model can be obtained under various conditions by coupling the Z-parameter 1  exp(Q / RT ) 1/ n as follows:   sinh 1.[( (10) 1/n ) ] 1  −1 ε exp( Q/RT ) A σ = sinh [( ) ] (10) α A

(a)

(b)

)]T−and 1 ; and1000 ln)]Z. Figure Therelations relations between and (b) (ασ Figure 4. The between (a) ln[(a) sinh(ln[sinh( ασ )] and 1000 (b) lnTZ−1;and ln[sinh ln[sinh( )] .

and

At this point, all the constitutive parameters of the material have been obtained. Further details thisanalysis point, all the constitutive parameters of can the be material have been obtained.[29–31]. FurtherAfter details relatingAt to the process of the material constants obtained from References relating to the analysis of the material can be obtained References [29–31]. this, the constitutive modelprocess of the 5754 alloy at theconstants true deformation strain of from 0.2 can be determined. After this, the constitutive model of the 5754 alloy at the true deformation strain of 0.2 Using the above modeling method, the constitutive constants at other true deformation strainscan can be determined. Using the above modeling method, the constitutive constants at other true deformation be calculated. The relationship between the strain and constitutive constants can be obtained by strains can be calculated. relationship the straincan andbeconstitutive polynomial fitting. Finally, theThe SA model withinbetween the test conditions determined.constants can be obtained by polynomial fitting. Finally, the SA model within the test conditions be determined. The predicted results using the SA model under the test conditions are shown incan Figure 5. Clearly, The predicted results using the SA model under the test conditions are shown Figure 5. the flow behavior of the 5754 aluminum alloy can be properly predicted by this model. In in addition, ◦ Clearly, the flowthat behavior of the aluminum alloy can predicted by this model. In it can be observed the model has5754 a large predicted error at be 400properly C (Figure 5b). addition, it can be observed that the model has a large predicted error at 400 °C (Figure 5b).

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Figure Figure 5.5. Comparisons Comparisons of of the the predicted predicted and and experimental experimental flow flow stress stress by by the thestrain-compensation strain-compensation ◦ C, (b) 400 ◦ C, and (c) 500 ◦ C. Arrhenius (SA) model at (a) 300 Arrhenius (SA) model at (a) 300 °C, (b) 400 °C, and (c) 500 °C.

3.2. 3.2. Correction Correction of of Temperature TemperatureRise RiseEffect Effect −1, the During when thethe strain raterate is less than 1 s−11 ,sthe change of Duringthe thecompression compressiondeformation deformationprocess, process, when strain is less than change temperature can be compensated by the heating system of the Gleeble-3500 thermo-simulation machine, of temperature can be compensated by the heating system of the Gleeble-3500 thermo-simulation with the tests being isothermal compression. the rise in temperature machine, with the considered tests beingunder considered under isothermal However, compression. However, the rise of in the deformation increases with increasing strainincreasing rate. When the actual deformation temperature is temperature of the deformation increases with strain rate. When the actual deformation more than 5 ◦ is C more higherthan than5 the pre-setthan temperature, thetemperature, temperaturethe risetemperature effect beginsrise to occur, temperature °C higher the pre-set effect which begins can decrease the flow stress. Correspondingly, softening effectthe willsoftening increase. effect Figurewill 6a–c shows to occur, which can decrease the flow stress. the Correspondingly, increase. the variation in truethe deformation temperature compared to the truecompared strain at the pre-set Figure 6a–c shows variation in true deformation temperature to the truedeformation strain at the temperature. It can betemperature. seen that the It temperature effect be corrected at a should strain rate of 10 s−1 . pre-set deformation can be seenrise that the should temperature rise effect be corrected −1. Similar Similar results be sfound in Reference [32].beInfound general, the temperature correction proposed at a strain ratecan of 10 results can in Reference [32]. In general, factor the temperature by Devadas [33] can be used to correct the flow stress data, as shown by Equation (11): correction factor proposed by Devadas [33] can be used to correct the flow stress data, as shown by Equation (11): Qde f 1 1 ∆σ = Q [ − ] (11) 1 nαR T1 def T + ∆T (11)   [  ] n R Tasmentioned T T previously, and ∆T is the absolute where Qde f , α, n, R, and T have the same meaning temperature (K). Where Qdef increment ,  , n, R , and T have the same meaning as mentioned previously, and T is the As shown in Figure 6d, after correcting for the temperature rise effect, the flow stress increased absolute temperature increment (K). significantly. The details of the temperature correction and the process of calculation can be found in As shown in Figure 6d, after correcting for the temperature rise effect, the flow stress increased References [32,34]. significantly. The details of the temperature correction and the process of calculation can be found in References [32,34].

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Figure6. Thevariation variationin truedeformation deformationtemperature temperatureaccording accordingto truestrain strainatatthe thepre-set pre-set Figure 6.6.The The variation inintrue true deformation temperature according tototrue true ◦ ◦ ◦ deformation temperatures of (a) 300 °C, (b) 400 °C, and (c) 500 °C; (d) the corrected flow curves with deformation temperatures 400 °C, C, andand (c) (c) 500500 °C; (d) flow flow curves with temperaturesof of(a) (a)300 °C,C,(b)(b) 400 C; the (d) corrected the corrected curves −1 − 1 −1 preset deformation temperature andaand astrain strain rateof of1010 preset deformation temperature and rate s s10 . .s . with preset deformation temperature a strain rate of

3.3.The TheBP–ANN BP–ANNModel Model 3.3. 3.3. The BP–ANN Model Artificialneural neuralnetwork network(ANN), (ANN),which whichhas hasseveral severalperformance performancecharacteristics characteristicsthat thatare aresimilar similar Artificial neural network (ANN), which several are similar Artificial has performance characteristics that to neural networks in the human brain, is a large-scale parallel distributed information processing to neural networks in the human brain, is a large-scale parallel distributed information processing to neural networks in the human brain, is a large-scale parallel distributed information processing system[35]. [35].The Theneurons neuronsare arethe thebasic basicelements elementsof ANN,which whichcan canbe beused usedto dealwith withcomplex complex system [35]. with system The neurons are the basic elements ofofANN, ANN, which can be used totodeal deal complex data information. The research shows that the performance of the ANN is roughly determined byits its data information. The research shows that the performance of the ANN is roughly determined data information. The research shows that the performance of the ANN is roughly determined by by its structure and the weights of the connections [36]. Phaniraj and Lahiri [37] and Lucon and Donovan structure andthe theweights weightsofofthe theconnections connections [36]. Phaniraj and Lahiri Lucon Donovan structure and [36]. Phaniraj and Lahiri [37][37] andand Lucon andand Donovan [38] [38]introduced introduced the BPalgorithm algorithm representative method reduce the errorsby created bythe the [38] the BP asasa arepresentative totothe reduce the errors created by introduced the BP algorithm as a representative methodmethod to reduce errors created the gradient gradient descent method. Therefore, a four-layered BP–ANN architecture model was established gradient descent method. Therefore, a four-layered BP–ANN architecture model was established to descent method. Therefore, a four-layered BP–ANN architecture model was established to determineto determine theflow flow behavior the5754 5754 aluminum alloy.The Theschematic schematic diagram forthe theANN ANN determine the behavior ofofthe aluminum alloy. for the flow behavior of the 5754 aluminum alloy. The schematic diagram for thediagram ANN architecture and architecture and neuron model is shown in Figure 7. architecture and neuron model is shown in Figure 7. neuron model is shown in Figure 7.

X1X1

TT

𝜎𝜎

.. .. ..

XX 22

WW i2i2

.. .. ..

𝜀̇𝜀̇

.. .. ..

𝜀𝜀

WW i1i1

XX 1 1

XX nn

𝑓(𝑥) ∑∑ 𝑓(𝑥)

WW inin

YiYi

WW i0i0==𝜃 𝜃 XX 00

Synapsescontaining containing Synapses

Neuroncontaining containing Neuron

i-input signals;WW ij-weights; 𝜃-threshold; XX i-input signals; ij-weights; 𝜃-threshold; i-output signal;f(x)-activate f(x)-activatefunction function YY i-output signal;

weights threshold weights threshold Figure7. Thestructure structureof theartificial artificialneural neuralnetwork network(ANN) (ANN)model modeland andthe theneuron neuronmodel. model. Figure model and the neuron model. Figure 7.7.The The structure ofofthe the artificial neural network (ANN)

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The mathematical form of the neuron model is shown in Equations (12) and (13). The 8 of 15 deformation parameters are used as the input signals for the input layer neurons in parallel, while the flow stress is calculated as the response to the output layer of neurons. The Levenberg–Marquardt algorithm has been usedis to traininthis ANN model. ofdeformation two hidden The mathematical form of the neuron model shown Equations (12) andThe (13).use The layers withare 10 used and 8asneurons, gives thelayer best neurons predictions. The hidden neurons parameters the inputrespectively, signals for the input in parallel, whilelayer the flow stressuse is the hyperbolic sigmoid function, while the output layers use the linear function:has calculated as thetangent response to thetransfer output layer of neurons. The Levenberg–Marquardt algorithm been used to train this ANN model. The use of two hidden layers with 10 and 8 neurons, respectively, n gives the best predictions. The hidden layer neurons use the hyperbolic tangent sigmoid transfer neti  Wij X j   (12) j 1 function, while the output layers use the linear function: Materials 2018, 11, 855



Yi n f (neti )

(13) (12) represent the weights and

neti = ∑ Wij X j − θ 1 neurons; represents all the input signals forj=the

Wij and  thresholds, respectively; Yi is the output signal the Yi = for f (net (13) i ) neurons; and f ( x) is the activate function. where X j

MATLAB–Neural Network was used develop the ANN model.and A total of 234 whereThe X j represents all the input signalstoolbox for the neurons; Wijtoand θ represent the weights thresholds, data points were obtained from the experimental data at true strains of 0–0.7 at appropriate intervals. respectively; Yi is the output signal for the neurons; and f ( x ) is the activate function. The The otherMATLAB–Neural data are shown in Network Table 2. Before establishing model, collected dataAshould be toolbox was usedthe to ANN develop the all ANN model. total of normalized by Equation (14) to reduce the order of magnitude difference between the various 234 data points were obtained from the experimental data at true strains of 0–0.7 at appropriate dimensions: intervals. The other data are shown in Table 2. Before establishing the ANN model, all collected data should be normalized by Equation (14) to reduce ( x the xminorder ) of magnitude difference between the XK  k (14) various dimensions: xmin ((xxkmax − xmin ) ) (14) XK = ( xmax − xmin ) where X K and xk are the experimental data and the processed data, respectively; and xmax and where XK and xk are the experimental data and the processed data, respectively; and xmax and xmin the maximum and minimum values of the experimental data, respectively. x theare aremin maximum and minimum values of the experimental data, respectively. After the theANN ANN training training isis completed, completed, the the flow flow stress stress of of the the5754 5754aluminum aluminum alloy alloy under under test test After conditions can canbe bepredicted predictedby bythe theBP–ANN BP–ANNmodel modelas asshown shownin inFigure Figure8.8.The Thepredicted predictedvalues valuesare are conditions ◦ consistent with the experimental data over the whole range except at 300 °C and true strains in the consistent with the experimental data over the whole range except at 300 C and true strains in the rangeof of0.4–0.6. 0.4–0.6. As AsBP–ANN BP–ANN lacks lacks the theability abilityto tooptimize optimize globally, globally,this thismay maycause causethe thealgorithm algorithmto to range fall into a local optimum. Through training of 100 times, we found that each training has different fall into a local optimum. Through training of 100 times, we found that each training has different predictionaccuracy, accuracy,which whichmeans meansthat thatthe theprediction predictionaccuracy accuracyfluctuates fluctuatesrandomly randomlyaccording accordingto tothe the prediction trainingtimes. times. training Table2.2.The Theparameters parametersofofthe theANN ANNmodel. model. Table

Total Total data 234

Training

Verificat

Training data data data dataVerification ion data

234136

136

98 98

Train Train epoch epoch 1000 1000

Figure 8. Cont.

Learning Training Learning rate rate targetTraining target 0.2 0.2

10−7

10−7

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Figure the Figure8.8.8.Comparisons Comparisonsof thepredicted predictedand andexperimental experimentalflow flowstress stressby bythe thenon-optimized non-optimizedANN ANN Figure Comparisons ofofthe predicted and experimental flow stress by the non-optimized ANN ◦ ◦ ◦ model at:at:(a) 300 C, (b) 400 C, and (c) 500 C. model (a) 300 °C, (b) 400 °C, and (c) 500 °C. model at: (a) 300 °C, (b) 400 °C, and (c) 500 °C.

3.4. 3.4.The TheANN–GA ANN–GAModel Model 3.4. The ANN–GA Model The Thechoice choiceof theANN ANNconnection connectionweights weightsand andthresholds thresholdshas hasaaagreat greatinfluence influenceon onthe thestability stability The choice ofofthe the ANN connection weights and thresholds has great influence on the stability and andtraining trainingefficiency efficiencyof thenetwork. network.The Theoverall overalldistribution distributionof theANN ANNweights weightsand andthresholds thresholds and training efficiency ofofthe the network. The overall distribution ofofthe the ANN weights and thresholds determines the fidelity of the model. However, the initial weights and thresholds of ANN are randomly determines the fidelity of the model. However, the initial weights and thresholds of ANN are are determines the fidelity of the model. However, the initial weights and thresholds of ANN generated by the system so that the network is not convergent, which leads to instability in ANN randomlygenerated generatedby bythe thesystem systemso sothat thatthe thenetwork networkisisnot notconvergent, convergent,which whichleads leadstotoinstability instability randomly for the prediction of flow behavior. This paper proposes an intelligent algorithm based on the GA on to in ANN for the prediction of flow behavior. This paper proposes an intelligent algorithm based on in ANN for the prediction of flow behavior. This paper proposes an intelligent algorithm based optimize the BP–ANN. Figure 9 shows the optimization process of BP–ANN using GA. the GA to optimize the BP–ANN. Figure 9 shows the optimization process of BP–ANN using GA. the GA to optimize the BP–ANN. Figure 9 shows the optimization process of BP–ANN using GA. TT

......



......



Output Output

ee

Target Target

GA GA

Initialthresholds thresholds Initial andweights weights and

Initial Initial thresholds thresholds andweights weights and

Weightsofof Weights optimization optimization

Thresholdsofof Thresholds

optimization optimization

Figure9. Schematicstructure structureof theoptimization optimizationusing usinggenetic geneticalgorithm algorithm(GA). (GA). Figure Figure 9.9.Schematic Schematic structure ofofthe the optimization using genetic algorithm (GA).

Geneticalgorithm algorithm(GA) (GA)isisisaaaheuristic heuristicparallel parallelsearch searchtechnique techniquebased basedon onthe thedynamics dynamicsof natural Genetic algorithm (GA) heuristic parallel search technique based on the dynamics ofofnatural natural Genetic selectionand and genetics, which has very very good good optimization optimization ability [36]. The GA GA selection and genetics, which ability The ofof aa selection genetics, which has has very good optimization ability [36]. The[36]. GA consists ofconsists aconsists population population of many individuals. Each individual has its own fitness. The GA always retains the most population of many individuals. Eachhas individual its own The GA always retains the most of many individuals. Each individual its own has fitness. Thefitness. GA always retains the most successful successful individual and eliminates the low-fitness individual to achieve the best solution. Further successful and individual andthe eliminates theindividual low-fitnesstoindividual achieve the best solution. individual eliminates low-fitness achieve thetobest solution. Further detailsFurther about details about the concepts and theories of GA can be obtained from reference [39]. details about the concepts and theories of GA can be obtained from reference [39]. the concepts and theories of GA can be obtained from reference [39]. Thisarticle articleused usedMATLAB MATLABto towrite writeall allthe thecodes codesfor for the ANN model and GA. The structure This article The structure ofof This to write all the codes forthe theANN ANNmodel modeland andGA. GA. The structure BP–ANN has been introduced Section 3.3.AA four-layered BP–ANN architecture hasbeen been selected. BP–ANN has been introduced ininSection 3.3. four-layered BP–ANN architecture has selected. of BP–ANN has been introduced in Section 3.3. A four-layered BP–ANN architecture has been The number of neurons in each layer is 3, 10, 8 and 1, respectively. There are 118 weights The number of neurons in eachinlayer is 3, is 10,3, 810,and 1, 1, respectively. weights selected. The number of neurons each layer 8 and respectively.There Thereare are 118 weights and19 19thresholds thresholds((10 ( 10+ Therefore,GA GAneeds needstotooptimize optimize 310 10+ 10× 111= 118 19 19 ( (3× ) )and ).).Therefore, optimize 10 10 888+888× 118 10 88+ 111= (3 10 118) and 19 thresholds 19). Therefore, 137parameters parameters that are connected an individual. individual. The individual individual boundary conditions are 137 parameters that connected an The boundary conditions are 137 that areare connected to antoto individual. The individual boundary conditions are between between −0.5 and 0.5. Each individual is encoded by binary methods, which is shown in Figure 10. between −0.5 and 0.5. Each individual is encoded by binary methods, which is shown in Figure 10. Thelength lengthofofeach eachvariable variableininthe theindividual individualisis10 10digits. digits.Therefore, Therefore,each eachindividual individualneeds needs1370 1370 The

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−0.5 and 0.5. Each individual is encoded by binary methods, which is shown in Figure 10. The length of each variable in the individual is 10 digits.GA. Therefore, eachrealize individualGA needs 1370 encoding encoding encodinglengths. lengths.Table Table33shows showsthe theparameters parametersof of GA.In Inorder orderto to realizethe the GAfor foroptimizing optimizingthe the lengths. Table 3 shows the parameters of GA. In order to realize the GA for optimizing the BP–ANN, BP–ANN, BP–ANN, the the predictive predictive error error norm norm of of the the neural neural network network isis used used to to measure measure the the prediction prediction the predictive error norm of the neural network is used to measure the prediction accuracy and accuracy accuracyand andgeneralization generalizationability abilityof ofthe themodel. model.Therefore, Therefore,the thefitness fitnessvalues valuesof ofeach eachindividual individualinin generalization ability of the model. Therefore, the fitness values of each individual in GA are calculated GA GAare arecalculated calculatedby bythe theerror errornorm. norm. by the error norm. Table Table3.3.The Theparameters parametersofofGA. GA. Table 3. The parameters of GA.

Encoding Encoding Encoding Length Length Length 1370 1370

Genetic Population Crossover Mutation Generation Genetic Population Crossover Mutation Generation Genetic Population Crossover Mutation Generation Algebra Size Probability Probability Gap Algebra Size Probability Probability Gap Algebra Size Probability Probability Gap 88 55 0.7 0.01 11 0.7 0.01

1370

8

5

0.7

118Weights 118Weights

00

00

11

1180digits 1180digits

0.01

1

19Thresholds 19Thresholds

… …

… … 11

00

11

190digits 190digits

Figure 10. The binary coding of the neural network initial weights and thresholds. Figure Figure10. 10.The Thebinary binarycoding codingofofthe theneural neuralnetwork networkinitial initialweights weightsand andthresholds. thresholds.

The The predicted results of ANN–GA model are shown in Figure 11. The results show that the Thepredicted predictedresults resultsof ofANN–GA ANN–GAmodel modelare areshown shownin inFigure Figure11. 11.The Theresults resultsshow showthat thatthe the predicted values can fit the experimental value faultlessly within the whole deformation range. Finally, predicted values can fit the experimental value faultlessly within the whole deformation range. predicted values can fit the experimental value faultlessly within the whole deformation range. an ANN–GA model with highwith accuracy and stability been has developed. Finally, an model high and stability Finally, anANN–GA ANN–GA model with highaccuracy accuracy andhas stability hasbeen beendeveloped. developed.

Figure 11. Comparison calculated stress data by ANN–GA atat(a) 300 °C, Figure11. 11.Comparison Comparisonbetween betweenthe theoriginal originaland and °C, (b) ◦(b) Figure between the original and calculated calculated stress stressdata databy byANN–GA ANN–GA at(a) (a)300 300 C, 400 °C and (c) 500 °C. 400 °C and (c) 500 °C. ◦ ◦ (b) 400 C and (c) 500 C.

3.5. Performance of the SA, BP–ANN, and ANN-GA Models The predictability of the SA, BP–ANN, and ANN–GA models were quantified by R (Equation (15)) and AARE (Equation (16)), which are expressed as follows: Materials 2018, 11, 855

R

iN1 ( Oi   O )( Ci   C )  (   O )  (   C ) N i 1

i O

2

N i 1

i C

11 of 15 2

(15)

3.5. Performance of the SA, BP–ANN, and ANN-GA Models i N

 O   Ci 1 AARE (%)  100 % quantified by R (Equation(16)  i The predictability of the SA, BP–ANN, and ANN–GA were (15)) N i 1 models O

and AARE (Equation (16)), which are expressed as follows: where  Oi is the original data;  Ci is the calculated value derived from the established SA, ∑ N=1 (σOi i− σO )(σCi − σC ) (15) R = q  Oi iand BP–ANN, and ANN–GA models; the mean values of  Oi and  Ci , respectively,  C are 2 N N i i − σ )2 ( σ − σ ) ( σ ∑ ∑ O C i =1 data.× i =1 O C and N is the amount of the experimental

between the experimental and predicted R is usually applied to evaluate the linear correlation N σi − σi 1evaluate O when C the prediction is biased towards a local values [14,22,26]. However, AARE it is unable to (16) (%) = ∑ × 100% i σO scope. Consequently, the AARE was usedNtoi=check the reliability of models. The R of the three 1 developed models are shown in Figure 12a, Figure 12b and Figure 12c, respectively. Most data where σOi is the original data; σCi is the calculated value derived from the established SA, BP–ANN, points could track the ideal regression line and achieved good correlation between the predictive i and σi , respectively, and N is the amount and and ANN–GA models; σOi and σCi arestress. the mean values of σR C BP–ANN, and ANN–GA models experimental values of flow Moreover, the O for SA, of the experimental data. were 0.9918, 0.9929, and 0.9999 (Table 4). The results reflected that the developed ANN–GA had R is usually applied evaluate linear correlation theBP–ANN experimental and predicted better predictive abilitytofor the flowthe behavior compared tobetween the SA and models. values [14,22,26]. is unable to evaluate the prediction is biased towards local scope. Figure 13However, shows theitvalues of AARE of thewhen developed models under the test straina conditions. Consequently, the AARE was used to check the reliability of models. The R of the three The AAREs obtained from the ANN–GA model were significantly lower than the other developed models in models are true shown in Figure 12a–c, 13 respectively. data pointsthat could the ideal regression the test strain range (Figure and Table 4),Most which indicates thetrack developed ANN–GA had line better and achieved good between the predictive experimental stability for thecorrelation prediction of flow behavior comparedand to other models. values of flow stress. Moreover, the R for SA, BP–ANN, and ANN–GA models were 0.9918, 0.9929, and 0.9999 (Table 4). Table 4. The obtained of correlation coefficient andpredictive average absolute relative error (AARE) The results reflected that thevalues developed ANN–GA had (R) better ability for the flow behavior for the established models. compared to the SA and BP–ANN models. Figure 13 shows the values models under Model of AARE of the developed AAREthe (%)test strain conditions. R The AAREs obtained from the ANN–GA model were significantly lower than the other models in the Arrhenius 0.9918 3.2499–5.6774 test true strain range (Figure 13 and Table 4), which indicates that the0.0567–5.4436 developed ANN–GA had better ANN 0.9929 stability for the prediction of flow behavior compared ANN–GA 0.9999to other models. 0.0232–1.0485

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Figure 12. Correlation between the predictedand andthe the experimental experimental values stress by:by: (a) (a) SA,SA, (b) Figure 12. Correlation between the predicted valuesofofflow flow stress BP–ANN, and (c) ANN–GA models. (b) BP–ANN, and (c) ANN–GA models.

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Table 4. The obtained values of correlation coefficient (R) and average absolute relative error (AARE) for the established models. Model

R

AARE (%)

Arrhenius 0.9918 3.2499–5.6774 ANN 0.9929 0.0567–5.4436 Figure 12. Correlation between the predicted and the experimental values of flow stress by: (a) SA, (b) ANN–GA 0.9999 0.0232–1.0485 BP–ANN, and (c) ANN–GA models.

Figure13. 13.The The AARE AARE of of the the (a) (a) BP–ANN –0.06; and (b) Figure BP–ANN and and ANN–GA ANN–GAmodels modelsatattrue truestrains strainsofof0.01 0.01–0.06; and Arrhenius, BP–ANN, and ANN–GA models at true strains of 0.1 – 0.7. (b) Arrhenius, BP–ANN, and ANN–GA models at true strains of 0.1–0.7.

After the above statistical analysis, the flow behavior of 5754 aluminum alloy could be After the above statistical analysis, the flow behavior of 5754 aluminum alloy could be accurately accurately and reliably predicted by the developed ANN–GA model. In addition, the SA model and reliably predicted by the developed ANN–GA model. In addition, the SA model contains a large contains a large number of model constants that need to be calculated repeatedly. The time it took to number of model constants that need to be calculated repeatedly. The time it took to build the ANN build the ANN model was shorter than the time required to build the phenomenological model. model was shorter than the time required to build the phenomenological model. 3.6.The TheDynamic DynamicSoftening SofteningMechanism Mechanismofof5754 5754Aluminum AluminumAlloy Alloy 3.6. Asshown shownininFigure Figure11, 11,the the5754 5754aluminum aluminumalloy alloyshowed showedthe thetypical typicalrheological rheologicalcharacteristics characteristics As during the hot compression tests. The flow stress rose rapidly with an increase in the during the hot compression tests. The flow stress rose rapidly with an increase in the strain strain until untilit did not not significantly significantly change change with withincreasing increasingstrain, strain, itreached reached the the peak. peak. After After this, this, the the flow flow stress stress did which demonstrated a steady flow softening phenomenon. Moreover, the flow stress increased with which demonstrated a steady flow softening phenomenon. Moreover, the flow stress increased with a decrease in the deformation temperature and an increase in strain rate. It can be seen that the 5754 a decrease in the deformation temperature and an increase in strain rate. It can be seen that the aluminum alloyalloy possessed the property of strain rate rate dependence. In addition, the required strain 5754 aluminum possessed the property of strain dependence. In addition, the required for reaching the steady state deformation increased with increasing strain rate and decreasing strain for reaching the steady state deformation increased with increasing strain rate and decreasing temperature.The Thecross-slip cross-slipofofthe thescrew screwdislocation dislocationand andthe theclimbing climbingofofthe theedge edgedislocation dislocationwere were temperature. difficult to carry out fully. Furthermore, the softening degree provided by the cross-slip and difficult to carry out fully. Furthermore, the softening degree provided by the cross-slip and climbing climbing was small under the conditions of a lower deformation temperature and higher strain rate, was small under the conditions of a lower deformation temperature and higher strain rate, which can which can cause steady-state stress hysteresis. cause steady-state stress hysteresis. The aluminum alloy had highstacking stackingfault faultenergy energy(SFE). (SFE).Perfect Perfectdislocations dislocationswere weredifficult difficult The aluminum alloy had a ahigh to transform into partial dislocations and stacking faults, while the dynamic recovery (DRV) process to transform into partial dislocations and stacking faults, while the dynamic recovery (DRV) process requireda alarge largeamount amountofofstored storedenergy energyduring duringthe theprocess processofofhot hot deformation.Therefore, Therefore,the the required deformation. driving force required for the dynamic recrystallization (DRX) of the alloy decreased, which driving force required for the dynamic recrystallization (DRX) of the alloy decreased, which increased increased the DRX. The microstructure deformation microstructure of the 5754 alloy the difficulty of difficulty DRX. The of deformation of the 5754 aluminum alloyaluminum deformed at a −1 and temperature of 500 C is shown in Figure 14. It can be seen deformed at a strain rate of 0.1 s − 1 strain rate of 0.1 s and temperature of 500 C is shown in Figure 14. It can be seen from Figure 14a that the original cast grains were elongated after deformation. To determine the morphology of the microstructure further, a local area was selected for electron backscattered diffraction (EBSD) analysis as shown in Figure 14b. The grain shape was substantially maintained in an elongated and compressed state, and the grain boundary was relatively smooth with no remotion. Besides, no obvious equiaxed recrystallization grains were observed around the grain boundary even under

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from Figure 14a that the original cast grains were elongated after deformation. To determine the morphology of the microstructure further, a local area was selected for electron backscattered diffraction Materials 2018, (EBSD) 11, 855 analysis as shown in Figure 14b. The grain shape was substantially maintained 13 of 15 in an elongated and compressed state, and the grain boundary was relatively smooth with no remotion. Besides, no obvious equiaxed recrystallization grains were observed around the grain ◦ C). This is consistent with the the conditions a low the strain rate (0.1 of s−a1 )low andstrain high temperature (500high boundary evenofunder conditions rate (0.1 s−1) and temperature (500 °C). This dynamic softening process of thesoftening 5754 aluminum shown in Figurealloy 11. Therefore, dynamic is consistent with the dynamic processalloy of theas5754 aluminum as shown the in Figure 11. softening mechanism of the 5754 aluminum alloy was mainly controlled by the DRV process during Therefore, the dynamic softening mechanism of the 5754 aluminum alloy was mainly controlled by the hot compression. Further details the dynamic softening mechanism been the single-pass DRV process during the single-pass hot about compression. Further details about thehave dynamic reported by the authors of Reference [40]. softening mechanism have been reported by the authors of Reference [40].

(a)

(b)

Figure 14. The deformed microstructures under the conditions of 500 °C and 0.1 s−1 using: (a) optical Figure 14. The deformed microstructures under the conditions of 500 ◦ C and 0.1 s−1 using: (a) optical micrograph; and (b) electron backscattered diffraction (EBSD) mapping. micrograph; and (b) electron backscattered diffraction (EBSD) mapping.

4. Conclusions 4. Conclusions An optimized constitutive model of the 5754 aluminum alloy was established to predict the An optimized constitutive model of the 5754 aluminum alloy was established to predict the flow flow behavior and study the dynamic softening mechanism at 300–500 °C and 0.1–10 s−1. The ◦ − 1 behavior and study the dynamic softening mechanism at 300–500 C and 0.1–10 s . The conclusions conclusions are as follows: are as follows: 1. Three models were established to predict the flow behavior of 5754 aluminum alloy during the 1. Three models were established to predict flowprediction behavior of during hot compression tests. The SA model had the a good of 5754 flow aluminum stress in thealloy steady statethe for hot compression tests. The SA model had a good prediction of flow stress in the steady state for the 5754 aluminum alloy. However, it resulted in a large predicted error◦ at 400 °C. The the 5754 aluminum alloy.predict However, resulted in aunder large predicted at 400 but C. The BP–ANN model could theitflow curves different error conditions, theBP–ANN network model could predict flowfind curves different conditions, but prediction the network training results training results couldthe easily the under local optimum. Therefore, the results fluctuated ◦ C and could easily find the local optimum. Therefore, the prediction results fluctuated at 300 at 300 °C and true strains in the range of 0.4–0.6. The BP–ANN model optimized by GA had the true in the range 0.4–0.6. The BP–ANN model optimized GA had best predictive best strains predictive ability ofof the flow behavior of 5754 aluminum alloyby under hot the compression tests. of the flowfrom behavior of 5754 aluminum alloy under hot compression 2. ability The R calculated the SA, BP–ANN, and ANN–GA models were 0.9918,tests. 0.9929, and 0.9999, 2. The R calculated from theAARE SA, BP–ANN, ANN–GA were 0.9918, 0.9929, and 0.9999, respectively, while the for theseand models weremodels 3.2499–5.6774%, 0.0567–5.4436%, and respectively, while the AARE for these models were 3.2499–5.6774%, 0.0567–5.4436%, and 0.0232–1.0485%, respectively. Compared with the SA and BP–ANN models, the ANN–GA was 0.0232–1.0485%, SA and BP–ANN models, the a better predictorrespectively. of the flow Compared behavior ofwith 5754the aluminum alloy. In addition, the ANN–GA ANN–GA was a better predictor of the flow behavior of 5754 aluminum alloy. In addition, the ANN–GA model could find the optimal weight and threshold for ANN at the same time. Thus, the model could find the optimal for ANN at thewere samestable time. Thus, theprediction predicted predicted values of the flowweight stress and fromthreshold the ANN–GA model and the values of the flow stress from the ANN–GA model were stable and the prediction accuracy was accuracy was very high. high. aluminum alloy experienced a steady flow softening phenomenon during the hot 3. very The 5754 3. The 5754 aluminum alloy experienced steadywith flowincreasing softening strain phenomenon during the hot compression tests. The flow stress rose arapidly until it reached a peak, compression tests.constant. The flowBesides, stress rose rapidly with increasing strain untiltoit reach reached peak, before remaining the flow stress and the required strain theasteady before remaining constant. Besides, the flow stress and the required strain to reach the steady state deformation increased with decreasing deformation temperature and increasing strain state with deformation temperature and increasing strain even rate. rate. deformation Through the increased observation ofdecreasing the microstructure, no recrystallized grains were found Through the observation thestrain microstructure, no recrystallized grains were mechanism found even at at high temperature and of low rate. Therefore, the dynamic softening of high 5754 temperature and low strain rate. Therefore, the dynamic softening mechanism of 5754 aluminum alloy was mainly controlled by the DRV process during the single-pass hot compression.

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Author Contributions: C.H. conceived and designed the experiments; X.J. performed the experiments; C.H., X.J., and Z.Z. analyzed the data; C.H. wrote the manuscript; and X.J. contributed to the revision of the paper. Funding: This research was funded by [National Natural Science Foundation of China] grant number [51275533]. Acknowledgments: The authors appreciate financial support by the State Key Laboratory of High-Performance Complex Manufacturing (Contract No. zzyjkt2013-10B), Central South University, China. Conflicts of Interest: The authors declare no conflicts of interest.

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