research article modeling material flow behavior during...

Research ArticleModeling Material Flow Behavior during Hot DeformationBased on Metamodeling Methods

Gang Xiao12 Qinwen Yang2 Luoxing Li12 and Zhengbing Xu3

1State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University Changsha 410082 China2College of Mechanical and Vehicle Engineering Hunan University Changsha 410082 China3Key Laboratory of Nonferrous Materials and New Processing Technology of Ministry of Education of ChinaGuangxi University Nanning 530004 China

Correspondence should be addressed to Luoxing Li llxly2000163com

Received 27 July 2015 Accepted 30 August 2015

Academic Editor Mohsen Asle Zaeem

Copyright copy 2015 Gang Xiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Modeling material flow behavior is an essential step to design and optimize the forming process In this context four popularmetamodel types Kriging radial basis function multivariate polynomial and artificial neural network are investigated as potentialmethods for modeling the flow behavior of 6013 aluminum alloy Based on the experimental data from hot compression tests themodeling performance of these four methods was tested and subsequently compared from different aspects It is found that all themethods are capable of constructing models for describing the hot deformation behaviorThemerits of Kriging method over otherthree methods are highlighted when the sample size for modeling is decreased Furthermore the applicability of Kriging methodis validated while decreasing the sample uniformity with respect to temperature or strain rate It is proved that Kriging method iscompetent in modeling the material flow behavior and is the most effective one among the four popular types of metamodelingmethod

1 Introduction

Hot forming technology of materials is widely applied inpractical manufacturing In order to achieve the requiredmicrostructure and mechanical property of products thedesign and optimization of thermomechanical processparameters have to be properly implemented [1 2] Forthis reason the designers strive to study the effect of theparameters with respect to the kinetics of the metallurgicaldeformation The flow behavior of materials is compactlyaffected by the factors of temperature strain rate andstrain [3] For exploring and describing the hot deformationbehavior many research groups have made use of regressionmethod to establish the phenomenological and physicallybased constitutive models [4ndash6] which would be committedto provide a complete mathematical description of the flowstress Lin et al [5] developed a newphenomenologicalmodelfor describing high-temperature flow behavior of inconel 718superalloy Saadatkia et al [6] developed a physically basedmodel of low and medium carbon steels to investigate the

deformation behaviors However the deformation behaviorof metal at elevated temperature is always associated withvarious metallurgical phenomena and thereby complicatedin nature [7] such as work hardening dynamic recoverydynamic recrystallization and flow instabilitiesThe effects ofthese factors on flow stress are so complex that the relation-ship among them is highly nonlinear It can be found that thephysical procedure is difficult to be systematically interpretedby the conventional constitutive model [8] which wouldreduce the prediction accuracy and limit the applicationrange Moreover the development of modeling flow behaviorin the conventional way is usually time-consuming and relieson amounts of experimental data

Considering the disadvantages of the abovementionedmethod artificial neural network (ANN)method is graduallyutilized as an alternative approach formodelingmaterial flowbehavior to improve processing scheme [8ndash10] In essenceANN is one type of metamodeling method which can getrid of the constraints from describing physical mechanismsMetamodeling method is widely accepted as a valuable

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 157892 8 pageshttpdxdoiorg1011552015157892

2 Mathematical Problems in Engineering

and efficient technique to model various complex nonlinearrelationships In previous researches [11] some other typesof metamodeling method were maturely developed suchas Kriging radial basis function (RBF) and multivariatepolynomial method Each metamodel type has its associatedfitting method and the corresponding characteristics [12]Due to the complicated nature of the relationships betweenmetamodeling methods and engineering problems there isno conclusion on which method is definitely superior to theothers [11] In order to investigate the applicability of meta-modelingmethods and search for themost appropriate one inmodeling flow behavior the four popular methods KrigingRBF multivariate polynomial and ANN were utilized andcompared for modeling the flow stress of 6013 aluminumalloy during hot compression

In this work the hot plane strain compression tests werecarried out on Gleeble-3500 thermomechanical simulatorConsidering deformation heating and heat transfer themeasured flow stress data at relatively high strain rates werecorrected Based on the measured and corrected data thefeasibility of these methods is firstly tested in modelingthe flow behavior The most effective modeling methodwith satisfactory accuracy stability and efficiency was thenselected on the basis of comparative study and detailedanalysis Finally the performance of Kriging method thatis considered as the best one was further assessed for thismodeling process

2 Metamodeling Methods

21 Kriging Method After being proposed by Krige [13] andimproved by others such as Matheron [14] Kriging methodwas gradually developed to be a popular metamodel typeand systematically introduced into the area of computerexperiment [15] In Kriging method the random outputis assumed to be obtained from a linear combination ofregression functions plus a random process factor as follows

119884 =

119873

sum

119895=1

120573119895119891119895 (x) + 119885 (x) (1)

where 119873 is the number of regression functions 119891119895(sdot) is a

regression function 120573119895is the coefficient for 119891

119895(sdot) x is the

design point and 119885(sdot) is the random process function Thecorrelation function is defined by the following equation

Cov (119885 (x1) 119885 (x

2)) = 119903 (120579 x

1 x2) (2)

where Cov(sdot) is the correlation function x1and x

2are two

design points and 120579 is a structural parameter to be optimizedThe correlation function could be defined in several differentways such as Gaussian exponential linear spherical andcubical [16]

22 Radial Basis Function Method Radial basis function(RBF) method was initially developed as an exact interpo-lation technique for data in multidimensional space [17] InRBF method a series of center points (x

1198951 x1198952 x

119895119898) are

chosen first from the design points (x1 x2 x

119899) based on

some criteria Then the basis functions are constructed byusing these center points as

119861119894 (x) = 119891 (

10038171003817100381710038171003817x minus x119895119894

10038171003817100381710038171003817) (3)

where x is the design point x119895119894is the center point for 119861

119894(sdot)

and xminusx119895119894 is the Euclidian distance between the two points

The function119891( sdot) could have different forms such as linearGaussian cubic thin-plate spline and multiquadratic [17]The relationship between inputs and outputs is constructedas a linear combination of radial basis functions

119892 (x) = 1205730 +119873

sum

119895=1

119861119895 (x) 120573119895 (4)

where119892 is the estimated output value119861119895(sdot) is a basis function

1205730is a constant and 120573

119895is the coefficient for 119861

119895(sdot)

23 Multivariate Polynomial Method Multivariate polyno-mial method is one of the most fundamental metamodelingmethods used in computer experiment It is mostly knownfrom response surface method [18] which employs quadraticpolynomial in the field of engineering design optimizationIn multivariate polynomial method basis functions arebuilt directly by using input variable components (x =

[1199091 1199092 119909

119899]) and their interactions such as

1198610 (119909) = 1 1198611 (119909) = 1199091 119861119899 (119909) = 119909119899

11986111 (119909) = 119909

2

1 11986112 (119909) = 11990911199092 1198611119899 (119909) = 1199091119909119899

(5)

A multivariate polynomial is constructed as the weightedsum of these basis functions

119892 (x) = 1198870 +119873

sum

119895=1

119861119895 (x) 119887119895 (6)




119895(sdot)

24 Artificial Neural Network Method Artificial neural net-work (ANN)method is based on a particular set of nonlinearfunctions to build the relationship between inputs and out-puts Its architecture consists of input layer output layer andhidden layer which are connected by the processing unitscalled neurons [19] as shown in Figure 1 Each neuron inthe input layer and output layer represents one independentvariable while the neurons in the hidden layers are only forcomputation purpose During training procedure the knowninput-output pairs are used to update the weights and biases[20] The objective is to minimize the errors between theANN outputs and the targets for the corresponding inputs

3 Model Construction and Discussion

31 Experiments and Modeling Preparation The materialused in the present experiment is 6013 aluminum alloy with

Mathematical Problems in Engineering 3

Input 1

Input 2

Input 3

Output

Figure 1 The architecture of the ANN model

Tem

pera

ture

T (K

)

Water quenching

Time t (s)

120 s

10Ks

613sim773K

Figure 2 Experimental procedure for the compression tests

Table 1 Experimental conditions

Level 1 2 3 4 5Temperature (K) 613 653 693 733 773Strain rate (sminus1) 0001 001 01 1 10

a nominal composition of 095 Mg 075 Si 09 Cu and035 Mn (mass fraction) The cubic specimens with a sizeof 20mm times 15mm times 10mm were prepared for testing Thehot plane strain compression tests were carried out 25 timesusing Gleeble-3500 thermomechanical simulator at differenttemperatures and strain rates (listed inTable 1) All specimenswere heated to the preset deformation temperature at a rateof 10 Ks and held for 120 s to eliminate thermal gradientThe specimens were compressed to a true strain of 08 andcooled down to room temperature inwaterThe experimentalprocedure is explained in Figure 2

It is widely accepted that the flow stress data measureddirectly from tests at relatively high strain rates are alwaysaffected by instantaneous temperature rise during defor-mation [21 22] According to the previous work of theauthors [23] the temperature rise during hot plane straincompression depends on deformation heating integratedwith heat transfer which does occur among the deformedregion the undeformed region and the press indenter Thistype of temperature rise considering deformation heating andheat transfer has been quantitatively described by a proposedmodel in this literature Based on the published work themeasured flow stress data at strain rates of 10 and 1 sminus1 are

Kriging

RBF

Polynomial

ANN

Temperature T

Strain 120576 Flow stress 120590

Strain rate 120576

Figure 3 Schematic of modeling flow behavior

corrected and then used for comparison and analysis ofdifferent modeling methods in this paper

Based on the 25 experimental (including measured andcorrected) flow stress curves the true stress data correspond-ing to the true strain between 005 and 08 with the intervalof 005 were adopted to construct models and test theirperformance It can be found that a total of 400 sample pointsare located on the 25 curves which are considered as 25 casesin the modeling and testing processes These data will bedivided into training and testing datasets according to therequirements in different sections of this paper

The schematic of modeling the flow behavior of 6013aluminum alloy is shown in Figure 3The inputs of themodelare temperature (119879) strain rate ( 120576) and strain (120576) whereasflow stress (120590) is the output In addition the parametersof temperature and strain rate are processed by naturallogarithm

32 Model Construction and Applicability Assessment Thefeasibility of metamodeling methods for modeling the flowbehavior is investigated in this section Three cases withtotally different temperatures and strain rates are successivelyselected for model testing as shown in Table 2 In eachtesting process 24 cases (excluding the testing one) areutilized for model training based on the schematic given inFigure 3 The average (120575ave) and the maximum (120575max) valuesof absolute relative errors (120575) from each testing are employedas quantitative criteria to evaluate the performance of modelsand the applicability of modeling methods

120575 =

1003816100381610038161003816100381610038161003816

119864 minus 119875

119864

1003816100381610038161003816100381610038161003816times 100

120575ave =1

119873

119873

sum

119894=1

120575119894times 100

(7)

where 119864 is the experimental value 119875 is the predicted valueand119873 is the number of testing points

As shown in Table 2 the detailed information on 120575aveand 120575max from the developed models is provided Exceptfew specific conditions the values of 120575 are all less than 10Meanwhile Figure 4 displays the comparisons between thepredicted values and experimental curves All the developedmodels can basically track the dynamic change of stress withthe strain increasing at different temperatures and strainrates Therefore it can be considered that the four metamod-eling methods Kriging RBF multivariate polynomial and


Table 2 Three testing cases to analyze the prediction accuracy of models

Number 119879 (K) 120576 (sminus1) Kriging RBF Polynomial ANN120575ave () 120575max () 120575ave () 120575max () 120575ave () 120575max () 120575ave () 120575max ()

1 653 1 287 837 824 1119 347 575 157 2792 693 01 379 619 591 856 242 426 344 5683 733 001 392 853 120 626 311 597 949 1385

30

60

90

120

150

180

0

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576

KrigingExperimental

(1) 653K 1 sminus1(2) 693K 01 sminus1

(3) 733K 001 sminus1

(a)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576

Experimental(1) 653K 1 sminus1(2) 693K 01 sminus1

(3) 733K 001 sminus1

RBF

30

60

90

120

150

180

0

(b)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

Polynomial

30

60

90

120

150

180

0

(c)

True

stre

ss120590

(MPa

)

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

ANN

30

60

90

120

150

180

1

2

3

0

(d)

Figure 4 Comparisons between the predicted values and experimental flow stress curves of three testing cases (a) Kriging (b) RBF (c)Polynomial (d) ANN

ANN are all capable of modeling the elevated temperatureflow behavior of 6013 aluminum alloy However it is reallydifficult to find out which one is themost appropriatemethodbased on the existing testing results In order to make areasonable selection from these modeling methods furthercomparison and analysis are still needed

33 Comparison of Metamodeling Methods Sample sizerefers to the number of data points in a datasetWhen samplesize is sufficiently large an accurate model can usually beconstructed However the experiments and simulations areusually expensive or time-consuming which may lead to theinsufficiency of samples Most of the modeling methods do

Mathematical Problems in Engineering 5Pr

edic

ted

flow

stre

ss120590

(MPa

)250

200

150

100

50

00 50 100 150 200 250

Data pointsBest linear fit

R = 09952

Experimental flow stress 120590 (MPa)

(a)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

814

0 50 100 150 200 250


R = 09814


(b)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

R = 09883

0 50 100 150 200 250


R = 09883


(c)

Pred

icte

d flo

w st

ress120590

(MPa

)250

200

150

100

50

0

R =

0 50 100 150 200 250


R = 09721


(d)

Figure 5 Correlations between the predicted and experimental flow stress (a) Kriging (b) RBF (c) Polynomial (d) ANN

not show good performance when sample size is not largeenough This could be an indication that the samples arelimited for modeling method to capture the general featuresof the problems The method which can provide the mostsatisfactory modeling performance with limited samples willthus be considered as the most effective one

In this section five cases with totally different temper-atures and strain rates are selected for each time of modeltesting to assess the developed models That is one-fifthof the data are evenly excluded from the training data fortesting each time There are 120 different combinations offive cases being utilized The general performances of themetamodeling methods are compared based on these 120times of testing In addition to 120575ave the standard deviation(119878) of 120575 and the correlation coefficient (119877) from all the testing

points are employed as criteria for comparison of these fourmetamodeling methods

119878 = radic1

119873 minus 1

119873

sum

119894=1

(120575119894minus 120575)2

119877 =

sum119873

119894=1(119864119894minus 119864) (119875

119894minus 119875)

radicsum119873

119894=1(119864119894minus 119864)2

sum119873

119894=1(119875119894minus 119875)2

(8)

where 120575 119864 and 119875 are the average values of 120575119894 119864119894 and 119875

119894

respectivelyFrom Table 3 it can be found that Kriging model has the

smallest 120575ave which indicates the highest prediction accuracy


Table 3 The predictability comparison of the models

Model 120575ave 119878 119877

Kriging 568 00485 09952RBF 773 00760 09814Polynomial 995 01098 09883ANN 968 01898 09721

25

20

15

10

5

0

()

minus03 minus02 minus01 00 01 02 03

Relative error

KrigingRBF

PolynomialANN

Figure 6 Distributions of the prediction errors

among the developed models Meanwhile the listed valuesof 119878 help to illustrate the prediction stability of models andKriging model performs the best Furthermore the values of119877 provide information on the strength of linear relationshipbetween the predicted and experimental flow stress Asshown in Figure 5 the fitting results of 119877 clearly demonstratethat the coincidence degree between the experimental andpredicted value from Kriging model is higher than that fromother models

In addition to the three criteria above related to the pre-diction error the distributions of all the errors are comparedamong themodels at amacro level As 80 points are predictedin each testing there are a total of 9600 values of errorscollected from 120 times of testing With the errors beingclassified into different groups according to their values thedistributions are reflected in Figure 6 while 119909-axis is relativeerror and119910-axis is the percentages of error distribution It canbe found that 8491 of predicted data from Kriging methodlocate in the relative error range of plusmn10 larger than 7296from RBF 6629 frommultivariate polynomial and 7153from ANN Based on the comparative analysis on 120575ave 119878 119877and the error distribution Krigingmethod is proved to be thebest one for modeling the material flow behavior

34 Assessing Performance of Kriging Method In the experi-ments the deformation conditions such as temperature andstrain rate are evenly distributed However these conditionsare not always evenly distributed in practical manufacturingUniformity is a measure to evaluate how uniform a point

00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0

Figure 7 Comparisons between the predicted values and experi-mental flow stress curves at 733 K

set is scattered in a space In this task the performance ofKriging method has been further assessed when the sampleuniformity is decreased The assessment is carried out by theway that one level of temperature and one level of strain rateare successively excluded frommodel training data for modeltesting

341 Decreasing Sample Uniformity Related to TemperatureAll the data related to temperature 733K are excluded frommodel training data and then used for model testing Thatis the training data cover four levels of temperature and fivelevels of strain rate The comparisons between the predictedvalues and experimental flow stress curves at 733 K are shownin Figure 7 which are quantified by the calculated values of120575ave (656) and 119878 (00480) It is indicated that the predictionresults fromKrigingmodel match well with the experimentaldata

342 Decreasing Sample Uniformity Related to Strain RateWith the information related to all the temperature levels thedata related to strain rate 001 sminus1 are excluded from modeltraining data for model testing The predictability of Krigingmodel is also analyzed based on the 120575ave and 119878 from all thetesting data The predicted values from Kriging model areall close to or on the experimental flow stress curves andthe deviations are shown in Figure 8 The high predictionaccuracy is also validated by the 120575ave 553 and the 119878 00360

From the analysis above it is found that Kriging modelcan be constructed with high accuracy and stability evenwhen the sample quality is decreased from the aspectsof sample size and sample uniformity The applicability ofKrigingmethod has been fully validated inmodeling the flowbehavior of 6013 aluminum alloy during hot deformation

Mathematical Problems in Engineering 7Tr

ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0

Figure 8 Comparisons between the predicted values and experi-mental flow stress curves at 001 sminus1

4 Conclusions

In order to investigate the applicability of metamodelingmethods and search for the appropriate method in model-ing material flow behavior four popular metamodel typesKriging radial basis function multivariate polynomial andartificial neural network are investigated and compared formodeling the elevated temperature flow behavior of 6013 alu-minum alloy These four metamodeling methods are provedto be capable ofmodeling the flowbehavior of 6013 aluminumalloyWhen sample size formodeling is decreased by 20 thesuperiority of Kriging model is revealed in prediction withsatisfactory accuracy and stability Furthermore the excellentperformance of Kriging model is again demonstrated whiledecreasing the sample uniformity related to temperature orstrain rate It is indicated that Kriging method can be takenas an appropriate option to model the material flow behavioreven if the sample size is not large enough or the samples arenot evenly distributed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (no 51475156) National Key Project ofScience and Technology of China (no 2014ZX04002071) andthe Opening Foundation of Key Laboratory for NonferrousMetal and Featured Material Processing Guangxi ZhuangAutonomous Region (no GXKFJ14-08)

References

[1] C X Wang F X Yu D Z Zhao X Zhao and L Zuo ldquoHotdeformation and processing maps of DC cast Al-15Si alloyrdquoMaterials Science and Engineering A vol 577 pp 73ndash80 2013

[2] R S Nalawade A J Puranik G Balachandran K N Mahadikand V Balasubramanian ldquoSimulation of hot rolling deforma-tion at intermediate passes and its industrial validityrdquo Interna-tional Journal of Mechanical Sciences vol 77 pp 8ndash16 2013

[3] A Rusinek and J R Klepaczko ldquoShear testing of a sheet steelat wide range of strain rates and a constitutive relation withstrain-rate and temperature dependence of the flow stressrdquoInternational Journal of Plasticity vol 17 no 1 pp 87ndash115 2001

[4] Y C Lin and X-M Chen ldquoA critical review of experimentalresults and constitutive descriptions for metals and alloys in hotworkingrdquoMaterials amp Design vol 32 no 4 pp 1733ndash1759 2011

[5] Y C Lin K K Li H B Li J Chen X M Chen and D XWenldquoNew constitutive model for high-temperature deformationbehavior of inconel 718 superalloyrdquoMaterials amp Design vol 74pp 108ndash118 2015

[6] S Saadatkia HMirzadeh and J M Cabrera ldquoHot deformationbehavior dynamic recrystallization and physically-based con-stitutive modeling of plain carbon steelsrdquoMaterials Science andEngineering A vol 636 pp 196ndash202 2015

[7] W Liu H Zhao D Li Z Zhang G Huang and Q LiuldquoHot deformation behavior of AA7085 aluminum alloy duringisothermal compression at elevated temperaturerdquo MaterialsScience and Engineering A vol 596 pp 176ndash182 2014

[8] G-Z Quan W-Q Lv Y-P Mao Y-W Zhang and J ZhouldquoPrediction of flow stress in a wide temperature range involvingphase transformation for as-cast Ti-6Al-2Zr-1Mo-1V alloy byartificial neural networkrdquoMaterials amp Design vol 50 pp 51ndash612013

[9] V Senthilkumar A Balaji and D Arulkirubakaran ldquoApplica-tion of constitutive and neural networkmodels for prediction ofhigh temperature flow behavior of AlMg based nanocompos-iterdquo Transactions of Nonferrous Metals Society of China (EnglishEdition) vol 23 no 6 pp 1737ndash1750 2013

[10] Y Han G J Qiao J P Sun and D N Zou ldquoA comparativestudy on constitutive relationship of as-cast 904L austeniticstainless steel during hot deformation based on Arrhenius-typeand artificial neural network modelsrdquo Computational MaterialsScience vol 67 pp 93ndash103 2013

[11] G G Wang and S Shan ldquoReview of metamodeling techniquesin support of engineering design optimizationrdquo Transactions ofthe ASMEmdashJournal of Mechanical Design vol 129 no 4 pp370ndash380 2007

[12] D Zhao and D Xue ldquoA comparative study of metamodelingmethods considering sample quality meritsrdquo Structural andMultidisciplinary Optimization vol 42 no 6 pp 923ndash938 2010

[13] D G Krige A statistical approach to some mine valuations andallied problems at the witwatersrand [MS thesis] University ofthe Witwatersrand Johannesburg South Africa 1951

[14] G Matheron ldquoPrinciples of geostatisticsrdquo Economic Geologyvol 58 no 8 pp 1246ndash1266 1963

[15] J SacksW JWelch T J Mitchell andH PWynn ldquoDesign andanalysis of computer experimentsrdquo Statistical Science vol 4 no4 pp 409ndash435 1989

[16] S N Lophaven H B Nielsen and J Soslashndergaard ldquoDacea Matlab kriging toolbox version 20rdquo Tech Rep IMMREP-2002-12 Technical University of Denmark Kongens LyngbyDenmark 2002


[17] M J D Powell ldquoRadial basis functions for multivariableinterpolation a reviewrdquo in Algorithms for Approximation J CMason andM G Cox Eds pp 143ndash167 Clarendon Press 1987

[18] R H Myers and D C Montgomery Response Surface Method-ology Process and Product Optimization Using Designed Exper-iments John Wiley amp Sons 1995

[19] N Haghdadi A Zarei-Hanzaki A R Khalesian and H RAbedi ldquoArtificial neural network modeling to predict the hotdeformation behavior of an A356 aluminum alloyrdquoMaterials ampDesign vol 49 pp 386ndash391 2013

[20] H-Y Li X-F Wang D-D Wei J-D Hu and Y-H Li ldquoAcomparative study on modified ZerillindashArmstrong Arrhenius-type and artificial neural network models to predict high-temperature deformation behavior in T24 steelrdquo MaterialsScience and Engineering A vol 536 pp 216ndash222 2012

[21] F J Humphreys and M Hatherly Recrystallization and RelatedAnnealing Phenomena Elsevier 2nd edition 2004

[22] J Zhang H Di X Wang Y Cao J Zhang and T Ma ldquoCon-stitutive analysis of the hot deformation behavior of Fe-23Mn-2Al-02C twinning induced plasticity steel in consideration ofstrainrdquoMaterials amp Design vol 44 pp 354ndash364 2013

[23] G Xiao L-X Li and T Ye ldquoModification of flow stresscurves and constitutive equations during hot plane compressiondeformation of 6013 aluminum alloyrdquo The Chinese Journal ofNonferrous Metals vol 24 no 5 pp 1268ndash1274 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and efficient technique to model various complex nonlinearrelationships In previous researches [11] some other typesof metamodeling method were maturely developed suchas Kriging radial basis function (RBF) and multivariatepolynomial method Each metamodel type has its associatedfitting method and the corresponding characteristics [12]Due to the complicated nature of the relationships betweenmetamodeling methods and engineering problems there isno conclusion on which method is definitely superior to theothers [11] In order to investigate the applicability of meta-modelingmethods and search for themost appropriate one inmodeling flow behavior the four popular methods KrigingRBF multivariate polynomial and ANN were utilized andcompared for modeling the flow stress of 6013 aluminumalloy during hot compression

In this work the hot plane strain compression tests werecarried out on Gleeble-3500 thermomechanical simulatorConsidering deformation heating and heat transfer themeasured flow stress data at relatively high strain rates werecorrected Based on the measured and corrected data thefeasibility of these methods is firstly tested in modelingthe flow behavior The most effective modeling methodwith satisfactory accuracy stability and efficiency was thenselected on the basis of comparative study and detailedanalysis Finally the performance of Kriging method thatis considered as the best one was further assessed for thismodeling process

2 Metamodeling Methods

21 Kriging Method After being proposed by Krige [13] andimproved by others such as Matheron [14] Kriging methodwas gradually developed to be a popular metamodel typeand systematically introduced into the area of computerexperiment [15] In Kriging method the random outputis assumed to be obtained from a linear combination ofregression functions plus a random process factor as follows

119884 =

119873

sum

119895=1

120573119895119891119895 (x) + 119885 (x) (1)

where 119873 is the number of regression functions 119891119895(sdot) is a

regression function 120573119895is the coefficient for 119891

119895(sdot) x is the

design point and 119885(sdot) is the random process function Thecorrelation function is defined by the following equation

Cov (119885 (x1) 119885 (x

2)) = 119903 (120579 x

1 x2) (2)

where Cov(sdot) is the correlation function x1and x

2are two

design points and 120579 is a structural parameter to be optimizedThe correlation function could be defined in several differentways such as Gaussian exponential linear spherical andcubical [16]

22 Radial Basis Function Method Radial basis function(RBF) method was initially developed as an exact interpo-lation technique for data in multidimensional space [17] InRBF method a series of center points (x

1198951 x1198952 x

119895119898) are

chosen first from the design points (x1 x2 x

119899) based on

some criteria Then the basis functions are constructed byusing these center points as

119861119894 (x) = 119891 (

10038171003817100381710038171003817x minus x119895119894

10038171003817100381710038171003817) (3)

where x is the design point x119895119894is the center point for 119861

119894(sdot)

and xminusx119895119894 is the Euclidian distance between the two points

The function119891( sdot) could have different forms such as linearGaussian cubic thin-plate spline and multiquadratic [17]The relationship between inputs and outputs is constructedas a linear combination of radial basis functions

119892 (x) = 1205730 +119873

sum

119895=1

119861119895 (x) 120573119895 (4)




119895(sdot)

23 Multivariate Polynomial Method Multivariate polyno-mial method is one of the most fundamental metamodelingmethods used in computer experiment It is mostly knownfrom response surface method [18] which employs quadraticpolynomial in the field of engineering design optimizationIn multivariate polynomial method basis functions arebuilt directly by using input variable components (x =

[1199091 1199092 119909

119899]) and their interactions such as

1198610 (119909) = 1 1198611 (119909) = 1199091 119861119899 (119909) = 119909119899

11986111 (119909) = 119909

2

1 11986112 (119909) = 11990911199092 1198611119899 (119909) = 1199091119909119899

(5)

A multivariate polynomial is constructed as the weightedsum of these basis functions

119892 (x) = 1198870 +119873

sum

119895=1

119861119895 (x) 119887119895 (6)




119895(sdot)

24 Artificial Neural Network Method Artificial neural net-work (ANN)method is based on a particular set of nonlinearfunctions to build the relationship between inputs and out-puts Its architecture consists of input layer output layer andhidden layer which are connected by the processing unitscalled neurons [19] as shown in Figure 1 Each neuron inthe input layer and output layer represents one independentvariable while the neurons in the hidden layers are only forcomputation purpose During training procedure the knowninput-output pairs are used to update the weights and biases[20] The objective is to minimize the errors between theANN outputs and the targets for the corresponding inputs

3 Model Construction and Discussion

31 Experiments and Modeling Preparation The materialused in the present experiment is 6013 aluminum alloy with


Input 1

Input 2

Input 3

Output


Tem

pera

ture

T (K

)

Water quenching

Time t (s)

120 s

10Ks

613sim773K






Kriging

RBF

Polynomial

ANN

Temperature T


Strain rate 120576






120575 =

1003816100381610038161003816100381610038161003816

119864 minus 119875

119864

1003816100381610038161003816100381610038161003816times 100

120575ave =1

119873

119873

sum

119894=1

120575119894times 100

(7)






1 653 1 287 837 824 1119 347 575 157 2792 693 01 379 619 591 856 242 426 344 5683 733 001 392 853 120 626 311 597 949 1385

30

60

90

120

150

180

0

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576

KrigingExperimental

(1) 653K 1 sminus1(2) 693K 01 sminus1

(3) 733K 001 sminus1

(a)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

RBF

30

60

90

120

150

180

0

(b)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

Polynomial

30

60

90

120

150

180

0

(c)

True

stre

ss120590

(MPa

)

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

ANN

30

60

90

120

150

180

1

2

3

0

(d)





edic

ted

flow

stre

ss120590

(MPa

)250

200

150

100

50

00 50 100 150 200 250


R = 09952


(a)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

814

0 50 100 150 200 250


R = 09814


(b)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

R = 09883

0 50 100 150 200 250


R = 09883


(c)

Pred

icte

d flo

w st

ress120590

(MPa

)250

200

150

100

50

0

R =

0 50 100 150 200 250


R = 09721


(d)





119878 = radic1

119873 minus 1

119873

sum

119894=1

(120575119894minus 120575)2

119877 =

sum119873

119894=1(119864119894minus 119864) (119875

119894minus 119875)

radicsum119873

119894=1(119864119894minus 119864)2

sum119873

119894=1(119875119894minus 119875)2

(8)


119894





Model 120575ave 119878 119877


25

20

15

10

5

0

()


Relative error

KrigingRBF

PolynomialANN





00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0







ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Input 1

Input 2

Input 3

Output


Tem

pera

ture

T (K

)

Water quenching

Time t (s)

120 s

10Ks

613sim773K






Kriging

RBF

Polynomial

ANN

Temperature T


Strain rate 120576






120575 =

1003816100381610038161003816100381610038161003816

119864 minus 119875

119864

1003816100381610038161003816100381610038161003816times 100

120575ave =1

119873

119873

sum

119894=1

120575119894times 100

(7)






1 653 1 287 837 824 1119 347 575 157 2792 693 01 379 619 591 856 242 426 344 5683 733 001 392 853 120 626 311 597 949 1385

30

60

90

120

150

180

0

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576

KrigingExperimental

(1) 653K 1 sminus1(2) 693K 01 sminus1

(3) 733K 001 sminus1

(a)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

RBF

30

60

90

120

150

180

0

(b)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

Polynomial

30

60

90

120

150

180

0

(c)

True

stre

ss120590

(MPa

)

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

ANN

30

60

90

120

150

180

1

2

3

0

(d)





edic

ted

flow

stre

ss120590

(MPa

)250

200

150

100

50

00 50 100 150 200 250


R = 09952


(a)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

814

0 50 100 150 200 250


R = 09814


(b)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

R = 09883

0 50 100 150 200 250


R = 09883


(c)

Pred

icte

d flo

w st

ress120590

(MPa

)250

200

150

100

50

0

R =

0 50 100 150 200 250


R = 09721


(d)





119878 = radic1

119873 minus 1

119873

sum

119894=1

(120575119894minus 120575)2

119877 =

sum119873

119894=1(119864119894minus 119864) (119875

119894minus 119875)

radicsum119873

119894=1(119864119894minus 119864)2

sum119873

119894=1(119875119894minus 119875)2

(8)


119894





Model 120575ave 119878 119877


25

20

15

10

5

0

()


Relative error

KrigingRBF

PolynomialANN





00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0







ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 653 1 287 837 824 1119 347 575 157 2792 693 01 379 619 591 856 242 426 344 5683 733 001 392 853 120 626 311 597 949 1385

30

60

90

120

150

180

0

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576

KrigingExperimental

(1) 653K 1 sminus1(2) 693K 01 sminus1

(3) 733K 001 sminus1

(a)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

RBF

30

60

90

120

150

180

0

(b)

True

stre

ss120590

(MPa

)

1

2

3

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

Polynomial

30

60

90

120

150

180

0

(c)

True

stre

ss120590

(MPa

)

00 01 02 03 04 05 06 07 08 09

True strain 120576


(3) 733K 001 sminus1

ANN

30

60

90

120

150

180

1

2

3

0

(d)





edic

ted

flow

stre

ss120590

(MPa

)250

200

150

100

50

00 50 100 150 200 250


R = 09952


(a)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

814

0 50 100 150 200 250


R = 09814


(b)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

R = 09883

0 50 100 150 200 250


R = 09883


(c)

Pred

icte

d flo

w st

ress120590

(MPa

)250

200

150

100

50

0

R =

0 50 100 150 200 250


R = 09721


(d)





119878 = radic1

119873 minus 1

119873

sum

119894=1

(120575119894minus 120575)2

119877 =

sum119873

119894=1(119864119894minus 119864) (119875

119894minus 119875)

radicsum119873

119894=1(119864119894minus 119864)2

sum119873

119894=1(119875119894minus 119875)2

(8)


119894





Model 120575ave 119878 119877


25

20

15

10

5

0

()


Relative error

KrigingRBF

PolynomialANN





00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0







ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










edic

ted

flow

stre

ss120590

(MPa

)250

200

150

100

50

00 50 100 150 200 250


R = 09952


(a)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

814

0 50 100 150 200 250


R = 09814


(b)

Pred

icte

d flo

w st

ress120590

(MPa

)

250

200

150

100

50

0

R = 09883

0 50 100 150 200 250


R = 09883


(c)

Pred

icte

d flo

w st

ress120590

(MPa

)250

200

150

100

50

0

R =

0 50 100 150 200 250


R = 09721


(d)





119878 = radic1

119873 minus 1

119873

sum

119894=1

(120575119894minus 120575)2

119877 =

sum119873

119894=1(119864119894minus 119864) (119875

119894minus 119875)

radicsum119873

119894=1(119864119894minus 119864)2

sum119873

119894=1(119875119894minus 119875)2

(8)


119894





Model 120575ave 119878 119877


25

20

15

10

5

0

()


Relative error

KrigingRBF

PolynomialANN





00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0







ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Model 120575ave 119878 119877


25

20

15

10

5

0

()


Relative error

KrigingRBF

PolynomialANN





00 01 02 03 04 05 06

1

2

3

4

5

07 08 09

(1) 10 sminus1

(2) 1 sminus1

(3) 01 sminus1

(4) 001 sminus1

(5) 0001 sminus1

KrigingExperimental

True

stre

ss120590

(MPa

)

True strain 120576

30

60

90

120

150

0







ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ue st

ress120590

(MPa

)

00 01 02 03 04 05 06 07 08

1

2

3

4

5

09

KrigingExperimental

True strain 120576

(1) 613K(2) 653K(3) 693K(4) 733K

(5) 773K

30

60

90

120

150

180

0


4 Conclusions




Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modeling material flow behavior during...

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