J Intell ManufDOI 10.1007/s10845-014-0879-6

Modeling, analysis and multi-objective optimization of twistextrusion process using predictive models and meta-heuristicapproaches, based on finite element results

Hamed Bakhtiari · Mahdi Karimi · Sina Rezazadeh

Received: 6 June 2013 / Accepted: 22 January 2014© Springer Science+Business Media New York 2014

Abstract Recently, twist extrusion has found extensiveapplications as a novel method of severe plastic deformationfor grain refining of materials. In this paper, two prominentpredictive models, response surface method and artificialneural network (ANN) are employed together with results offinite element simulation to model twist extrusion process.Twist angle, friction factor and ram speed are selected asinput variables and imposed effective plastic strain, strainhomogeneity and maximum punch force are considered asoutput parameters. Comparison between results shows thatANN outperforms response surface method in modelingtwist extrusion process. In addition, statistical analysis ofresponse surface shows that twist extrusion and friction fac-tor have the most and ram speed has the least effect on out-put parameters at room temperature. Also, optimization oftwist extrusion process was carried out by a combinationof neural network model and multi-objective meta-heuristicoptimization algorithms. For this reason, three prominentmulti-objective algorithms, non-dominated sorting geneticalgorithm, strength pareto evolutionary algorithm and multi-objective particle swarm optimization (MOPSO) were uti-lized. Results showed that MOPSO algorithm has relativesuperiority over other algorithms to find the optimal points.

Keywords Twist extrusion · FE simulation ·Multi-objective optimization · Artificial neural network ·Multi-objective meta-heuristic algorithms · Responsesurface method

H. Bakhtiari · M. Karimi (B)Bu-Ali Sina University, P.O.B. 651744161, Hamedan, Irane-mail: karimi@basu.ac.ir

S. RezazadehIslamic Azad University, Qazvin Branch, Qazvin, Iran

Introduction

Grain refining of materials often enhances their mechanicalproperties. Severe plastic dseformation (SPD) methods area series of methods where enforcing large plastic strains onthe grain structure of a bulk solid causes grain refinement ofthe material. Methods such as equal channel angular press-ing, accumulative roll-bonding, multi-directional forging andtwist extrusion (TE) are among these methods, each of whichleads to grain refinement based on the type of stress appliedto the material. However, it has been proved that generatingsimple shear stress in the material is the most effective wayto achieve ultra-fine grained structures (Beygelzimer 2005).Twist extrusion is one of the novel SPD methods in whichtwo shear planes parallel with and perpendicular to directionof extrusion leads to simple shear stress in the material andgrain refine it. Of the benefits of twist extrusion process is itscapability in the extrusion of hollow parts and parts with rec-tangular section with more homogeneous strains and withoutany significant change in dimensions (Mousavi et al. 2008).

TE has not been thoroughly studied yet and studies sofar have mostly addressed qualitative analysis and compar-ison with other methods. Orlov et al. (2008, 2009a) studiedthe effect of twist extrusion on microstructure of pure alu-minum billets. They showed that when the applied strain issmall, the hardness increases with the increase of strain but itdecreases in higher strains due to reduction of dislocations.Orlov et al. (2009b) analyzed variations of vickers hardness,yield strength and ultimate tensile strength of pure aluminumspecimen after TE at room temperature. They observed grainsize reduction and formation of a homogenous microstruc-ture after one pass. Mousavi et al. (2008) investigated theeffects of temperature and ram speed on Ti-6Al-4V speci-mens. They concluded that increasing temperature and ramspeed leads to higher plastic strains in the billet. By study-

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ing the mechanism of billet deformation in twist extrusionprocess, Beygelzimer et al. (2009) stated that twist angle androtation angle of cross section are two main factors in increas-ing or decreasing of imposed plastic strain in the part. Someother processes such as rolling (Stolyarov et al. 2005), directextrusion of the specimen (Mousavi et al. 2010) , natural andartificial aging heat treatments (Bahadori and Mousavi 2012)and sequential TE passes (Orlov et al. 2009a) are complemen-tary thermo-mechanical processes proposed to be performedafter twist extrusion process to produce more homogenousstructures.

The common goal of all studies carried out up to nowis analysis of methods and providing a solution for achiev-ing proper and homogenous mechanical properties. However,according to the best of the author’s knowledge, there is nota study in the literature to optimize geometry of the die andtwist extrusion parameters. Using the optimized conditionsin experiments, it is possible to obtain proper mechanicalproperties which are as homogenous as possible. The mainpurposes of this paper are successful modeling of twist extru-sion process using predictive models, investigating the effectof various parameters on the process and finally optimizingthe process regarding cost and time.

Sibalija and Majstorovic (2012) proposed a hybrid, inte-grated approach to optimiza parameter design of a multi-response system by incorporating taguchi’s method into arti-ficial neural network (ANN) and genetic algorithm mod-els. Ashhab et al. (2012) utilized artificial neural networkto model a combined deep drawing–extrusion process. Theyalso tried to find the inputs or geometrical parameters that willproduce the desired or optimum values of equivalent plasticstrain, contact ratio and forming force by using a complexconstrained optimization method. In this paper, ANN andresponse surface (RS) model as two well-known predictivemodels are utilized and compared To develop an appropriatemodel of TE process. In order to reach this goal, a num-ber of experiments were selected using design of experi-ments method. After numerical simulation of these exper-iments using finite element method, they were employedto develop ANN and RS models. Twist angle, friction fac-tor and ram speed were designated input variables whileimposed effective plastic strain in the billet after one TE pass,homogeneity of strain distribution and maximum punch forcewere selected as output variables. Then the importance andeffectiveness of each parameter was determined by statisti-cal analysis. Finally, three prominent multi-objective meta-heuristic algorithms were used to optimize the process. Forthis reason, ANN as the evaluating function was integratedwith non-dominated sorting genetic algorithm (NSGA-II),strength pareto evolutionary algorithm (SPEA2) and multi-objective particle swarm optimization (MOPSO) to decide onthe optimized process parameters and the best optimizationalgorithm.

Modeling

Due to continuous change in parameters during the optimiza-tion and also requirement for significant number of iterationswhen executing the algorithm, the process needs to be eval-uated by a mathematical relationship. This equation mustconsider nonlinear effect of parameters and possess exten-sion and adapting capabilities as well. Employing theoret-ical equation is not reliable because of their simplifyingassumptions. In this paper, two prominent predictive mod-els including response surface method (RSM) and ANN areused. These two models are extensively applied for modelingcomplicated and nonlinear processes. Steps for developingthese models include:

1. Designing experiments (selecting the values of inputparameters).

2. Evaluating Experiments (determining output values forinput data).

3. Developing predictive model using results of experimentevaluation (determination of coefficients).

4. Investigating performance of predictive models.

Design of experiments

It is impractical to evaluate all design combinations by FEM,so the appropriate number of designs must be selected basedon design of experiments method. Designing experiments issimply a procedure for selection of values of input parame-ters so that by the least number of experiments, the relation-ship between inputs and outputs of the problem is possible.First, input parameters and theirs span of variation shouldbe specified. Figure 1 shows the TE process studied in thispaper schematically in which β is the twist angle (the anglebetween extrusion direction and twisted path of die) and L islength of the twisted part. Billet is made of pure aluminumwith a rectangular section of 28 × 18 mm2 and length ofis 50 mm.Billet is pushed by the punch into the die with aconstant velocity and passes the die without any change incross section. Stress types applied include simple shear stressalong transversal cross section at input and output sectionsand simple shear stress along longitudinal cross section ofthe specimen at twisted section of the die (Beygelzimer et al.2009).

In TE process, twist angle (Beygelzimer et al. 2009), fric-tion factor (Latypov et al. 2012) and ram speed (Mousavi et al.2008) significantly influence the final properties of material.Considering these three parameters, it is possible to inves-tigate the effects of geometry of the die and conditions ofexperiment simultaneously. Table 1 shows design variablesand theirs respective span of variation. These spans are alsoconsidered as constraints of the problem. For designing train-ing data, different designs such as Box-Behnken (15 runs),

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Fig. 1 Schematic representation of twist extrusion process

Table 1 Design variables and their ranges

Design variables Symbol Variable range

Twist angle (◦) A [25–52.58]

Friction factor B [0.2–0.8]

Ram speed (mm/s) C [2–8]

Central composite (16–17 runs), 3-level factorial (27 runs)and taguchi were investigated and 3-level factorial methodproduced smallest errors for both ANN and RSM models.Design of experiments is done using 3-level factorial methodand collectively, 27 experiments, which were subsequentlyused as training data, were selected for evaluation.

Evaluating experiments

Output parameters considered for each experiment include:(1) Imposed effective plastic strain in the specimen after oneTE pass, (2) strain homogeneity level (strain standard devi-ation) and (3) Maximum punch force (the highest value offorce that punch applies to the billet during the extrusionprocess). Effective plastic strain is in fact the equivalent VonMises strain which is calculated in different locations on thebillet after one pass of extrusion. The equivalent Von Misesstrain is defined as Eq. (1):

ε =√

2

3

√(ε1 − ε2)2 + (ε2 − ε3)2 + (ε3 − ε1)2 (1)

where ε1, ε2 and ε3 are principal strains, and ε is equivalentVon Mises strain.

With the increasing values of plastic strain, grain structurebecomes finer and mechanical properties improve. Homoge-neous strain distribution inside the part also leads to decreas-ing gradient of properties and their more homogenous dis-tribution throughout the material. In order to evaluate straindistribution, strain standard deviation as Eq. (2) is used.

SD =√∑N

i=1 (εi − εave)2

N

εave =∑N

i=1 (εi )

N(2)

where SD is strain standard deviation, N is the number ofselection points on cross section of the billet and εi is theimposed effective plastic strain at ith point. Also εave is theaverage imposed effective plastic strain on cross section.Lower values of SD denote more homogenous strain dis-tribution. To compute strain and its standard deviation in thespecimen more closely, these two parameters were measuredat four sections of equal distance (5 mm) along the specimenand arithmetic average of these measurements were used.Maximum punch force is usually applied when the billet iscompletely enters the twist section of the die and creates themost contact area. In this condition, friction force reaches itspeak and results in the maximum value of force applied topunch. Numerical value of this force is extracted using themaximum recorded value.

Designs can be evaluated numerically or experimentally.Experimental evaluation of designs is a very costly and time-consuming process. In addition, it is very difficult to evaluatestrain and its homogeneity along the sections experimentally.Finite element method is not limited by such constraints andits accuracy for twist extrusion process is investigated bydifferent researchers (Mousavi et al. 2008, 2010; Latypov etal. 2012). In this paper, Deform-3D finite element softwarewas employed to simulate TE process. In order to assess theaccuracy of finite element model, simulation was carried outfor twist extrusion of a pure aluminum billet with a twistangle of β = 52.58◦. 15,608 quadrilateral elements wereused along the billet. Punch and die were considered to berigid bodies and die was completely constrained. Johnson–Cook constitutive model was utilized to describe behavior ofthe billet. Johnson–Cook model is defined as Eq. (3):

σ = [A + B(ε)n]

[

1 + C ln

(ε

ε0

)] [1 −

(T − T0

Tm − T0

)m]

(3)

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Table 2 Johnson–Cook constitutive parameters for pure aluminum

A (MPa) B (MPa) C n m Tm(◦K)

80 120 0.008 0.73 1.7 993

where σ is yield stress in a non-zero strain rate, ε is theequivalent (or effective) plastic strain and ε is the equiva-lent plastic strain rate for ε0 = 1 s−1. A, B, C, n and m arephysical coefficients obtained from experiments. T and Tm

are current and melting temperatures respectively. Johnson-Cook parameters for pure aluminum are listed in Table 2(Skrotzki et al. 2007).

Shear friction model as Eq. (4) is used for modeling thefriction between contact surfaces.

f = mk (4)

where m is the friction factor which is usually between 0(frictionless) and one (full sticking). k is the shear flowstress in deformed material. Friction factor for all contactsurfaces is m = 0.01 and ram speed is considered to be3 (mm/s).

In Fig. 2, variations of strain and heterogeneity of imposedeffective plastic strain in the billet is clearly illustrated. Asobserved, the highest strains are induced in the corners ofthe billet which are in contact with internal surface of thedie and strain is reduced moving toward the center of spec-imen cross section. This heterogeneity of strain inside thepart leads to a gradient of mechanical properties which isundesirable in metal forming processes, but can be min-imized through modification of die geometry and processconditions.

After finishing simulation, strain in points shown in thecross section in Fig. 2 is determined. Based on the experi-mental studies (Berta et al. 2007), minimum equivalent plas-tic strain (at center of cross section) and strain at an arbitraryradius from the center can be calculated with acceptable accu-racy using Eq. (5).

εmin = 0.4 + 0.1 tan (β)

εn = 2r√3R

tan (β) (5)

where εmin is strain at the center of the cross section withedge radius R and εn is strain at a point of distance r fromthe center of the cross section. It should be noted that this rela-tionship is valid only in frictionless condition and to evaluatebillet strain at the output of the die. Figure 3 provides a com-parison between simulation results and values determined bytheoretical equations. As it is obvious, results are compatiblewith good accuracy.

Developing predictive models using experiments evaluationresults

Response surface method

Response surface model which was introduced by Box andWilson (1951) is simply a polynomial that describes the rela-tionship between input and output variables. Second orderpolynomial usually used in RSM which includes two factorinteractions is written as Eq. (6):

Y = β0 +3∑

i=1

βi Xi +3∑

i=1

βi i X2i +

∑ ∑

i< j

βi j Xi X j (6)

where β0, βi , βi i and βi j are coefficients of polynomialwhich are determined using least squares regression analy-sis so that the difference between output of polynomial andresults of finite element simulation for training data is theleast possible. Xi and X j are independent design variablesand Y is response variable (here, objective function). In thispaper, statistical software Statgraphics Centurion XV.I isused for developing response surface model and analyzingthe effects of parameters.

Fig. 2 Deformation mechanism and strain heterogeneity formation during TE process

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Fig. 3 Effective strain comparison between FEM and theoreticalresults

Artificial Neural Network (ANN) model

Artificial neural network is simply a nonlinear mappingbetween input variables and output values inspired byhuman’s brain system. Coefficients of this mapping are deter-mined using training data and through multiple iterations ina process called training. Previous studies have shown thatneural network is suitably capable of modeling metal formingprocesses (Hans Raj et al. 2000; Lorenzo et al. 2006; Kimand Kim 2000; Chan et al. 2008; Katherasan et al. 2012).Lucignano et al. (2010) implemented the ANN to predict theoptimal conditions of the aluminium alloy extrusion process.Hsiang et al. (2006) used the ANN to investigate the influ-ence of billet temperature on tensile strength of hot extrudedmagnesium alloy for various extrusion speeds and extru-sion ratios. Chan et al. (2008) used a combination of finiteelement simulation results and artificial neural network formodeling metal forming process. They prepared a number ofpoints for simulation according to orthogonal arrays methodand used the results of simulation to train ANN. They con-cluded that combining finite element simulation and ANNnot only remarkably reduced simulation time but could prop-erly model complicated nonlinear processes such as metalforming.

In this paper, MATLAB 2012 software is used for creatingand training ANN. Training step was carried out using train-ing data listed in Table 3. Also, Table 4 shows nine pointsoutside training data set called validation data. In order tooptimally determine neural network structure, various net-work structures (radial-basis, back propagation and varioustransfer functions such as linear and sigmoid functions) withdifferent number of neurons and layers were studied and thenetwork with the most accurate result compared to evaluationdata points in Table 4 was selected as optimal network. Theoptimal neural network used in this paper is a feed-forward

network with error back propagation algorithm and a hiddenlayer composed of 3 neurons. Levenberg–Marquardt methodwas employed to train the network and tangent sigmoid andlinear functions were used as transfer functions (which cal-culate a layer’s output based on its net input ) of hidden andoutput layer, respectively. As to scattering of data, every datapoint was normalized in the interval between 0 and 1. Learn-ing rate of the network is set to be 0.01. If the learning rate istoo high, the network becomes unstable. On the other hand,if it is too low, training time will be very long. Training willstop when the number of iterations exceeds 1,000 or meansquare error (MSE) falls below the goal value (1 × 10−4).MSE of this network after 1,000 iterations was 0.0033 thatindicated the network was converged.

Comparison between RSM and ANN results

Training data and corresponding results are listed in Table 3.Also Table 4 shows the accuracy of predictive models inevaluation of validation data. R-square equation is used fordetermining accuracy values of methods. R-square is calcu-lated for each output function as Eq. (7).

R − square = 100 ×(∑n

i=1

(yi − y

)2

∑ni=1 (yi − y)2

)

% (7)

where yi is predictive model (RS or ANN) response, yi isfinite element response, y is arithmetic average of yi valuesand n is number of points. Regarding the values of R-squarepresented in Tables 3 and 4, it is possible to conclude thatANN model maintained better accuracy in training and pre-diction stages compared to response surface method and canprovide a more appropriate model.

Statistical analysis

For a more detailed investigation and exploring the effect ofcounteraction of design variables on final response, Paretodiagram shown in Fig. 4 was plotted for each of the objectivefunctions using statistical software Statgraphics XV.I. Lengthof each of the bands is proportional to effectiveness of thatparameter on final response. If the band surpasses the verticallevel line shown in the figure, it shows that the parameter hasa considerable influence on final response. Parameters whichare directly proportional to final response are specified withpositive sign and those which are reversely proportional withnegative sign next to their band.

Studying Pareto diagrams show that increasing twist angleand friction factor leads to significant increase of all threeobjective functions, but changing ram speed does not resultin any noticeable alteration in the response. Twist angle andfriction factor are most effective on average effective strainand strain standard deviation. Most effective parameters on

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Table 3 Comparison between ANN and RSM results for training data

Test no A B C Effective plastic strain SD Max punch force (KN)

FEM RSM ANN FEM RSM ANN FEM RSM ANN

1 25 0.2 2 0.41 0.48 0.40 0.24 0.23 0.25 50.45 52.79 45.79

2 52.58 0.2 2 1.17 1.24 1.17 0.52 0.51 0.52 167 164.52 173.04

3 38.79 0.2 5 0.78 0.65 0.69 0.35 0.33 0.36 82 76.82 86.86

4 52.58 0.2 5 1.20 1.09 1.20 0.46 0.54 0.46 155 162.31 155.09

5 38.79 0.8 8 1.37 1.19 1.21 0.54 0.62 0.58 365 368.53 368.38

6 52.58 0.5 2 1.58 1.43 1.58 0.68 0.69 0.68 303 295.34 298.21

7 38.79 0.5 2 0.83 0.86 0.77 0.39 0.41 0.42 165 175.47 166.10

8 38.79 0.8 2 1.32 1.25 1.20 0.57 0.60 0.59 375 372.76 373.64

9 25 0.5 8 0.48 0.42 0.43 0.24 0.28 0.25 104 110.89 108.92

10 25 0.2 8 0.41 0.41 0.41 0.24 0.24 0.25 48 51.24 45.37

11 52.58 0.8 8 1.87 1.82 1.85 0.96 0.97 0.97 522 523.84 518.53

12 25 0.5 2 0.42 0.46 0.42 0.26 0.27 0.25 111 114.14 111.53

13 38.79 0.8 5 0.96 1.16 1.20 0.63 0.62 0.59 367 368.51 371.00

14 38.79 0.5 8 0.71 0.76 0.77 0.46 0.43 0.42 169 172.93 163.96

15 52.58 0.5 8 1.25 1.28 1.25 0.75 0.71 0.76 309 293.51 306.78

16 38.79 0.2 8 0.64 0.65 0.70 0.36 0.32 0.36 95 78.54 87.16

17 52.58 0.8 5 1.85 1.82 1.85 0.97 0.97 0.96 523 523.47 519.07

18 25 0.8 8 0.64 0.74 0.73 0.44 0.40 0.42 275 271.75 275.38

19 52.58 0.5 5 1.29 1.30 1.26 0.77 0.71 0.76 298 292.29 307.40

20 52.58 0.8 2 1.83 1.94 1.86 0.96 0.94 0.96 522 527.37 519.56

21 38.79 0.5 5 0.68 0.75 0.77 0.42 0.43 0.42 165 172.06 165.02

22 25 0.2 5 0.41 0.39 0.41 0.24 0.25 0.25 52.2 49.88 45.57

23 38.79 0.2 2 0.75 0.77 0.76 0.34 0.31 0.34 82 79.38 84.29

24 52.58 0.2 8 0.97 1.06 0.97 0.50 0.53 0.49 148 164.38 156.56

25 25 0.8 2 0.88 0.75 0.73 0.38 0.38 0.42 283 276.69 280.50

26 25 0.5 5 0.41 0.38 0.42 0.25 0.29 0.25 113 110.38 110.22

27 25 0.8 5 0.65 0.68 0.73 0.46 0.41 0.42 273 272.09 277.93

R-Sq (%) 96.37 97.09 97.7 99.06 99.76 99.9

Table 4 Comparison between ANN and RSM results for validation data

Test no A B C Effective plastic strain SD Max punch force (KN)

FEM RSM ANN FEM RSM ANN F E M RSM ANN

1 25 0.5 4 0.42 0.39 0.42 0.26 0.29 0.25 105.37 111.16 110.65

2 27 0.3 3 0.45 0.43 0.42 0.27 0.25 0.25 64.04 62.38 61.94

3 37 0.7 6 1.05 0.92 0.99 0.49 0.52 0.47 277.5 277.07 275.96

4 40 0.2 6 0.7 0.67 0.72 0.39 0.34 0.37 96.34 82.09 92.47

5 48 0.2 5 1.03 0.93 0.97 0.43 0.45 0.43 119.37 127.43 130.13

6 52.58 0.5 3 1.3 1.38 1.35 0.76 0.7 0.73 307.58 293.85 304.96

7 52.58 0.6 4 1.42 1.47 1.43 0.82 0.79 0.85 379.83 358.74 377.68

8 52.58 0.8 6 1.84 1.81 1.85 0.94 0.97 0.96 541.6 523.12 518.88

9 52.58 0.2 8 0.97 1.06 0.97 0.5 0.53 0.49 147.64 164.38 156.56

R-Sq (%) 97.49 99.39 97.78 99.68 99.57 99.82

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Standardized effect of design variables on effective plastic strain

Standardized effect of design variables onstrain standard deviation

Standardized effect of design variables onMax. punch force

A B C

Fig. 4 Standardized effect of design parameters on objective functions a twist angle, b friction factor and c ram speed

maximum extrusion force successively are friction factor andtwist angle. It is also obvious that counteraction of these twoparameters on all three objective functions is significant. Forexample, counteraction between twist angle and friction fac-tor, which is specified with AB sign in all three diagrams,is significant. In all three diagrams, the role of ram speedin final response is small and the two are reversely related.Therefore, it is possible to say that ram speed at room tem-perature is less effective on producing strain and extrusionforce. One of the reasons behind this lack of significance forram speed could be that effect of strain rate on yield behaviorof metals in room temperature is small.

Research conducted by Mousavi et al. (2008) on titaniumalloy shows that at operational temperature of 700 − 900 ◦C,increasing ram speed causes significant rise in plastic strainand extrusion force. In addition, crystal structure of metalsaffects their behavior. Metals such as aluminum and copperthat are of FCC crystal structure, are less sensitive to strainrate compared to metals with BCC crystal structure such aspure iron and tantalum.

Multi-objective optimization and selection of the bestdesign

Problem statement

A multi-objective optimization problem is posed as Eq. (8):

Minimize f (x) = { f1 (x) , f2 (x) , . . . , fm (x)} ; x ∈ D

(8)

where x = (x1, x2, . . . , xn) is design variables vector, D isdesign space (or feasible solution space) which is determinedby problem constraints and fi (x), i = 1, 2, . . . , m repre-sent objective functions. In this paper, twist angle, frictionfactor and ram speed are selected as design variables andtheir variation span is considered as design space. Objec-tive functions are the output functions mentioned in section

“Evaluating experiments”. Imposed effective plastic strain inthe part after one pass, strain standard deviation and maxi-mum punch force are objective functions which are estimatedby well-trained neural network in each iteration. Evaluationof objective functions using numerical simulation method isearlier discussed in section “Evaluating experiments”. Thegoal of optimization is to find values for design variablesthat maximize imposed effective plastic strain and minimizestrain standard deviation and maximum punch force.

Meta-heuristic algorithms

In multi-objective optimization problems, objective functionare usually conflicting and opposing each other; i.e. improv-ing one function causes other function to deteriorate. Hence,it is not possible to have all objective functions in their bestpossible condition.In order to optimize all objective functionssimultaneously, the concept of Pareto optimal set is utilized.Non-dominated Pareto set points are those which are notdominated by any other point. In other words, x1 dominatex2 if and only if:

fi (x1) ≤ fi (x2) , for all i ∈ {1, 2, . . . , n}f j (x1) ≤ f j (x2) , for some j ∈ {1, 2, . . . , n} (9)

where n is the number of objective functions to be mini-mized. If there is no point in design space in which x2 candominate x1 as Eq. (9), then x1 is a Pareto point. The setof all Pareto points form a Pareto front in objective func-tions space. As mentioned in section “Problem statement”,imposed effective plastic strain, strain standard deviation andmaximum punch force are selected as objective functions. Bymultiplying imposed effective plastic strain function by −1,it is possible to obtain maximum value of the function in aminimization problem.

For optimizing nonlinear and complicated problems likemetal forming processes, using stochastic methods whosesearch algorithms are based on probability and statistics ismore efficient than classical methods that are based on gradi-

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ent of functions. Classical methods are only applied to con-tinuous and regular functions and there is a chance of beingtrapped in local minima for these methods ( Deb 2001). Meta-heuristic algorithms are among stochastic methods and mostof them are based on dominance criteria. In spite of the vari-ety of algorithms, there is a lack of comparative investiga-tion about their superiority particularly for metal formingprocesses. In this paper three prominent meta-heuristic algo-rithms for multi-objective optimization, namely NSGA-II(Kalyanmoy 2001), SPEA2 (Zitzler et al. 2001) and MOPSO(Coello et al. 2004) were utilized. In all these algorithms, a setof initial points are randomly selected and through iterationsthey are guided toward optimal locations. In each iteration,after evaluation of points by well-trained neural network, anew set of improved solutions are produced by randomizedoperators. This cycle goes on until stopping criteria is met sothat a set of solutions are introduced as Pareto set.

Optimization process in NSGA-II algorithm starts withan initial population in the feasible space. Then individualsare evaluated using objective functions and ranked in sepa-rate groups in a way that members of any group are dom-inated only by members of higher ranking groups. Mem-bers of the same group are sorted using a second indexcalled crowding distance. After that some of the members areselected as parents using binary tournament and recombina-tion, crossover and mutation operators are applied to themto produce offspring population. This is performed using theparameters listed in Table 5. Parents and offspring collec-tively are ranked again using non-dominated sorting pro-cedure described above and a specified number of the bestmembers are selected as the new generation. This processis continued until the maximum number of generations isreached or the average change in the spread of the Paretosolutions is less than the specified tolerance (0.001).

Contrary to NSGA-II, in SPEA2 and MOPSO algorithmsthere is a repository or archive besides the main populationthat has the responsibility of storing non-dominated solu-tions. In SPEA2 algorithm, first an initial population is ini-tiated and a null set is designated as repository. After calcu-lating fitness values of all members of population and repos-itory, non-dominated individuals of these two sets are addedto the repository. Next step is selection of some membersof the repository via binary tournament and using recombi-nation and mutation operators to create new generation thatreplaces the current generation. When the number of itera-tions reaches the specified value, members of the repositoryare introduced as non-dominated optimal solutions and thealgorithm is terminated.

In MOPSO algorithm, a finite number of particles equalto population size are selected in search space and a ran-dom position and zero velocity are assigned to them. Thenfitness values of all particles are calculated using objectivefunctions and a number of non-dominated particles equal to

repository size are stored in the repository. Then the positionand velocity of all particles of the population are updatedusing Eq. (10).

V ′ = W · V + C1r1 (Pbest − X) + C2r2 (gbest − X)

X ′ = X + V (10)

where X, V, X ′, V ′ and W are particle’s current position vec-tor, current velocity vector, new position vector,new veloc-ity vector and inertia weights, respectively. Pbest and gbest

also represent the best position of the particle and the bestglobal position among repository population, respectively.C1 and C2 are acceleration coefficients which indicate par-ticle’s proximity to Pbest and gbest . Finally, r1 and r2 arerandom numbers between 0 and 1. After updating positionsand velocities, particles’ fitness values are evaluated againand the process continues until the maximum number of iter-ations is reached. Stopping criteria and other parameters ofany algorithm significantly affect its results. In this paper,different sets of parameter values were used to run the algo-rithms. Analysis of norm values of the sets helped us to decideon the best parameters for any algorithm. Due to the smallnumber of variables, increasing the population size to over100 didn’t cause any improvement in the results. Parametersfor all three algorithms are listed in Table 5.

Finally, in order to select the best design between Paretopoints, distance of each point to the best feasible design (idealdesign) is calculated using the norm of normalized matrix ofoutputs as in Eq. (11) (Blasco et al. 2008).

norm =√√√√

s∑

i=1

fi (x)2, 0 ≤ norm ≤ 1

fi (x) = fi (x) − fimin

fimax − fi

min , i = 1, . . . , m

fi (x) = fimax − fi (x)

fimax − fi

min , i = 1, . . . , n

0 ≤ fi (x) ≤ 1, i = 1, . . . , s (11)

where norm is distance of Pareto point to the best design,m is the number of objective functions to be minimized, nis the number of objective functions to be maximized ands = m + n is total number of objective functions. Finally,the point with the least value of norm is introduced as thebest design. It should be noted again that none of the Paretopoints dominates other points. The designer can select anyof the points as the best solution based on the importanceof each of the objective functions. The best design in thispaper was chosen on the basis of norm value which is a rel-ative index of the distance to the best possible design. Also,the norm index is used in this paper to compare the perfor-mance of optimization algorithms in finding the best design.In this paper, program codes to execute the aforementioned

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Table 5 Best configurations of parameters for three meta-heuristicalgorithms

NSGA-II

Population size 100

Generations 100

Crossover fraction 0.7

Migration fraction 0.4

SPEA2

Population size 100

Generations 100

Individual mutation probability 1

Individual recombination probability 1

Variable mutation probability 1

Variable swap probability 0.5

Variable recombination probability 1

MOPSO

Population size 100

Repository size 100

Maximum number of iterations 100

C1 1.49

C2 1.49

Inertia weight damping ratio 1

algorithms are developed in MATLAB environment. Each ofalgorithms were evaluated 50 times with different configura-tions of parameters and best configurations were determinedand listed in Table 5.

Comparison of multi-objective meta-heuristic algorithms

Figure 5 illustrates the Pareto front curves created by threemeta-heuristic algorithms. Comparison between algorithms

was done by means of the norm value of the optimal pointand hypervolume indicator of the Pareto front presented byeach algorithm. Hypervolume indicator measures the qualityof the Pareto front quantitatively by estimating the closenessof the estimated points to the true Pareto front. The hypervol-ume of a set of solutions measures the size of the portion ofobjective space that is dominated by those solutions collec-tively. Generally, hypervolume is favored because it capturesin a single scalar both the closeness of the solutions to theoptimal set and, to some extent, the spread of the solutionsacross objective space. In this paper, Monte-Carlo approachhas been used to calculate the percentage of a set of randompoints in the objective functions space to be dominated bythe Pareto front. the interested reader is encouraged to viewthe reference (While et al. 2006).

Table 6 presents the optimum design with its norm andhypervolume indicator provided by each algorithm. As itis obvious, Pareto front and optimal points achieved by allthree algorithms are near each other. Comparison among theresults indicates that the optimal design by SPEA2 algorithmhas the least norm and the highest hypervolume. Hence, thisalgorithm’s performance to find the global optimal point isbetter than the other two algorithms. Based on this result,twist angle, friction factor and ram speed in optimal condi-tion are 46.3, 0.26 and 2 mm/s, respectively. In this condition,imposed effective plastic strain, strain standard deviation andmaximum punch force are 1.19, 0.39 and 131.9 kN, respec-tively. Comparison between algorithms shows that NSGA-II and SPEA2 suggest an average strain less than MOPSOalgorithm, while they produce more homogenous strain dis-tribution with a smaller punch force. It should be noted thatthe best design and best optimization algorithm is selectedbased on Eq. (11) but this criterion can be altered accordingto designer’s strategy.

Fig. 5 Pareto front curves developed by meta-heuristic algorithms (NSGA-II, MOPSO and SPEA2)

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Table 6 Performance analysis and comparison between optimum design provided by each optimization algorithm

Algorithm Twist angle Friction factor Ram speed Effective plasticstrain

SD Max punchforce (kN)

Norm Hypervolumeindicator (%)

NSGA-II 46.3 0.26 2.19 1.18 0.39 128.1 1 54

SPEA2 46.3 0.27 2 1.19 0.39 131.9 0.9 55

MOPSO 47.9 0.32 2 1.27 0.43 155.42 1.41 50

Conclusion

In this paper, modeling of twist extrusion process was car-ried out using artificial neural network and response surfacemethod. Also, multi-objective optimization of twist extrusionprocess with a combination of artificial neural network andthree prominent meta-heuristic algorithms, namely NSGA-II, SPEA2 and MOPSO was investigated. Finally, with thehelp of response surface analysis, relations between designvariables and output functions and also importance of eachof deign parameters were analyzed. Most important resultsof this study include:

1. Artificial neural network model is more efficient thanresponse surface model in training and evaluation stagesof modeling of twist extrusion process.

2. Comparison between results of meta-heuristic algorithmsshows that SPEA2 algorithm exhibits relative superiorityover NSGA-II algorithm and MOPSO is the least effi-cient.

3. Optimal design values for die twist angle, friction factorand ram speed are 46.3◦, 0.27 and 2 mm/s respectively.These values may be used to manufacture twist extrusiondie, select the type of lubricant and adjust punch velocityfor twist extrusion of aluminum.

4. Effective plastic strain, strain homogeneity and max-imum punch force are directly and friction factor isinversely proportional to ram speed.

5. Twist angle and friction factor are most effective on aver-age effective strain and strain standard deviation respec-tively. Most effective parameters on maximum punchforce successively are friction factor and twist angle.Also, ram speed at room temperature is the least effectiveparameter among design variables.

6. It is foreseeable that at higher extrusion temperature andwith the change in billet material (to materials which aremore sensitive to strain rate), extrusion speed would bemore effective on mechanical properties of final extrudedmaterial.

The results obtained in this paper can be directly appliedto twist extrusion of pure aluminum at room temperature.Using the optimal parameters can yield high plastic strain(> 1) and also a homogenous structure. Generating proper

mechanical properties with only one TE pass can give rise totwist extrusion as a competitor for other extrusion method inindustry. One of the limitations of this method is the temper-ature of the process. Effects of temperature can be studiedin materials such as magnesium which need higher temper-atures for extrusion. Also, finding optimal parameters of theprocess, especially ram speed in the extrusion of materialsthat are more sensitive to strain rate is another interestingfuture research topic.

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