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The International Journal ofAdvanced Manufacturing Technology ISSN 0268-3768Volume 73Combined 1-4 Int J Adv Manuf Technol (2014)73:241-249DOI 10.1007/s00170-014-5788-5

Integration of artificial neural network andsimulated annealing algorithm to optimizedeep drawing process

Mohsen Manoochehri & Farhad Kolahan

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ORIGINAL ARTICLE

Integration of artificial neural network and simulated annealingalgorithm to optimize deep drawing process

Mohsen Manoochehri & Farhad Kolahan

Received: 5 December 2012 /Accepted: 14 March 2014 /Published online: 13 April 2014# Springer-Verlag London 2014

Abstract Deep drawing is characterized by very complicateddeformation affected by the process parameter values includ-ing die geometry, blank holder force, material properties, andfrictional conditions. The aim of this study is to model andoptimize the deep drawing process for stainless steel 304(SUS304). To achieve the purpose, die radius, punch radius,blank holder force, and frictional conditions are designated asinput parameters. Thinning, as one of the major failure modesin deep drawn parts, is considered as the process outputparameter. Based on the results of finite element (FE) analysis,an artificial neural network (ANN) has been developed, as apredictor, to relate important process parameters to processoutput characteristics. The proposed feed forward back prop-agation ANN is trained and tested with pairs of input/outputdata obtained from FE analysis. To verify the FE model, theresults obtained from the FEmodel were compared with thoseof several experimental tests. Afterward, the ANN is integrat-ed into a simulated annealing algorithm to optimize the pro-cess parameters. Optimization results indicate that byselecting the proper process parameter settings, uniform wallthickness with minimum thinning can be achieved.

Keywords Deep drawing . Finite element analysis .

Optimization . Artificial neural networks

NomenclatureBHF Blank holder forceFE Finite elementσy Yield stress

σUTS Ultimate stressFLDs Forming limit diagramsFLSDs Forming limit stress diagramsDOE Design of experimentANN Artificial neural networkRp Punch radiusRd Die radiusμ1 Friction coefficient between punch and blankμ2 Friction coefficient between die and blankN Number of possible designL Number of factorsSTHm Minimum section thicknessBPN Back propagation networkFEM Finite element methodMSE Mean square errorSA Simulated annealing algorithmPr Probability numberΔC Difference between the present and the new

objective function valueTk Temperature in k-th iteration

1 Introduction

Sheet metal forming is a widely used manufacturing processfor mass production due to its high speed and low cost. Deepdrawing is one of the most common sheet metal formingprocesses which is widely used for mass production of cup-shaped parts in automobile, petrochemical, and packagingindustries [1–4]. Cup drawing is also used as a basic test forsheet metal formability. Figure 1 shows deep drawing toolsand the states of forming parts. In this figure,Do,Dp, c, Rp, andT are the diameter of initial blank, diameter of punch, clear-ance between punch and die, punch radius, and thickness ofblank, respectively.

M. Manoochehri (*) : F. KolahanDepartment of Mechanical Engineering, Ferdowsi University ofMashhad, Mashhad, Irane-mail: [email protected]

F. Kolahane-mail: [email protected]

Int J Adv Manuf Technol (2014) 73:241–249DOI 10.1007/s00170-014-5788-5

Author's personal copy

The limits of sheet metal drawability are limited by theonset of process failures: wrinkling, earring, and fracture.Effective parameters on the different types of defects, whichmay appear in the formed parts, can be divided into three maincategories [5]: (1) material properties such as yield stress,work hardening coefficient, anisotropic coefficient, blank di-mensions, and thickness of blank; (2) tool properties such aspunch radius, die radius, and clearance; and (3) process pa-rameters such as blank holder force (BHF), friction coeffi-cient, type and position of lubricant, strain rate, and pressure.

In order to survive in the daily increasing competitivemarket, lighter, safer, and cheaper products are required. Sub-sequently, it is essential to choose the proper drawing param-eters which influence the drawing operation. In the past fewdecades, researchers have focused on the optimization of deepdrawing process.

Delamézière et al. [6] optimized the material properties ofthin sheets applied in deep drawing. They considered twomaterial parameters: the strain hardening exponent and theaverage anisotropic coefficient. They defined two objectivefunctions, namely, failure and wrinkling to control the formingdefects. Response surface, built on a limited number of eval-uations of the objective function, has been employed to opti-mally determine material properties. Browne and Hillery [2]investigated the variation and effects of punch and die geom-etry, blank holding pressure, top ram pressure, lubrication, anddrawing speed in the deep drawing of C.R.1 steel cups of0.9 mm thickness. A series of cups were deep drawn usingDOE, where a screening experiment was conducted and thedesired factors were varied at different levels. Based on theirfindings, top ram pressure, punch and die profile, lubrication,and position of lubrication are significant factors when

measuring the punch load induced. Blank holding pressure,speed, and top ram are the effective parameters on thicknessdistribution.

Ozek and Bal [7] investigated the effect of die/punch radiusand die/blank holder angle on drawability in the deep drawingprocess. They collected their data using experimental testswith altering variables in five levels. Their results showed thatthe limit drawing ratio increases with increasing radius ofpunch and die/blank holder angle.

Padmanabhan et al. [8] studied the significance of threeimportant process parameters, namely, die radius, blank hold-er force, and friction coefficient on the deep drawing charac-teristics of a stainless steel axisymmetric cup. They combinedthe finite element method with Taguchi technique to form apredictive tool to determine the influence of forming processparameters. Their results denote that die radius has the greatestinfluence on the thickness distribution, followed by the blankholder force and the friction coefficient. Further, they reportedthat a blank holder force application and local lubricationscheme improved the quality of the formed part. Agrawalet al. [9] performed an attempt to predict the minimum blankholding pressure required to avoid wrinkling in the flangeregion during axisymmetric deep drawing process.

Singh et al. [10] identified optimum values for die radius,punch radius, friction coefficients, and drawing ratios in deepdrawing of St-14 steel plates with initial uniform thickness of1 mm. Their data have been gathered from 28 experimental tests.Then, artificial neural network (ANN) and genetic algorithmhave been used to find the optimal levels of process parameters.Chamekh et al. [11] optimized initial blank shape in a deepdrawing process of AISI 304 stainless steel. They built ametamodel by combining the ANN and FEM. Then, the

a) b)

AfterBefore

Punch

Pressure plate

Blank holder

Die

Drawn cupBlank

Blank

Die(draw ring)

Blank holderforce

Blank holderT

F

Do

Dp

µ1

µ2

Rpc

Fig. 1 Deep drawing process. aDeep drawing tools. b States offorming part

242 Int J Adv Manuf Technol (2014) 73:241–249


metamodel is coupled to an optimization algorithm to determinethe initial blank shape for deep drawing of a rectangular cup.

For angular squared deep drawing process, Özek and Ünal[12] have studied the effects of die/blank holder angles, blankholder force, and die/punch diameter on the limit drawingratio, punch force, and minimum wall thickness of DIN EN10130-1999 steel sheets. They used a full factorial experimen-tal design for data collection. To model the process and todetermine the optimal conditions, they employed regressionanalysis, signal-to-noise ratio, and analysis of variance(ANOVA). Their results indicate that the deep drawing ratiolimits would increase at higher die/blank holder angles anddie/punch diameter and that more uniform cup surfaces wouldbe obtained with lower blank holder forces.

A rich body of knowledge in this area can be found inrelated literature. For instance, Wifi and Abdelmaguid [13]have presented a concise literature review on the optimizationtechniques used for the single-stage and multistage deepdrawing process and identified directions for future research.

In spite of the extensive research work on deep drawing inmost current papers, the optimization process is performed basedon statistical analysis, which would limit the process variablesettings to be chosen solely from a set of pre-specified values.

With regards to this item, the objective of this research is todevise an efficient modeling and optimization procedure fordeep drawing process using finite element (FE) modeling,ANN, and simulated annealing (SA) algorithm. The researchpresented here is twofold. First, an accurate FE model, whichcan substitute the real deep drawing process, has been

developed. Then, to further reduce computational times,an ANN model has been proposed based on the simula-tion results of D-Optimal matrix. In the second stage, asimulated annealing optimization method has been imple-mented to optimally determine deep drawing process pa-rameters. The proposed modeling and optimization proce-dure would treat the process parameters as continuousvariables, and hence they can take any values within theirfeasible ranges.

The important process parameters considered here are dieand punch radiuses, BHF, and frictional conditions (frictioncoefficient between punch-–blank, die–blank, and blank hold-er–blank). Thinning, as a major failure mode or defect en-countered in deep drawn parts, has been considered as theprocess output characteristic. In drawing process, thinning canbe defined as the maximum decrease in the blank thicknessafter drawing compare to its initial value. Figure 2 shows thesteps of the proposed procedure tomodel and to optimize deepdrawing process.

2 Material and equipment

In this study, stainless steel 304 (AISI 304) with 0.5 mmthickness has been used for simulation and experimentalefforts. Table 1 shows the chemical composition of this mate-rial. Mechanical properties and anisotropic coefficients weredetermined using the results of uniaxial tensile tests. Rolledmaterials, such as sheet metals, may exhibit anisotropic

ModelingSimulation Optimization

Optimization of deep drawingprocess with combined ANN and

simulated annealing algorithm

Verification of optimization results with FE model and

experimental test

Fig. 2 Flowchart of proposed procedure to modeling and optimization of deep drawing process

Table 1 Chemical compositionof SUS 304 Element Carbon Manganese Phosphorus Sulfur Silicon Chromium Nickel Nitrogen

Percentage 0.08 2 0.045 0.03 0.75 18–20 8–10.5 0.10

Int J Adv Manuf Technol (2014) 73:241–249 243


mechanical properties, that is, their properties may vary indifferent directions. To obtain anisotropic coefficients andmechanical properties, tensile tests were performed on threespecimens which were cut from the blank in three differentdirections (0°, 45°, and 90°). Each specimen was preparedaccording to ASTM B0557 M-02 Standard. The tests werethen performed using a Zwick-Z250 tensile test machine. Themechanical properties and the anisotropic coefficient obtainedfrom the uniaxial tests are reported in Table 2. These materialproperties have been used in the simulation of deep drawingprocess with the FE method.

Drawing ratio with respect to punch radius (65 mm) andblank diameter (140 mm) is equal to 2.15. A set die has beendesigned and manufactured to perform the experiments. Theset die was mounted on a 60-ton hydraulic press to form parts(Fig. 3). Blank holder force was supplied by eight standardsprings with 134 N/mm stiffness. Friction coefficient betweenthe die–blank and punch–blank was obtained by several ex-periments using a force gauge of 0.01 N resolution.

3 Finite element simulation

In this study, ABAQUS/EXPLICIT software V 6.10 was usedto develop a 3D FE model of the deep drawing process for

SUS 304 alloy as shown in Fig. 4. Tools (die, punch, andblank holder) were modeled as rigid bodies using R3D4elements, and the sheet is modeled as a formable body usingC3D8R. A surface-to-surface contact model was adopted inwhich the coulomb friction law has been considered. Threecontact pairs, namely, punch–blank, die–blank, and blankholder–blank, were created. Material behavior obtained fromthe tensile testing experiments and the Hill’s anisotropy pa-rameters [14] were incorporated into the FE model to accountfor the specimen orientation and sheet anisotropy. A constantuniform pressure was applied on the blank holder in Y-direction as the BHF, while the freedom of the blank holderin Y-direction was unfastened.

3.1 Verification of FE model

To validate the developed FE model, the final shape of thedrawn parts and the predicted punch loads versus punchstrokes were compared with the experimental results underthe same process conditions. The comparisons are presentedin Fig. 5. With respect to the force–displacement curves, themaximum error of the punch load for the case with wrinklingis less than 9 %. The difference for the parts with no suchdefect is much less. This is an indication that the FE model iscapable of modeling the process under various circumstances

Table 2 Mechanical propertiesand anisotropic coefficient ofSUS 304

Mechanical properties Anisotropic coefficient

σy (MPa) σUTS (MPa) Young's modulus (GPa) Angular offset from rolling direction

0° 45° 90°

265 573 191.7 0.913 1.120 1.424

Fig. 3 Hydraulic press and die set used for experiments Fig. 4 FE model for the deep drawing of cup-shaped parts



and the simulated formed shapes are consistent with those ofexperiments.

3.2 Fracture detection in FEM model

In general, fracture can be predicted by: (1) strain-basedcriteria, such as forming limit diagrams (FLDs) and maximumpart thinning; (2) stress-based criteria, using forming limitstress diagrams (FLSDs); and (3) ductile damage criteria,including the Cockroft and Latham criterion [15]. Amongthese, thinning in the part wall is commonly used in industry

to indicate probability of fracture. Usually, a maximum thin-ning of 25 % thickness is used as the critical fracture criterion

(a)

(b)

Fig. 5 Simulation and experimental comparisons a with wrinkling and b without wrinkling

Table 3 Variable parameters with their levels

Levels BHF (KN) Rp (mm) Rd (mm) μ1 μ2

1 5 3 3 0.01 0.01

2 11 5 5 0.1 0.1

3 16 8 8 0.2 0.2

4 22 12 12 – –



[16]. Therefore, in the present study, we also have selectedwall thinning as the process quality index as well. The max-imum allowed thinning of drawn parts for 0.5-mm thicknessof blank would be 0.125 mm, which has been used as theacceptance criterion in the analysis.

4 Design of experiment and ANN

One of the main objectives in this research is to predict theproduct quality under various sets of process parameters.According to the results, the FE model could be used for thispurpose. However, it is a time-consuming process as each runof the FE model may take several hours. Therefore, as a moreefficient prediction technique, ANNwas employed to estimatethe process response (thinning) for any given set of processparameters. To gather the required data, a set of simulationruns was performed based on design of experiment (DOE)approach.

Blank holder force (BHF), punch radius (Rp), die radius(Rd), friction coefficient between punch–blank (μ1), die–blank(μ2), and holder–blank were considered as input parameters.The ranges of these parameters and their levels are given inTable 3. As shown, the first three parameters are taken to havefour levels, while the last two can vary in three levels. Basedon the previous researches [17–19], punch and die radius areconsidered in the range 3–12 mm. All simulation experimentsare carried out for the depth of 30 mm.

For a full factorial design, the number of possible designs(N) is N=Lm, where L is the number of levels for each factorand m is the number of factors. Thus, a full factorial for input

parameters in this study would consist of 576 experiments. Tofurther reduce the number of simulation runs, D-Optimaldesign of experiments (DOE) has been implemented. In thisway, only 81 runs are needed to generate the required data forANN development. Table 4 illustrates part of the designmatrix and measured outputs. In this table, the last columnshows the minimum section thickness under each set of inputparameters.

4.1 The proposed artificial neural network

ANNs are widely accepted as a tool to simulate complex andill-defined problems. They are particularly useful in processmodeling and have been used in diverse applications such ascontrol, robotics, pattern recognition, forecasting, manufactur-ing optimization, etc. [20]. Due to the inherent complexity ofdeep drawing process, the relationships between part qualityindexes and process parameters could not be established byany analytical model. On the other hand, the characteristics of

Table 4 Design matrix and out-put values

aMinimum section thickness(mm)b Maximum thinning exceedsfrom 0.125 mm and hence thedrawn part is failed

Number of experiment Inputs Output

BHF (kN) Rp (mm) Rd (mm) Mu1 Mu2 STHma

1 5 3 3 0.01 0.01 0.391

2 22 8 3 0.1 0.1 0.374b

3 16 12 5 0.1 0.1 0.419

4 22 5 8 0.01 0.01 0.432

5 16 8 12 0.1 0.1 0.444

6 16 12 5 0.01 0.01 0.424

– – – – – – –

– – – – – – –

– – – – – – –

75 11 5 12 0.1 0.01 0.454

76 16 3 3 0.2 0.1 0.374b

77 22 12 12 0.1 0.1 0.449

78 5 8 8 0.2 0.2 0.442

79 5 5 8 0.1 0.1 0.431

80 11 5 8 0.01 0.1 0.399

81 22 5 12 0.2 0.1 0.448

BHF

Rp

Rd

Mu1

Mu2

Input Layer Hidden Layers Output Layer

Minimum STH

Fig. 6 Schematic illustration of the proposed ANN



the ANN techniquemake it suitable for predicting the thinningin deep drawn parts; therefore, it is employed in this study.

The back propagation network (BPN) is a typical ANN thathas been widely used in many research fields. BPNs havehierarchical feed forward network architecture, and the out-puts of each layer are sent directly to each neuron in the nextlayer. This kind of ANN is trained by repeatedly presenting aseries of input/output pattern sets to the network. The neuralnetwork gradually “learns” the governing relationship in thedata set by adjusting the weights between its neurons tominimize the error between the actual and predicted outputsof the training set. A separate set of data called the test set isusually used to evaluate the network’s performance. Trainingis considered completed when the mean squared error (MSE)of the test set reaches a pre-determined value. The MSEformula is:

MSE ¼ 1

n

Xi¼n

i¼1

t−t0ð Þ2 ð1Þ

where n is the number of experiments in the test set, t is thepredicted thickness, and t0 is the actual thickness.

The number of neurons for the input layer is equal to thenumber of process variables. Likewise, the number of neuronsat the output layer is specified by the number of measuredoutput responses. Therefore, for the problem under consider-ation, input and output layers have five and one neuron,respectively. The number of middle (hidden) layers and theircorresponding neurons are usually defined through trial runs.Based on the results of several test runs, a four-layer ANN(two hidden layers) with one neuron in the first hidden layerand three neurons in the second hidden layer was found to bethe best architecture for the problem under study. The networkwas trained using resilient back propagation algorithm. Otherinitial parameters of training algorithm, such as training timeand termination criterion, were set according to the defaults ofMATLAB software. A schematic illustration of the proposedANN architecture is presented in Fig. 6.

To train and to test the ANN, 81 sets of input/outputsamples were randomly divided into two data sets. The 60set points were assigned to the training set and the remaining21 set points were used as the testing set. The MSE betweenthe ANN and FEM models was found to be less than 3 %.

Therefore, it can be concluded that the proposed ANN iscapable to predict the output with high accuracy.

After being trained, the ANN can map the complex andnon-linear relation between quality index and input variablesof deep drawing process, and hence there is no need to use FEmodel for this purpose anymore. The developed ANN predic-tor may now be used in the optimization process using simu-lated annealing algorithm.

5 Optimization of deep drawing process parameters

The aim of this section is to present an optimization procedurein order to find the process parameter values in such a way thatthe minimum wall thicknesses of drawn parts are maximized.Simulated annealing (SA) algorithm has been employed todetermine the best values of process variables with respect tominimum wall thickness. One of the reasons for selecting SAis because it has fewer parameters compared to other heuris-tics, such as genetic algorithm (GA), which makes it easier forcomputer programming.

Simulated annealing is a powerful stochastic technique forobtaining optimum or near-to-optimum solutions to combina-torial and function optimization problems. This algorithm,first proposed by Kirkpatrick et al., is inspired by the thermo-dynamic process of cooling (annealing) a molten metal toattain the lowest free energy state [21]. A standard SA proce-dure begins by generating an initial solution at random. Ateach stage, a small random change is made to the current

Table 5 Optimum parameters

Variables BHF (kN) Rp (mm) Rd (mm) Mu1 Mu2 STHm predictedwith ANN

STHm obtainedfrom FEM

STHm obtainedfrom experiment

Optimum value 8 12 12 0.2 0.01 0.466 0.477 0.43

0 15 30 45 60 75 90 105 1200.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

Number of Iteration

Max

imum

Wal

l Thi

ckne

ss

Fig. 7 Simulated annealing performance



solution. Then, the objective function value of the new solu-tion is calculated and compared with that of the current solu-tion. A move is then made to the new solution if it has a bettervalue. A non-improving solution is also accepted with the

certain probability which is given by Pr ¼ exp −ΔcTk

� �. The

acceptance probability of non-improving solutions decreasesas the difference in the costs (Δc) increases and as the tem-perature of the method decreases. As the number of iterations(k) increases, the temperature (Tk) is gradually decreased froma relatively high value to near zero [22].

In order to find the optimum values of the deep drawingparameters, the proposed ANN has been used as the fitnessfunction in SA algorithm. In optimization process, punch anddie radius vary from 3 to 12 mm with an increment of 1 mm;BHF is increased from 5 KN with an increment of 1 KN toreach the maximum of 22 KN, and friction coefficients varyaccording to Table 5.

The algorithm was run for 120 iterations (about 2.0 min).As shown in Fig. 7, the algorithm was converged to the finalsolution after 75 iterations, and no further improvement wasobserved thereafter. The optimization results are summarizedin Table 5.

Nevertheless, we cannot claim that the results obtained bythe proposed SA are global optimum. Computational resultsmerely demonstrate that SA is quite capable of obtaining high-quality solutions (optimal or near-optimal) within reasonablecomputational times.

To validate the optimization procedure, experimental testswere performed based on the optimum values of processparameters. In addition, FE model was run using the sameinput values. The last three columns of Table 5 show that theSTHm values were obtained by ANN, FEM, and the experi-ments, respectively. The error between ANN prediction andFE simulation is less than 2.5 %, while ANN prediction andexperimental test show about 8 % error. Figure 8 displaydrawn parts in simulation and experimental test. It can beconcluded that the optimization procedure proposed in thisstudy is capable of achieving the optimal values of the variableparameters with high accuracy.

The proposed procedure for optimization of deep drawingprocess is efficient and reliable. The optimum set of parameter

settings results in a wall thickness better than all 81 othercombinations used in data collection.

6 Conclusion

In this study, a combined FE–ANN–SA procedure was pro-posed to model and to optimize the deep drawing process ofSUS304. First, deep drawing process was simulated inABAQUS software and verified against experimental data.In FE modeling, some of the most important process param-eters of deep drawing have been simultaneously considered.This made it possible to develop a model as close to actualprocess as possible. Next, based on D-Optimal design matrix,a total of 81 simulation runs were performed to gather therequired data needed for ANN modeling. Verification testsreveal that the developed ANN is in good agreement with bothFEM model and experimental results. Hence, in our futureinvestigations, the FE model and ANN predictor could accu-rately replace the actual deep drawing process. This wouldeliminate the need for additional time-consuming and costlyexperiments. Finally, to determine the optimum values ofprocess parameters, the proposed ANNwas used as the fitnessfunction in a simulated annealing algorithm.

Optimization results show that in order to minimize thin-ning, BHF should be set at 8 KN, and die and punch radiusmust be at their highest possible levels of 12 mm. Computa-tional results also illustrate that the frictional conditions be-tween blank and die components should not be the same. Thisis in line with the study previously done in this area [2]. Usingoptimal parameters, an average improvement of about 14% inminimum wall thickness can be achieved. The findings of thisstudy show that the combination of ANN, as the processpredictor, and SA, as an optimization technique, is quietefficient in modeling and optimization of deep drawingprocess.

The approach presented in this work was applied to macro-scale deep drawing process. In the future, the same approachmay be used to investigate deep drawing process in micro-scales. It is certainly worthwhile to carry out comparativestudies by employing other modeling techniques, such as

Fig. 8 Formed parts insimulation and experimental test



regression analysis or response surface method, to model deepdrawing process. Another interesting area of research, whichcalls for further investigation, is the possible implementationof more efficient optimization techniques, such as hybrid GA/SA, for deep drawing optimization. In this case, the resultsmay be compared with those reported in this work.

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