Modeling and Simulated Annealing Optimization of Surface Roughness in CO2 Laser Nitrogen Cutting of Stainless Steel

Abstract

This paper presents a systematic methodology for empirical modeling and optimization of surface roughness in CO2 laser nitrogen cutting of stainless steel. The surface roughness prediction model was developed in terms of laser power, cutting speed, assist gas pressure and focus position by using the artificial neural network (ANN). To cover a wider range of laser cutting parameters and obtain an experimental database for the ANN model development, Taguchi's L27 orthogonal array was implemented in the experimental plan. The developed ANN model was expressed as an explicit nonlinear function, while the influence of laser cutting parameters and their interactions on surface roughness were analyzed by generating 2D and 3D plots. The final goal of the experimental study focuses on the determination of the optimal laser cutting parameters for the minimization of surface roughness. Since the solution space of the developed ANN model is complex, and the possibility of many local solutions is great, simulated annealing (SA) was selected as a method for the optimization of surface roughness.

Full text

167
 Vol.35,No.3(2013)167‐176

TribologyinIndustry

www.tribology.fink.rs



ModelingandSimulatedAnnealingOptimizationof
SurfaceRoughnessinCO2LaserNitrogenCuttingof
StainlessSteel


M.Madića,M.Radovanovića,B.Nedićb

aUniversityofNiš,FacultyofMechanicalEngineering,Serbia.
bUniversityofKragujevac,FacultyofEngineering,Serbia.

Keywords:
SurfaceRoughness
CO2LaserNitrogenCutting
A
rtificialNeuralNetworks
SimulatedAnnealing
Optimization

ABSTRACT
Thispaperpresentsasystematicmethodologyforempiricalmodelingand
optimizationofsurfaceroughnessinCO2lasernitrogencuttingofstainless
s
teel.Thesurfaceroughnesspredictionmodelwasdevelopedintermso
f

laserpower,cuttingspeed,assistgaspressureandfocuspositionbyusing
theartificialneuralnetwork(ANN).Tocoverawiderrangeoflasercutting
parametersandobtainanexperimentaldatabasefortheANNmodel
development,Taguchi’sL27orthogonalarraywasimplementedinthe
experimentalplan.ThedevelopedANNmodelwasexpressedasanexplicit
nonlinearfunction,whiletheinfluenceoflasercuttingparametersandtheir
interactionsonsurfaceroughnesswereanalyzedbygenerating2Dand3D
plots.Thefinalgoaloftheexperimentalstudyfocusesonthedetermination
oftheoptimallasercuttingparametersfortheminimizationofsurface
roughness.SincethesolutionspaceofthedevelopedANNmodeliscomplex,
andthepossibilityofmanylocalsolutionsisgreat,simulatedannealing(SA)
wasselectedasamethodfortheoptimizationofsurfaceroughness.
©2013PublishedbyFacultyofEngineering
Correspondingauthor:
M.Madić
UniversityofNiš,
FacultyofMechanicalEngineering,
Serbia
E‐mail:madic@masfak.ni.ac.rs


1. INTRODUCTION

Lasercuttingisoneofthemostusednon‐
conventionalmachiningprocesses for straight
and contour cutting of sheet stock. By
directing the focused laser beam onto the
workpiece surface it comes to rapid heating
which results, depending on the workpiece
characteristics and beamintensity,inmelting
orevaporationofworkpiecematerial.The
molten material is then removed using a
coaxialjetofanassistgas.
Lasercuttingtechnologyrequiresrelativelyhigh
capital cost of equipment, however, low
operational costs justifies its use for both large
batch processing and processing of customized
products. The other main advantages over the
competing machining processes include better
productivity, higher quality, applicability for
bothverysoftandthinmaterialsaswellas
difficulttocutmaterials.

Laser cutting is a complex, multifactor
machining process. The principal factors that
RESEARCH

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
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affectthecuttingprocessinclude[1]:beam
powerandcharacteristics,cuttingspeed,typeof
assist gas and flow and focus position. The
effectsoftheseparametersonthelasercutting
performanceshavebeenwidelystudied[2,3].As
reported in many experimental studies,
depending on materials characteristics,
workpiece thickness as well as varying interval
of process factors, the main process factors
differentlyaffecttheprocessperformances.This
makes prediction of process performance
characteristicsandidentificationofnearoptimal
factorsquitedifficult[4].

Inthispapermathematicalmodelforsurface
roughness prediction in CO2lasernitrogen
cuttingofstainlesssteelwasdeveloped.Detailed
reviewedaboutsurfaceroughnessinlaser
cuttingisavailableinliterature[5].Asseenfrom
previous studies, the mechanism of surface
roughnessformation in laser cuttingiscomplex,
requiring modeling of multiple non‐linearities
which justifies the use of artificial neural
networks (ANNs). The back propagation (BP)
ANN trained with gradient descent with
momentumalgorithmwasappliedtoconstructa
mathematicalmodelwhereinthesurface
roughness was expressed as an explicit
nonlinear function of the four laser cutting
parameters. For conducting the laser cutting
experiment,Taguchi’sL27orthogonalarray(OA)
wasusedwherethelasercuttingparameters,
namelythelaserpower,cuttingspeed,assistgas
pressure, and focus position, were arranged.
Statisticallyassessedasadequate,theANN
model was then used to study the effect of the
laser cutting parameters on surface roughness.
Furthermore,inordertodeterminetheoptimal
lasercuttingparametersforachievingminimum
surface roughness, the ANN model was
integratedwithSA.


2. EXPERIMENTALPROCEDURE

2.1.Experimentaldetails

Thelaser cutting experimentwas performedby
means of ByVention 3015 (Bystronic) CO2laser
cutting machine delivering a maximum output
powerof2.2kWatawavelengthof10.6µm,
operating in the continuous wave mode. The
cuts were performed with a Gaussian
distributionbeammode (TEM00) on 3mm thick
AISI 304 stainless steel. In consideration of the
numerousparameters that influencethecutting
process and final cut quality, i.e. surface
roughness,someoftheprocessparameterswere
kept constant throughout the experimentation.
Ontheotherhand,thelasercuttingparameters
such as laser power (P),cuttingspeed(v), assist
gaspressure(p)andfocusposition(f)weretaken
as controllable input parameters. The laser
cuttingconditionsaresummarizedinTable1.

Table1.Laser‐cuttingconditions.
Constantparameters:
Workpiecematerial AISI304stainlesssteel
Materialthickness,mm 3
Laser CO2
Operatingmode continuouswave
Max.power,kW 2.2
Lensfocallength,mm 127
Nozzle conicalshape,=2mm
Standoffdistance,mm 1
Typeofassistgas N2
Controllableparameters:
 Level1 Level2 Level3
A:Laserpower‐P,kW 1.6 1.8 2
B:Cuttingspeed‐v,m/min 2 2.5 3
C:Assistgaspressure‐p,bar 9 10.5 12
D:Focusposition‐f,mm 2.5 1.5 0.5

The value range for each of the laser cutting
parameter was chosen such that wider
experimental range was covered, a full cut for
each parameter combination was achieved, and
byconsidering manufacturer's recommendation
forparametersettings.Twostraightcuts,eachof
60 mm in length, were made in each
experimental trial to ascertain surface finish.
Surface roughness on the cut edge was
measuredintermsoftheaveragesurface
roughness(Ra) using SurftestSJ‐301 (Mitutoyo)
profilometer. Each measurement was taken
alongthecutatapproximatelythemiddleofthe
thicknessandthe measurementswererepeated
threetimestoobtainaveragedvalues.

2.2.Experimentalplan

Taguchi experimental design provides an
efficient plan to study the entire experimental
regionof interest for theexperimenter,withthe
minimumnumberofexperimenttrials,therefore
itwaschosentoperformthelasercutting
experiment. To this aim, Taguchi’s L27

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
169
orthogonalarraywith 4inputparametersand3
levelswasused.Table2showsthe27conducted
trialswiththecombinationofthelasercutting
parametersandtheexperimentalresults.


3. SURFACEROUGHNESSANNMODEL

Theobjectiveofthesurfaceroughnessmodeling
is to quantify the relationships that exist
between process parameters and surface
roughness, so as to be able to identify the near
optimal laser cutting conditions in which the
required surface roughness will be obtained.
MATLABsoftwarewasusedforthedevelopment
oftheANNmodelfortheaveragesurface
roughness (Ra)intermsoffourlasercutting
parameters,thatis,laserpower(P), cutting
speed (v), assist gas pressure (p),andfocus
position(f). Allexperimental datawere usedto
generate an experimental database for the ANN
modeldevelopment,i.e.ANNtraining.

The ANN architecture consisted of four input
neurons,eachtorepresentP, v, pandf,and one
outputneuronfor estimating Ra.The number of
hiddenneuronswasselectedbyconsideringthat
the total number of weights and biases in the
ANNdoesnotexceedthenumberofdatafor
training. Considering the total number of
weightsandbiasesintheANNmodel,itiseasy
tocalculate that for fourinputsand one output,
theupperlimitofthenumberofhiddenneurons
is4 for 27 available trainingdata. Therefore,4‐
4‐1 ANN architecture was selected for surface
roughnessmodeling.

Table2.ExperimentallayoutusinganL27orthogonalarrayandexperimentalresults.
Exp.
trial
NaturalfactorCodedfactorExperimental
results
Pvpf
A B C D Ra
(kW) (m/min) (bar) (mm) (µm)
1 1.6 2 9 2.5 1 1 1 1 1.84
2 1.6 2 10.5 1.5 1 1 2 2 1.98
3 1.6 2 12 0.5 1 1 3 3 2.17
4 1.6 2.5 9 1.5 1 2 1 2 2.34
5 1.6 2.5 10.5 0.5 1 2 2 3 2.08
6 1.6 2.5 12 2.5 1 2 3 1 1.67
7 1.6 3 9 0.5 1 3 1 3 2.20
8 1.6 3 10.5 2.5 1 3 2 1 1.83
9 1.6 3 12 1.5 1 3 3 2 2.30
10 1.8 2 9 1.5 2 1 1 2 1.71
11 1.8 2 10.5 0.5 2 1 2 3 1.96
12 1.8 2 12 2.5 2 1 3 1 2.20
13 1.8 2.5 9 0.5 2 2 1 3 1.70
14 1.8 2.5 10.5 2.5 2 2 2 1 1.77
15 1.8 2.5 12 1.5 2 2 3 2 1.69
16 1.8 3 9 2.5 2 3 1 1 2.09
17 1.8 3 10.5 1.5 2 3 2 2 2.15
18 1.8 3 12 0.5 2 3 3 3 1.91
19 2 2 9 0.5 3 1 1 3 1.89
20 2 2 10.5 2.5 3 1 2 1 3.02
21 2 2 12 1.5 3 1 3 2 1.83
22 2 2.5 9 2.5 3 2 1 1 2.294
23 2 2.5 10.5 1.5 3 2 2 2 1.47
24 2 2.5 12 0.5 3 2 3 3 2.16
25 2 3 9 1.5 3 3 1 2 1.60
26 2 3 10.5 0.5 3 3 2 3 2.21
27 2 3 12 2.5 3 3 3 1 1.93


M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
170
The hyperbolic tangent sigmoid transfer function
was used in the hidden layer, and linear transfer
functionwasusedintheoutputlayer.PriortoANN
training, the initial values of weights and biases
weresetaccordingtoNguyen‐Widrowmethod.In
ordertostabilizeandenhanceANNtraining,the
inputandoutputdatawasnormalizedinthe[1,
1]rangeusingthefollowingequation:



12 



minmax
mini
norm pp
pp
p. (1)
wherepnormandpirepresentthenormalizedand
original (raw) data, and pminandpmax are the
minimum and maximum values of the original
data.TotraintheANN,thegradientdescentwith
momentumalgorithmwasused.TheANN
training process performance was followed
accordingtothemeansquarederror(MSE)[6]:




 N
i
ii dy
N
MSE
1
2
1. (2)
where Nisthenumberofdata;diisthe
experimentalvalue(target);andyiisthe
predictedvalueofANNforthetrainingsamplei.
ItwasfoundthattheselectedANNarchitecture
provided the best data fitting capability when
learning rate (α) and momentum(µ) werekept
at 0.1 and 0.9, respectively. The MSE achieved
during the training was 0.0131. Regarding the
architecture of the developed ANN, the used
transferfunctionsinhiddenandoutputlayer,and
byusingtheweightsandbiasesfromtrainedANN,
the mathematical model for surface roughness in
termsofthelasercuttingparameterscanbe
expressedbythefollowingequation:


kkj
bwX
norm|a bw
e
Rjji 







 1
1
2
2. (3)
where Xis the column vector which contains
normalized values of P, v, p, and f, and norma
R|is
thenormalizedvalueoftheRa.Inordertoobtain
the actual values for Ra, one needs to perform
thedenormalizationbythefollowingequation:




minminmaxnorm|aactual|a pppRR  1
2
1. (4)
Inordertocheckthereliabilityofthedeveloped
ANNmodel, the predictionaccuracyof theANN
model was tested. Initially, the ANN model for
surface roughness was tested by presenting 27
input data patterns, which were employed for
thetrainingpurpose.UsingEqs.3and4the
predictedandexperimentallymeasuredaverage
surfaceroughnessvaluesarecomparedinFig.1.

Inaddition,theabsolutepercentageerrorswere
foundtobe max= 11.02 %, min=0.06%,aver=
3.37%.Inordertotestthegeneralizationability
(i.e. model robustness) of the developed ANN
model,3newexperimenttrialswereconducted,
withthelasercuttingparameterlevelswhich
didnotbelongtothetrainingdataset(Table3).

Table3. Experiment trials for testing the ANN
predictionmodel.
P
(kW)
v
(m/min)
p
(bar)
f
(mm)
Exp.
measuredRa
(µm)
ANN
predicted
Ra(µm)
1.8 2.5 12 2.5 2.068 1.847
2 2.5 10.5 0.5 1.733 1.760
1.8 3 10.5 0.5 1.879 2.093


Fig.1.ComparisonofANNpredictedandexperimentallymeasuredaveragesurfaceroughness.

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
171
TheresultsfromFig.1and Table 3 suggestthat
theANNpredictionsareingoodagreementwith
experimental values for Rawithinthescopeof
cutting conditions investigated in the study.
Thus,theANNmodelcanbeusedtoanalyzethe
effectsofthelasercuttingparametersonsurface
roughness.Furthermore, the ANNmodelcan be
used in conjunction with the SA algorithm for
theoptimizationpurpose.


4. EFFECTOFTHELASERCUTTING
PARAMETERSONSURFACEROUGHNESS

4.1.Maineffects–2Dplots

Initially,the effectof the laser cutting parameters
on surface roughness was analyzed by changing
one parameter at a time, while keeping all of the
otherparameters constantatlow,centerandhigh
level(Fig.2).AsshowninFig.2a,theeffectofthe
laser power on surface roughness is variable and
dependable on the values of other parameters.
Whenallotherparametersarekeptatlowlevel,an
increaseinthelaserpowerincreasessurface
roughness. However, an increase in the laser
powerdecreasessurfaceroughnesswhenallother
parameters are kept at high level. The figure
shows no significant change in surface roughness
withthelaserpower,whenallotherparameters
arekeptatcenterlevel.Fig.2asuggestthatthebest
surfacefinishcanbeobtainedusingthelaserpower
of2kW,however,theeffectofthisparameteristo
beconsideredthroughtheinteractions.

Fig.2bshowsthatanincreaseinthecuttingspeed
resultsinnonlinear increase in surfaceroughness
and this functional dependence is constant, apart
fromthevaluesofotherparameters.Theeffectof
thecuttingspeedcanbeexplainedbythefactthat
asthecuttingspeedincreases,theinteractiontime
between the laser beam and material decreases,
i.e. the heat generation decreases, which leads to
minimumsideburning.

FromFig.2citcanbeseenthat,inrespecttoother
parametervalues,theassistgaspressurebetween
9.75 bar and 11.25 bar negatively affects the
surfacefinish.Adecreaseintheassistgaspressure
shows a good decrease in surface roughness. The
pressure that is too high expels the melt more
efficiently,andhasapositiveeffectonsurface
quality,particularlyforimpendingburrformation,
i.e.rathercreateshighgasconsumption.
(a)

(b)

(c)

(d)
Fig.2. Effect of the laser cutting parameters on
surfaceroughness(······· other parameters at level 1;
——— other parameters at level 2; ‐‐‐‐ other
parametersatlevel3).

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
172
Inthecaseofthefocusposition,Fug.2dsuggests
that focusing the laser beam deep into the bulk
of the material is beneficial for achieving good
surface finish. The functional dependence
between the focus position and surface
roughness is nonlinear and follows the same
trendapartfromthevaluesofotherparameters.

The results from Fig. 2 indicate that the
mechanismbehindsurfaceroughnessformation
iscomplexandfurthercomplicatedhavingin
mindthattheinteractionsbetweenthelaser
cuttingparametershaveahugeimpacton
surface roughness. Thus, it is necessary to
investigate the interaction effects of the laser
cuttingparametersonsurfaceroughness.

4.2.Interactioneffects–3Dplots

In order to determine the interaction effects of
the laser cutting parameters on surface
roughness, 3D surface plots were generated
consideringtwoparametersatatime,whilethe
third and fourth parameter were kept constant
at center level. Since there were six possible
two‐way interactions, six plots were generated
(Fig.3)usingEqs.3and4.

Fig.3a showssurfaceroughnessasafunctionof
thelaserpowerandcuttingspeed.Itcanbeseen
thataparallelincreaseinthelaserpowerand
cutting speed linearly increases surface
roughness. High cut quality can be obtained
usinghighlaserpowerandcuttingspeedinthe
2.25‐2.5m/minrange.

FromFig.3bitcanbeseenthatwhenusinglow
assist gas pressure, increasing the laser power
improvessurfacefinish,andviceversa.Using
the laser power of up to 1.8 kW with the
combinationoftheassistgaspressureofupto
11barproducesroughsurfacefinish.

Fig.3cshowsthatwhen thefocuspositionis set
to 2.5mm,theeffectofthelaserpoweron
surface roughness is negligible. When the focus
position is shifted in positive direction (moves
towards workpiece surface), an increase in the
laser power has a positive effect on surface
finish. Using the laser power of up to 1.9 kW,
while focusing the laser beam at the half of the
materialthickness,resultsinhighsurface
roughness.

Inthecaseofinteractionbetweentheassistgas
pressureandcuttingspeed,Fig.3dsuggeststhat
usingtheassistgaspressureofupto10.5bar
allowsthecuttingspeedofupto2.25m/minfor
goodsurfacefinish.

From Fig. 3e it can be seen that the interaction
effect of the focus position and cutting speed
produces highly nonlinear change in surface
roughness.Usingthelowcuttingspeedofupto
2.25m/minwhilefocusingthelaserbeam
approximately at the half of the material
thickness, is beneficial for obtaining low
roughnessvalues.

Finally,Fig.3fshowsthatalowfocuspositionin
conjunction with low assist gas pressure is
beneficialforsurfacefinish.Ontheotherhand,
focusingthelaserbeamnearthetopsurface,and
increasing the assist gas pressure, results in
surfaceroughnessincrease.

The results from Fig. 3 indicate that there are
highly nonlinear interactions between the laser
cuttingparametersand surface roughness.Note
thattheplotsinFig.3weregeneratedby
keepingthe two parametersconstant.However,
findinganoptimalsetoflasercuttingparameter
values to meet the desired surface roughness
call for the parameter optimization in a four‐
dimensionallasercuttingparameterhyperspace.

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
173
(a) (b)
(c) (d)
(e) (f)
Fig.3.Interactioneffectofthelasercuttingparametersonsurfaceroughness.


5. OPTIMIZATIONMETHODOLOGY

Since the solution space of the developed ANN
model is complex, and the possibility of many
local solutions is great, SA was selected as the
methodforsurfaceroughnessoptimization.The
SA optimization procedure was done using the
MATLAB Optimization Toolbox on the basis of

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
174
the code of developed ANN model written in
MATLAB. The details about the SA algorithm,
optimization problem formulation and results
arediscussedbellow.

5.1.Simulatedannealing(SA)

InitiallypresentedbyKirkpatricketal.[7],SAis
arandomsearchtechniqueforglobal
optimization problems able to escape local
optima.ThesalientfeaturesofSAareitsgeneral
applicabilityandabilitytoavoidlocaloptima[8].

The concept of simulated annealing mimics the
metals recrystallization in the process of
annealing.Annealingistheslowcoolingofmetal
thatproducesgoodlowenergystate
crystallization, whereas fast cooling produces
poor crystallization. At high temperatures, the
movementoftheatomsinmoltenmetalsisfree.
Withtemperaturedecreasing,themovementof
the atoms becomes limited, the atoms tend to
getorderedand,finally,formcrystalshavingthe
minimumpossibleenergywhichdependsonthe
cooling rate. Slow cooling produces good low
energystatecrystallization,whereasfastcooling
produces poor crystallization (high energy
polycrystalline state). If the temperature is
reducedataveryfastrate,thesystemmay
achieve the high energy polycrystalline state
insteadofthelowenergycrystallinestate.

SAusesasinglepointsearchmethod.Itisa
memoryless search algorithm in the sense that
noinformation is savedfrom previoussearches
[9]. The SA algorithm starts with a random
initial design vector (solution) Xiandahigh
temperatureT.Aseconddesign point is created
atrandominthevicinityoftheinitialpointand
thedifferenceinthe function values ∆Eatthese
twopointsiscalculatedas[10]:


iiii XfXffffE   11


. (5)
Iftheobjectivefunctionvalueofthenew
solution is smaller, the new solution is
automaticallyacceptedandbecomesthecurrent
solution from which the search continues.
Otherwise,thepointisacceptedwitha
probability e(−∆E/kT) where k is the Boltzmann’s
constant.ThiscompletesoneiterationofSA.Due
totheprobabilisticacceptanceofanon
improving solution, SA can escape from local
optima.AtacertaintemperatureT,a
predeterminednumberofnewpointsaretested.
The algorithm is terminated when the current
value of temperature is small enough or when
changes in function values (

f) are sufficiently
small.FurtherdetailsofSAcanbefound
elsewhere[9,10].

5.2.Definitionoftheobjectivefunctionand
constraints

The goal of the optimization process in this
study is to determine the optimal laser cutting
parameter values that contribute to the
minimum value of average surface roughness
(Ra).Toformulatetheoptimizationproblem,the
ANNmodelwhichisproposedinEq.4istaken
to be the fitness function of the optimization
solutionandisformulatedasfollows:






mm 5052
bar 129
m/min 3 2
kW 2 61 :subject to
function :minimize to
:Find
a
.f. -
p
v
P.
f,p,v,PR
, f,p,vP optoptoptopt





. (6)

5.3.Optimizationresults

Forsolvingtheoptimizationproblemformulatedin
Eq.6,thecomputercodewaswritteninMATLABto
integrate the ANN based process models and SA.
The SA algorithm was implemented with the
followingparameters(Table4):
 Annealing function is selected as the
Boltzmannannealingwhich takes random
steps, with size proportional to square
rootoftemperature.
 Reannealing interval is the number of
points to accept before reannealing.
Defaultvalueof100wasused.
 Exponential temperature update which
decreasesas0.95iterationwasused.
 Initial temperature of 100 C was set at
thebeginningoftheoptimization.

Table4.SAparametersused.
Startpoint [1111]
Initialtemperature,T 100
Annealingfunction Boltzmannannealing
Temperatureupdate
function
Exponential
temperature
Reannealinginterval 100

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
175
(a) (b)
Fig.4.Optimizationresults.

NotethatsincetheANNfunctionwasdeveloped
usingthenormalizedvaluesofthelasercutting
parametersinthe[1,1]range,theinitialpoints
for the SA solution in Table 4 are also given as
normalizedvalues.

Theoptimizationsolution resultsof the MATLAB
optimization toolbox are given in Fig. 4a. It is
indicated that the near optimal solution was
foundatthe3646‐thiteration.Thecombination
ofthelasercuttingparametersettingsleadto
minimumRavalueof1.082µmwiththefollowing
values:laserpowerP=2 kW, cutting speed v=2
m/min, assist gas pressure p=11.06barand
focuspositionf=0.739mm.Optimizationofthe
lasercutting parameters wastestedbyusingthe
Monte‐Carlo method, and identical results were
obtained. The optimization results can be
confirmedfromFig.4b.

Apart from the obtained optimization results,
thenearoptimallasercuttingparametervalues
for obtaining minimal Ra were determined
considering the following constraints: (a)
maximalcuttingspeedwasused,and(b)
minimalassistgaspressurewasused.Theabove
optimization formulations are of practical
importance since they assure maximal
productivityandminimalcosts,respectively.

The solution of the optimization problem
formulated in Eq. 6, with the constraint v= 3
m/min,isobtainedas:minimalRa= 1.232 µm
forP=2kW,p=9barandf=2.5mm.Ithasto
be noticed that the obtained solution is in the
same time the optimal one for the optimization
problem when the constraint is p=9bar.In
other words, the obtained solution, which is
actuallyaboundarypointinthehyperspaceof
the laser cutting parameters, simultaneously
satisfies both goals, i.e. maximal productivity
andminimalcosts.


6. CONCLUSION

Inthispaper,empiricalmodelingand
optimization of surface roughness in CO2laser
nitrogencuttingofstainlesssteelwaspresented.
The applied methodology integrates surface
roughness modeling usingtheartificialneural
network (ANN), and single‐objective
optimization of laser cutting parameters using
the simulated annealing (SA) algorithm. To
obtain an experimental database for the ANN
model development, Taguchi’s L27 orthogonal
arraywasimplementedintheexperimentalplan
in which four laser cutting parameters (laser
power,cuttingspeed,assistgaspressureand
focus position) were arranged at three levels.
The mathematical model of the surface
roughness developed byusingtheANNwas
expressed as an explicit nonlinear function of
the selected input parameters. The statistical
results indicated good agreement between the
predicted and experimental values so that the
ANNmodelwasusedforanalyzingtheeffectof
the laser cutting parameters and their
interactions on surface roughness. From the
analysis of the effect of the laser cutting
parameters on surface roughness the following
conclusionscanbedrawn:

M.Madićetal.,TribologyinIndustryVol.35,No.3(2013)167‐176
176
 Surfaceroughness is highlysensitiveto the
selectedlasercuttingparameters,
 Thefunctionaldependencebetweensurface
roughnessandthelasercuttingparameters
ishighlynonlinear,
 The effect of a given parameter on surface
roughness must be considered through the
interactionwithotherparameters.

Inaddition to modeling,optimization of surface
roughness based on the integrated ANN‐SA
approach was conducted. Based on the
optimizationresults,highlaserpower(2kW),
lowcuttingspeed(2m/min),mediumassistgas
pressure(11.06bar)andfocusposition(0.739
mm)yieldedtheminimumsurfaceroughness
(1.082µm).However,usinghigh laser power (2
kW),highcuttingspeed(3m/min),lowassist
gaspressure(9bar)andfocusposition(2.5
mm)aminimalsurfaceroughnessof1.232µm
wasobtainedandthisturnedouttobebeneficial
forbothproductivityandcosts.

FindingsinthispaperindicatethattheANN‐SA
approach can be efficiently used for
mathematical modeling and optimization of the
CO2lasercuttingprocess.


Acknowledgement

ThisworkwascarriedoutwithintheprojectTR
35034 financially supported by the Ministry of
Science and Technological Development of the
RepublicofSerbia.




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