Design of pressure vessels using shape optimization

15 Pages·2012·1.36 MB·English

InternationalJournalofPressureVesselsandPiping88(2011)198e212
ContentslistsavailableatScienceDirect
International Journal of Pressure Vessels and Piping
journal homepage: www.elsevier.com/locate/ijpvp
Design of pressure vessels using shape optimization: An integrated approach
R.C. Carbonaria, P.A. Muñoz-Rojasb, E.Q. Andradec, G.H. Paulinod,e, K. Nishimotof, E.C.N. Silvaa,*
aDepartmentofMechatronicEngineering,EscolaPolitécnicadaUniversidadedeSãoPaulo,Av.Prof.MelloMoraes,2231SãoPaulo,SP05508-900,Brazil
bDepartmentofMechanicalEngineering,UniversidadedoEstadodeSantaCatarina,BomRetiro,Joinville,SC89223-100,Brazil
cCENPES,PDP/MétodosCientíficos,Petrobras,Brazil
dNewmarkLaboratory,DepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisatUrbana-Champaign,205NorthMathewsAv.,Urbana,IL61801,USA
eDepartmentofMechanicalScienceandEngineering,UniversityofIllinoisatUrbana-Champaign,158MechanicalEngineeringBuilding,1206WestGreenStreet,Urbana,
IL61801-2906,USA
fDepartmentofNavalArchitectureandOceanEngineering,EscolaPolitécnicadaUniversidadedeSãoPaulo,Av.Prof.MelloMoraes,2231SãoPaulo,SP05508-900,Brazil
a r t i c l e i n f o a b s t r a c t
Articlehistory: Previouspapersrelatedtotheoptimizationofpressurevesselshaveconsideredtheoptimizationofthe
Received7September2010 nozzleindependentlyfromthedishedend.Thisapproachgeneratesproblemssuchasthicknessvariation
Receivedinrevisedform fromnozzletodishedend(couplingcylindricalregion)and,asaconsequence,itreducestheoptimality
10May2011 ofthefinalresultwhichmayalsobeinfluencedbytheboundaryconditions.Thus,thisworkdiscusses
Accepted11May2011
shapeoptimizationofaxisymmetricpressurevesselsconsideringanintegratedapproachinwhichthe
entirepressurevesselmodelisusedinconjunctionwithamulti-objectivefunctionthataimstominimize
Keywords:
the von-Mises mechanical stress from nozzle to head. Representative examples are examined and
Shapeoptimization
solutionsobtainedfortheentirevesselconsideringtemperatureandpressureloading.Itisnoteworthy
Pressurevessel
Multi-objectivefunction thatdifferentshapesfromtheusualonesareobtained.Eventhoughsuchdifferentshapesmaynotbe
profitable considering present manufacturing processes, they may be competitive for future
manufacturingtechnologies,andcontributetoabetterunderstandingoftheactualinfluenceofshapein
thebehaviorofpressurevessels.
(cid:1)2011ElsevierLtd.Allrightsreserved.
1. Introduction astrength/weightratiohigherthansteeltanks,theyhaveahigher
manufacturing cost. Thus, this work emphasizes homogeneous
Inthepressurevesselliterature,theoptimizationofnozzleand tanks,andfocusesonCNG(CompressedNaturalGas)tankdesign
headshasbeenconductedindependently.Althoughthisisaprac- bymeansofshapeoptimizationtechniques.
tical and widely used approach, it leads to undesirable problems This paper is organized as follows. The literature review and
such as thickness variation from nozzle to head (among others) related work are presented in Section 2. The formulation and
and, as a consequence, reduces the optimality of the final result numerical implementation of the optimization problem for the
(which may also be influenced by the adopted boundary condi- pressure vessel is described in Sections 3 and 4, respectively. In
tions). Thus, this work investigates the optimization of pressure Section5,theoptimizedshapesfortheentirepressurevesselare
vesselsconsideringamodeloftheentirevessel.Amulti-objective presentedandtheresultsarediscussed.Finally,inSection6,some
function basedon ap-rootof summationofp-exponenttermsof conclusionsareinferred.
von-Mises stress values is defined in order to minimize the tank
maximum von-Mises stresses. Mechanical and thermal loads are 2. Relatedwork
considered. Shape optimization techniques are applied, and the
design optimizationprocedure is implemented bycombining the The design of pressure vessels is an important and practical
commercial finiteelement analysis systemANSYSwith ourMAT- topic which has been explored for decades. Even though optimi-
LAB optimization algorithm. Although composite tanks have zation techniques have been extensivelyapplied to design struc-
turesingeneral,fewpiecesofworkcanbefoundwhicharedirectly
relatedtooptimalpressurevesseldesign.Thesefewreferencesare
mainly related to the design optimization of homogeneous and
* Correspondingauthor.Tel.:þ551130919754;fax:þ551130915722.
compositepressurevessels.
E-mailaddresses:[email protected](R.C.Carbonari),[email protected](P.A.
A pioneering work on optimization techniques for designing
Muñoz-Rojas), [email protected] (E.Q. Andrade), [email protected]
(G.H.Paulino),[email protected](K.Nishimoto),[email protected](E.C.N.Silva). pressurevesselswaspresentedbyMiddletownandOwen[1],who
0308-0161/$eseefrontmatter(cid:1)2011ElsevierLtd.Allrightsreserved.
doi:10.1016/j.ijpvp.2011.05.005
R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212 199
y spline Sh } y midsurface
y y
1 2 head top
y bottom
nh variables
integration
points
}
x1 nozzle nodes
x
2 variables
x
x
nnspline S
n
x Fig.3. Detailsoftheshellelement(employedinthiswork).
Fig.1. Definitionofdesignvariables.
dimensions of a torispherical shell; optimization of the elliptical
profileofthetorisphericalshellwithuniformthickness;thesame
used parametric optimization techniques to minimize the maxi- problem,butincludingthethicknessasadesignvariable;andagain
mum shear stress in the design of a pressure vessel dished end the same problem, however, considering the thickness varying
(head)modeledwithaxisymmetricfiniteelements.Followingthis alongthemeridiandefinedbyapolynomial.Theweightreduction
work, Middletown [2] applied these parametric optimization obtainedwas8%,19%,27%,and31%,respectively.Asanimportant
techniques to design a pressure vessel nozzle considering the conclusion,theyshowedthattheboundaryconditionsbetweenthe
minimization of the maximum shear stress. Mechanical and dishedendandthecylinderdoinfluencethedesign.A“FullyStressed
thermal loads were taken into account by specifying different Design”-(FSD),whichkeepsthestressconstantalongthebody,was
temperaturesintheinternalandexternalwallsofthenozzleand alsoperformed,however,itproducedinferiorresultsinrelationto
dishedend(head).Later,Blachut[3]alsoappliedparametricopti- the best result among the four cases discussed, showing that, as
mizationtechniquestominimizetheweightofthedishedendby expected,thedesignconsideringstressconstraintsisnotasintui-
considering the limit pressure instead of mechanical stress. The tiveandthusamoreformalapproachisneeded.
limitpressurewasdefinedasthevaluethatfirstcausesafullplastic Anotherrelevantworkconsideringpressurevesseldesignusing
regionalongtheentirethicknessofthepressurevessel.Theopti- computationalmodelingwaspresentedbyZhuandBoyle[4],who
mization problem was defined as the minimization with appliedshapeoptimizationtechniquestooptimizeseparatelythe
constraintsofthelimitpressure,thestrainatthepeakofthedished head and nozzle of the pressure vessel, considering as objective
end, and box constraints for parametric design variables. A zero functioneitherthemechanicalstressesorthelimitpressure.The
ordermethod(nogradients)wasappliedtosolvetheoptimization limit pressure is calculated by using the elastic compensation
problem. Four situations were studied: optimization of the main method [5e7], and performing successive linear analyses, which
has as main advantage the fact that it precludes the need to
Initialization and
Data Input y
Creation of Spline Curve
and
Solution of FEM
Calculation of Objective Function and Constraints
Y y P, Ti Te
Converged? END etr
m
m
N y
s
Calculate Sensitivities and Moving Limits
Solve LP problem with
respect to x and y
i i
x
Update x and y
i i
Fig.4. Computer-aidedEngineering(CAE)modelincludingpressure(P),temperature
Fig.2. Flowchartofoptimizationprocedure. (Ti,Te),andboundaryconditions.
200 R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212
y Table2
Initialvaluesfordesignvariables(mm).
symmetry point
initialshape1 initialshape2 initialshape3 ellipticalhead
h y1(0;3066) y1(0;3066) y1(0;3066) y1(0;2066)
1 y2(500;2932) y2(382.7;2989.9) y2(258.8;3.31.9) y2(223;2053)
y3(866;2566) y3(707.1;2773.1) y3(500;2932) y3(434;2016)
x1(900;893) y4(923.9;2448.7) y4(707.1;2773.1) y4(624;1957)
x2(800;719) x1(900;893) y5(866;2560) y5(782;1877)
x3(700;546) x2(800;719) y6(965.9,2324.8) y6(901;1783)
x4(600;373) x3(700;546) x1(900;893) y7(975;1677)
d0 h x4(600;373) x2(800;719) x1(900;893)
2 x3(700;546) x2(800;719)
x4(600;373) x3(700;546)
x4(600;373)
thethicknessisuniformfortheproposedgeometry.Anequivalent
ellipsoidalheadwouldhaveanequivalentstressvalueinthehead
h
3 20% larger than the equivalent stress in the cylinder and a head
thickness50%largerthanthecylinderthickness.Theminimization
d
1 h of stress concentration in pressure vessels with ellipsoidal heads
4
x wasalsodiscussedbyMagnuckietal.[11]bymeansofanalytical
techniques.Asnoticedintheirpreviouswork,ifthesamethickness
isconsideredforheadandcylinder,thereisastressconcentration
Fig.5. Initialpressurevesselshape.
inthejunctionofheadandcylinderwithheadstressesbeinglarger
than cylinder stresses. A stress equalization can be obtained by
performnon-linearanalysis,thusreducingthecomputationalcost increasingtheheadthicknessorbychangingtheheadgeometryb/
in the overall optimization procedure. Their optimization proce- a > 0.5, where b and a are the minor and major ellipse axes,
dure was implemented using the software ANSYS with the APDL respectively.Inthecaseofthicknessvariation,anincreaseof70%
(ANSYSParametricDesignLanguage)languageandsplinecurvesto from cylinder to head would be necessary, which is impractical
perform shape optimization. The obtained results showed satis- fromamanufacturingpointofview.Aninterestingplotofthickness
factory improvements. The work of Malinowski and Magnuki [8] ratio as a function of (b/a) ratio related to the non-dimensional
employed parametric discrete optimization techniques to design equivalent stress equalization between head and cylinder could
internal reinforcements of pressure vessels by minimizing the bebuilt.Fromthisplot,byconsideringthesamethicknessforhead
reinforcement mass considering stress constraints. Following an andcylinder,theratio(b/a)mustbeequalto0.6forstressequal-
analyticalapproach,Banichuketal.[9]usedvariationalcalculusto ization. In another plot, the ratio between the maximum non-
findtheheadmeridianprofileoptimalcurveinordertomaximize dimensional equivalent stress and the cylinder equivalent stress
the ratio between head volume and mass subjected to stress (subjectedtopressureonly)isgivenasafunctionoftheratio(b/a).
constraints. Itshowsthat(b/a)valueslessthan0.48arenotofinterestbecause
Aninterestingissuewhendesigningthehead(ornozzle)isthe the stress concentration would increase. For (b/a) values ranging
stressconcentrationinthetransitionbetweenthehead(ornozzle) from0.48to0.86,highstressvaluesaregenerated(althoughequal
and the cylinder. At the head and cylinder junction, bending forheadandcylinder).Thus,othercriteria,suchasminimummass,
moments and shear forces appear in order to compensate the mustbeselectedtodecideona(b/a)valuebetween0.48and0.86.
stiffnessdifferencebetweenthecomponents(headandcylinder). Fromthesepreviouspiecesofwork,weconcludethatheadswith
These bending moments and shear forces generate a stress hemispherical,torisphericalorellipticalgeometriesarenotoptimal
concentration that can be estimated by thin shell theory. When becausetheycanonlyavoidthestressconcentrationinthejunction
a non-hemispherical head is considered, the stressconcentration between head and cylinder with largethickness variation, which
along the junction increases. Such stress concentration can be may be impractical (e.g. from a manufacturing point of view).
reducedeitherbytransitioningthethicknessfromtheheadtothe Meanwhile,itisalsoconcludedthattheminimizationofthestress
cylinder or by changing the head geometry. Following this idea, concentrationinthejunctioncanbeachievedbychangingthehead
Magnucki and Lewinsky [10] analytically assessed the head meridian profile shape, suggesting that shape optimization tech-
geometry (shape) design that generates an equivalent stress less niquescanbeappliedtosearchforthisoptimumshape.Thesame
than or equal to the cylinder stress. In this case, the internal study could be developed for the junction between nozzle and
bendingmomentsandshearforceswouldbenull.Theyconsidered cylinder, however, no analytical work has been found, probably,
a head geometry partially composed by a sphere and a generic duetodifficultiestodeveloptheanalyticalmodel.However,similar
curve (to be determined). The sphere radius is calculated to conclusions may be applied, that is, by changing the nozzle
generate equivalent stresses in the head that do not exceed the meridian profile shape the stress concentration in the transition
equivalentstressatthecylindersurface.Thestressisuniformand canbereduced.
Table3
Table1 Representativeoptimizationparameters.
Materialproperties.
p 1to8
Young’sModulus E 200.0(cid:2)109 Pa w 0.5
Poisson’sRatio n 0.3 ah 0.25
Density r 7810.0 Kg/m3 an 0.5
Thermalexpansion a 1.15(cid:2)10(cid:3)5 1/◦C fh¼fn¼0.01
R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212 201
250
B C x
number of elements:
200 140 A D E
a) 280 y
P 700
M
( 150 1400
s
s
e
r
st
s 100
e
s
Mi
on 50
v
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
Fig.6. von-Misesstressvaluesalongvesselmeridionalprofile,consideringdifferentdiscretizationsandnothermalload.
Although most of previous optimization literature assumed optimization approach, the head and nozzle meridian profile
specific geometries, Hammerand Olhoff [12] used topologyopti- curves will be optimized by using spline curves, thus no pre-
mizationtodesignstructuressubjectedtopressureloads.Forsuch defined shape will be assumed. Shape here refers to the
design-dependent problem, the boundary where the pressure is geometricshapeofthemidsurfacewithauniformthickness.The
applied must move as the material is removed. This problem is shapeofthemidwallsurfaceismoreimportantthanthicknessfor
addressed by using a spline function to represent the boundary either maximum stress or load-carrying capacity [4]. A typical
where the pressure is applied. A follow up of this work was optimizationproblemisdefinedas
providedbyDuandOlhoff[13]andZhengetal.[14],whoextended
the investigation to three-dimensional structures. However, they Min FðxÞ
considered as objective function the minimization of mean gðxÞ(cid:4)0
Subjectedto
compliance without stress constraints, which is not realistic for x (cid:4)x(cid:4)x
L U
pressurevesseldesign.Infact,thesolutionoftopologyoptimiza-
tionconsideringstressconstraintsisstillanopenproblem[15e17]. whereF(x)istheobjectivefunction,xarethedesignvariables,g(x)
Recently,BlachutandMagnuki[18]publishedanextensivereview are inequalityconstraints, and xL and xU are boxconstraints. The
pressure vessel optimized design can be achieved by direct
about modeling and optimization of pressure vessels where they
mentionthat,intheoptimizationfield,therearestillfewpiecesof enforcementofmaximumstressasconstraintsor,alternatively,by
definingastressmeasuretobeminimizedasobjectivefunction.In
workrelatedtopressurevesseldesign.Thisworkprovidesacontri-
this work, we deal with the stress minimization problem by
bution along those lines by presenting an integrated approach in
definingamulti-objectivefunctionbasedonap-rootofsummation
whichtheentirevesselisconsideredintheoptimizationprocess.
ofp-exponenttermsofvon-Misesstresses.Thesestressesarethe
usual choice of stress measurement for pressure vessel design in
3. Formulationoftheshapeoptimizationproblem theliterature[10,11].Thevon-Misesstressesarechoseninsteadof
limitpressure[5e7]becausetheintentionistodesignthevesselfor
In this section, an approach to design CNG tanks based on actual working conditions [4]. If different levels of mechanical
the shape optimization technique is presented. In this shape stresses occur in nozzle and head regions in this work, then an
600
500
)
a
P
M
( 400
s x
s B C
e
r
st 300
s E
e number of elements: A D
s
Mi y
200 140
n
o 280
v
A 700 B C D E
100
1400
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
Fig.7. von-Misesstressvaluesalongvesselmeridionalprofile,consideringdifferentdiscretizationsandthermalload.
202 R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212
analysisoftheentirevesselisperformed.Apropermathematical 3.1. Designvariables
setting must be formulated in the sense that the optimization
process should guaranteethat nozzle and head stressvalues will Intheadoptedshapeoptimizationapproachtheshapesofhead
haveequalinfluenceintheobjectivefunction.Thisisachievedby andnozzlearegivenbycubicsplineinterpolationwhoseshapeis
using a logarithmic function. Thus, a multi-objective function is changed during the optimization process by varying the coordi-
definedasthesumofthelogarithmicofthep-rootofsummationof nates of spline knots. Separate spline curves are defined for the
p-exponenttermsofheadandnozzlevon-Misesstresses,allowing head and for the nozzle. The optimization must be performed in
the control of the head and nozzle stresses through a weighting a way that the spline knots move in the normal direction to the
coefficient,whichitisgivenby: spline curve, because analytical shape sensitivity in the direction
tangenttothesplinecanbeshowntobenull.Inprinciple,xandy
(cid:1)X (cid:3)1 (cid:1)X (cid:3)1 coordinates of spline knots could be chosen as design variables,
F ¼ wln nh sp pþð1(cid:3)wÞln nn sp p (1) however,consideringtheconstraintthatknotsmustmoveinthe
i¼1 hj j¼1 nj normaldirection,itsufficestochooseonlyxorycoordinates.Fig.1
wpohinertesosfhithanediesntjharaendthjeevothn-fiMniisteeselmemecehnatnsicthalatstmreosdseesltahteGhaeuasds svharoiwabsletshafotrxnoaznzdleyankdnohteacdoosrpdliinnaetse,sreasrpeeccthivoesleyn.Tahsusth,tehedehseigand
and nozzle, respectively; w is the weighting coefficient which designvariablesetisdefinedbythecoordinatesoftheheadspline
allowscontroloftheinfluenceofstressesintheheadandnozzle knots (yh) in the Cartesian direction y, while the nozzle design
regionsintheoptimizationproblem;pisanexponentcoefficient variablesetisdefinedbythecoordinatesofthenozzlesplineknots
whichcanassumeevenoroddvalues(asvon-Misesstressvalues (xn)intheCartesianxdirection,asillustratedinFig.1.
arealwayspositive);nhandnnarethenumberoffiniteelements
used to discretize the head and nozzle regions, respectively. 3.2. Complexityconstraints
Therefore, the minimization of the multi-objective function
consists of minimizing simultaneously the von-Mises mechanical In order to reduce the numerical oscillations during the opti-
stressesintheheadandnozzleregions. mization process, and to control the spline shape, complexity
a b
3000 initial 3000 initial
optimized optimized
m)2500 m) 2500
m m
n y (2000 n y ( 2000
n i1500 n i 1500
o o
si si
n n
e1000 e 1000
m m
Di Di
500 500
0 0
0 500 1000 0 500 1000
Dimension in x (mm) Dimension in x (mm)
head spline knots=3 head spline knots=4
c
3000 initial
optimized
m) 2500
m
(
y 2000
n
n i 1500
o
si
n
e 1000
m
Di
500
0
0 500 1000
Dimension in x (mm)
head spline knots=6
Fig.8. Optimizationresultobtainedforinitialshapes1,2,and3correspondingtodifferentnumberofheadsplineknots(nothermalload).Detailedviewsofheadandnozzleare
givenbesideeachfigure.
R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212 203
a
350
optimized B C x
initial
300
a) semi-spherical head
P D E
M 250 analytical (8 cylinder) y A
(
s
s
re 200
st
s
se 150
Mi
n 100
o
v
50
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=3
b
350
optimized B C x
initial
300
a) semi-spherical head
P D E
M 250 analytical (8 cylinder) y A
(
s
s
re 200
st
s
se 150
Mi
n 100
o
v
50
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=4
c
350
optimized B C x
) 300 initial
Pa semi-spherical head
(M 250 analytical (8 cylinder) y A D E
s
s
e
r 200
st
s
se 150
Mi
n 100
o
v
50
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=6
Fig.9. von-Misesstressvaluesalongvesseloptimizedmeridionalprofileforinitialshapes1,2,and3correspondingtodifferentnumberofheadsplineknots(nothermalload).
204 R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212
TMaabxleim4umvon-Misesstressvalues(MPa),wheresubscriptsh,n,0,ss,andh2indicate SMuibnje:ctedtoF:ðyh;xnÞ
(hseaad¼,no8z6z:l6e,Minpiatiafolrshaanpien,fisntaintedcayrdlinsdpehre)r.ical,andcylindricalregionh2,respectively y1(cid:4)ymax x1(cid:4)xmax
h2 (cid:3)y þy (cid:4)(cid:3)e (cid:3)x þx (cid:4)(cid:3)d
Example sh sh0 sss sn sn0 sh2 sh20 (cid:3)y1iþyiþ21(cid:4)(cid:3)e (cid:3)1xjþx2jþ1(cid:4)(cid:3)d
Fig.9(a) 80.4 101.3 67.0 142.9 318.6 89.0 91.5 (cid:3)y (cid:4)(cid:3)y ði ¼ 2.nh(cid:3)1Þ
nh min
FFiigg..99((bc)) 7735..96 9729..94 6677..00 110033..34 331188..66 8899..00 9911..55 (cid:3)xnn(cid:4)(cid:3)xmin ðj ¼ 2.nn(cid:3)1Þ
Fig.12(a) 303.5 292.0 261.6 276.6 452.7 276.5 278.0
Fig.12(b) 302.1 286.5 261.6 276.6 452.7 276.5 278.0 Thenumericalimplementationofthisdefinedshapeoptimiza-
Fig.12(c) 256.6 276.2 261.6 276.6 452.7 276.5 278.0 tionproblemisdescribedinthenextsection.Mechanical(pressure)
Fig.9(a,bandc) thermalload and thermal loads (temperature difference) are considered. In
Fig.12(a,bandc) nothermalload
addition, in the case of vessels subjected to internal pressure,
bucklinginstabilitymayoccurforheadandnozzle(mainlyhead)if
hoopstressesreachnegativevalues[19].Theheadbucklingismore
constraintsareintroducedintheoptimizationproblem.Thus,the
susceptibletooccurforvesselswithlargediameter/thicknessratio
complexityconstraintsfortheheadregionaredefinedas:
andhighyieldstressvalues[20].Becauseduringtheoptimization
processtheheadandnozzlegeometryischanged,thenageometry
y1(cid:4)ymax susceptibletobucklingproblemsmaybeobtained.Inthiswork,no
y1(cid:3)y2(cid:5)e (2) constraint is specified for the hoop stress in the optimization
yi(cid:3)yiþ1(cid:5)e problem,itsvalueisonlyverified“aposteriori”inthefinalresult.
y (cid:5)y ði ¼ 2.nh(cid:3)1Þ
nh min
4. Numericalimplementationofshapeoptimizationprogram
whereyiareshowninFig.1andnhisthenumberofknotsinthe
headregion;ymaxandyminarethemaximumandminimumvalues TheflowchartoftheprogramisillustratedinFig.2.Theprogram
allowedforthesecoordinates,respectively;andeisasmallnumber. isimplementedusingthecommercial softwareMATLAB[21]and
Forthenozzle,thecomplexityconstraintsaredefinedas: ANSYS[22].Allprocesses,includingtheoptimizationprocessand
fileinputandoutputfromANSYSarecontrolledbyMATLAB.ANSYS
x1(cid:4)xmax input files contain all the information of the pressure vessel
x (cid:3)x (cid:5)d computer-aidedengineering(CAE)model.Afterobtainingtheinput
1 2 (3)
xj(cid:3)xjþ1(cid:5)d files, the MATLAB routine executes ANSYS, and an output file is
xnn(cid:5)xmin ði ¼ 2.nn(cid:3)1Þ generated,whichisinturnreadbyMATLAB.
The pressurevessel ismodeled byusingthe2D-axisymmetric
wherexiareshowninFig.1andnnisthenumberofknotsinthe shellfiniteelementtypeSHELL208[22].Thedefaultoptionforthis
nozzle region; xmax and xmin are the maximum and minimum elementemploystwonodes(lineardisplacementinterpolationof
valuesallowedforthesecoordinates,respectively;anddisasmall themidsurface)andthreeintegrationpointsthroughthethickness
number. (atthebottom,atthetopandatthemidsurface),asillustratedby
A material volume constraint is not specified because the Fig. 3. This element has the capability of modeling bending and
changeofsplinelengthisnotsignificantduringtheoptimization. membranestiffness,andtorsion.Itgives,asaresult, thelongitu-
Althoughmaterialvolumeconstraintisnotincluded,itisverified“a dinal, meridional, and von-Mises stress values at the Gauss inte-
posteriori”. grationpoints.Thevon-Misesstressconsideredandpresentedfor
allresultsinthisworkarethemaximumvaluesamongtheGauss
3.3. Summaryandremarks integrationpointsthroughthethickness.
Fig. 4 illustrates the mechanical and thermal boundarycondi-
Basedontheaboveconsiderations,shapeoptimizationproblem tionsappliedtothemodel.Inthisfigure,TiandTeareinternaland
canbedefinedas: externaltemperature,respectively,andPistheinternalpressure.
10
9
p = 1
n
o p = 5
cti 8
n
u
f
e
v 7
cti
e
obj 6
-
ulti
M 5
4
0 50 100 150 200 250 300
Iterations
Fig.10. Convergenceofmulti-objectivefunctionforinitialshape1,correspondingtoheadsplineknotnumberequalto3(nothermalload).
R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212 205
FollowingtheflowchartinFig.2,theinitialdataoftheoptimi- containtheresultsofvon-Misesmechanicalstresses(shandsn)at
zation problem is the geometry of the pressure vessel, which eachfiniteelementintegrationpoint.
contains the spline knot coordinates (or design variable initial Thesensitivityanalysisofthemulti-objectivefunctioninrela-
values), material properties, load and boundary conditions. The tion to design variables is obtained by using the central finite
spline curves are obtained using the MATLAB function spline, differencescheme.Thus,forthegradientnumericalcalculationsit
considering the design variables yh and xn and discretization is necessary to obtain the perturbations dh and dn defined in the
parametersnhandnnforheadandnozzle,respectively.Theused expressionsbelow,
functionsplineconsidersacubicsplinedatainterpolation,whichis
generatedusingthefollowingparameters: vFyFðyiþdhÞ(cid:3)Fðyi(cid:3)dhÞ (6)
vy 2d
i h
S ¼ fðx ;y Þ ¼ ½splineðh;x;1:nhÞ;splineðh;y;1:nhÞ(cid:6) (4)
h h h i i
h (cid:4) (cid:5) (cid:1) (cid:3)i vFyFðxiþdnÞ(cid:3)Fðxi(cid:3)dnÞ (7)
Sn¼fðxn;ynÞ¼ spline n;xj;1:nn ;spline n;yj;1:nn (5) vyi 2dn
Noticethatyandxcoordinatesaredefinedasdesignvariablesfor
wcohoerrdeinSahteasnfdorStnhearheemadatarnicdefsorthtahtecnoonztzalien,rtehsepeccutbivicelsyp.lTihneehkenaodt head and nozzle, respectively. Moreover, dh and dn are calculated
splineShisobtainedconsideringthetangentatthesymmetrypoint consideringafractionofthesplinelengthLhorLn,whichisgiven
by:
equal to zero (see Fig. 5). As mentioned before, xh and yn are
calculated as functions of yh and xn, respectively, considering the Hnh S (cid:1)x;y(cid:3)ds
constraintthatknotsmustmoveinthenormaldirection. L ¼ j¼1 hP j j 0d ¼ f L (8)
Inthesequence,theCAEmodelisdefinedincludingalgorithmic h y h h h
i
parameters and mesh attributes such as the finite element type,
number of nodes and elements, loads and boundary conditions. H
OrunncneintgheaCliAnEearmsotdateilciasnbaluyisltis,,twhehicAhNSgYenSesroaltveesrouistpeuxtecfiuletesdthbayt Ln ¼ in¼n1SPnðxxij;yiÞds0dn ¼ fnLn (9)
a b
3000 initial 3000 initial
optimized optimized
)2500 )2500
m m
m m
y (2000 y (2000
n n
n i1500 n i1500
o o
si si
n n
e1000 e1000
m m
Di Di
500 500
0 0
0 500 1000 0 500 1000
Dimension in x (mm) Dimension in x (mm)
head spline knots=3 head spline knots=4
c
3000 initial
optimized
)2500
m
m
y (2000
n
n i1500
o
si
n
e1000
m
Di
500
0
0 500 1000
Dimension in x (mm)
head spline knots=6
Fig.11. Optimizationresultobtainedforinitialshapes1,2,and3correspondingtodifferentnumberofheadsplineknotandwiththermalload.Detailedviewsofheadandnozzle
aregivenbesideeachfigure.
206 R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212
500
optimized B C x
initial
a) 400 semi-spherical head D E
P A
M y
(
ss 300
e
r
st
s
e
s 200
Mi
n
o
v
100
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=3
500
optimized B C x
initial
a) 400 semi-spherical head D E
P A
M y
(
ss 300
e
r
st
s
e
s 200
Mi
n
o
v
100
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=4
500
optimized B C x
initial
a) 400 semi-spherical head D E
P A
M y
(
ss 300
e
r
st
s
e
s 200
Mi
n
o
v
100
A B C D E
0
0 500 1000 1500 2000 2500 3000 3500 4000
Along length (mm)
head spline knots=6
Fig.12. von-Misesstressvaluesalongvesseloptimizedmeridionalprofileforinitialshapes1,2,and3correspondingtodifferentnumberofknotsatheadsplineandwith
thermalload.
R.C.Carbonarietal./InternationalJournalofPressureVesselsandPiping88(2011)198e212 207
12
11
n
io p = 1 p = 7
t
c 10
n
u
f
e
v 9
i
t
c
e
j
b 8
o
-
i
t
l
u
M 7
6
0 50 100 150 200 250 300
Iterations
Fig.13. Convergenceofmulti-objectivefunctionforinitialshape3,correspondingtoheadsplineknotnumberequalto6andthermalload.
where f and f are percentage values specified for head and
h n y ¼ y (cid:3)q l (11)
nozzle,respectively. iL i hi i
Tosolvetheoptimizationproblem,thegradientbasedsequen- y ¼ y þq l ði ¼ 1.nhÞ (12)
tiallinearprogramming(SLP)optimizationalgorithmisused[23]. iU i hi i
x ¼ x (cid:3)q l (13)
TheSLPisimplementedusingtheavailablefunctionlinprog(Linear jL j nj j
Programming)fromMATLAB.TheSLPrequiresthatamovinglimit x ¼ x þq l ðj ¼ 1.nnÞ (14)
scheme be defined fordesignvariables. These movinglimits also jU j nj j
helptocontrolthenumericaloscillationsofmulti-objectivefunc- wheresubscriptsLandUarelowerandupperbounds,respectively.
tion gradients. The amplitude of the moving limits is defined as The parameters li and lj are updated during the optimization
afunctionofcoefficientsqhandqn.Thesecoefficientsarecalculated process,andthusthemovinglimitstepsdecreaseby5%withthe
byusingaschemesimilartotheoneusedforcalculatingdhanddn. convergenceofmulti-objectivefunction.
Thus, qh and qn are obtained considering a fraction of the spline Thus, the following optimization problem at each iteration is
length(LhorLn),whichisgivenby: defined:
qh ¼ ahLh;qn ¼ anLn (10) SMuibnje:ctedto: Fðyh;xnÞ
whereqhandqnarethemovinglimitscoefficientsfortheheadand (cid:3)yy1þ(cid:4)yym(cid:4)ax(cid:3)e (cid:3)x1x(cid:4)þxmxax(cid:4)(cid:3)d
ninogzlzyl,e,arheasnpdecatinvealyr,ewpheircchenartaeguepdvaatluedesatspeeaccihfieitderafotironh.eAadccoarndd- (cid:3)(cid:3)yy1iþ(cid:4)yi(cid:3)2þ1y(cid:4)(cid:3)e i(cid:3)¼xj1þ2.xjþ2n1h(cid:4)(cid:3)d
nh min
nozzle, respectively. Therefore, the upper and lower bounds (cid:3)xnn(cid:4)(cid:3)xmin j ¼ 2.nn
obtainedusingthemovinglimitsroutineforthedesignvariables y (cid:4)y (cid:4)y k ¼ 1.nh
arecalculatedby: xkL(cid:4)x k(cid:4)x kU l ¼ 1.nn
lL l jU
2500 initial 2500 initial
optimized optimized
m)2000 m)2000
m m
mension in y (11050000 mension in y (11050000
Di 500 Di 500
0 0
0 500 1000 0 500 1000
Dimension in x (mm) Dimension in x (mm)
Fig.14. Optimizationresultforexample2,consideringh2¼1000mmandfreetangent Fig.15. Optimizationresultforexample2,consideringh2¼1000mmandnulltangent
angleatheadsymmetryknot(nothermalload).Detailedviewsofheadandnozzleare angleatheadsymmetrypoint(nothermalload).Detailedviewsofheadandnozzleare
givenbesidefigure. givenbesidefigure.