美国大学生数学建模竞赛2013 获奖论文

美国大学生数学建模竞赛

1

Team#1111111Page1of22

TheUltimateBrowniePan

JiWang

FengHuang

CongliFan

UniversityofElectronicScienceandTechnologyofChina

Advisor:YongZhang

Summary

Makingdeliciousbrownieshasacloserelationshipwiththeshapeofbakingpans,howevertheproblemdepressingussomuchisthatwhenbakinginrectangularpans,theheatconcentratesonthecornercausingthefoodgetovercooked.Meanwhile,bakinginroundpansisnotefficientwithrespecttousingthespaceinarectangularoven.Nowtherefore,wedevelopoptimalmodelstoselectthebesttypeofpans.

Maximizeevendistributionofheatforthepans,itisnecessaryforustofigureouttheheatdistributionfordifferentshapes’pansprimarily.Basedonthesearchingresults,weuseTwo-DimensionalHeatConductionEquationtodescribethedistributionofheatacrosstheouteredgeofpans.Moreover,wesimulatetheheatdistributionofrectangular,roundandpolygonalpansbyusingthePEDtoolboxinMATLAB.

Forthecaseofthemaximumnumberofpansthatcanfitintheoven,weadoptRectangularPackingAlgorithmtosimplifytheproblem.Fromthiswedemonstrateageneralmethodforchoosingtheshapeofpanstomaximizethespaceutilization.Foranovenwithdeterminedradioofwidthtolength,itcancontainmorerectangularpansthanroundandpolygonalpans.

Inaddition,wesetupanindex definedasthestandarddeviationofthetemperatureinthedomainofpanstorepresenttheuniformityofheatdistribution.Andwedrawaconclusionthattheheatdistributionofroundpansisevenerthanregularpolygonalpans.Meanwhile,forrectangularpans,whentheratioofthelengthtothewidthisnearby1:1,itisaworsechoicethanroundpans.Butwhentheratioislesserthanthe1.0:1.8,therectangularpanscanbeabetterchoice.

Whentwofactorsabovearetakenintoaccount,wedevelopanoptimalmodeltodeterminethefinalbestbakingpan.Finally,wecometoaconclusionthattheoptimalshapeofpansvarieswithvaluesofW/Landp.

Finally,wecommentthestrengthsandweaknessesonourmodels.

KeywordsHeatConductionEquation,PEDtoolbox,RectangularPacking

美国大学生数学建模竞赛

Introduction

Whenbakinginarectangularpan,heatisconcentratedinthe4cornersandtheproductgetsovercookedatthecorners(andtoalesserextentattheedges).Inaroundpan,theheatisdistributedevenlyovertheentireouteredgeandtheproductisnotovercookedattheedges.So,heatcannotcirculateevenlyinsidethepan,leadingtotheboundarytemperaturehigherthaninner.

However,sincemostovensarerectangularinshape,usingroundpansisnotefficientwithrespecttousingthespaceinanoven.Thus,searchingforthemostsuitableshapeofthepansintheovenisverybeneficial.

Toexplorethemostsuitableshape,weshouldtaketwoaspectsintoconsideration:

1.Maximizenumberofpansthatcanfitintheoven.

2.Maximizeevendistributionofheatforthepan.

Optimizeacombinationofconditions1and2whereweightspand(1-p)areassignedtoillustratehowtheresultsvarywithdifferentvaluesofW/Landp.

MaximizingthenumberofpansinarectangularovenisequivalenttotheproblemthathowtoarraymoredifferentgeometricfigureswiththesameareaofAinarectangle.ThroughtheLeiHuangetc[4],weuseRectangularPackingAlgorithmtosolveit.

Tomaximizeevendistribution,figuringouttheheatdistributioninthepansfordifferentshapesiscrucial.ApreliminaryresearchofheatconductionhasbeencarriedoutbyFranklinC.daSilvaetc[1]:

TheTwo-DimensionalHeatConductionEquationisusuallyusedtodescribe

theheatconductionprogress.

PDEToolboxinMATLABcanbeusedtosimulatetheheatpartialdifferential

equation,especiallyfortheconductionequation.

Sowecanapplytheresearchabovetodevelopthemathematicmodelofheatdistributionandsimulateit.

VariablesandAssumption

Variables

Variable

u(x,y,t)

Q

Tw

Tf

S

ATable1.VariablesusedinthemodelDefinitionthetemperatureoftheanypointx,yofpanatthetimet( C)theheattransferringtothepanintheunitareaandunittime(W/m2)theouteredgetemperatureofthepan( C)theairtemperatureintheoven( C)theareaoftherectangularoven(cm2)theareaofthepan(cm2)

anindextodescribethedegreeoftheuniformityofheatdistribution

anindextomeasuretheoptimizenumberandtheevendistributioncomprehensively

美国大学生数学建模竞赛

GeneralAssumptions

Theheatcanonlytransfertothepanfromitsouteredgethroughtheair.Sincethefoodplacedonitpreventstheheatfromconductingtoit,thisisareasonable.Thetemperatureinovenisevensincetheairisflowing.Thatthereisonlyonekindofpansintheovens.Initiallytherearetworacksintheoven,evenlyspaced.Wesupposethatthe

temperatureandheatareequivalentandconstant,sowejustconsiderateonerackandtheotheroneisthesamewithit

Theratiooftheovenplane’swidthandlengthisW/L.

EverypansharesthesameareaofA.

Thedatawecitedinthemodelsaretrue.TheareaoftheovenisS 750cm2,andtheratiobetweenwidthandlengthisW/L 22:34.Moreover,theareaofpanisA 100cm2.[2]

TheMathematicalModelofHeatDistribution

Definitions

TheouteredgeandouteredgeandinnerofthepansareillustratedintheFigure1.

Figure1.Theillustrationofdifferentedges.

1.Overview

Firstly,weassumethatthetemperatureinsidetheovenisaconstantandthe

motionstateoftheinternalairisstable.Asaresult,weignoretheinfluencefromtheinternalenvironmentoftheovenwhenheatconductinginthepan.Theairflows

circularlyinsidetheoventoensurethesteadytemperature(intheoven).Accordingtothis,weassumedthatthereisnocertainflowdirectionofair,andtheflowvelocitytowardsanydirectionisconsistent.

Decomposetheairflowintothreedirectionsparallelingtothex,y,andz-axis.Whentakingthepanasathinplane,thereisnoheattransferringtothepanfromthez-axis’directionbecauseofthefoodplacedonit.Sotheprocessofheattransfercanbedescribedbyatwo-dimensionalheatconductionequation.

Theschematicdiagramoftwo-dimensionalheatconductionprocessshowsin

Figure2.

美国大学生数学建模竞赛

HeatdirectionConductionHeatConduction

direction

HeatConduction

direction

Figure2.Diagramoftwo-dimensionalheatconductionprocess

2.Two-dimensionalHeat-conductionEquation

Asexplainedintheoverview,wehaveignoredtheconvectioninfluencefromz-axisandchosetwo-dimensionalheatconductionmodeltodescribethedistributionofheatacrosstheouteredgeofdifferentshapes’pans.TheTwo-DimensionalHeatConductionEquationshowsasfollows:

2 u 2u2 u a(2 2 t x y(1)where

isaconstant;

u(x,y,t)isthetemperatureoftheanypoint x,y inthetimet;

cistheheatcapacityofthepan;

isthedensityofthepan;

isaconstantandrepresentsthermalconductivity.

Theheattransferisthroughairconvectionintheoven.Inaddition,accordingtotheNewton’scoolingformula:

(2)Q h(Tw Tf)

where

Qistheheattransferringtothepanintheunitareaandunittime;

histhermalconductivity;

Twistheouteredgetemperatureofthepan;

Tfistheairtemperatureintheoven.

From(1)and(2),wecanobtaintheboundarycondition:

u (w h(uw uf) n(3)a2 3.Heat-ConductionEquationSolutionbyFiniteDifferenceMethod

美国大学生数学建模竞赛

Finitedifferencemethodisanefficientnumericalmethodwhichisusedtosolvepartialdifferentialequation,especiallyheat-conductionequation[1].Usingfinitedifferencemethodtosolvethesystemofdifferenceequation,weshouldsubstitutefinitediscretepointsforthecontinuousarea.

SolvingstepsofFiniteDifferenceEquation’numericalcalculationisasshownbelow:

Step1:Dispersethearea

Firstly,wegivethemeshgenerationofthecomputationaldomain.Inthearea

D:{(x,y,t)|0 x X,0 y Y,0 t T},wesetinitialconditionandboundaryconditionsofthetwo-dimensionalheat-conductionequationasfollows:

2 u 2u2 u a(2 2),0 x X,0 y Y,0 t T t x y u(x,y,0) u0(x,y),0 x X,0 y Y

u(x,y,0) f(y,t),u(x,y,t) g(y,t)X 0

u(x,y0,0) h(x,t),u(x,yY,t) k(x,t)

where(4)a2 isaconstant.

Throughtheknownequation,wecanbuildafunctionaboutthetimetandstep-size.Inthisway,wefirstparttheinitialareaDintomeshDkasfollows:

Dk {(xi,yj,tn)|xi i x,yj j y,tn n t;i 0,1 ,Nx;j 0,1, ,Ny;n 0,1, Nt}where

x Xand y x

tyarethestep-sizeofx-axisandy-axis. t T

isthestep-sizeoftime.Step2:stabilityanalysis

Itisbasedonthedrivingtimesequenceusingthefinitedifferencemethodtocompute.Theresulterrorfromtheupperlevelmustinfluencetheresultofthenextstep.Hence,itisnecessarytoanalyzethetransmissionoftheerror.Firstly,weusenormtomeasuretheerror.

un n (ui)k i 0 N1kk(5)

Letthatui0haserror i0,thenuinhaserror in.

AslongasaconstantKexists,andwhen t t0,n t T,thereexistsuniformlyn k 0,thenweconsiderthedifferenceasstable.Wecandrawaconclusionthatthedifferenceschemeisstable.[3]

美国大学生数学建模竞赛

Step3:Establishingthedifferencescheme

WeusetheForwardDifferenceFormatandSecondCentralDifferenceFormattoestablishthedifferencescheme.

unuin 1 uin()i (6) t t

2unuin 1 2uin uin 1(2)i (7)2 x( x)

2unuin 1 2uin uin 1(2)i y( y)2

Letrx (8) t2 t2a,r athenwecangety x2 y2

1 2(rx ry)nr ruin 1 ui xyuin 11 rx ry1 rx ry(9)

4.TheSimulationofHeat-conductionEquation

AfterestablishingthemathematicalmodelofHeat-conductionandthesolutionalgorithm,weapplythemtothespecificbrowniepanofdifferentshapes(rectangulartocircularandseveralothershapesinbetween).

ByusingthePDEtoolboxinMATLAB,wecompare(1)withtheparabolicequationdefinedinthetoolbox

ud C u au f(10) t

ThenitiseasytoknowtheparameterofPDEmodel

C ;d c (11) a f 0

Moreover,theboundaryconditionofferedbythePDEtoolboxwhichissuitableforbothrectangleandcirculariscorrespondingtotheNeumannboundarycondition. (12)n (C u) qu g

where nistheunitvectorwhichisperpendiculartotheboundary.

Atthesametime,wecanobtain

q h(13) g huf

Moreover,weknowtheheatcapacityofthepanisc 480J(kg K),thedensityofthepanis 7.8 g3,thethermalconductivityis 48 W/(m K)andh 348W/(m2 K).[3]

Thenwesettheinitialtemperatureofthepanas25 Candthetemperatureintheovenis200 C.TakingtheseparametersintoMATLABprogramandsettingtheobservingtimet 5s,wecangettheheatdistributionacrosstheedgeofpansintheshapeofrectangular,circularandwechooseregularoctagonastheshapesinbetweentoanalyzetheheatdistribution.ThetemperatureprofilesareshowedinFigure3-5.

美国大学生数学建模竞赛

Team

#111111

Page7of22

(a)heatmap(b)surfaceplot

Figure

3.

Thetemperatureprofilesofrectangularpan

(a)heatmap(b)surfaceplot

Figure4.Thetemperatureprofilesofcircularpan

美国大学生数学建模竞赛

Team

#111111

Page8of22

(a)heatmap(b)surfaceplot

Figure5.Thetemperatureprofilesofregularoctagonpan

ObservingtheheatdistributionacrosstheouteredgeinFigure3-5,wecanfindthatthetemperatureofthecornersisobviouslyhigherthanotherspotsforrectangularandpolygonalpans,meanwhile,forroundpans,thetemperatureisequalonthespotswhichhavethesamedistancefromthecentreofthepan.Andthetemperaturegraduallyreducesfromtheedgetotheinner.

Sincehavingfoundtheheatdistributionforthepans,itisthefoundationtodevelopamodelaimingtomaximizeevendistributionofheatforthepan.

MaximizeNumberforDifferentShape’sPans

1.Overview

Inthissection,wesupposethatthewidthtolengthratioofW/Lfortheovenisacertaindeterminateconstant.Wedevelopamodeltomaximizethenumberofpansputintotheoven.Here,weonlyconsideronerackduetothetworacks’equivalence.

Obviously,thisisanarrangementproblemabouthowtoarraymorepanswiththesameareaintherectangularplane.

Sincemostovensarerectangularshape,itisnotefficientwithrespecttousingthespacebyusingroundpansinanoven.Sowefirstlyconsidertherectangularpansanddeterminethemaximalnumberofitthatcanbeputintotheovenononerack.Andthenwediscusstheroundpans.Finallyforthepolygonalpans,weutilizeRectangularPackingAlgorithmtotransformittothecaseofrectangularpans.

2.MaximizeNumberofRectangularPans

Consideringthehabitofpeopletoarrangepansintheoven,weassumethatwhenputtingpansintotheoven,thelongeredgeofpansiscorrespondingtotheoven’slongeredgeaswellastheshortertotheshorter.

Meanwhile,wesupposethatthewaytoputpansintotheovenisfromlefttoright.However,whentheremainingblackspaceoftheovencannotaccommodatethelongersideoftherectangularpanafterputtingenoughpans,itmaybeenoughtocontaininvertedpans.Sowecancontinuetoarraythepansintheoven.TheschematicdiagramshowsinFigure6.

美国大学生数学建模竞赛

Length

...

...

Width

Figure6.Theschematicdiagramofarrayforrectangularpans

Inthisway,wedevelopanoptimizationmodelwiththegoalofmaximumpans.......

maxn1n2 n3

s.t. ab A

a b 0L n1a b 1L n1a b

where

representstorounddown;

LandWrespectivelyrepresentthelengthandthewidthoftheoven;

aandbrespectivelyrepresentthelengthandthewidthoftherectangularpans;Aistheareaofthepans.

Fromthemodelassumption,weknowthatthelengthandthewidthoftheovenrespectivelyare34cmand22cm[2].What’smore,theareaofthepansis100cm2[2].WeuseLINGOsoftwaretosolvetheaboveoptimizationmodelinordertogetthemaximumnumbernofrectangularpanswithdifferentsize.Whenthepanissquarewithalengthoften,itfollowsthatthemaximumthenumberoftherectangularpansissix. LWWn1 [n2 [],n3 [aba......(1)(2)

3.MaximizeNumberofRoundPans

SinceallthepanshavethesameareaA,wecaneasilygetthediameterdofthe

d24Aroundpansviatheexpression:A .Then,itisexpressedbyd .4

Supposingthatthewaytoputpansintotheovenisfromlefttorightandfromthetopdown，wearraytheroundpansbythewayshowninFigure7.

美国大学生数学建模竞赛

Length

...

......…...

...

Figure7.Theschematicdiagramofarrayforroundpans

Inthisway,wedevelopanoptimizationmodelwiththegoalofmaximumnumber: max n1n2 1 n1 1 2 n2 1 s.t. L n 1 d W n 2 d 0 W-nd d 0 2 2 1 (3) 1 W-nd d 0 2 2 0 L-nd d 0 1 2 2 1 L-nd d 0 1 2 d2

A4

Accordingtosolvingthemodelabove,themaximumoftheroundpansis5.

4.Maximizenumberofpolygonalpans

WeadoptRectangularPackingAlgorithm[4]tosolvethepolygonalproblem,whichcantransformtheproblemofpolygontotheproblemofrectangle.

RectangularPackingAlgorithm

Definition

Theminimumpackingrectangle:Thepackingrectangleofgeometryisarectangle

美国大学生数学建模竞赛

whichincludesallofthepoints,lineofthegeometry,andeachsideofwhichcontactswiththegeometry.Thenumberofpackingrectangleofgeometryisinfinite,thesmallestpackingrectangleisonewithasmallestarea[4].

ForthecaseofPolygon

Firstofall,weillustratetheprinciplesoftheRectangularPackingAlgorithmwithRegularpentagonandRegularhexagoninFigure

8.

(a)(b)

Figure8.ThesmallestpackingrectanglesofRegularpentagon(a)andRegularhexagon(b)Forequilateralpolygon,alloftheedgesandtheanglesarethesame.Sothepackingrectangleiswell-determined,andwedevelopamodeltogetit.

ForaregularpentagonwithanareaofA,itsareacanbeexpressedas:

a12A 4

where,a1isthevalueofedges.

Theparametersofthepackingrectangleis:25 105(4)

108 L1 2a12 1 cos 180

W1 a1 cos sin 189

where,L1,W1isthelengthandwidthoftherectangle.(5)(6)

ForaregularhexagonwithanareaofA,itsareacanbeexpressedas:

32(7)A a22

where,a2isthelengthedgeofregularofhexagon.

Thelengthandthewidthofthepackingrectangleare:

(8)L2 a

(9)W2 2a2

Sincewehavegotthepackingrectangleoftheregularpolygon,wecansolvethemaximumnumberofitviatherectangleswhichhavebeendiscussedabove.

WeobtainsomeresultsbasedonthedataZhangWentingprovidedinhispaper[2].

美国大学生数学建模竞赛

Table2showstheparametersofthepackingrectanglesontheconditionthatthepans’areais100cm2.

Table2.Theparametersofthepackingrectangles

Shape

Regularpentagon

RegularhexagonMaximumnumber(m)35Lengthoflength/cm7.66.2Packingrectangles’length/cm12.311.7Packingrectangles’width/cm12.410.7

Accordingtotheanalysisofmaximumnumberfordifferentshape’spans,wecometoaconclusionthatweshouldchooserectangularpanwithalengthof10cm.

MaximizetheEvenDistributionofHeatforthePan

Todescribethedegreeoftheuniformityofheatdistributionfordifferentpans,wesetupanindex ,whichisafunctionabouttemperatureofpans’edgeandgeometricalcenter.However,wheneachpanmusthaveaareaofA,theshapeofregularpolygonalpansandcircularpansisonlydetermined,butasfor

therectangularpans,itscanvarywiththewidthtolengthratioa.So,weneedtorespectivelyanalyzetheuniformityofheatdistributionspecifictorectangularpans,regularpolygonalandcircularpans.

FirstlywetakethedifferentshapepanstodisplayinCartesiancoordinates.

Figure9.TherepresentationofdifferentshapepansinCartesiancoordinates

Ithasbeenknownthatwhenbakinginarectangularpan,heatisconcentratedinthefourcornersaccordingtotheresultofheatdistributionmodel.Therefore,thetemperatureincornerisgreatlyhigherthanthetemperatureinthegeometricalcentreofthepan.Andasforthecircularandregularpolygonalpans,thetemperatureoftheedgeandcornerishighest.Sowecandefineindexwiththestandarddeviationofthetemperatureinthedomainofdifferentshapepanstodescribethedegreeoftheuniformityofheatdistribution.

Where,ux,y,t ,2ij0NxNy(xi,yj) ,i 0,1, Nx,j 0,1, ,Ny(1)

美国大学生数学建模竞赛

Team#111111

NxNyPage13of22NxNy

representsthedomainofdifferentshapepans.

t0representsacertainmoment.

Inthisway,wehavetheconclusionthatthesmaller is,theevenertheheatdistributionis.What’smore,theu(x,y,t)mustmeettheTwo-DimensionalHeat-ConductionEquationinviewofthediscussionabove.Sowecandeveloptheoptimizationmodeltodecidethebesttypeofthepan. u(x,y,t)ij0i 0j 0(2)

min s.t. 2 u 2u2 u a(2 2 x y t

u(x,y,0) u(x,y)ij0ij u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t)

0 x W,0 y L,0 t Tij

(3)SomeResults

Thispart,wegivesomeresultsbasedonthedatainthemodelassumptions.

Wechoosethefifthsecondaftertheheat-conductionprocessbeginningtocomputethevaluesof .AccordingtotheresultdataofthePDEtoolboxofMATLAB,wegetthe forthecircularpanswiththeradiusof1cmis41.7197,fortheregularhexagonwiththelengthof1.1cmis43.0405,whoseareaisnearlythesameasthecircularpans.

Butasfortherectangularpanswhoseareaisnearlyequaltoothershapes,consideringthefeasibility,wechooseninekindsofratiosofthewidthtothelengthandrespectivelycomputethestandarddeviationofthetemperature.Wecangetthedatagraphasfellows:

Table3.Datagraphoftemperaturestandarddeviation

1:1.21:1.41:1.61:1.81:2.01:3.01:4.0

43.6944.8043.5742.3442.1239.0034.90RatioStd1:1.045.111:5.031.97

Inordertoobtaintherelationalexpressionthattheindex iswithrespecttotheratioofthelengthtothewidth,weadoptthepolynomialfittingtoapproximateit.Theexpressionis

a(4) 3.33 48.6b

ThefittedcurveshowsintheFigureX.

美国大学生数学建模竞赛

123

Rtion45

Figure10.Thestandarddeviationandfittedfigureofdifferentratioforrectangular

Resultanalysis

Inviewoftheresultabove,wecandrawaconclusionthattheheatdistributionofroundpansisevenerthantheregularhexagonundertheconditionofthesamearea.Inaddition,forrectangularpans,whentheratioofthelengthtothewidthisnearby1:1,theuniformityisworsethantheothershapes,butitisbetterthantheminthecontextofratiolesserthanthe1.0:1.8.

SelecttheBestTypeofPan

Accordingtothecontentabove,wehaverespectivelydevelopedamathematicalmodeltodescribethemaximumnumberandthemaximumevendistribution.Thus,inthissection,wecanusesomeconclusionswehavegot.

Furthermore,wehavetransformedtheregularpolygonintorectangletodiscussthemaximumthenumberofthepansbasedonRectangularPackingAlgorithm.Sowejustanalyzetheoptimalcombinationofthetwoconditionsfortherectangularandcircularpans.

Weuseobjectivefunctionbelowtooptimizethecombinationofthetwoconditions.

(2)max pm (1 p)

Where,

mand respectivelyrepresentsthenumberofthepansandthedegreeoftheuniformityofheatdistributionforsomeashapepan.

Addingtheconditionsthemodelshavetoconsider,wedevelopthemodelindetailasthefollowingwhichillustratestherectangularpans:

美国大学生数学建模竞赛

max p(n1n2 n3) (1 p) s.t. ab A a 1 0L n1a b 1L n1a b LWWn1 [n2 [n3 [ aba WL S 1

2 u 2u2 u a(2 )2 t x y u(xi,yj,0) u0(xi,yj) u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t) 0 xi W,0 yj L,0 t T (3)

Inthesameway,wecandevelopthesimilartheoptimalmodelbelowwhichillustratestheroundpans.

max p(n1n2 1(n1 1) 2(n2 1)) (1 p)i s.t. L W n1 ,n2 d d 3 0 W-nd d 0 2 2 1 1 W-nd d 0 2 2 3 0 L-nd d 0 1 2 2 1 L-nd 3d 0 1 2 d2 A 4 WL S L 1

2 u 2u2 u a( t x y u(xi,yj,0) u0(xi,yj) u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t) 0 xi W,0 yj L,0 t T (4)

美国大学生数学建模竞赛

Where

a,brespectivelyrepresentthelengthandthewidthoftherectangularpan;

W,Lrespectivelyrepresentthewidthandthelengthoftheoven;

Sistheareaofpan;

Aistheareaofthepans,includingrectangularpanandroundpan;

m,arespectivelyrepresentthenumberofthepansandthedegreeoftheuniformityofheatdistributionforsomeashapepan;

u(xi,yi,t)isthetemperatureoftheanypoint xi,yj inthetimet.

Result

Thebesttypeofpansbeingselectedcountonthedataweassumedinthemodelassumption.Asweassumedabove,theareaofthepansis100cm2andtheacreageis750cm2[2].Moreover,wetakeintoaccountthefactorthatwhenbakingintheoven,peopleusuallythinktheuniformityofheatdistributionmoreimportantthanthe

numberthepanscontainedintheoven.Andweassumethattheweightvaluepislessthan0.5.ThenwetrytoacquirethebesttypeofthepansandratioofthelengthtothewidthbelongingtotheovenbymeansoftheLINGOsoftware.

Throughthesolvingprocess,wegetseveraldifferentlocaloptimumsolutionsforthevariousweightvaluepbelow.Table4.Thebesttypeofthepansandthesizeoftheovenforrectangle

p0.100.150.200.300.410.50

a34.4417.1513.7020.0345.6515.20

b2.905.837.305.002.196.58

L34.4434.3027.3940.0545.6530.40

W21.7721.874.4218.7216.4324.67

W/L0.630.640.160.470.360.81

Optimalvalue-7.47-32.09-31.80-22.8715.14-17.45

Table5.Thebesttypeofthepansandthesizeoftheovenforroundness

p0.100.150.200.300.410.50

L27.3988.7376.0576.0588.7388.73

W27.398.459.869.868.458.45

W/L1.000.100.130.130.100.10

Optimalvalue-37.15-35.61-32.38-27.70-25.02-21.36 Analysis

AstheTable4andTable5reveal,theopticalchoiceofthebakingpansgreatlydependsonthesizeoftheoven.

Wefoundthatselectingrectangularpanswitha 15.20,b 6.58isfarbetterthanselectingroundpanswhenp 0.50.Whenp 0.5,selectingrectanglepansinbetterthanroundpanstoadegree.When W/L 1,weshouldselectrectangularpansortheshapesbetweenrectangleandroundness.

美国大学生数学建模竞赛

Conclusion

Heat-conductionequationisveryapplicabletodescribingthedistributionofheatacrosstheouteredgeofpans.Thesimulationresultconformswellwiththerealitiesofthesituation.Inviewofthediscussionaboutthedistributionofheatofapan,werespectivelydevelopanoptimalmodeltomaximizenumberandevenheatdistributionforpans.Inaddition,inordertoillustratehowtheresultsvarywithdifferentshapesofpans,werespectivelymakeanalysisfortherectangular,circularpansandothershapesinbetween.

Fromourdiscussion,wedrawthefollowingrecommendation:

Theheatdistribution:Fortherectangularandregularpolygonalpans,thetemperatureatthecornersisgreatlyhigherthanotherregionsofthepans.However,itdoesnotappeartobemuchdifferentforthesameradiusareaoftheroundpans.Soweshouldtrytoselecttheapproximatecircularshapeofthepans.

Maximizethenumberofthepans:Whenonlyconsideringmaximizingthenumberofthepans,wefoundthattherectangularpanswiththelengthandwidthbothare10cmisanoptimalshapeforthepans.

Sensitivitytomodel:TheweightedvaluepandthevalueoftheW/Lcanenormouslyinfluencetheresults.Hence,weconstantlyalterthemtosimulateourmodelinordertoillustratehowtheresultsvarywithdifferentvaluesofthepandW/L.WefoundtheoptimalshapeofthepansisdifferentwithdifferentpandW/L.

Strength&Weakness

Strengths

Forpolygons,whendiscusshowmanypanscanfittheoven,itisfilledwithdifficulty;tosomeextent,therectangulariseasierthanotherpolygons,thus,weuserectangularpackingalgorithm,aimingatchangethepolygonsintorectangular,whichdoesn’taltertheanswerbutmakethecalculationmoresimpleandconvenient.

Ourmodelisflexibilityandrealistic,thatistosay,ourmodeleasilyadaptstoproblemswithdifferentshapeofpans,suchastriangleandoctagonwhichmaynotberegular.Ontheotherhand,basedonthereality,wesetarightratioaboutthewidthtolengthofthepan.

Weaknesses

Inourmodel,someindexisfullofsubjectivity,whichmaymakethemodelnottootypical.

Someassumptionsmakeforsimplicitythatlikelydonothold.Forinstance,formostovens,wewillputdifferentshapepansinit,ratherthanonlyonekindofshapes.

References[1]FranklinC.daSilva,AntBnioM.Soares,andLeonardoR.A.X.deMenezes.”NewAbsorbingBoundaryConditionsfortheFiniteDifferenceMethodbasedonDiscreteSolutionsofLaplaceEquation”.IeeeConferencePublications,2003

1037-1041.

美国大学生数学建模竞赛

[2]ZhangWenting,ChengJie,”Astudyofanalyzingtheheatdistributiononvaryingtypesofpansanditsoptimal“,20136thInternationalConferenceonInformationManagement,2013.

[3]LuoXuanCuiGuozhongLeFulong.2014.HeatDistributionMathematicalModelandNumericalSimulationofanElectricOven.Proceedingsofthe33rdChineseControlConferenceJuly28-30,2014,Nanjing,China.

[4]LeiHuang,ZhongLiu,ZhiLiu,”Animprovedlowest-levelbest-fitalgorithmwithmemoryforthe2Drectangularpackingproblem”,InformationScience,ElectronicsandElectricalEngineering(ISEEE),2014InternationalConferenceonVolume:2PublicationYear:2014,Page(s):1279–1282.

[5]Hoshi,F.,Murai,Y.,Tsuji,H.Tokumasu,S.“Anewdeterministicalgorithmfortwo-dimensionalrectangularpackingproblemsbasedonpolyominopacking

models”SystemsManandCybernetics(SMC),2010IEEEInternational

ConferenceonPublicationYear:2010,Page(s):2760-2766

[6]Zecova,M.,Terpak,J.,Dorcak,L.”Usageoftheheatconductionmodelfortheexperimentaldeterminationofthermaldiffusivity“CarpathianControlConference(ICCC),201314thInternationalPublicationYear:2013,Page(s):436-441

[7]Nakayama,W.“Heatconductionincompositesofthermallydissimilarmaterials-amethodologytoeconomizenumericalheattransferanalysisofelectronic

components“ElectronicsMaterialsandPackaging,2005.EMAP2005.

InternationalSymposiumonPublicationYear:2005,Page(s):282-287

AdvertisingSheet

To:theeditorofBrownieGourmetMagazine.

From:afoodlover

Date:February20,2015

Subject:anExcellentBakingPanforBrownie

美国大学生数学建模竞赛

Team

#111111Page19of22Areyoustill

D

epressedwithyourDissatisfyingbakingpans?Grieving

cakes?fortheUnevenheating

BoredtoRepeatagainand

againforanenjoyabletastingofcake?

youmustinterestinourdesign,themostdesirablepans,withwhich,youcanmakemorefavoriteandlusciouscakes.Moreover,thereisonereasonaboutwhycan’tmissourpans:youcanownyouruniquepansbecauseourpansaredesignedaccordingtoyouroven’ssize.

Thebakingpanswedesignedarescience-based.Wehave

本文来源：https://www.bwwdw.com/article/liii.html