美国大学生数学建模竞赛2013 获奖论文(2)

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

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(a)heatmap(b)surfaceplot

Figure

3.

Thetemperatureprofilesofrectangularpan

(a)heatmap(b)surfaceplot

Figure4.Thetemperatureprofilesofcircularpan

美国大学生数学建模竞赛

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(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

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