美国数学建模A题论文 最佳布朗尼锅The Ultimate Brownie Pan.docx

美国数学建模A题论文 最佳布朗尼锅The Ultimate Brownie Pan.docx

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美国数学建模A题论文最佳布朗尼锅TheUltimateBrowniePan

TheUltimateBrowniePan

Summary

Thebrowniesaredelicious,butselectingpansplaysanimportantroleinordertobaketheonesthattastebest.Rectangularpancanmakefulluseofthespaceyetwiththeuneventemperaturedistribution,whilecircularpansareonthecontrary,whichisquitedisturbing.Therefore,wearetaskedtodevelopamodeltofindoutthebestchoicewiththemostuniformtemperaturedistributionandthemostpansintheoven.

WethinkoftheheatconductionequationandapplyittoModelOne.Inordertosimplifytheproblem,butnottodivorcefromreality,wemake​​areasonableassumption.Whenconsideringtheproblemofheatconductionunderthestateofthermalequilibriuminatwo-dimensionalplane,wegetthesimulationdiagramsoftemperaturedistributionofdifferent-shapedpansundertheassumedcircumstancewiththeaidofPDEtoolofMATLAB.

AsforModelTwo,weconverttheproblemofoptimizationofpan’snumbersintotheoneofspaceutilization.Thepanwithmaximumutilizationofspaceiscorrespondingtothelargestnumberofbakingpans.Weintroducethetemperaturevariancetocharacterizetheuniformityofthetemperaturedistribution.Thebestoptionshouldtakethetwofactorsaboveintoaccount.Weights（p）and（1-p）areassignedtoillustratehowtheresultsvarywithdifferentvalues.Herewecreativelyintroducetheconceptofrelativeperformance;itmeansthatwescorethebakingpansbyexaminingtherelativedegreesofspaceutilizationandtheunevennessoftemperaturedistribution.Fromthescores,wecanfindoutthebestshape.

KeyWord:

HeatConductionEquation,Variance,Spaceutilization,RelativelyComprehensivePerformance

1Introduction

Abrownieisasquareorbarofveryrichchocolatecakeusuallywithnuts.Owingtoitstastyflavorandconveniencetobetakenalong,itenjoysagreatpopularityinwesterncountries.Inrecentlyyears,itiswidespreadinChinanow.Aplentyofgirlsliketocookthedessertsthemselves,butintheprocessofmakingbrownies,theyareconfrontedwithadisturbingproblem.

Whenbakinginarectangularpan,heatisconcentratedinthefourcornersandtheproductgetsovercookedatthecorners（andtoalesserextentattheedges）.Inaroundpantheheatisdistributedevenlyovertheentireouteredgeandtheproductisnotovercookedattheedges.However,sincemostovensarerectangularinshape,usingroundpansisnotefficientwithrespecttousingthespaceinanoven.

Themostcommonlyusedshapesofpansinthemarketaresquareandround.Inordertomakefulluseofthespaceoftheovenwithtakingtheevendistributionofheatintoaccount,shapesbetweensquareandround（regularpolygonmainly）maypresentuswithabetteroutcome.Inthemeantime,theselectionofthebesttypeisalsoinfluencedbythewidthtolengthration,theareaofthepan,themaximumutilizationofspaceandhowtobalancetheweightsoftheevendistributionofheat.Consequently,wearetaskedtodevelopamodeltohelptheminthisdilemmaandapplyittotheselectionofthemostappropriateshape.

2Assumptions

ØWeonlystudypanprobleminthetwo-dimensionaltemperaturedistribution,donotconsiderthethree-dimensionalcase;

ØThereisnoheatsourceinthepan,norheatexchangewiththeoutsideworld;

ØPot temperature tends to equilibrium steady state, it does not change with time; 

ØInthesteadystate,thetemperatureinsidetheoventemperature,theedgeofthecakeandthehotplatetemperatureisconsistent;

ØBakeware and oven are made of iron; 

ØOven temperature equilibrium within 180 degrees Celsius.

3ParameterDefinitions

X

Lengthofeachinapolygon

μ

RatioofWandL

a

NumberofpansinalengthofL

b

NumberofpansinalengthofW

N

Maximumnumberofpansinanoven

S

Varianceofheatinapan

Q

Relativeoccupationofspace

U

Relativeinhomogeneousdegree

R

Relativeperformancescore

A

Areaofapan

W

Widthofapan

L

Lengthofapan

p

Weightgiventooccupationofspace

4ModelDevelopment

4.1Distributionofheatacrosstheouteredge

Ifthetemperatureofeachplaceinanobjectisnotthesamewitheachother,thentheheatwillflowfromthepartsofhighertemperaturetothepartsoflowerones,asiscalledheatconduction.Sincetheconductionofheatisshownasthechangesoftemperaturevarywithtimeandlocation,thesolutiontotheproblemofheatconductionisattributedtothedistributionoftemperatureintheobject.Weregardthepanasahomogeneousandisotropicheatconductor;thedifferentialequationofheattransferprocessinthetwo-dimensionalplanethatitmeetsisasfollow:

（4.1）

Ifweconsidersteadystationarytemperaturefield,wherethetemperatureoftheobjectintheheatconductionequationtendstoastateofequilibriumandthetemperature（u）hasnothingtodowithtime（t）,sothat

=0,theequation

（1）becomestheLaplaceEquationnow,as

=0（4.2）

Thisshowsthatthetemperature（u）ofthestationarytemperaturefieldalsomeetsLaplaceEquation.

TheLaplaceEquation,asadescriptionofphysicalphenomenasuchasstabilityandbalance,cannotmentiontheinitialconditions.Asfortheboundaryconditions,weassumethatwhenthebakingtemperaturetendstoastateofdynamicbalance,theheatofcakeisthesamewiththeironpan,soitmeetsthefirstboundaryvalueproblem,whichistheDirichletProblem.

Inthetwo-dimensionalplane,theproblemboilsdowntothedefinitesolutiontothetwo-dimensionalLaplaceEquationwithinthecircledomainwhenthepaniscircular.Assumingthattheradiusisrepresentedbyrandthetemperatureisrepresentedbyu,thentheradiusofthecircleis

theproblemisshownas

（4.3）

Weassumethatinthesteadystate,thetemperatureinsidetheoventemperature,theedgeofthecakeandthehotplatetemperatureisconsistentboundaryconditionsareobtainedbyDirichletProblemconditionsas

（4.4）

u（0,

）=limitedvalue（4.5）

ThePDEtooloftheMATLABisusedtogetthefollowingimage.

Figure4-1Distributionofheatincircle

Fromtheimageshownabove,wecanfindtheheatdistributionofthecircle-fromtheinsideout,thetemperatureisgraduallyincreased,withthetemperatureattheedgesappearsasthemaximumandeven-distributed.

Next,weconsiderthecaseofarectangle.TheproblemisconvertedintothedefinitesolutiontotheLaplaceEquationwithintherectangulardomainwhenthepanisarectangle.Wesolvetheproblemintherectangularcoordinatesystem,withwidthrepresentedbyxandlengthrepresentedbyy,supposingthewidthoftherectangleisa,thelengthofitisb,andustandsfortemperature,wecangettheLaplaceEquationas

=0

（4.6）

Thesameasabove,inthesteadystate,thetemperatureinsidetheoventemperature,theedgeofthecakeandthehotplatetemperatureisconsistent;boundaryconditionsareobtainedbyDirichletProblemconditions.

ThePDEtooloftheMATLABisusedtogetthefollowingimage.

Figure4-2Distributionofheatinrectangle

Fromtheimageabove,wecandrawtheconclusionoftheheatdistributionoftherectanglefromtheinsideout;thetemperatureisgraduallyincreasedaswell.Thetemperaturedistributionoffouredgesoftherectangleisnoteven,itincreasesgraduallyfromthemiddletothecorners.However,thetemperaturedistributionsofthecorrespondingsidesarethesame,andthemaximumvalueofthefinaltemperatureappearedinthefourcornersoftherectangle.

Withthecomparisonofthetwocases,wefindthatthetemperaturedistributionofthecircleismoreuniform,sowesuspectthatthetemperaturedistributionofcircularpanisthemostuniformoneamongallkindsofshapes.Inordertoverifyourconjecture,weusetheMATLABsoftwaretofindoutthetemperaturedistributionofregularhexagonandregularoctagonunderthecircumstancethattheboundaryconditionsremainunchanged.Theimagesareasfollows.

Figure4-3Distributionofheatinregularhexagon

Figure4-4Distributionofheatinregularoctagon

Finally,wedrawtheconclusionthatsurelythetemperatureofthearbitrarypolygonisnotuniform,andthemaximumtemperaturesallappearinthecorners.Whenthenumberofpolygonedgesismorethantwohundred,approximatelytheshapecanberegardedasacircle.Itmeansthatthecloseritgetstoacircle,themoreuniformthetemperaturedistributionwillbe.Consequently,theshapewiththemostuniformdistributionoftemperaturewillbethecircle.

4.2Maximizenumberofpans

Whenconsideringtheissueofdifferentshapesarrangedinapanonthegrill,combinedwiththeactualsituation,westudythecasesthatthebakewareisasquare,hexagonal,octagonal,andcircularshape.WeassumethatthewidthoftheovenisW,thelengthofitisL,andwehavetheequationthatW/L=μ.SupposethatthesidelengthoftheregularpolygonisX,itstandsfortheradiusinthecircle.

Forthisproblem,wedivideitintotwostepstodescribethemaximizationofthenumberofpans.Firstofall,wetrytofindthelargestnumbersofsidesinaregularpolygonthattheoven’slengthLandwidthWcanaccommodate,andsetthemasp（L）andm（W）,sothenumberofpanswhichcanbecontainedbythewholeovenplaneis

N=p*m（4.7）

Figure4-5

Squarepan:

Inarectangularoven,squarebakewarescanbeplacedliketheexampleinthepicture.WeassumethatthelengthofsquareisX,soeverybakeware'sareais"X2=A"Whenitcomestothesidelengthoftheovenplane,thelongsideaccommodateasmuchasp=[

]lengthofsquare,whiletheshortsidem=[

].“[ ]”isstandforroundingarithmetic.sowecanworkoutthemaxnumberofSquarebakewarethatovenplanecan