2013年美赛a题h奖论文

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1、SummaryWebuildtwobasicmodelsforthetwoproblemsrespectively:oneistoshowthedistributionofheatacrosstheouteredgeofthepanfordifferentshapes,rectangular,circularandthetransitionshape;anotheristoselectthebestshapeforthepanundertheconditionoftheoptimizationofcombinationsofmaximal

2、numberofpansintheovenandthemaximalevenheatdistributionoftheheatforthepan.Wefirstusefinite-differencemethodtoanalyzetheheatconductandradiationproblemandderivetheheatdistributionoftherectangularandthecircular.Intermsofourisothermalcurveoftherectangularpan,weanalyzetheheatdi

3、stributionofroundedrectanglethoroughly,usingfinite-elementmethod.Wethenusenonlinearintegerprogrammingmethodtosolvethemaximalnumberofpansintheoven.Intheevenheatdistribution,wedefineafunctiontoshowthedegreeoftheevenheatdistribution.Weusepolynomialfittingwithmultiplevariable

4、stosolvetheobjectivefunctionForthelastproblem,combiningtheresultsabove,weanalyzehowresultsvarywiththedifferentvaluesofwidthtolengthratioW/Landtheweightfactorp.Atlast,wevalidatethatourmethodiscorrectandrobustbycomparingandanalyzingitssensitivityandstrengths/weaknesses.Base

5、dontheworkabove,weultimatelyputforwardthattheroundedrectangularshapeisperfectconsideringoptimalnumberofthepansandevenheatdistribution.AndanadvertisementispresentedfortheBrownieGourmetMagazine.Contents1Introduction31.1Browniepan31.2Background31.3ProblemDescription32.Modelf

6、orheatdistribution32.1Problemanalysis32.2Assumptions42.3Definitions42.4Themodel43Resultsofheatdistribution73.1Basicresults73.2Analysis93.3Analysisofthetransitionshape—roundedrectangular94Modeltoselectthebestshape114.1Assumptions114.2Definitions114.3Themodel125Comparisiona

7、ndDegreeoffitting196Sensitivity207Strengths/weaknesses218Conclusions219AdvertisementfornewBrownieMagazine2310References241Introduction1.1BrowniepanTheBrowniePanisusedtomakeBrownieswhichareakindofpopularcakesinAmerica.Itusuallyhasmanylatticesinitandismadeofmetalorothermate

8、rialstoconductheatwell.Itistrivially9×9inchor9×13inchinsize.OneexampleoftheconcreteshapeofBrowni