2013年美赛a题h奖论文

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1、SummaryWebuildtwobasicmodelsforthetwoproblemsrespectively:oneistoshowthedistributionofheatacrosstheouteredgeofthepanfordifferentshapes,rectangular,circularandthetransitionshape;anotheristoselectthebestshapeforthepanundertheconditionoftheoptimizationofcombinationsofmaximalnumberofpansintheovenandthe

2、maximalevenheatdistributionoftheheatforthepan.Wefirstusefinite-differencemethodtoanalyzetheheatconductandradiationproblemandderivetheheatdistributionoftherectangularandthecircular.Intermsofourisothermalcurveoftherectangularpan,weanalyzetheheatdistributionofroundedrectanglethoroughly,usingfinite-ele

3、mentmethod.Wethenusenonlinearintegerprogrammingmethodtosolvethemaximalnumberofpansintheoven.Intheevenheatdistribution,wedefineafunctiontoshowthedegreeoftheevenheatdistribution.WeusepolynomialfittingwithmultiplevariablestosolvetheobjectivefunctionForthelastproblem,combiningtheresultsabove,weanalyzeh

4、owresultsvarywiththedifferentvaluesofwidthtolengthratioW/Landtheweightfactorp.Atlast,wevalidatethatourmethodiscorrectandrobustbycomparingandanalyzingitssensitivityandstrengths/weaknesses.Basedontheworkabove,weultimatelyputforwardthattheroundedrectangularshapeisperfectconsideringoptimalnumberofthepa

5、nsandevenheatdistribution.AndanadvertisementispresentedfortheBrownieGourmetMagazine.Contents1Introduction31.1Browniepan31.2Background31.3ProblemDescription32.Modelforheatdistribution32.1Problemanalysis32.2Assumptions42.3Definitions42.4Themodel43Resultsofheatdistribution73.1Basicresults73.2Analysis9

6、3.3Analysisofthetransitionshape—roundedrectangular94Modeltoselectthebestshape114.1Assumptions114.2Definitions114.3Themodel125ComparisionandDegreeoffitting196Sensitivity207Strengths/weaknesses218Conclusions219AdvertisementfornewBrownieMagazine2310References241Introduction1.1BrowniepanTheBrowniePanis

7、usedtomakeBrownieswhichareakindofpopularcakesinAmerica.Itusuallyhasmanylatticesinitandismadeofmetalorothermaterialstoconductheatwell.Itistrivially9×9inchor9×13inchinsize.OneexampleoftheconcreteshapeofBrowni