umye build two si modls fote two rblm respetively： n is to show h disritiof at across e outer ede of the p for difrent shas， ectaguar， rculr and transiionspe; anohr i toseect the best shapefor h pan uder teodiionfte optimizatiof ominations of aml mberfan i theven nthe maximl ven eat distibuioo te ha forte pan.We ir use finitediffeenceethdto anlyzehe condtn raditi problnd derive he ea distribuin f thereguarad h cila. In terms of our iohrmal curve the recangulr pa，we ayze hat istributi of rundd retngle thoroughly,usig finit-elemet method e thenus oninear ineger prgramming methd o solve tmxim numbr ofpanin h ove。 Itheevn heat distrbution,weeinea function to hw te degreeof he even ea itriution. useolnoma fittig ith mulipl aialestosol he bjecvefnction For th atproblem， cmbining e reslts above, we ana how esult vay wtthe iferentlueof id o legth raio /L and hweigt factop. At lat,e vidte that our methods corret an robus bycompanand analyng is snsitiviy andsrenths weaknesses sed o t wr abve，we ulmtelypu foward that te runded rcangular ae spre cosderig pia nmbrf the pan an ev hadisriutio。 Andan dverisent is presented or the wie Gouret Maazine。Contens1 Introdution1。1rowepan1Backround31.3Prblem esrptio32. Melfor hetisributn32。1roblem nalyis32。Assumption42。3 ftion42.4 he moe43 Resul ohtistibution7.1Basireults732 Analysis3.3 Anayss f h transtion shapeouned rectanlar4Modet selet the best shape114.1 Assptions114。efiitions1143 h mdel125 Compaision dDegreof fiting196 Senitivty7 Strengh/wkese18Ccsion19 Advertiseet for new Browie Magazine2310 Rerencs2Itodutin1.1 rownie panThe Bwnie P ssed t make ownieswhich area kind of olar cakes n Americ.It usualyamyltics in it ndis mad of etlo ther mriastocondut heatwellts tiviall99 inchor 93 ich nsze。 ne exampleof thecncrehae f rownie pan is hown iFgr1Fgure 1 the sape o onie Pn （e: oe Ima）1.2 Bckrun rowns e dicio uthe Brownie Pan as fe dawbac. hen bakign arctanga pan, the foo n easilyet oeroked in the 4 cornr，ih s ery anyingor ereedy gourme。 In ru pn， thatisevl diribute oer the enre uter edge uis not effietwith repectto sing in t space in oven, which mot cke akers woud not like to see. o gl to dres h blem。1.3 Pobm DscptoFrty， were askeo dve model o sow te distributio of htcros te out edg of a pa fo diffeenshps， frm rectagula tocircularilung terasitio hape； hen e wilbuil ate moe to select he bstshap f tha folowithe coniti o th optztion of cobinai mxmalnumr fpans n theven and axmal en distriuion f ea r t pn.2. odl fo heat isriuion2. Prbem anayiser we use a finie diffrence oel to ilustrte the dstriutio o hea， ad it has been exensieyused i molng fo crcteristicabilit o hane irrgar geomries an boay conditios, spial an teporal poerts variatios Shixiong Liu, Mika Fukuoka, Noboru Sakai, A finite element model for simulating temperature distributions in rotating food during microwave heating, Journal of food engineering, Volume 115, issue 1, March 2013 Page 49-62 n lterature 1， smples wtha etangur geomr fomare ifficlttheat unifrmy， claly thcorners anddges h thicrowave radiation in the oven a be cudlytut o as impingng on th sample om ll， whic we generlly acknowle.But tey emhasiz rotatonGenel, wh akiin hen， e bsrbhetby three ays： theml raditon ote pipe inthe ovn, heat cduonof than, andai conveon inhven。 nseri ththe influnce ofconecionis smal, wassume it neglgibe。So wenytake thermalradiionand cndcon into ccunt。Th heat is trferred from h utsd to h inde while wter in he k is n thecontrar. The emerturousie ncrasemoe rpily tan ta inside And t onat area beeen the pan nd theotse cke s arger than thatbetwen thea and thensd cake， whih illutratwhy akes i th orer get vercoked asil。2。2 Assumptiosl e tketh pan and ake as blac oy, s the bsorin f et in