美国大学生数学建模竞赛2013 获奖论文(3)

Figure9.TherepresentationofdifferentshapepansinCartesiancoordinates

Ithasbeenknownthatwhenbakinginarectangularpan,heatisconcentratedinthefourcornersaccordingtotheresultofheatdistributionmodel.Therefore,thetemperatureincornerisgreatlyhigherthanthetemperatureinthegeometricalcentreofthepan.Andasforthecircularandregularpolygonalpans,thetemperatureoftheedgeandcornerishighest.Sowecandefineindexwiththestandarddeviationofthetemperatureinthedomainofdifferentshapepanstodescribethedegreeoftheuniformityofheatdistribution.

Where,ux,y,t ,2ij0NxNy(xi,yj) ,i 0,1, Nx,j 0,1, ,Ny(1)

美国大学生数学建模竞赛

Team#111111

NxNyPage13of22NxNy

representsthedomainofdifferentshapepans.

t0representsacertainmoment.

Inthisway,wehavetheconclusionthatthesmaller is,theevenertheheatdistributionis.What’smore,theu(x,y,t)mustmeettheTwo-DimensionalHeat-ConductionEquationinviewofthediscussionabove.Sowecandeveloptheoptimizationmodeltodecidethebesttypeofthepan. u(x,y,t)ij0i 0j 0(2)

min s.t. 2 u 2u2 u a(2 2 x y t

u(x,y,0) u(x,y)ij0ij u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t)

0 x W,0 y L,0 t Tij

(3)SomeResults

Thispart,wegivesomeresultsbasedonthedatainthemodelassumptions.

Wechoosethefifthsecondaftertheheat-conductionprocessbeginningtocomputethevaluesof .AccordingtotheresultdataofthePDEtoolboxofMATLAB,wegetthe forthecircularpanswiththeradiusof1cmis41.7197,fortheregularhexagonwiththelengthof1.1cmis43.0405,whoseareaisnearlythesameasthecircularpans.

Butasfortherectangularpanswhoseareaisnearlyequaltoothershapes,consideringthefeasibility,wechooseninekindsofratiosofthewidthtothelengthandrespectivelycomputethestandarddeviationofthetemperature.Wecangetthedatagraphasfellows:

Table3.Datagraphoftemperaturestandarddeviation

1:1.21:1.41:1.61:1.81:2.01:3.01:4.0

43.6944.8043.5742.3442.1239.0034.90RatioStd1:1.045.111:5.031.97

Inordertoobtaintherelationalexpressionthattheindex iswithrespecttotheratioofthelengthtothewidth,weadoptthepolynomialfittingtoapproximateit.Theexpressionis

a(4) 3.33 48.6b

ThefittedcurveshowsintheFigureX.

美国大学生数学建模竞赛

123

Rtion45

Figure10.Thestandarddeviationandfittedfigureofdifferentratioforrectangular

Resultanalysis

Inviewoftheresultabove,wecandrawaconclusionthattheheatdistributionofroundpansisevenerthantheregularhexagonundertheconditionofthesamearea.Inaddition,forrectangularpans,whentheratioofthelengthtothewidthisnearby1:1,theuniformityisworsethantheothershapes,butitisbetterthantheminthecontextofratiolesserthanthe1.0:1.8.

SelecttheBestTypeofPan

Accordingtothecontentabove,wehaverespectivelydevelopedamathematicalmodeltodescribethemaximumnumberandthemaximumevendistribution.Thus,inthissection,wecanusesomeconclusionswehavegot.

Furthermore,wehavetransformedtheregularpolygonintorectangletodiscussthemaximumthenumberofthepansbasedonRectangularPackingAlgorithm.Sowejustanalyzetheoptimalcombinationofthetwoconditionsfortherectangularandcircularpans.

Weuseobjectivefunctionbelowtooptimizethecombinationofthetwoconditions.

(2)max pm (1 p)

Where,

mand respectivelyrepresentsthenumberofthepansandthedegreeoftheuniformityofheatdistributionforsomeashapepan.

Addingtheconditionsthemodelshavetoconsider,wedevelopthemodelindetailasthefollowingwhichillustratestherectangularpans:

美国大学生数学建模竞赛

max p(n1n2 n3) (1 p) s.t. ab A a 1 0L n1a b 1L n1a b LWWn1 [n2 [n3 [ aba WL S 1

2 u 2u2 u a(2 )2 t x y u(xi,yj,0) u0(xi,yj) u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t) 0 xi W,0 yj L,0 t T (3)

Inthesameway,wecandevelopthesimilartheoptimalmodelbelowwhichillustratestheroundpans.

max p(n1n2 1(n1 1) 2(n2 1)) (1 p)i s.t. L W n1 ,n2 d d 3 0 W-nd d 0 2 2 1 1 W-nd d 0 2 2 3 0 L-nd d 0 1 2 2 1 L-nd 3d 0 1 2 d2 A 4 WL S L 1

2 u 2u2 u a( t x y u(xi,yj,0) u0(xi,yj) u(x0,yj,0) f(yj,t),u(xX,yj,t) g(yj,t) u(xi,y0,0) h(xi,t),u(xi,yY,t) k(xi,t) 0 xi W,0 yj L,0 t T (4)

美国大学生数学建模竞赛

Where

a,brespectivelyrepresentthelengthandthewidthoftherectangularpan;

W,Lrespectivelyrepresentthewidthandthelengthoftheoven;

Sistheareaofpan;

Aistheareaofthepans,includingrectangularpanandroundpan;

m,arespectivelyrepresentthenumberofthepansandthedegreeoftheuniformityofheatdistributionforsomeashapepan;

u(xi,yi,t)isthetemperatureoftheanypoint xi,yj inthetimet.

Result

Thebesttypeofpansbeingselectedcountonthedataweassumedinthemodelassumption.Asweassumedabove,theareaofthepansis100cm2andtheacreageis750cm2[2].Moreover,wetakeintoaccountthefactorthatwhenbakingintheoven,peopleusuallythinktheuniformityofheatdistributionmoreimportantthanthe

numberthepanscontainedintheoven.Andweassumethattheweightvaluepislessthan0.5.ThenwetrytoacquirethebesttypeofthepansandratioofthelengthtothewidthbelongingtotheovenbymeansoftheLINGOsoftware.

Throughthesolvingprocess,wegetseveraldifferentlocaloptimumsolutionsforthevariousweightvaluepbelow.Table4.Thebesttypeofthepansandthesizeoftheovenforrectangle

p0.100.150.200.300.410.50

a34.4417.1513.7020.0345.6515.20

b2.905.837.305.002.196.58

L34.4434.3027.3940.0545.6530.40

W21.7721.874.4218.7216.4324.67

W/L0.630.640.160.470.360.81

Optimalvalue-7.47-32.09-31.80-22.8715.14-17.45

Table5.Thebesttypeofthepansandthesizeoftheovenforroundness

p0.100.150.200.300.410.50

L27.3988.7376.0576.0588.7388.73

W27.398.459.869.868.458.45

W/L1.000.100.130.130.100.10

Optimalvalue-37.15-35.61-32.38-27.70-25.02-21.36 Analysis

AstheTable4andTable5reveal,theopticalchoiceofthebakingpansgreatlydependsonthesizeoftheoven.

Wefoundthatselectingrectangularpanswitha 15.20,b 6.58isfarbetterthanselecting …… 此处隐藏：5556字，全部文档内容请下载后查看。喜欢就下载吧 ……