有限差分作业华中科技大学版.docx
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有限差分作业华中科技大学版.docx
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有限差分作业华中科技大学版
华中科技大学
研究生课程考试答题本
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Q1.Pleasededucethesecond-orderforwarddifference,second-orderbackwarddifferenceandsecond-ordercentraldifferenceofy=f(x),thendeducetheprecisionofy=f(x)second-ordercentraldifference,basedonTaylorseriesexpansion.
Answer:
One-orderforwarddifference:
One-orderbackwarddifference:
One-ordercentraldifference:
Differentiateagain,andwecangetthesecond-orderdifferenceasfollows:
Second-orderforwarddifference:
Second-orderbackwarddifference:
Second-ordercentraldifference:
AccordingtotheTaylorseriesexpansion:
Addthetwoformulasabove,andwecanget:
Sothesecond-ordercentraldifferencequotienthassecond-orderprecision.
Q2.PleasewritetheFTCSformatoftheconvectionequation
Andpointoutthetruncationerrorprecision.
Answer:
At
replacetimederivativewithfirst-orderforwarddifferencequotient,
Replacespacederivativewithfirst-ordercentraldifferencequotient,
Sotheconvectionequationat
canbeapproximateas
Giventheinitialconditions,wecangettheFTCSformatoftheconvectionequation:
AccordingtotheTaylorseriesexpansion,
SothetruncationerroroftheFTCSformatis
Therefore,theFTCSformathasone-orderprecisionof
andtwo-orderprecisionof
.
Q3.PleasesimulatethetemperaturefieldofanH-shapedcastingusingFDM.Thegeometricconditionsandinitialconditionsareasfollows:
1)ThematerialoftheH-shapedcastingisZG25,theenvironmenttemperatureis25℃andthemoldmaterialisresinsand;
2)Pouringtemperatureis1560℃;
3)CastingsizeasshowninFigure2andFigure3,themoldthicknessis40mm.
Requirements:
1)Writeoutthe2Dor3Dmathematicalmodelthatdescribesthetemperaturefieldofthecastingcoolingprocess;
2)DeducetheFDMformatofthemathematicalmodel;
3)DrawuptheFDMgridmap,describesitusedatastructure;
4)ProvidethethermalpropertiesofZG25,resinsandandtheair;
5)Programtosimulatethisphysicalprocess,assumingthecavitywasfilledveryfastandtheinitialtemperatureevenlydistributed.Pleaseprovidethemaincodeoftheprogram;
6)Howlongdoesittakewhilethehighesttemperatureofthecastingdroppedto1450℃?
Figure1.3DmodelofH-shapedcasting
Figure2.DimensionofH-shapedcasting
Figure3.A2DsliceoftheH-shapedcasting
Answer:
1)TheFourierheatconductiondifferentialequationonthethreedimensionaloccasionsis:
Inwhich:
T—temperature;t—time;x,y,z—spacecoordinates;
—density;
—specificheat;
—coefficientofheatconduction;L—latentheat.
Ifthereisnoinnerheatsource,andset
isthermaldiffusivity(m2/s),thenwecangetthethreedimensionaltemperaturefieldmathematicalmodel:
2)Unitiisaregularhexahedronelementwithalengthofoneside
anditistransferringheatwiththesixunitsadjacenttoit.Thethermalequilibriumrelationshipdiagramofthreedimensionaldifferenceunitiisshownbelow.
Inaveryshorttime
heatabsorptionQofunitiis:
Heatsummation
flowingfromtheadjacentunit1,2,3,4,5,6tounitiis:
Accordingtothelawofconservationofenergy:
Then,
Transformtheformulaabove,andwecangettheFDMformatofthemathematicalmodel:
Inaddition,fromthephysicalmeaning,wecangetthestabilizationcondition:
inwhich,
Transform:
and
.
3)TheFDMgridmapisshownbelow:
4)ThethermalpropertiesofZG25,resinsandandtheairareshowninthetablebelow:
ZG25
Resinsand
Air
7750
1610
1.205
470
1054.9
0.001005
27.2
0.41
0.0259
1833.15
298.15
298.15
5)Themaincodeoftheprogramisasfollows:
Meshing:
for(intz=0;z<=z_boundary;z++)
{
for(inty=-y_boundary;y<=y_boundary;y++)
{
for(intx=-x_boundary;x<=x_boundary;x++,node++)
{
Node[z][y+y_boundary][x+x_boundary]=node;
if(z==0||z==z_boundary)
{
for(intk=0;k { Temp[k][z][y+y_boundary][x+x_boundary]=298.15;//air Material[z][y+y_boundary][x+x_boundary]=2; Density[z][y+y_boundary][x+x_boundary]=1.205; Cond[z][y+y_boundary][x+x_boundary]=0.001005; Thermal[z][y+y_boundary][x+x_boundary]=0.0259; } } elseif(z>0&&z<=H_shape_cycleHigh/spaceLength) { if((x*x+y*y)<=Radius*Radius) { Temp[0][z][y+y_boundary][x+x_boundary]=1833.15;//casting Material[z][y+y_boundary][x+x_boundary]=0; Density[z][y+y_boundary][x+x_boundary]=7750; Cond[z][y+y_boundary][x+x_boundary]=470; Thermal[z][y+y_boundary][x+x_boundary]=27.2; } elseif(x==-x_boundary||x==x_boundary||y==-y_boundary||y==y_boundary) { for(intk=0;k { Temp[k][z][y+y_boundary][x+x_boundary]=298.15;//air Material[z][y+y_boundary][x+x_boundary]=2; Density[z][y+y_boundary][x+x_boundary]=1.205; Cond[z][y+y_boundary][x+x_boundary]=0.001005; Thermal[z][y+y_boundary][x+x_boundary]=0.0259; } } else { Temp[0][z][y+y_boundary][x+x_boundary]=298.15;//mold Material[z][y+y_boundary][x+x_boundary]=1; Density[z][y+y_boundary][x+x_boundary]=1610; Cond[z][y+y_boundary][x+x_boundary]=1054.9; Thermal[z][y+y_boundary][x+x_boundary]=0.41; } } elseif(z>H_shape_cycleHigh/spaceLength&&z<=(H_shape_cycleHigh+H_shape_long)/spaceLength||z> H_shape_cycleHigh+H_shape_long+H_shape_h_long)/spaceLength &&z<=(H_shape_cycleHigh+2*H_shape_long+H_shape_h_long)/spaceLength) { if(x==-x_boundary||x==x_boundary||y==-y_boundary||y==y_boundary) { for(intk=0;k { Temp[k][z][y+y_boundary][x+x_boundary]=298.15;//air Material[z][y+y_boundary][x+x_boundary]=2; Density[z][y+y_boundary][x+x_boundary]=1.205; Cond[z][y+y_boundary][x+x_boundary]=0.001005; Thermal[z][y+y_boundary][x+x_boundary]=0.0259; } } elseif(x>=-((H_shape_length/spaceLength))/2&&x<=(H_shape_length/spaceLength)/2 &&y>=-((H_shape_width/2)/spaceLength)&&y<=(H_shape_width/2)/spaceLength) { Temp[0][z][y+y_boundary][x+x_boundary]=1833.15;//casting Material[z][y+y_boundary][x+x_boundary]=0; Density[z][y+y_boundary][x+x_boundary]=7750; Cond[z][y+y_boundary][x+x_boundary]=470; Thermal[z][y+y_boundary][x+x_boundary]=27.2; } else { Temp[0][z][y+y_boundary][x+x_boundary]=298.15;//mold Material[z][y+y_boundary][x+x_boundary]=1; Density[z][y+y_boundary][x+x_boundary]=1610; Cond[z][y+y_boundary][x+x_boundary]=1054.9; Thermal[z][y+y_boundary][x+x_boundary]=0.41; } } else { if(x==-x_boundary||x==x_boundary||y==-y_boundary||y==y_boundary) { for(intk=0;k { Temp[k][z][y+y_boundary][x+x_boundary]=298.15;//air Material[z][y+y_boundary][x+x_boundary]=2; Density[z][y+y_boundary][x+x_boundary]=1.205; Cond[z][y+y_boundary][x+x_boundary]=0.001005; Thermal[z][y+y_boundary][x+x_boundary]=0.0259; } } else { Temp[0][z][y+y_boundary][x+x_boundary]=298.15;//mold Material[z][y+y_boundary][x+x_boundary]=1; Density[z][y+y_boundary][x+x_boundary]=1610; Cond[z][y+y_boundary][x+x_boundary]=1054.9; Thermal[z][y+y_boundary][x+x_boundary]=0.41;} } } } } Calculate: for(inti=1;i {for(intz=1;z {for(inty=-y_boundary+1;y { for(intx=-x_boundary+1;x {Temp[i][z][y+y_boundary][x+x_boundary]=Temp[i-1][z][y+y_boundary][x+x_boundary]+ 1.0e6*(time_space/spaceLength)/(Density[z][y+y_boundary][x+x_boundary]*Cond[z][y+y_boundary][x+x_boundary])* ((Temp[i-1][z][y+y_boundary][x+x_boundary+1]-Temp[i-1][z][y+y_boundary][x+x_boundary])/ ((spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary+1]+(spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary]) +(Temp[i-1][z][y+y_boundary][x+x_boundary-1]-Temp[i-1][z][y+y_boundary][x+x_boundary])/ ((spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary-1]+(spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary])+ (Temp[i-1][z][y+y_boundary+1][x+x_boundary]-Temp[i-1][z][y+y_boundary][x+x_boundary])/ ((spaceLength/2)/Thermal[z][y+y_boundary+1][x+x_boundary]+(spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary])+ (Temp[i-1][z][y+y_boundary-1][x+x_boundary]-Temp[i-1][z][y+y_boundary][x+x_boundary])/ ((spaceLength/2)/Thermal[z][y+y_boundary-1][x+x_boundary]+(spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary])+ (Temp[i-1][z+1][y+y_boundary][x+x_boundary]-Temp[i-1][z][y+y_boundary][x+x_boundary])/ ((spaceLength/2)/Thermal[z-1][y+y_boundary][x+x_boundary+1]+(spaceLength/2)/Thermal[z][y+y_boundary][x+x_boundary])); } } } } } } Savetheresult: fstreamfs; charszFileName[20]; doublecount=0.0; for(inttime=0;time { sprintf(szFileName,"G: \\%d.plt",time); fs.open(szFileName,ios_base: : out|ios: : _Noreplace); fs<<"Title=\"finite-elementdataforbrick\""< fs<<"Variables=\"x\",\"y\",\"z\",\"Temp\",\"w\""<
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