限元与有限差分法应用实验报告Word文档下载推荐.docx
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second-ordercentraldifferencebasedonTaylorseriesexpansion.（10points）
Answer:
（1.1）Second-orderforwarddifferenceformula:
Weknowthatthefirstorderforwarddifferenceformulais:
So,thesecond-orderforwarddifferenceformula:
（1.2）Second-ordercentraldifferenceformula:
Weknowthatthefirstordercentraldifferenceformulais:
So,thesecond-ordercentraldifferenceformula:
\
（1.3）Precisionofsecond-ordercentraldifferenceformula:
Taylorseriesexpansionfor
and
are
（1）
（2）
Addingequation
（1）and
（2）weget
or,
thus,second-ordercentraldifferenceformulahastwo-orderprecision.
Q2
Pleasecombinetheresearchdirectionofyoursubject,illustratethetypicalapplicationofthefinitedifferencemethodinthisresearcharea.（35points）
Mymajoristhemathematicalcalculation.Theexactdirectionisthenumericalsolutionofpartialdifferentialequationsanditsapplication.Firstofall,FDMiswidelyusedinthenumericalsolutionofpartialdifferentialequations.
Concreteexampleisasfollow:
Finitedifferencemethodforsolvingboundaryvalueproblems
（exactsolutionis
.）.Bystep
，wecangetrectangularsectionandwecanstructuredifferencescheme.
Gridnodeis
.Differenceequationisasfollow:
Boundaryvalueconditionisasfollow:
Successively，Weorder
andsolveitwiththeeliminationmethods.
.Wecanmakealistofexactsolutionsandnumericalsolutions.
Q3
PleasesimulatethetemperaturefieldofanH-shapedcastingusingFDM.
Thegeometricconditionsandinitialconditionsareasfollows:
1）ThematerialoftheH-shapedcastingisZG25,theenvironmenttemperatureis25℃andthemoldmaterialisresinsand;
2）Pouringtemperatureis1560℃;
3）CastingsizesareshowninFigure2andFigure3,themoldthicknessis40mm.
Requirements:
1）Writeoutthe2Dor3Dmathematicalmodelthatdescribesthetemperaturefieldofthecastingcoolingprocess;
（10points）
2）DeducetheFDMformatofthemathematicalmodel;
3）DrawtheFDMgridmap,anddescribeitusingdatastructure;
4）ProvidethethermalpropertiesthermalpropertiesthermalpropertiesthermalpropertiesparametersofZG25,resinsandandtheair;
5）Programtosimulatethisphysicalprocess,assumingthecavitywasfilledveryfastandtheinitialtemperatureevenlydistributed.Pleaseprovidethemaincodeoftheprogram.（25points）
6）Howlongdoesittakewhilethehighesttemperatureofthecastingdropsto1450℃?
F
igure1:
3DmodeloftheH-shapedcasting
Figure2:
DimensionofH-shapedcasting
Figure3:
A2DsliceoftheH-shapedcasting
Q4
（3.1）The3Dmathematicalmodelis
WhereTistemperature（K）,
isdensity（kg/m3）,Cisheatcapacity（J/（kg•K））,
isthermalconductivity（W/（m•K））,Listhelatentheat（J/kg）
Ifwedonottakethelatentheatintoconsideration,thenthelasttermoftheaboveequationshouldbeomitted.
（3.2）WediscretizetheFourierdifferentialequationofheatconductionbasedonFDM.Andtheheatexchangeprocessisshownasfollows.
Thededucingprocess:
Thequantityofheatwhichtheelementihasabsorbedis:
ThequantityofheatflowintoIfromthe6elementadjacenttoitis:
Accordingtothelawofconservationofenergy
Afterrearrangementwegettheexplictschemoftemperatureevolution
（3.3）Hereweintroduceafunctionofflag
WhereZG25wouldbecutintofoursmallpiecestobehandledeasily.
for（i=8;
i<
=88;
i++）{
for（j=8;
j<
=28;
j++）{
for（k=8;
k<
=24;
k++）{
flag[i][j][k]=1;
//ZG
}
}
for（i=38;
=58;
for（k=24;
=100;
flag[i][j][k]=1;
for（k=100;
=116;
k++）{flag[i][j][k]=1;
}
for（k=116;
=132;
if（（（i-48.0）*（i-48.0）+（j-18.0）*（j-18.0））<
=6.0*6.0）
（3.4）ThethermalpropertiesparametersofZG25,resinsandandtheairare:
heat
conduct
coefficient
density
specific
ZG25
27.2
7750
470
sand
0.73
1610
1054.9
air
0.0259
1.205
0.001005
（3.5）Theresultsofthissimulationislistingfollowing:
Ourprogrammaininclude5sectionsbelow:
4invokingfunctionsandthemainfunction
Init（）isusedtoinitializevariablessuchastemperatureandthermalpropertiesparameters.
voidinit（）
{
for（i=0;
=NX+1;
for（j=0;
=NY+1;
for（k=0;
=NZ+1;
if（flag[i][j][k]==-1）{//airlayer
CP[i][j][k]=0.001005;
rho[i][j][k]=1.205;
therm[i][j][k]=0.0259;
tem[i][j][k]=25;
tem_0[i][j][k]=25;
if（flag[i][j][k]==1）{//ZG
CP[i][j][k]=470;
rho[i][j][k]=7750;
therm[i][j][k]=27.2;
tem[i][j][k]=1560;
tem_0[i][j][k]=1560;
if（flag[i][j][k]==0）{//sand
CP[i][j][k]=1054.9;
rho[i][j][k]=1610;
therm[i][j][k]=0.73;
calculate_temperature（）isusedtocalculatetheevolutionofthetemperaturefield.
voidcaculate_temperature（）
doubleTEMPLE,t1,t2,t3,t4,t5,t6;
TEMPLE=0;
Tmax=0;
for（i=1;
=NX;
for（j=1;
=NY;
for（k=1;
=NZ;
if（flag[i][j][k]==1||flag[i][j][k]==0）{t1=（tem_0[i+1][j][k]-tem_0[i][j][k]）/（dx/（2*therm[i+1][j][k]）+dx/（2*therm[i][j][k]））;
t2=（tem_0[i-1][j][k]-tem_0[i][j][k]）/（dx/（2*therm[i-1][j][k]）+dx/（2*therm[i][j][k]））;
t3=（tem_0[i][j+1][k]-tem_0[i][j][k]）/（dx/（2*therm[i][j+1][k]）+dx/（2*therm[i][j][k]））;
t4=（tem_0[i][j-1][k]-tem_0[i][j][k]）/（dx/（2*therm[i][j-1][k]）+dx/（2*therm[i][j][k]））;
t5=（tem_0[i][j][k+1]-tem_0[i][j][k]）/（dx/（2*therm[i][j][k+1]）+dx/（2*therm[i][j][k]））;
t6=（tem_0[i][j][k-1]-tem_0[i][j][k]）/（dx/（2*therm[i][j][k-1]）+dx/（2*therm[i][j][k]））;
TEMPLE=dt/rho[i][j][k]/CP[i][j][k]/（dx）*（t1+t2+t3+t4+t5+t6）;
tem[i][j][k]=tem_0[i][j][k]+TEMPLE;
if（tem[i][j][k]>
1560）printf（"
x=%d,y=%d\n"
i,j）;
Tmax）
Tmax=tem[i][j][k];
}}}
tem_0[i][j][k]=tem[i][j][k];
data（）isusedtooutputtheresultsfile.
voiddata（intn）
sprintf（fName,"
temperature%d.dat"
n）;
if（（fp=fopen（fName,"
w"
））==NULL）return;
fprintf（fp,"
VARIABLES=X,Y,Z,T\n"
）;
Zone,I=%d,J=%d,K=%d,F=BLOCK\n"
NX+2,NY+2,NZ+2）;
for（k=0;
for（j=0;
j<
j++）{
for（i=0;
i<
i++）{
fprintf（fp,"
%d\t"
i）;
}
fprintf（fp,"
\n"
）;
j）;
k）;
%.5e\t"
tem[i][j][k]）;
fclose（fp）;
output（）isusedtodrawthegridmap.
voidoutput（）
FILE*file;
file=fopen（"
flag.dat"
"
if（file==0）printf（"
cannotopenflagfile"
fprintf（file,"
Title=\"
flag\"
variables=\"
x\"
\"
y\"
z\"
zoneT=\"
BOX\"
I=%d,J=%d,K=%d\n"
NX+2,NY+2,NZ+2）;
for（j=0;
for（i=0;
fprintf（file,"
%d,%d,%d,%d\n"
i,j,k,flag[i][j][k]）;
fclose（file）;
（3.6）Throughtheresultsoftheprocedure,wecaneasilygetthetimeofdecresingfrom1560to1450is133.62,whereitcost6681stepscoupledwithourTimeintervalofdtis0.02
Whatdoyouthinkofthiscourse？
Firstofall,thiscourseisaninternationalcourseofferedbytheInstituteofmaterialsscience,materialsscience,materialsscience,andotherprofessionalgraduatestudentsordoctoralstudentsinHuazhongUniversityofScienceandTechnology.Thiscourseemphasizestheconnotationandextensionofteachingguide,basedon"
student-centered,teacherled"
thestartingpointtothecurriculumasthecarrierandplatform,thepositioninginthecoursewiththehelpofavarietyofteachingmethodstocultivatestudents'
comprehensiveability,solidfoundationoftheoreticalknowledgeandanalysisofengineeringproblems,makethecourseaswiththecharacteristicsof"
research"
teaching.
Meanwhile,teachersinthiscourseareverydetailed,fromtheorytothemodel,fromtheprincipletopractice,fromdiscretetoprogramming,stepbystepisverydetailed.Wenotonlystudytheproblemsinphysicsandengineeringbackground,butalsolearnhowtousethefiniteelementmethodandfinitedifferencetothediscretemodel.What’smore,welearntousesomesoftwaressuchasMATLABprogrammingtosimulatethephysicalprocess,soIthinkitisveryusefulforthiscourseandtheirgreatharvest.
Finally,thiscoursehasbeenfinishedononlyafewweeks,leavingaverydeepimpressiononme.Ialsolearnedalotofacknowledgefromit.Thankprofessorsintheclassforgivingussuchawonderfulcourse.
升级会员 