numerical analysis chapra 5ESM29.docx
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numericalanalysischapra5ESM29
CHAPTER29
29.1HerearetheresultsofusingLiebmann’smethodtoobtainthesolution.Noticethatafter6iterationsalltherelativeerrorestimateshavefallenbelow1%andthecomputationisterminated.
iteration=1
185.416.62
23.48.6422.578
61.0256.89874.8428
ea:
100100100
100100100
100100100
iteration=2
23.0413.4122.4724
41.1338.476851.222
71.204479.977675.39132
ea:
21.87559.7315436226.04261227
43.1072210177.5449101855.92128382
14.3030486928.857580120.727563863
iteration=6
37.8066606632.4163214834.29437766
59.0195280157.5172739454.73884006
80.7192162883.8443383477.14262488
ea:
0.6881817460.1302044540.019913358
0.0109336940.0208740120.049145839
0.0337676720.0371761680.02465349
Therefore,theresultsare:
29.2ThefluxesforProb.29.1canbecalculatedasinExample29.2.Forexample,fori=j=1,
Alltheresultsaresummarizedbelow:
qx:
0.6758001240.086050934-0.430800124
0.0608267880.1048768550.184173212
-0.5841862890.0876264890.829186289
qy:
-1.445978436-1.409173212-1.341101582
-1.051357613-1.259986413-1.049782057
-1.494021564-1.530826788-1.598898418
qn:
1.5961075921.411798111.408595825
1.0531157241.264343671.065815246
1.6041739471.5333326641.801118001
theta:
-64.95024572-86.50558167-107.8085014
-86.68881693-85.24186843-80.04932455
-111.356292-86.72389108-62.588814
29.3Theplateisredrawnbelow
After15iterationsoftheLiebmannmethod,theresultis
100
100
100
100
100
100
100
75
85.32617
88.19118
88.54443
87.79909
86.06219
82.39736
73.69545
50
75
78.10995
78.88691
78.1834
76.58771
74.05069
69.82967
62.38146
50
75
73.23512
71.06672
68.71675
66.32057
63.72554
60.48986
55.99875
50
75
68.75568
63.42793
59.30269
56.25934
54.04625
52.40787
51.1222
50
75
63.33804
54.57569
48.80562
45.37425
43.79945
43.97646
46.08048
50
75
54.995
42.71618
35.95756
32.62971
31.80514
33.62176
39.22063
50
75
38.86852
25.31308
19.66293
17.3681
17.16645
19.48972
27.17735
50
0
0
0
0
0
0
0
withpercentapproximateerrorsof
0
0
0
0
0
0
0
0
0.012%
0.011%
0.007%
0.005%
0.004%
0.004%
0.003%
0
0
0.011%
0.012%
0.008%
0.005%
0.006%
0.006%
0.005%
0
0
0.024%
0.010%
0.001%
0.001%
0.004%
0.007%
0.006%
0
0
0.054%
0.016%
0.007%
0.011%
0.002%
0.007%
0.008%
0
0
0.101%
0.040%
0.008%
0.007%
0.003%
0.008%
0.011%
0
0
0.234%
0.119%
0.063%
0.033%
0.012%
0.010%
0.015%
0
0
0.712%
0.292%
0.219%
0.126%
0.030%
0.001%
0.014%
0
0
0
0
0
0
0
0
29.4ThesolutionisidenticaltoProb.29.3,exceptthatnowthebottomedgemustbemodeled.Thismeansthatthenodesalongthebottomedgearesimulatedwithequationsoftheform
Theresultingsimulation（after15iterations）yields
100
100
100
100
100
100
100
75
86.4529
90.2627
91.2337
90.6948
88.7323
84.4436
74.8055
50
75
80.5554
83.3649
83.9691
82.8006
79.7785
74.2277
64.7742
50
75
77.4205
78.6771
78.4736
76.7481
73.3375
67.8999
60.0537
50
75
75.5117
75.4794
74.5080
72.3743
68.9073
63.9608
57.5247
50
75
74.2631
73.2996
71.7406
69.3405
65.9436
61.4870
56.0597
50
75
73.4348
71.8433
69.8853
67.3320
64.0357
59.9572
55.1934
50
75
72.8998
70.9218
68.7359
66.1171
62.9167
59.0892
54.7153
50
75
72.5345
70.4401
68.1797
65.5685
62.4467
58.7477
54.5354
50
withpercentapproximateerrorsof
0
0
0
0
0
0
0
0
0.009%
0.018%
0.024%
0.024%
0.018%
0.011%
0.004%
0
0
0.042%
0.066%
0.074%
0.069%
0.053%
0.034%
0.015%
0
0
0.079%
0.140%
0.155%
0.140%
0.110%
0.071%
0.032%
0
0
0.113%
0.224%
0.260%
0.239%
0.187%
0.124%
0.057%
0
0
0.133%
0.327%
0.388%
0.359%
0.284%
0.190%
0.089%
0
0
0.173%
0.471%
0.544%
0.502%
0.395%
0.261%
0.124%
0
0
0.289%
0.628%
0.709%
0.651%
0.508%
0.330%
0.153%
0
0
0.220%
0.665%
0.779%
0.756%
0.620%
0.407%
0.180%
0
29.5ThesolutionisidenticaltoExamples29.1and29.3,exceptthatnowheatbalancesmustbedevelopedforthethreeinteriornodesonthebottomedge.Forexample,usingthecontrol-volumeapproach,node1,0canbemodeledas
UsingLiebmann’smethodanditeratingtoahighlevelofprecision,theresultsare
100
100
100
75
81.6571
80.1712
72.5078
50
75
71.4579
66.5204
59.8602
50
75
62.6549
54.5930
50.4129
50
75
49.5694
38.7843
37.1984
50
Thefluxesforthecomputednodescanbedeterminedas
qx:
-0.1267
0.2242
0.7392
0.2077
0.2841
0.4048
0.5000
0.2999
0.1125
0.8873
0.3031
-0.2748
qy:
-0.6993
-0.8202
-0.9834
-0.4656
-0.6267
-0.5413
-0.5363
-0.6795
-0.5552
-0.6412
-0.7746
-0.6475
qn:
0.7107
0.8503
1.2303
0.5098
0.6881
0.6759
0.7332
0.7428
0.5665
1.0947
0.8318
0.7034
（degrees）:
-100.269
-74.7153
-53.0695
-65.9517
-65.6094
-53.2145
-47.0062
-66.1845
-78.5428
-35.8536
-68.6311
-112.995
29.6ThesolutionisidenticaltoExample29.4,exceptthatnowheatbalancesmustbedevelopedfortheinteriornodesatthelowerleftandtheupperrightedges.Thebalancesfornodes1,1and3,3canbewrittenas
Usingtheappropriateboundaryconditions,simpleLaplacianscanbeusedfortheremaininginteriornodes.Theresultingsimulationyields
75
100
100
100
50
75
86.02317
94.09269
100
50
63.97683
75
86.02317
100
50
55.90731
63.97683
75
100
50
50
50
75
29.7Thenodestobesimulatedare
SimpleLaplaciansareusedforallinteriornodes.Balancesfortheedgesmusttakeinsulationintoaccount.Forexample,node1,0ismodeledas
Thecornernode,0,0wouldbemodeledas
Theresultingsetofequationscanbesolvedfor
0
25
50
75
100
23.89706
32.16912
45.58824
60.29412
75
31.25
34.19118
39.88971
45.58824
50
32.72059
33.45588
34.19118
32.16912
25
32.72059
32.72059
31.25
23.89706
0
Thefluxescanbecomputedas
Jx
-1.225
-1.225
-1.225
-1.225
-1.225
-0.40533
-0.53143
-0.68906
-0.72059
-0.72059
-0.14412
-0.21167
-0.27923
-0.2477
-0.21618
-0.03603
-0.03603
0.031526
0.225184
0.351287
0
0.036029
0.216176
0.765625
1.170956
Jy
1.170956
0.351287
-0.21618
-0.72059
-1.225
0.765625
0.225184
-0.2477
-0.72059
-1.225
0.216176
0.031526
-0.27923
-0.68906
-1.225
0.036029
-0.03603
-0.21167
-0.53143
-1.225
0
-0.03603
-0.14412
-0.40533
-1.225
Jn
1.694628
1.274373
1.243928
1.421222
1.732412
0.866299
0.577174
0.732232
1.019066
1.421222
0.259812
0.214008
0.394888
0.732232
1.243928
0.050953
0.050953
0.214008
0.577174
1.274373
0
0.050953
0.259812
0.866299
1.694628
（degrees）
136.2922
163.999
-169.992
-149.534
-135
117.8973
157.0362
-160.228
-135
-120.466
123.6901
171.5289
-135
-109.772
-100.008
135
-135
-81.5289
-67.0362
-73.999
0
-45
-33.6901
-27.8973
-46.2922
29.8Node0,3:
Therearetwoapproachesformodelingthisnode.OnewouldbetoconsideritaDirichletnodeandnotmodelitatall（i.e.,setitstemperatureat50oC）.Thesecondalternativeistouseaheatbalancetomodelitasshownhere
Node2,3:
Node2,2:
Node5,3:
29.9Node0,0:
Node1,1:
Node2,1:
29.10Thecontrolvolumeisdrawnasin
Afluxbalancearoundthenodecanbewrittenas（notex=y=h）
Collectingandcancelingtermsgives
29.11AsetupsimilartoFig.29.11,butwith>45ocanbedrawnasin
Thenormalderivativeatnode3canbeapproximatedbythegradientbetweennodes1and7,
Whenisgreaterthan45oasshown,thedistancefromnode5to7isycot,andlinearinterpolationcanbeusedtoestimate
ThelengthL17isequaltoy/sin.Thislength,alongwiththeapproximationforT7canbesubstitutedintothegradientequationtogive
29.12ThefollowingVBAprogramimplementsLiebmann’smethodwithrelaxation.
OptionExplicit
SubLiebmann（）
DimnxAsInteger,nyAsInteger,lAsInteger
DimiAsInteger,jAsInteger
DimT（20,20）AsDouble,ea（20,20）AsDouble,Told（20,20）AsDouble
Dimqy（20,20）AsDouble,qx（20,20）AsDouble,qn（20,20）AsDouble
Dimth（20,20）AsDouble
DimTritAsDouble,TlefAsDouble,TtopAsDouble,TbotAsDouble
DimlamAsDouble,emaxAsDouble,esAsDouble
DimpiAsDouble
DimkAsDouble,xAsDouble,yAsDouble,dxAsDouble,dyAsDouble
nx=4
ny=4
pi=4*Atn
（1）
x=40
y=40
k=0.49
lam=1.5
es=1
dx=x/nx
dy=y/ny
Tbot=0
Tlef=75
Trit=50
Ttop=100
Fori=1Tonx-1
T（i,0）=Tbot
Nexti
Fori=1Tonx-1
T（i,ny）=Ttop
Nexti
Forj=1Tony-1
T（0,j）=Tlef
Nextj
Forj=1Tony-1
T（nx,j）=Trit
Nextj
l=0
Sheets（"sheet1"）.Select
Range（"a5:
z5000"）.ClearContents
Range（"a5"）.Select
Do
l=l+1
emax=0
Forj=1Tony-1
Fori=1Tonx-1
Told（i,j）=T（i,j）
T（i,j）=（T（i+1,j）+T（i-1,j）+T（i,j+1）+T（i,j-1））/4
T（i,j）=lam*T（i,j）+（1-lam）*Told（i,j）
ea（i,j）=Abs（（T（i,j）-Told（i,j））/T（i,j））*100
If（ea（i,j）>emax）Thenemax=ea（i,j）
Nexti
Nextj
ActiveCell.Value="iteration="
ActiveCell.Offset（0,1）.Select
ActiveCell.Value=l
ActiveCell.Offset（1,-1）.Select
Forj=1Tony-1
Fori=1Tonx-1
ActiveCell.Value=T（i,j）
ActiveCell.Offset（0,1）.Select
Nexti
ActiveCell.Offset（1,-（nx-1））.Select
Nextj
ActiveCell.Value="ea="
ActiveCell.Offset（1,0）.Select
Forj=1Tony-1
Fori=1Tonx-1
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