568 lines
124 KiB
Plaintext
568 lines
124 KiB
Plaintext
Section 1: Analytical
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Analytical Temperature Results x_bar = 0.5:
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x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
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k2/k1 = 0.0625 0.0 0.011667 0.025181 0.042681 0.066937 0.066937 0.847794 4.228179 20.573130 100.0
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k2/k1 = 0.125 0.0 0.097641 0.210739 0.357195 0.560194 0.560194 3.303981 10.626282 32.674022 100.0
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k2/k1 = 0.25 0.0 0.464798 1.003171 1.700341 2.666666 2.666666 8.626608 20.264910 45.242343 100.0
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k2/k1 = 0.5 0.0 1.470842 3.174511 5.380689 8.438605 8.438605 17.437015 32.027881 56.890833 100.0
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k2/k1 = 1.0 0.0 3.440413 7.425426 12.585848 19.738549 19.738549 30.015765 45.044332 67.203196 100.0
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k2/k1 = 2.0 0.0 6.351689 13.708819 23.235988 36.441301 36.441301 45.815586 58.792577 76.392720 100.0
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k2/k1 = 4.0 0.0 9.653341 20.834759 35.314219 55.383747 55.383747 62.464754 71.993744 84.344156 100.0
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k2/k1 = 8.0 0.0 12.567937 27.125318 45.976504 72.105546 72.105546 76.701212 82.797389 90.513336 100.0
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k2/k1 = 16.0 0.0 14.652807 31.625083 53.603455 84.066993 84.066993 86.741990 90.264767 94.669752 100.0
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Analytical Temperature Results x_bar = 2/pi:
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x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.637 x = 0.750 x = 0.875 x = 1.000
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k2/k1 = 0.0625 0.0 0.043265 0.093379 0.158275 0.248224 0.377466 0.392138 4.086217 20.545109 100.0
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k2/k1 = 0.125 0.0 0.217770 0.470012 0.796654 1.249403 1.899926 1.973771 10.004385 32.490342 100.0
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k2/k1 = 0.25 0.0 0.723845 1.562272 2.647999 4.152891 6.315166 6.560619 18.929061 44.739811 100.0
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k2/k1 = 0.5 0.0 1.775153 3.831304 6.493931 10.184517 15.487261 16.089210 30.677503 56.308955 100.0
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k2/k1 = 1.0 0.0 3.440413 7.425426 12.585848 19.738549 30.015765 31.182398 45.044332 67.203196 100.0
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k2/k1 = 2.0 0.0 5.484175 11.836467 20.062415 31.464146 47.846498 49.706164 60.645947 77.284348 100.0
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k2/k1 = 4.0 0.0 7.434698 16.046273 27.197894 42.654810 64.863775 67.384857 74.763593 85.702464 100.0
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k2/k1 = 8.0 0.0 8.921798 19.255874 32.638061 51.186694 77.837932 80.863285 85.279652 91.742445 100.0
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k2/k1 = 16.0 0.0 9.877475 21.318505 36.134152 56.669658 86.175696 89.525116 91.966805 95.516633 100.0
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Analytical Heat Convection Results x_bar = 0.5:
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Heat Convection
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k2/k1 = 0.0625 -1.241829
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k2/k1 = 0.125 -1.756437
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k2/k1 = 0.25 -2.486946
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k2/k1 = 0.5 -3.529489
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k2/k1 = 1.0 -4.985127
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k2/k1 = 2.0 -6.832006
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k2/k1 = 4.0 -8.797049
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k2/k1 = 8.0 -10.486214
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k2/k1 = 16.0 -11.680625
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Analytical Heat Convection Results x_bar = 2/pi:
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Heat Convection
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k2/k1 = 0.0625 -1.241978
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k2/k1 = 0.125 -1.758799
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k2/k1 = 0.25 -2.501202
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k2/k1 = 0.5 -3.564213
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k2/k1 = 1.0 -4.985127
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k2/k1 = 2.0 -6.610807
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k2/k1 = 4.0 -8.118713
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k2/k1 = 8.0 -9.254592
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k2/k1 = 16.0 -9.980639
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Section 2: FDM and FEM with Varying k_ratio
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FDM Temperature Results x_bar = 0.5:
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x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
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k2/k1 = 0.0625 0.0 0.023566 0.050814 0.086002 0.134628 1.249213 5.486829 23.441518 100.0
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k2/k1 = 0.125 0.0 0.131160 0.282813 0.478656 0.749289 3.850963 11.766340 34.389643 100.0
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k2/k1 = 0.25 0.0 0.522237 1.126073 1.905858 2.983433 9.158380 21.057315 46.117073 100.0
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k2/k1 = 0.5 0.0 1.534785 3.309379 5.601064 8.767916 17.841592 32.490766 57.293304 100.0
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k2/k1 = 1.0 0.0 3.487383 7.519670 12.726906 19.922720 30.231460 45.263865 67.368749 100.0
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k2/k1 = 2.0 0.0 6.373779 13.743460 23.260557 36.412116 45.832593 58.833740 76.431273 100.0
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k2/k1 = 4.0 0.0 9.664493 20.839062 35.269736 55.211305 62.353389 71.931152 84.318726 100.0
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k2/k1 = 8.0 0.0 12.587755 27.142347 45.937931 71.911316 76.562507 82.709060 90.471024 100.0
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k2/k1 = 16.0 0.0 14.689868 31.675027 53.609410 83.920263 86.634225 90.194225 94.635027 100.0
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FEM Temperature Results x_bar = 0.5:
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x = 0.000 x = 0.125 x = 0.250 x = 0.375 x = 0.500 x = 0.625 x = 0.750 x = 0.875 x = 1.000
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k2/k1 = 0.0625 0.0 0.008040 0.017369 0.029485 0.046331 0.431694 2.667176 16.333414 100.0
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k2/k1 = 0.125 0.0 0.092421 0.199668 0.338947 0.532604 2.747608 9.300941 30.539969 100.0
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k2/k1 = 0.25 0.0 0.461952 0.998014 1.694185 2.662150 8.197855 19.452994 44.279989 100.0
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k2/k1 = 0.5 0.0 1.463251 3.161248 5.366397 8.432466 17.199749 31.637279 56.504681 100.0
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k2/k1 = 1.0 0.0 3.408346 7.363486 12.499933 19.641717 29.852520 44.852497 67.048062 100.0
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k2/k1 = 2.0 0.0 6.282365 13.572595 23.040247 36.204196 45.614623 58.635706 76.298138 100.0
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k2/k1 = 4.0 0.0 9.556088 20.645239 35.046460 55.070107 62.227195 71.830962 84.259013 100.0
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k2/k1 = 8.0 0.0 12.463187 26.925816 45.708094 71.823222 76.490410 82.656431 90.442107 100.0
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k2/k1 = 16.0 0.0 14.553527 31.441845 53.374310 83.869497 86.594484 90.166500 94.620483 100.0
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FDM Temperature Results x_bar = 2/pi:
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x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
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k2/k1 = 0.0625 0.0 0.077055 0.173629 0.314183 0.534322 3.515595 11.139042 33.471059 100.0
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k2/k1 = 0.125 0.0 0.319719 0.720423 1.303613 2.217011 8.401972 20.134133 45.159372 100.0
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k2/k1 = 0.25 0.0 0.980497 2.209358 3.997855 6.799021 16.282888 31.141948 56.281369 100.0
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k2/k1 = 0.5 0.0 2.330072 5.250358 9.500576 16.157318 27.426136 43.221817 66.151538 100.0
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k2/k1 = 1.0 0.0 4.469024 10.070065 18.221884 30.989359 41.796025 56.052044 74.933939 100.0
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k2/k1 = 2.0 0.0 7.113289 16.028395 29.003540 49.325368 57.925784 68.916457 82.750907 100.0
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k2/k1 = 4.0 0.0 9.660074 21.767073 39.387737 66.985431 72.825251 80.167607 89.163985 100.0
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k2/k1 = 8.0 0.0 11.617645 26.178075 47.369488 80.559732 84.071349 88.450249 93.741604 100.0
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k2/k1 = 16.0 0.0 12.883057 29.029430 52.529043 89.334419 91.281472 93.699357 96.600545 100.0
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FEM Temperature Results x_bar = 2/pi:
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x = 0.000 x = 0.159 x = 0.318 x = 0.477 x = 0.637 x = 0.727 x = 0.818 x = 0.909 x = 1.000
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k2/k1 = 0.0625 0.0 0.051867 0.117450 0.214096 0.367363 2.434575 8.623646 29.412987 100.0
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k2/k1 = 0.125 0.0 0.275874 0.624708 1.138757 1.953971 7.451625 18.477330 43.210596 100.0
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k2/k1 = 0.25 0.0 0.917965 2.078702 3.789190 6.501798 15.648464 30.261653 55.446231 100.0
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k2/k1 = 0.5 0.0 2.241155 5.075024 9.251075 15.873740 27.039252 42.794017 65.812018 100.0
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k2/k1 = 1.0 0.0 4.339198 9.825975 17.911409 30.733838 41.542855 55.828147 74.785094 100.0
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k2/k1 = 2.0 0.0 6.933484 15.700652 28.620142 49.108745 57.744459 68.779447 82.672214 100.0
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k2/k1 = 4.0 0.0 9.434441 21.363990 38.943632 66.822621 72.697960 80.078383 89.116689 100.0
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k2/k1 = 8.0 0.0 11.358900 25.721867 46.887445 80.453254 83.990156 88.394995 93.713291 100.0
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k2/k1 = 16.0 0.0 12.604001 28.541357 52.026991 89.272101 91.234400 93.667692 96.584539 100.0
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FDM Heat Convection Results x_bar = 0.5:
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Heat Convection
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k2/k1 = 0.0625 -1.583037
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k2/k1 = 0.125 -2.012353
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k2/k1 = 0.25 -2.674530
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k2/k1 = 0.5 -3.665089
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k2/k1 = 1.0 -5.082312
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k2/k1 = 2.0 -6.905215
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k2/k1 = 4.0 -8.864016
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k2/k1 = 8.0 -10.561319
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k2/k1 = 16.0 -11.768665
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FEM Heat Convection Results x_bar = 0.5:
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Heat Convection
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k2/k1 = 0.0625 -1.365065
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k2/k1 = 0.125 -1.845516
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k2/k1 = 0.25 -2.549900
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k2/k1 = 0.5 -3.572301
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k2/k1 = 1.0 -5.014775
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k2/k1 = 2.0 -6.861111
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k2/k1 = 4.0 -8.842519
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k2/k1 = 8.0 -10.559111
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k2/k1 = 16.0 -11.780305
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FDM Heat Convection Results x_bar = 2/pi:
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Heat Convection
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k2/k1 = 0.0625 -1.432463
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k2/k1 = 0.125 -1.898803
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k2/k1 = 0.25 -2.603336
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k2/k1 = 0.5 -3.639854
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k2/k1 = 1.0 -5.047653
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k2/k1 = 2.0 -6.678556
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k2/k1 = 4.0 -8.208090
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k2/k1 = 8.0 -9.370579
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k2/k1 = 16.0 -10.118258
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FEM Heat Convection Results x_bar = 2/pi:
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Heat Convection
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k2/k1 = 0.0625 -1.308439
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k2/k1 = 0.125 -1.805860
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k2/k1 = 0.25 -2.533474
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k2/k1 = 0.5 -3.587897
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k2/k1 = 1.0 -5.013420
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k2/k1 = 2.0 -6.664558
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k2/k1 = 4.0 -8.214918
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k2/k1 = 8.0 -9.394792
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k2/k1 = 16.0 -10.154415
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Section 3: Convergence of FDM and FEM with Varying k Ratio
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Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 0.0625
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True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 0.066937 0.400229 NaN 4.979197 NaN NaN NaN -0.363773 NaN 6.434572 NaN NaN NaN
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N = 8 0.066937 0.134628 NaN 1.011272 2.299743 NaN NaN 0.046331 NaN 0.307843 4.385579 NaN NaN
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N = 16 0.066937 0.082453 0.069697 0.231796 2.125242 0.183010 2.347809 0.062673 0.063352 0.063696 2.272928 0.010706 4.649294
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N = 32 0.066937 0.070718 0.067312 0.056482 2.036995 0.050588 2.152561 0.065888 0.066675 0.015670 2.023150 0.011806 2.345888
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N = 64 0.066937 0.067876 0.066968 0.014026 2.009682 0.013561 2.045906 0.066671 0.066924 0.003966 1.982240 0.003772 2.036753
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N = 128 0.066937 0.067171 0.066939 0.003501 2.002449 0.003469 2.012080 0.066870 0.066937 0.001003 1.984061 0.001005 1.981623
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Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 0.0625
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True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 -1.241829 -2.323339 NaN 0.870901 [None] NaN NaN -1.681631 NaN 0.354157 [None] NaN NaN
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N = 8 -1.241829 -1.583037 NaN 0.274763 1.664323 NaN NaN -1.365065 NaN 0.099238 1.835423 NaN NaN
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N = 16 -1.241829 -1.335328 -1.210764 0.075292 1.867617 0.102880 1.579472 -1.273756 -1.236742 0.025710 1.948582 0.029928 1.793664
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N = 32 -1.241829 -1.265851 -1.238768 0.019344 1.960577 0.021863 1.834029 -1.249887 -1.241439 0.006489 1.98625 0.006805 1.935639
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N = 64 -1.241829 -1.247878 -1.241605 0.004871 1.989623 0.005052 1.950668 -1.243848 -1.241803 0.001626 1.9965 0.001647 1.982806
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N = 128 -1.241829 -1.243344 -1.241814 0.001220 1.997371 0.001232 1.987025 -1.242334 -1.241827 0.000407 1.999121 0.000408 1.995625
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Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 0.125
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True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 0.560194 1.321586 NaN 1.359157 NaN NaN NaN 0.284007 NaN 0.493020 NaN NaN NaN
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N = 8 0.560194 0.749289 NaN 0.337552 2.009531 NaN NaN 0.532604 NaN 0.049252 3.323401 NaN NaN
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N = 16 0.560194 0.607201 0.560273 0.083911 2.008170 0.083760 2.009981 0.553343 0.555230 0.012231 2.009667 0.003400 3.583391
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N = 32 0.560194 0.571925 0.560274 0.020940 2.002601 0.020794 2.010016 0.558340 0.559926 0.003310 1.885500 0.002833 2.053153
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N = 64 0.560194 0.563125 0.560201 0.005233 2.000690 0.005220 2.003237 0.559704 0.560217 0.000875 1.920215 0.000915 1.872828
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N = 128 0.560194 0.560927 0.560195 0.001308 2.000175 0.001307 2.000862 0.560068 0.560200 0.000226 1.955459 0.000236 1.907766
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Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 0.125
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True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 -1.756437 -2.635211 NaN 0.500316 [None] NaN NaN -2.090293 NaN 0.190075 [None] NaN NaN
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N = 8 -1.756437 -2.012353 NaN 0.145701 1.779824 NaN NaN -1.845516 NaN 0.050716 1.906073 NaN NaN
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N = 16 -1.756437 -1.823780 -1.741900 0.038341 1.926064 0.047006 1.723786 -1.779124 -1.754414 0.012916 1.973249 0.014085 1.882382
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N = 32 -1.756437 -1.773512 -1.755242 0.009721 1.97963 0.010409 1.907407 -1.762136 -1.756295 0.003245 1.993049 0.003326 1.966545
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N = 64 -1.756437 -1.760722 -1.756356 0.002439 1.99477 0.002486 1.974522 -1.757864 -1.756428 0.000812 1.998239 0.000817 1.991312
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N = 128 -1.756437 -1.757509 -1.756432 0.000610 1.998684 0.000613 1.993462 -1.756794 -1.756437 0.000203 1.999555 0.000203 1.997799
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Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 0.25
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True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 2.666666 3.862069 NaN 0.448276 NaN NaN NaN 2.715719 NaN 0.018395 NaN NaN NaN
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N = 8 2.666666 2.983433 NaN 0.118788 1.916002 NaN NaN 2.662150 NaN 0.001694 3.441210 NaN NaN
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N = 16 2.666666 2.747136 2.660209 0.030176 1.976907 0.032677 1.894660 2.661848 2.661847 0.001807 -0.093302 6.420173e-07 7.472074
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N = 32 2.666666 2.686866 2.666231 0.007575 1.994067 0.007739 1.971109 2.664898 2.662123 0.000663 1.446493 1.042592e-03 -3.337586
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N = 64 2.666666 2.671721 2.666638 0.001896 1.998506 0.001906 1.992582 2.666147 2.667012 0.000195 1.767683 3.244563e-04 1.288577
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N = 128 2.666666 2.667930 2.666664 0.000474 1.999626 0.000475 1.998132 2.666526 2.666692 0.000052 1.891986 6.204964e-05 1.718994
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Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 0.25
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True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
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N = 4 -2.486946 -3.171745 NaN 0.275357 [None] NaN NaN -2.730060 NaN 0.097756 [None] NaN NaN
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N = 8 -2.486946 -2.674530 NaN 0.075427 1.868146 NaN NaN -2.549900 NaN 0.025314 1.949262 NaN NaN
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N = 16 -2.486946 -2.535136 -2.480834 0.019377 1.960724 0.021889 1.834707 -2.502844 -2.486207 0.006392 1.98551 0.006691 1.936807
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N = 32 -2.486946 -2.499080 -2.486500 0.004879 1.989662 0.005059 1.950854 -2.490931 -2.486894 0.001602 1.996036 0.001624 1.981971
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N = 64 -2.486946 -2.489985 -2.486917 0.001222 1.99738 0.001234 1.987073 -2.487943 -2.486943 0.000401 1.998875 0.000402 1.995088
|
|
N = 128 -2.486946 -2.487706 -2.486944 0.000306 1.999343 0.000306 1.996725 -2.487196 -2.486946 0.000100 1.999654 0.000100 1.998615
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 0.5
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 8.438605 9.674633 NaN 0.146473 NaN NaN NaN 8.652638 NaN 0.025364 NaN NaN NaN
|
|
N = 8 8.438605 8.767916 NaN 0.039024 1.908190 NaN NaN 8.432466 NaN 0.000727 5.123736 NaN NaN
|
|
N = 16 8.438605 8.522370 8.431179 0.009926 1.975035 0.010816 1.884659 8.429022 8.428967 0.001136 -0.642537 0.000006 5.998275
|
|
N = 32 8.438605 8.459639 8.438114 0.002493 1.993614 0.002551 1.968752 8.435145 8.431226 0.000410 1.469764 0.000465 -0.830074
|
|
N = 64 8.438605 8.443869 8.438574 0.000624 1.998394 0.000628 1.992015 8.437603 8.439252 0.000119 1.788181 0.000195 1.316727
|
|
N = 128 8.438605 8.439921 8.438603 0.000156 1.999598 0.000156 1.997993 8.438337 8.438650 0.000032 1.903464 0.000037 1.743720
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 0.5
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -3.529489 -4.044925 NaN 0.146037 [None] NaN NaN -3.695537 NaN 0.047046 [None] NaN NaN
|
|
N = 8 -3.529489 -3.665089 NaN 0.038419 1.926443 NaN NaN -3.572301 NaN 0.012130 1.955519 NaN NaN
|
|
N = 16 -3.529489 -3.563869 -3.527097 0.009741 1.979723 0.010426 1.907890 -3.540313 -3.529099 0.003067 1.983847 0.003178 1.945806
|
|
N = 32 -3.529489 -3.538115 -3.529327 0.002444 1.994793 0.002490 1.974639 -3.532207 -3.529456 0.000770 1.993661 0.000779 1.980542
|
|
N = 64 -3.529489 -3.531648 -3.529479 0.000612 1.998689 0.000614 1.993491 -3.530170 -3.529486 0.000193 1.997282 0.000194 1.992449
|
|
N = 128 -3.529489 -3.530029 -3.529489 0.000153 1.999672 0.000153 1.998362 -3.529660 -3.529489 0.000048 1.998756 0.000048 1.996790
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 1.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 19.738549 20.447284 NaN 0.035906 NaN NaN NaN 19.679634 NaN 0.002985 NaN NaN NaN
|
|
N = 8 19.738549 19.922720 NaN 0.009331 1.944198 NaN NaN 19.641717 NaN 0.004906 -0.716842 NaN NaN
|
|
N = 16 19.738549 19.785060 19.736081 0.002356 1.985391 0.002482 1.930009 19.703350 19.665192 0.001783 1.459966 0.001940 -0.700878
|
|
N = 32 19.738549 19.750206 19.738390 0.000591 1.996303 0.000599 1.981722 19.728328 19.745349 0.000518 1.784051 0.000862 1.303049
|
|
N = 64 19.738549 19.741465 19.738539 0.000148 1.999073 0.000148 1.995378 19.735813 19.739016 0.000139 1.901696 0.000162 1.738548
|
|
N = 128 19.738549 19.739278 19.738548 0.000037 1.999768 0.000037 1.998841 19.737842 19.738597 0.000036 1.952962 0.000038 1.883408
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 1.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -4.985127 -5.363661 NaN 0.075933 [None] NaN NaN -5.095071 NaN 0.022055 [None] NaN NaN
|
|
N = 8 -4.985127 -5.082312 NaN 0.019495 1.961618 NaN NaN -5.014775 NaN 0.005947 1.890731 NaN NaN
|
|
N = 16 -4.985127 -5.009594 -4.984248 0.004908 1.989885 0.005085 1.951981 -4.992802 -4.984523 0.001540 1.949706 0.001661 1.869549
|
|
N = 32 -4.985127 -4.991254 -4.985069 0.001229 1.997436 0.001241 1.987353 -4.987077 -4.985061 0.000391 1.976014 0.000405 1.940629
|
|
N = 64 -4.985127 -4.986659 -4.985123 0.000307 1.999357 0.000308 1.996795 -4.985618 -4.985119 0.000099 1.988326 0.000100 1.971841
|
|
N = 128 -4.985127 -4.985510 -4.985126 0.000077 1.999839 0.000077 1.999196 -4.985250 -4.985126 0.000025 1.994248 0.000025 1.986336
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 2.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 36.441301 36.351342 NaN 0.002469 NaN NaN NaN 35.832852 NaN 0.016697 NaN NaN NaN
|
|
N = 8 36.441301 36.412116 NaN 0.000801 1.624043 NaN NaN 36.204196 NaN 0.006507 1.359606 NaN NaN
|
|
N = 16 36.441301 36.433533 36.445188 0.000213 1.909614 0.000320 1.504701 36.370470 36.505289 0.001944 1.743070 0.003693 1.159187
|
|
N = 32 36.441301 36.439329 36.441479 0.000054 1.977577 0.000059 1.885733 36.422094 36.445339 0.000527 1.882741 0.000638 1.687447
|
|
N = 64 36.441301 36.440806 36.441312 0.000014 1.994404 0.000014 1.971895 36.436309 36.441710 0.000137 1.943854 0.000148 1.860649
|
|
N = 128 36.441301 36.441177 36.441302 0.000003 1.998602 0.000003 1.993004 36.440029 36.441348 0.000035 1.972518 0.000036 1.933921
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 2.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -6.832006 -7.120389 NaN 0.042211 [None] NaN NaN -6.930078 NaN 0.014355 [None] NaN NaN
|
|
N = 8 -6.832006 -6.905215 NaN 0.010716 1.977901 NaN NaN -6.861111 NaN 0.004260 1.752561 NaN NaN
|
|
N = 16 -6.832006 -6.850381 -6.831629 0.002690 1.994255 0.002745 1.972378 -6.839891 -6.830460 0.001154 1.88406 0.001381 1.700471
|
|
N = 32 -6.832006 -6.836605 -6.831982 0.000673 1.998549 0.000677 1.992819 -6.834055 -6.831842 0.000300 1.943973 0.000324 1.862416
|
|
N = 64 -6.832006 -6.833156 -6.832005 0.000168 1.999636 0.000169 1.998186 -6.832528 -6.831987 0.000076 1.972497 0.000079 1.934089
|
|
N = 128 -6.832006 -6.832294 -6.832006 0.000042 1.999909 0.000042 1.999545 -6.832138 -6.832004 0.000019 1.986381 0.000020 1.967780
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 4.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 55.383747 54.738450 NaN 0.011651 NaN NaN NaN 54.410234 NaN 0.017578 NaN NaN NaN
|
|
N = 8 55.383747 55.211305 NaN 0.003114 1.903854 NaN NaN 55.070107 NaN 0.005663 1.634093 NaN NaN
|
|
N = 16 55.383747 55.339880 55.387898 0.000792 1.974909 0.000867 1.878789 55.295589 55.412632 0.001592 1.830954 0.002112 1.549177
|
|
N = 32 55.383747 55.372732 55.384006 0.000199 1.993656 0.000204 1.968568 55.360424 55.386591 0.000421 1.918370 0.000472 1.798167
|
|
N = 64 55.383747 55.380990 55.383763 0.000050 1.998410 0.000050 1.992066 55.377752 55.384071 0.000108 1.959856 0.000114 1.903734
|
|
N = 128 55.383747 55.383057 55.383748 0.000012 1.999602 0.000012 1.998012 55.382227 55.383786 0.000027 1.980090 0.000028 1.952921
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 4.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -8.797049 -9.060288 NaN 0.029924 [None] NaN NaN -8.946633 NaN 0.017004 [None] NaN NaN
|
|
N = 8 -8.797049 -8.864016 NaN 0.007612 1.974855 NaN NaN -8.842519 NaN 0.005169 1.71798 NaN NaN
|
|
N = 16 -8.797049 -8.813868 -8.796658 0.001912 1.993338 0.001956 1.968603 -8.809529 -8.794228 0.001419 1.865265 0.001740 1.658091
|
|
N = 32 -8.797049 -8.801259 -8.797023 0.000479 1.998309 0.000481 1.991674 -8.800315 -8.796744 0.000371 1.934016 0.000406 1.840087
|
|
N = 64 -8.797049 -8.798102 -8.797047 0.000120 1.999576 0.000120 1.997886 -8.797884 -8.797013 0.000095 1.967339 0.000099 1.922386
|
|
N = 128 -8.797049 -8.797312 -8.797049 0.000030 1.999894 0.000030 1.999469 -8.797260 -8.797045 0.000024 1.983751 0.000025 1.961743
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 8.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 72.105546 71.363029 NaN 0.010298 NaN NaN NaN 71.167084 NaN 0.013015 NaN NaN NaN
|
|
N = 8 72.105546 71.911316 NaN 0.002694 1.934658 NaN NaN 71.823222 NaN 0.003915 1.732945 NaN NaN
|
|
N = 16 72.105546 72.056411 72.108626 0.000681 1.982937 0.000724 1.917936 72.028173 72.121272 0.001073 1.867448 0.001291 1.678719
|
|
N = 32 72.105546 72.093226 72.105743 0.000171 1.995685 0.000174 1.978646 72.085293 72.107363 0.000281 1.933672 0.000306 1.843217
|
|
N = 64 72.105546 72.102464 72.105559 0.000043 1.998918 0.000043 1.994605 72.100365 72.105768 0.000072 1.966785 0.000075 1.922112
|
|
N = 128 72.105546 72.104776 72.105547 0.000011 1.999729 0.000011 1.998647 72.104236 72.105574 0.000018 1.983374 0.000019 1.961125
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 8.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -10.486214 -10.779857 NaN 0.028003 [None] NaN NaN -10.732095 NaN 0.023448 [None] NaN NaN
|
|
N = 8 -10.486214 -10.561319 NaN 0.007162 1.967076 NaN NaN -10.559111 NaN 0.006952 1.754021 NaN NaN
|
|
N = 16 -10.486214 -10.505104 -10.485636 0.001801 1.991276 0.001857 1.958852 -10.506062 -10.482597 0.001893 1.876861 0.002238 1.705232
|
|
N = 32 -10.486214 -10.490944 -10.486176 0.000451 1.997786 0.000455 1.989096 -10.491393 -10.485787 0.000494 1.938138 0.000535 1.854587
|
|
N = 64 -10.486214 -10.487397 -10.486211 0.000113 1.999444 0.000113 1.997232 -10.487537 -10.486161 0.000126 1.968962 0.000131 1.927410
|
|
N = 128 -10.486214 -10.486510 -10.486214 0.000028 1.999861 0.000028 1.999305 -10.486548 -10.486207 0.000032 1.98445 0.000033 1.963687
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.500 and k_ratio = 16.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 84.066993 83.499938 NaN 0.006745 NaN NaN NaN 83.389648 NaN 0.008057 NaN NaN NaN
|
|
N = 8 84.066993 83.920263 NaN 0.001745 1.950326 NaN NaN 83.869497 NaN 0.002349 1.778069 NaN NaN
|
|
N = 16 84.066993 84.029979 84.068733 0.000440 1.986999 0.000461 1.937740 84.013509 84.075262 0.000636 1.884628 0.000735 1.736399
|
|
N = 32 84.066993 84.057718 84.067105 0.000110 1.996710 0.000112 1.983738 84.053063 84.068040 0.000166 1.940876 0.000178 1.864285
|
|
N = 64 84.066993 84.064673 84.067000 0.000028 1.999175 0.000028 1.995887 84.063437 84.067126 0.000042 1.970029 0.000044 1.930747
|
|
N = 128 84.066993 84.066413 84.066994 0.000007 1.999794 0.000007 1.998969 84.066095 84.067010 0.000011 1.984906 0.000011 1.964966
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.500 and k_ratio = 16.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -11.680625 -12.024326 NaN 0.029425 [None] NaN NaN -12.024123 NaN 0.029407 [None] NaN NaN
|
|
N = 8 -11.680625 -11.768665 NaN 0.007537 1.964919 NaN NaN -11.780305 NaN 0.008534 1.784929 NaN NaN
|
|
N = 16 -11.680625 -11.702777 -11.679901 0.001896 1.990732 0.001959 1.956136 -11.707559 -11.676625 0.002306 1.887865 0.002649 1.744873
|
|
N = 32 -11.680625 -11.686172 -11.680577 0.000475 1.997649 0.000479 1.988414 -11.687633 -11.680115 0.000600 1.942452 0.000644 1.868167
|
|
N = 64 -11.680625 -11.682012 -11.680622 0.000119 1.99941 0.000119 1.997062 -11.682413 -11.680560 0.000153 1.970807 0.000159 1.932611
|
|
N = 128 -11.680625 -11.680972 -11.680625 0.000030 1.999852 0.000030 1.999263 -11.681077 -11.680617 0.000039 1.985293 0.000039 1.965880
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 0.0625
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 0.392138 0.964920 NaN 1.460667 NaN NaN NaN 0.149088 NaN 0.619806 NaN NaN NaN
|
|
N = 8 0.392138 0.534322 NaN 0.362587 2.010228 NaN NaN 0.367363 NaN 0.063178 3.294328 NaN NaN
|
|
N = 16 0.392138 0.427450 0.392169 0.090051 2.009513 0.089965 2.010465 0.386079 0.387834 0.015450 2.031788 0.004526 3.543820
|
|
N = 32 0.392138 0.400947 0.392207 0.022465 2.003074 0.022284 2.011646 0.390525 0.391911 0.004112 1.909892 0.003535 2.073563
|
|
N = 64 0.392138 0.394339 0.392144 0.005613 2.000819 0.005597 2.003825 0.391716 0.392151 0.001076 1.934512 0.001110 1.901067
|
|
N = 128 0.392138 0.392688 0.392138 0.001403 2.000209 0.001402 2.001022 0.392029 0.392142 0.000276 1.962964 0.000286 1.924564
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 0.0625
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -1.241978 -1.892412 NaN 0.523708 [None] NaN NaN -1.490265 NaN 0.199912 [None] NaN NaN
|
|
N = 8 -1.241978 -1.432463 NaN 0.153372 1.771728 NaN NaN -1.308439 NaN 0.053512 1.90142 NaN NaN
|
|
N = 16 -1.241978 -1.292229 -1.230720 0.040461 1.922447 0.049978 1.713643 -1.258921 -1.240389 0.013642 1.971793 0.014941 1.876530
|
|
N = 32 -1.241978 -1.254729 -1.241041 0.010267 1.978528 0.011030 1.902870 -1.246235 -1.241866 0.003428 1.992665 0.003518 1.964720
|
|
N = 64 -1.241978 -1.245178 -1.241914 0.002577 1.99448 0.002628 1.973144 -1.243044 -1.241971 0.000858 1.998145 0.000864 1.990831
|
|
N = 128 -1.241978 -1.242779 -1.241974 0.000645 1.998612 0.000648 1.993098 -1.242245 -1.241978 0.000215 1.999539 0.000215 1.997680
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 0.125
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 1.973771 2.884832 NaN 0.461584 NaN NaN NaN 1.950437 NaN 0.011822 NaN NaN NaN
|
|
N = 8 1.973771 2.217011 NaN 0.123236 1.905166 NaN NaN 1.953971 NaN 0.010032 0.236949 NaN NaN
|
|
N = 16 1.973771 2.035705 1.968139 0.031379 1.973570 0.034330 1.881035 1.965537 1.948881 0.004172 1.265896 0.008546 -1.710453
|
|
N = 32 1.973771 1.989328 1.973387 0.007882 1.993183 0.008078 1.966930 1.971213 1.976683 0.001296 1.686822 0.002767 1.026937
|
|
N = 64 1.973771 1.977665 1.973746 0.001973 1.998282 0.001985 1.991477 1.973063 1.973958 0.000358 1.854433 0.000453 1.617231
|
|
N = 128 1.973771 1.974744 1.973769 0.000493 1.999570 0.000494 1.997852 1.973585 1.973790 0.000094 1.929744 0.000104 1.826653
|
|
|
|
|
|
|
|
Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 0.125
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -1.758799 -2.268096 NaN 0.289571 [None] NaN NaN -1.940183 NaN 0.103130 [None] NaN NaN
|
|
N = 8 -1.758799 -1.898803 NaN 0.079602 1.86304 NaN NaN -1.805860 NaN 0.026757 1.946449 NaN NaN
|
|
N = 16 -1.758799 -1.794813 -1.754052 0.020477 1.958837 0.023238 1.828323 -1.770688 -1.758212 0.006760 1.984849 0.007096 1.933234
|
|
N = 32 -1.758799 -1.767870 -1.758449 0.005158 1.989132 0.005358 1.948491 -1.761780 -1.758758 0.001695 1.995926 0.001718 1.981123
|
|
N = 64 -1.758799 -1.761071 -1.758776 0.001292 1.997244 0.001305 1.986411 -1.759545 -1.758796 0.000424 1.998878 0.000426 1.994939
|
|
N = 128 -1.758799 -1.759367 -1.758797 0.000323 1.999308 0.000324 1.996554 -1.758985 -1.758799 0.000106 1.999671 0.000106 1.998614
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 0.25
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 6.560619 7.452341 NaN 0.135920 NaN NaN NaN 6.572238 NaN 0.001771 NaN NaN NaN
|
|
N = 8 6.560619 6.799021 NaN 0.036338 1.903201 NaN NaN 6.501798 NaN 0.008966 -2.339890 NaN NaN
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|
N = 16 6.560619 6.621332 6.554951 0.009254 1.973315 0.010127 1.878442 6.537168 6.525344 0.003575 1.326678 0.001812 0.993864
|
|
N = 32 6.560619 6.575870 6.560239 0.002325 1.993146 0.002383 1.966601 6.553589 6.567820 0.001072 1.738076 0.002167 1.106969
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|
N = 64 6.560619 6.564437 6.560595 0.000582 1.998275 0.000586 1.991430 6.558712 6.561034 0.000291 1.881813 0.000354 1.680666
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|
N = 128 6.560619 6.561574 6.560618 0.000146 1.999568 0.000146 1.997843 6.560124 6.560661 0.000076 1.943682 0.000082 1.859435
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 0.25
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True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -2.501202 -2.888923 NaN 0.155014 [None] NaN NaN -2.626242 NaN 0.049992 [None] NaN NaN
|
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N = 8 -2.501202 -2.603336 NaN 0.040834 1.924555 NaN NaN -2.533474 NaN 0.012903 1.954046 NaN NaN
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|
N = 16 -2.501202 -2.527108 -2.499354 0.010358 1.979071 0.011105 1.905549 -2.509362 -2.500895 0.003263 1.983522 0.003386 1.943932
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|
N = 32 -2.501202 -2.507703 -2.501075 0.002599 1.994616 0.002650 1.973826 -2.503251 -2.501176 0.000819 1.99364 0.000830 1.980114
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|
N = 64 -2.501202 -2.502828 -2.501194 0.000650 1.998644 0.000654 1.993269 -2.501715 -2.501199 0.000205 1.997315 0.000206 1.992409
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|
N = 128 -2.501202 -2.501608 -2.501201 0.000163 1.99966 0.000163 1.998305 -2.501330 -2.501201 0.000051 1.998785 0.000051 1.996825
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 0.5
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 16.08921 16.377482 NaN 0.017917 NaN NaN NaN 15.648397 NaN 0.027398 NaN NaN NaN
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N = 8 16.08921 16.157318 NaN 0.004233 2.081543 NaN NaN 15.873740 NaN 0.013392 1.032680 NaN NaN
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N = 16 16.08921 16.105900 16.090233 0.001037 2.028836 0.000974 2.098246 16.020446 16.294141 0.004274 1.647759 0.016797 0.619196
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N = 32 16.08921 16.093360 16.089315 0.000258 2.007904 0.000251 2.035696 16.070057 16.095406 0.001190 1.844072 0.001575 1.564197
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|
N = 64 16.08921 16.090246 16.089218 0.000064 2.002023 0.000064 2.009856 16.084171 16.089784 0.000313 1.926462 0.000349 1.813482
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N = 128 16.08921 16.089469 16.089211 0.000016 2.000509 0.000016 2.002527 16.087919 16.089273 0.000080 1.964289 0.000084 1.913195
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 0.5
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -3.564213 -3.858649 NaN 0.082609 [None] NaN NaN -3.651412 NaN 0.024465 [None] NaN NaN
|
|
N = 8 -3.564213 -3.639854 NaN 0.021222 1.960716 NaN NaN -3.587897 NaN 0.006645 1.88039 NaN NaN
|
|
N = 16 -3.564213 -3.583260 -3.563514 0.005344 1.989586 0.005541 1.950868 -3.570374 -3.563697 0.001728 1.942759 0.001874 1.857806
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|
N = 32 -3.564213 -3.568983 -3.564167 0.001338 1.997356 0.001351 1.986980 -3.565783 -3.564153 0.000441 1.972281 0.000457 1.932522
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|
N = 64 -3.564213 -3.565406 -3.564210 0.000335 1.999336 0.000336 1.996695 -3.564609 -3.564206 0.000111 1.986436 0.000113 1.967471
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|
N = 128 -3.564213 -3.564511 -3.564213 0.000084 1.999834 0.000084 1.999170 -3.564312 -3.564212 0.000028 1.993303 0.000028 1.984125
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 1.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 31.182398 30.514477 NaN 0.021420 NaN NaN NaN 29.932292 NaN 0.040090 NaN NaN NaN
|
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N = 8 31.182398 30.989359 NaN 0.006191 1.790787 NaN NaN 30.733838 NaN 0.014385 1.478678 NaN NaN
|
|
N = 16 31.182398 31.132234 31.193717 0.001609 1.944144 0.001971 1.732826 31.050756 31.258002 0.004222 1.768681 0.006630 1.338677
|
|
N = 32 31.182398 31.169733 31.183078 0.000406 1.985785 0.000428 1.929804 31.146901 31.188771 0.001138 1.890838 0.001342 1.720830
|
|
N = 64 31.182398 31.179224 31.182440 0.000102 1.996430 0.000103 1.982208 31.173192 31.183087 0.000295 1.947032 0.000317 1.870631
|
|
N = 128 31.182398 31.181604 31.182401 0.000025 1.999106 0.000026 1.995536 31.180055 31.182479 0.000075 1.973922 0.000078 1.937733
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 1.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -4.985127 -5.229953 NaN 0.049111 [None] NaN NaN -5.078401 NaN 0.018711 [None] NaN NaN
|
|
N = 8 -4.985127 -5.047653 NaN 0.012543 1.969225 NaN NaN -5.013420 NaN 0.005676 1.720991 NaN NaN
|
|
N = 16 -4.985127 -5.000849 -4.984681 0.003154 1.991676 0.003243 1.961604 -4.992897 -4.983423 0.001559 1.864391 0.001901 1.662746
|
|
N = 32 -4.985127 -4.989063 -4.985096 0.000790 1.997874 0.000796 1.989601 -4.987161 -4.984935 0.000408 1.933604 0.000446 1.839028
|
|
N = 64 -4.985127 -4.986111 -4.985125 0.000197 1.999466 0.000198 1.997343 -4.985647 -4.985104 0.000104 1.967248 0.000109 1.921860
|
|
N = 128 -4.985127 -4.985373 -4.985126 0.000049 1.999866 0.000049 1.999332 -4.985258 -4.985124 0.000026 1.983749 0.000027 1.961621
|
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|
|
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|
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 2.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 49.706164 48.318359 NaN 0.027920 NaN NaN NaN 47.861343 NaN 0.037115 NaN NaN NaN
|
|
N = 8 49.706164 49.325368 NaN 0.007661 1.865716 NaN NaN 49.108745 NaN 0.012019 1.626666 NaN NaN
|
|
N = 16 49.706164 49.608556 49.719351 0.001964 1.963950 0.002228 1.830245 49.536981 49.760850 0.003404 1.820156 0.004499 1.542449
|
|
N = 32 49.706164 49.681606 49.707000 0.000494 1.990813 0.000511 1.954806 49.661198 49.711952 0.000905 1.911699 0.001021 1.785533
|
|
N = 64 49.706164 49.700015 49.706216 0.000124 1.997692 0.000125 1.988508 49.694576 49.706841 0.000233 1.956261 0.000247 1.895902
|
|
N = 128 49.706164 49.704626 49.706167 0.000031 1.999422 0.000031 1.997114 49.703223 49.706246 0.000059 1.978235 0.000061 1.948711
|
|
|
|
|
|
|
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 2.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -6.610807 -6.872679 NaN 0.039613 [None] NaN NaN -6.783079 NaN 0.026059 [None] NaN NaN
|
|
N = 8 -6.610807 -6.678556 NaN 0.010248 1.950595 NaN NaN -6.664558 NaN 0.008131 1.680312 NaN NaN
|
|
N = 16 -6.610807 -6.627905 -6.610024 0.002586 1.986349 0.002705 1.938323 -6.625791 -6.606947 0.002267 1.842832 0.002852 1.612237
|
|
N = 32 -6.610807 -6.615092 -6.610753 0.000648 1.996494 0.000656 1.982940 -6.614761 -6.610374 0.000598 1.922142 0.000664 1.813308
|
|
N = 64 -6.610807 -6.611879 -6.610804 0.000162 1.999117 0.000163 1.995618 -6.611822 -6.610755 0.000154 1.961274 0.000161 1.908370
|
|
N = 128 -6.610807 -6.611075 -6.610807 0.000041 1.999779 0.000041 1.998897 -6.611064 -6.610801 0.000039 1.980691 0.000040 1.954625
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|
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 4.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 67.384857 65.895105 NaN 0.022108 NaN NaN NaN 65.562359 NaN 0.027046 NaN NaN NaN
|
|
N = 8 67.384857 66.985431 NaN 0.005928 1.899072 NaN NaN 66.822621 NaN 0.008344 1.696669 NaN NaN
|
|
N = 16 67.384857 67.283107 67.394899 0.001510 1.972907 0.001659 1.872943 67.228445 67.421196 0.002321 1.845831 0.002859 1.634795
|
|
N = 32 67.384857 67.359297 67.385506 0.000379 1.993096 0.000389 1.966071 67.343581 67.389183 0.000613 1.921976 0.000677 1.817524
|
|
N = 64 67.384857 67.378459 67.384898 0.000095 1.998265 0.000096 1.991365 67.374253 67.385391 0.000157 1.960712 0.000165 1.908339
|
|
N = 128 67.384857 67.383257 67.384859 0.000024 1.999566 0.000024 1.997832 67.382169 67.384923 0.000040 1.980281 0.000041 1.954008
|
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|
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|
|
|
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 4.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -8.118713 -8.460794 NaN 0.042135 [None] NaN NaN -8.433519 NaN 0.038775 [None] NaN NaN
|
|
N = 8 -8.118713 -8.208090 NaN 0.011009 1.936363 NaN NaN -8.214918 NaN 0.011850 1.710279 NaN NaN
|
|
N = 16 -8.118713 -8.141329 -8.117358 0.002786 1.982599 0.002953 1.920359 -8.145364 -8.112905 0.003283 1.851968 0.004001 1.652077
|
|
N = 32 -8.118713 -8.124385 -8.118622 0.000699 1.995544 0.000710 1.978241 -8.125732 -8.118012 0.000864 1.924895 0.000951 1.824974
|
|
N = 64 -8.118713 -8.120132 -8.118708 0.000175 1.998879 0.000175 1.994429 -8.120515 -8.118626 0.000222 1.962136 0.000233 1.911811
|
|
N = 128 -8.118713 -8.119068 -8.118713 0.000044 1.999719 0.000044 1.998599 -8.119170 -8.118703 0.000056 1.980985 0.000058 1.955685
|
|
|
|
|
|
|
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 8.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 80.863285 79.714831 NaN 0.014202 NaN NaN NaN 79.498914 NaN 0.016873 NaN NaN NaN
|
|
N = 8 80.863285 80.559732 NaN 0.003754 1.919673 NaN NaN 80.453254 NaN 0.005071 1.734431 NaN NaN
|
|
N = 16 80.863285 80.786251 80.869227 0.000953 1.978385 0.001026 1.899150 80.750296 80.884534 0.001397 1.859558 0.001660 1.683834
|
|
N = 32 80.863285 80.843953 80.863675 0.000239 1.994488 0.000244 1.972949 80.833578 80.866025 0.000367 1.927308 0.000401 1.834600
|
|
N = 64 80.863285 80.858447 80.863310 0.000060 1.998615 0.000060 1.993108 80.855665 80.863637 0.000094 1.962954 0.000099 1.914803
|
|
N = 128 80.863285 80.862075 80.863286 0.000015 1.999653 0.000015 1.998268 80.861355 80.863330 0.000024 1.981291 0.000024 1.956682
|
|
|
|
|
|
|
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 8.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -9.254592 -9.698320 NaN 0.047947 [None] NaN NaN -9.722544 NaN 0.050564 [None] NaN NaN
|
|
N = 8 -9.254592 -9.370579 NaN 0.012533 1.935704 NaN NaN -9.394792 NaN 0.015149 1.738869 NaN NaN
|
|
N = 16 -9.254592 -9.283942 -9.252810 0.003171 1.982521 0.003365 1.919493 -9.293169 -9.247500 0.004169 1.86165 0.004939 1.689379
|
|
N = 32 -9.254592 -9.261952 -9.254472 0.000795 1.995531 0.000808 1.978140 -9.264727 -9.253673 0.001095 1.928326 0.001195 1.837124
|
|
N = 64 -9.254592 -9.256433 -9.254584 0.000199 1.998876 0.000200 1.994413 -9.257191 -9.254473 0.000281 1.963456 0.000294 1.916010
|
|
N = 128 -9.254592 -9.255052 -9.254591 0.000050 1.999719 0.000050 1.998595 -9.255250 -9.254577 0.000071 1.98154 0.000073 1.957273
|
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|
|
|
|
|
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Convergence of FDM and FEM Temperature at x_bar = 0.637 and k_ratio = 16.0
|
|
True T FDM T FDM Extrap T FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM T FEM Extrap T FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 89.525116 88.797458 NaN 0.008128 NaN NaN NaN 88.671094 NaN 0.009539 NaN NaN NaN
|
|
N = 8 89.525116 89.334419 NaN 0.002130 1.931977 NaN NaN 89.272101 NaN 0.002826 1.755051 NaN NaN
|
|
N = 16 89.525116 89.476830 89.528233 0.000539 1.981624 0.000574 1.914748 89.455749 89.536559 0.000775 1.866912 0.000903 1.710436
|
|
N = 32 89.525116 89.513005 89.525323 0.000135 1.995309 0.000138 1.977014 89.506913 89.526672 0.000203 1.930093 0.000221 1.843749
|
|
N = 64 89.525116 89.522085 89.525129 0.000034 1.998821 0.000034 1.994135 89.520450 89.525321 0.000052 1.964098 0.000054 1.918185
|
|
N = 128 89.525116 89.524358 89.525116 0.000008 1.999705 0.000008 1.998526 89.523934 89.525142 0.000013 1.981797 0.000013 1.958049
|
|
|
|
|
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|
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Convergence of FDM and FEM Heat Convection with x_bar = 0.637 and k_ratio = 16.0
|
|
True Q FDM Q FDM Extrap Q FDM Exact % Error FDM Exact B FDM Extrap % Error FDM Extrap B FEM Q FEM Extrap Q FEM Exact % Error FEM Exact B FEM Extrap % Error FEM Extrap B
|
|
N = 4 -9.980639 -10.508396 NaN 0.052878 [None] NaN NaN -10.567911 NaN 0.058841 [None] NaN NaN
|
|
N = 8 -9.980639 -10.118258 NaN 0.013789 1.939192 NaN NaN -10.154415 NaN 0.017411 1.756795 NaN NaN
|
|
N = 16 -9.980639 -10.015440 -9.978646 0.003487 1.983481 0.003687 1.923889 -10.028253 -9.972858 0.004771 1.867755 0.005555 1.712595
|
|
N = 32 -9.980639 -9.989364 -9.980505 0.000874 1.995777 0.000888 1.979342 -9.993130 -9.979579 0.001252 1.930509 0.001358 1.844764
|
|
N = 64 -9.980639 -9.982822 -9.980630 0.000219 1.998938 0.000220 1.994720 -9.983840 -9.980499 0.000321 1.964305 0.000335 1.918678
|
|
N = 128 -9.980639 -9.981185 -9.980638 0.000055 1.999734 0.000055 1.998673 -9.981449 -9.980621 0.000081 1.9819 0.000083 1.958292
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