Note
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Heat transfer through a thick cylinder
This problem demonstrates the use of prescribed temperature on both sides of the wall.
Using
mesher.PatchMesher(see PatchMesher class)diffusion.Triangle(see Triangle class for Diffusion)materials.Thermal(see Diffusion Material classes)
Theory
We shall consider a stationary heat transfer problem on a thick ring. The inner surface of the ring, \(r_i\), is heated to \(200~K\), the outer surface of the ring, \(r_o\), to \(300~K\).
The thermal equation for an axi-symmetric problem can be expressed as
\[\Delta T = \frac{1}{r} \: \frac{\partial }{\partial r}
\left( r \frac{\partial T}{\partial r} \right) = 0\]
where \(\Delta\) is the Laplace operator.
The analytic solution follows as
\[T(r) = {\displaystyle \frac{T_i \log(r_o/r) + T_o \log(r/r_i)}{\log(r_o/r_i)}}\]
This solution will be compared against the finite element solution in the last figure.
import matplotlib.pyplot as plt
import math
import sys
import numpy as np
from femedu.examples.Example import *
from femedu.domain import *
from femedu.mesher import PatchMesher
from femedu.elements.diffusion import Triangle
from femedu.materials import Thermal
class ExampleThermal01(Example):
def problem(self):
# ========== setting mesh parameters ==============
Nx = 8 # number of elements through the wall
Ny = 4 # number of elements parallel to the wall
Lx = 10.00 # wall thickness in m
Ly = 5.00 # wall thickness in m
Ri = 5.00
Ro = Ri + Lx
alpha = np.radians(45.0)
# ========== setting material parameters ==============
params = dict(
E=20000., # Young's modulus
nu=0.250, # Poisson's ratio
t=1.00 # thickness of the plate
)
# ========== setting load parameters ==============
qn = 1.00 # uniform flux normal to x=const
# ========== setting analysis parameters ==============
target_load_level = 1.00 # reference load
max_steps = 2 # number of load steps: 2 -> [0.0, 1.0]
# define a list of target load levels
load_levels = np.linspace(0, target_load_level, max_steps)
#
# ==== Build the system model ====
#
model = System()
# create nodes
# 2 -------- 3
# | |
# | |
# | | |
# 0 -------- 1
pts = (
( Ri, 0), # 0
( Ro, 0), # 1
( Ri*np.cos(alpha), Ri*np.sin(alpha)), # 2
( Ro*np.cos(alpha), Ro*np.sin(alpha)), # 3
( Ri*np.cos(alpha/2), Ri*np.sin(alpha/2)), # 4
( Ro*np.cos(alpha/2), Ro*np.sin(alpha/2)), # 5
)
mesher = PatchMesher(model, pts[0], pts[1], pts[3], pts[2], None, pts[5], None, pts[4])
nodes, elements = mesher.triangleMesh(Nx, Ny, Triangle, Thermal(params))
model.plot(factor=0.0,
title='Radial diffusion',
show_reactions=0, show_bc=0, show_loads=0)
model.report()
# boundary condition(s)
## find nodes at y==0 and x==0
for node in nodes:
X = node.getPos()
R = np.linalg.norm(X)
if math.isclose(R, Ri, rel_tol=0.02):
node.setDOF(['T'],[200.]) # prescribed temperature at x=0.0
if math.isclose(R, Ro, rel_tol=0.02):
node.setDOF(['T'],[300.]) # prescribed temperature at x=0.0
# perform the analysis
model.setLoadFactor(1.0)
model.solve()
model.report()
model.valuePlot('T', show_mesh=True)
# creating a path plot
R_list = []
T_list = []
for node in nodes:
X = node.getPos()
R = np.linalg.norm(X)
T = node.getDisp('T')
R_list.append(R)
T_list.append(T)
# the analytic solution for comparison
r = np.linspace(Ri, Ro, 21)
T = (200. * np.log(Ro/r) + 300. * np.log(r/Ri)) / np.log(Ro/Ri)
fig, axs = plt.subplots()
axs.plot(r,T,'-b',label="analytic solution")
axs.plot(R_list,T_list,'ro',label="FEM")
axs.set_title('Nodal Temperature for ALL Nodes')
axs.set_xlabel("Radial distance")
axs.set_ylabel('T')
axs.legend()
axs.grid(True)
plt.show()
Run the example by creating an instance of the problem and executing it by calling Example.run()
if __name__ == "__main__":
ex = ExampleThermal01()
ex.run()
System Analysis Report
=======================
Nodes:
---------------------
Node_494:
x: [5. 0.]
u: [0.]
Node_495:
x: [6.25 0. ]
u: [0.]
Node_496:
x: [7.5 0. ]
u: [0.]
Node_497:
x: [8.75 0. ]
u: [0.]
Node_498:
x: [10. 0.]
u: [0.]
Node_499:
x: [11.25 0. ]
u: [0.]
Node_500:
x: [12.5 0. ]
u: [0.]
Node_501:
x: [13.75 0. ]
u: [0.]
Node_502:
x: [15. 0.]
u: [0.]
Node_503:
x: [4.89760651 0.99312113]
u: [0.]
Node_504:
x: [6.00662931 1.19360994]
u: [0.]
Node_505:
x: [7.14861749 1.40775346]
u: [0.]
Node_506:
x: [8.32357105 1.63555168]
u: [0.]
Node_507:
x: [9.53148999 1.87700461]
u: [0.]
Node_508:
x: [10.7723743 2.13211225]
u: [0.]
Node_509:
x: [12.046224 2.40087459]
u: [0.]
Node_510:
x: [13.35303907 2.68329164]
u: [0.]
Node_511:
x: [14.69281953 2.9793634 ]
u: [0.]
Node_512:
x: [4.61939766 1.91341716]
u: [0.]
Node_513:
x: [5.62040864 2.32804949]
u: [0.]
Node_514:
x: [6.66537346 2.76088809]
u: [0.]
Node_515:
x: [7.75429212 3.21193296]
u: [0.]
Node_516:
x: [8.88716462 3.68118411]
u: [0.]
Node_517:
x: [10.06399095 4.16864154]
u: [0.]
Node_518:
x: [11.28477112 4.67430525]
u: [0.]
Node_519:
x: [12.54950514 5.19817523]
u: [0.]
Node_520:
x: [13.85819299 5.74025149]
u: [0.]
Node_521:
x: [4.16537346 2.76088809]
u: [0.]
Node_522:
x: [5.091338 3.40331863]
u: [0.]
Node_523:
x: [6.05026792 4.05940389]
u: [0.]
Node_524:
x: [7.04216322 4.72914385]
u: [0.]
Node_525:
x: [8.06702389 5.41253852]
u: [0.]
Node_526:
x: [9.12484995 6.10958789]
u: [0.]
Node_527:
x: [10.21564138 6.82029197]
u: [0.]
Node_528:
x: [11.33939819 7.54465076]
u: [0.]
Node_529:
x: [12.49612038 8.28266426]
u: [0.]
Node_530:
x: [3.53553391 3.53553391]
u: [0.]
Node_531:
x: [4.41941738 4.41941738]
u: [0.]
Node_532:
x: [5.30330086 5.30330086]
u: [0.]
Node_533:
x: [6.18718434 6.18718434]
u: [0.]
Node_534:
x: [7.07106781 7.07106781]
u: [0.]
Node_535:
x: [7.95495129 7.95495129]
u: [0.]
Node_536:
x: [8.83883476 8.83883476]
u: [0.]
Node_537:
x: [9.72271824 9.72271824]
u: [0.]
Node_538:
x: [10.60660172 10.60660172]
u: [0.]
Elements:
---------------------
Triangle_755: nodes ( Node_494 Node_495 Node_503 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_756: nodes ( Node_504 Node_503 Node_495 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_757: nodes ( Node_495 Node_496 Node_504 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_758: nodes ( Node_505 Node_504 Node_496 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_759: nodes ( Node_496 Node_497 Node_505 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_760: nodes ( Node_506 Node_505 Node_497 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_761: nodes ( Node_497 Node_498 Node_506 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_762: nodes ( Node_507 Node_506 Node_498 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_763: nodes ( Node_498 Node_499 Node_507 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_764: nodes ( Node_508 Node_507 Node_499 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_765: nodes ( Node_499 Node_500 Node_508 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_766: nodes ( Node_509 Node_508 Node_500 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_767: nodes ( Node_500 Node_501 Node_509 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_768: nodes ( Node_510 Node_509 Node_501 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_769: nodes ( Node_501 Node_502 Node_510 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_770: nodes ( Node_511 Node_510 Node_502 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_771: nodes ( Node_503 Node_504 Node_512 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_772: nodes ( Node_513 Node_512 Node_504 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_773: nodes ( Node_504 Node_505 Node_513 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_774: nodes ( Node_514 Node_513 Node_505 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_775: nodes ( Node_505 Node_506 Node_514 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_776: nodes ( Node_515 Node_514 Node_506 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_777: nodes ( Node_506 Node_507 Node_515 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_778: nodes ( Node_516 Node_515 Node_507 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_779: nodes ( Node_507 Node_508 Node_516 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_780: nodes ( Node_517 Node_516 Node_508 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_781: nodes ( Node_508 Node_509 Node_517 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_782: nodes ( Node_518 Node_517 Node_509 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_783: nodes ( Node_509 Node_510 Node_518 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_784: nodes ( Node_519 Node_518 Node_510 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_785: nodes ( Node_510 Node_511 Node_519 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_786: nodes ( Node_520 Node_519 Node_511 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_787: nodes ( Node_512 Node_513 Node_521 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_788: nodes ( Node_522 Node_521 Node_513 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_789: nodes ( Node_513 Node_514 Node_522 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_790: nodes ( Node_523 Node_522 Node_514 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_791: nodes ( Node_514 Node_515 Node_523 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_792: nodes ( Node_524 Node_523 Node_515 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_793: nodes ( Node_515 Node_516 Node_524 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_794: nodes ( Node_525 Node_524 Node_516 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_795: nodes ( Node_516 Node_517 Node_525 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_796: nodes ( Node_526 Node_525 Node_517 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_797: nodes ( Node_517 Node_518 Node_526 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_798: nodes ( Node_527 Node_526 Node_518 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_799: nodes ( Node_518 Node_519 Node_527 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_800: nodes ( Node_528 Node_527 Node_519 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_801: nodes ( Node_519 Node_520 Node_528 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_802: nodes ( Node_529 Node_528 Node_520 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_803: nodes ( Node_521 Node_522 Node_530 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_804: nodes ( Node_531 Node_530 Node_522 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_805: nodes ( Node_522 Node_523 Node_531 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_806: nodes ( Node_532 Node_531 Node_523 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_807: nodes ( Node_523 Node_524 Node_532 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_808: nodes ( Node_533 Node_532 Node_524 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_809: nodes ( Node_524 Node_525 Node_533 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_810: nodes ( Node_534 Node_533 Node_525 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_811: nodes ( Node_525 Node_526 Node_534 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_812: nodes ( Node_535 Node_534 Node_526 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_813: nodes ( Node_526 Node_527 Node_535 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_814: nodes ( Node_536 Node_535 Node_527 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_815: nodes ( Node_527 Node_528 Node_536 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_816: nodes ( Node_537 Node_536 Node_528 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_817: nodes ( Node_528 Node_529 Node_537 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_818: nodes ( Node_538 Node_537 Node_529 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
System Analysis Report
=======================
Nodes:
---------------------
Node_494:
x: [5. 0.]
fix: ['T']
u: [200.]
Node_495:
x: [6.25 0. ]
u: [220.29127216]
Node_496:
x: [7.5 0. ]
u: [236.88141125]
Node_497:
x: [8.75 0. ]
u: [250.92121811]
Node_498:
x: [10. 0.]
u: [263.08788071]
Node_499:
x: [11.25 0. ]
u: [273.81918105]
Node_500:
x: [12.5 0. ]
u: [283.41522016]
Node_501:
x: [13.75 0. ]
u: [292.09006017]
Node_502:
x: [15. 0.]
fix: ['T']
u: [300.]
Node_503:
x: [4.89760651 0.99312113]
fix: ['T']
u: [200.]
Node_504:
x: [6.00662931 1.19360994]
u: [218.45274717]
Node_505:
x: [7.14861749 1.40775346]
u: [234.2457741]
Node_506:
x: [8.32357105 1.63555168]
u: [248.08728543]
Node_507:
x: [9.53148999 1.87700461]
u: [260.43574977]
Node_508:
x: [10.7723743 2.13211225]
u: [271.60396263]
Node_509:
x: [12.046224 2.40087459]
u: [281.81558241]
Node_510:
x: [13.35303907 2.68329164]
u: [291.23744346]
Node_511:
x: [14.69281953 2.9793634 ]
fix: ['T']
u: [300.]
Node_512:
x: [4.61939766 1.91341716]
fix: ['T']
u: [200.]
Node_513:
x: [5.62040864 2.32804949]
u: [217.83610312]
Node_514:
x: [6.66537346 2.76088809]
u: [233.3463883]
Node_515:
x: [7.75429212 3.21193296]
u: [247.11989949]
Node_516:
x: [8.88716462 3.68118411]
u: [259.53850452]
Node_517:
x: [10.06399095 4.16864154]
u: [270.86739278]
Node_518:
x: [11.28477112 4.67430525]
u: [281.29848706]
Node_519:
x: [12.54950514 5.19817523]
u: [290.97413032]
Node_520:
x: [13.85819299 5.74025149]
fix: ['T']
u: [300.]
Node_521:
x: [4.16537346 2.76088809]
fix: ['T']
u: [200.]
Node_522:
x: [5.091338 3.40331863]
u: [218.44785871]
Node_523:
x: [6.05026792 4.05940389]
u: [234.24484664]
Node_524:
x: [7.04216322 4.72914385]
u: [248.08987905]
Node_525:
x: [8.06702389 5.41253852]
u: [260.4394968]
Node_526:
x: [9.12484995 6.10958789]
u: [271.60738824]
Node_527:
x: [10.21564138 6.82029197]
u: [281.8181251]
Node_528:
x: [11.33939819 7.54465076]
u: [291.23892713]
Node_529:
x: [12.49612038 8.28266426]
fix: ['T']
u: [300.]
Node_530:
x: [3.53553391 3.53553391]
fix: ['T']
u: [200.]
Node_531:
x: [4.41941738 4.41941738]
u: [220.31826658]
Node_532:
x: [5.30330086 5.30330086]
u: [236.90296194]
Node_533:
x: [6.18718434 6.18718434]
u: [250.92789434]
Node_534:
x: [7.07106781 7.07106781]
u: [263.08217289]
Node_535:
x: [7.95495129 7.95495129]
u: [273.80727206]
Node_536:
x: [8.83883476 8.83883476]
u: [283.40330541]
Node_537:
x: [9.72271824 9.72271824]
u: [292.0827652]
Node_538:
x: [10.60660172 10.60660172]
fix: ['T']
u: [300.]
Elements:
---------------------
Triangle_755: nodes ( Node_494 Node_495 Node_503 )
material: Thermal
grad phi: x=1.623e+01 y=1.674e+00
flux: x=-1.623e+01 y=-1.674e+00
Triangle_756: nodes ( Node_504 Node_503 Node_495 )
material: Thermal
grad phi: x=1.632e+01 y=1.786e+00
flux: x=-1.632e+01 y=-1.786e+00
Triangle_757: nodes ( Node_495 Node_496 Node_504 )
material: Thermal
grad phi: x=1.327e+01 y=1.166e+00
flux: x=-1.327e+01 y=-1.166e+00
Triangle_758: nodes ( Node_505 Node_504 Node_496 )
material: Thermal
grad phi: x=1.355e+01 y=1.509e+00
flux: x=-1.355e+01 y=-1.509e+00
Triangle_759: nodes ( Node_496 Node_497 Node_505 )
material: Thermal
grad phi: x=1.123e+01 y=9.313e-01
flux: x=-1.123e+01 y=-9.313e-01
Triangle_760: nodes ( Node_506 Node_505 Node_497 )
material: Thermal
grad phi: x=1.153e+01 y=1.274e+00
flux: x=-1.153e+01 y=-1.274e+00
Triangle_761: nodes ( Node_497 Node_498 Node_506 )
material: Thermal
grad phi: x=9.733e+00 y=8.050e-01
flux: x=-9.733e+00 y=-8.050e-01
Triangle_762: nodes ( Node_507 Node_506 Node_498 )
material: Thermal
grad phi: x=1.001e+01 y=1.085e+00
flux: x=-1.001e+01 y=-1.085e+00
Triangle_763: nodes ( Node_498 Node_499 Node_507 )
material: Thermal
grad phi: x=8.585e+00 y=7.299e-01
flux: x=-8.585e+00 y=-7.299e-01
Triangle_764: nodes ( Node_508 Node_507 Node_499 )
material: Thermal
grad phi: x=8.808e+00 y=9.342e-01
flux: x=-8.808e+00 y=-9.342e-01
Triangle_765: nodes ( Node_499 Node_500 Node_508 )
material: Thermal
grad phi: x=7.677e+00 y=6.807e-01
flux: x=-7.677e+00 y=-6.807e-01
Triangle_766: nodes ( Node_509 Node_508 Node_500 )
material: Thermal
grad phi: x=7.844e+00 y=8.163e-01
flux: x=-7.844e+00 y=-8.163e-01
Triangle_767: nodes ( Node_500 Node_501 Node_509 )
material: Thermal
grad phi: x=6.940e+00 y=6.454e-01
flux: x=-6.940e+00 y=-6.454e-01
Triangle_768: nodes ( Node_510 Node_509 Node_501 )
material: Thermal
grad phi: x=7.053e+00 y=7.257e-01
flux: x=-7.053e+00 y=-7.257e-01
Triangle_769: nodes ( Node_501 Node_502 Node_510 )
material: Thermal
grad phi: x=6.328e+00 y=6.184e-01
flux: x=-6.328e+00 y=-6.184e-01
Triangle_770: nodes ( Node_511 Node_510 Node_502 )
material: Thermal
grad phi: x=6.395e+00 y=6.593e-01
flux: x=-6.395e+00 y=-6.593e-01
Triangle_771: nodes ( Node_503 Node_504 Node_512 )
material: Thermal
grad phi: x=1.578e+01 y=4.769e+00
flux: x=-1.578e+01 y=-4.769e+00
Triangle_772: nodes ( Node_513 Node_512 Node_504 )
material: Thermal
grad phi: x=1.581e+01 y=4.840e+00
flux: x=-1.581e+01 y=-4.840e+00
Triangle_773: nodes ( Node_504 Node_505 Node_513 )
material: Thermal
grad phi: x=1.310e+01 y=3.915e+00
flux: x=-1.310e+01 y=-3.915e+00
Triangle_774: nodes ( Node_514 Node_513 Node_505 )
material: Thermal
grad phi: x=1.317e+01 y=4.039e+00
flux: x=-1.317e+01 y=-4.039e+00
Triangle_775: nodes ( Node_505 Node_506 Node_514 )
material: Thermal
grad phi: x=1.114e+01 y=3.313e+00
flux: x=-1.114e+01 y=-3.313e+00
Triangle_776: nodes ( Node_515 Node_514 Node_506 )
material: Thermal
grad phi: x=1.122e+01 y=3.440e+00
flux: x=-1.122e+01 y=-3.440e+00
Triangle_777: nodes ( Node_506 Node_507 Node_515 )
material: Thermal
grad phi: x=9.649e+00 y=2.871e+00
flux: x=-9.649e+00 y=-2.871e+00
Triangle_778: nodes ( Node_516 Node_515 Node_507 )
material: Thermal
grad phi: x=9.729e+00 y=2.977e+00
flux: x=-9.729e+00 y=-2.977e+00
Triangle_779: nodes ( Node_507 Node_508 Node_516 )
material: Thermal
grad phi: x=8.480e+00 y=2.531e+00
flux: x=-8.480e+00 y=-2.531e+00
Triangle_780: nodes ( Node_517 Node_516 Node_508 )
material: Thermal
grad phi: x=8.545e+00 y=2.611e+00
flux: x=-8.545e+00 y=-2.611e+00
Triangle_781: nodes ( Node_508 Node_509 Node_517 )
material: Thermal
grad phi: x=7.539e+00 y=2.261e+00
flux: x=-7.539e+00 y=-2.261e+00
Triangle_782: nodes ( Node_518 Node_517 Node_509 )
material: Thermal
grad phi: x=7.586e+00 y=2.313e+00
flux: x=-7.586e+00 y=-2.313e+00
Triangle_783: nodes ( Node_509 Node_510 Node_518 )
material: Thermal
grad phi: x=6.769e+00 y=2.040e+00
flux: x=-6.769e+00 y=-2.040e+00
Triangle_784: nodes ( Node_519 Node_518 Node_510 )
material: Thermal
grad phi: x=6.794e+00 y=2.066e+00
flux: x=-6.794e+00 y=-2.066e+00
Triangle_785: nodes ( Node_510 Node_511 Node_519 )
material: Thermal
grad phi: x=6.131e+00 y=1.854e+00
flux: x=-6.131e+00 y=-1.854e+00
Triangle_786: nodes ( Node_520 Node_519 Node_511 )
material: Thermal
grad phi: x=6.129e+00 y=1.853e+00
flux: x=-6.129e+00 y=-1.853e+00
Triangle_787: nodes ( Node_512 Node_513 Node_521 )
material: Thermal
grad phi: x=1.458e+01 y=7.812e+00
flux: x=-1.458e+01 y=-7.812e+00
Triangle_788: nodes ( Node_522 Node_521 Node_513 )
material: Thermal
grad phi: x=1.456e+01 y=7.732e+00
flux: x=-1.456e+01 y=-7.732e+00
Triangle_789: nodes ( Node_513 Node_514 Node_522 )
material: Thermal
grad phi: x=1.213e+01 y=6.539e+00
flux: x=-1.213e+01 y=-6.539e+00
Triangle_790: nodes ( Node_523 Node_522 Node_514 )
material: Thermal
grad phi: x=1.208e+01 y=6.416e+00
flux: x=-1.208e+01 y=-6.416e+00
Triangle_791: nodes ( Node_514 Node_515 Node_523 )
material: Thermal
grad phi: x=1.033e+01 y=5.587e+00
flux: x=-1.033e+01 y=-5.587e+00
Triangle_792: nodes ( Node_524 Node_523 Node_515 )
material: Thermal
grad phi: x=1.027e+01 y=5.460e+00
flux: x=-1.027e+01 y=-5.460e+00
Triangle_793: nodes ( Node_515 Node_516 Node_524 )
material: Thermal
grad phi: x=8.956e+00 y=4.843e+00
flux: x=-8.956e+00 y=-4.843e+00
Triangle_794: nodes ( Node_525 Node_524 Node_516 )
material: Thermal
grad phi: x=8.894e+00 y=4.733e+00
flux: x=-8.894e+00 y=-4.733e+00
Triangle_795: nodes ( Node_516 Node_517 Node_525 )
material: Thermal
grad phi: x=7.867e+00 y=4.247e+00
flux: x=-7.867e+00 y=-4.247e+00
Triangle_796: nodes ( Node_526 Node_525 Node_517 )
material: Thermal
grad phi: x=7.815e+00 y=4.162e+00
flux: x=-7.815e+00 y=-4.162e+00
Triangle_797: nodes ( Node_517 Node_518 Node_526 )
material: Thermal
grad phi: x=6.986e+00 y=3.762e+00
flux: x=-6.986e+00 y=-3.762e+00
Triangle_798: nodes ( Node_527 Node_526 Node_518 )
material: Thermal
grad phi: x=6.948e+00 y=3.704e+00
flux: x=-6.948e+00 y=-3.704e+00
Triangle_799: nodes ( Node_518 Node_519 Node_527 )
material: Thermal
grad phi: x=6.259e+00 y=3.360e+00
flux: x=-6.259e+00 y=-3.360e+00
Triangle_800: nodes ( Node_528 Node_527 Node_519 )
material: Thermal
grad phi: x=6.237e+00 y=3.329e+00
flux: x=-6.237e+00 y=-3.329e+00
Triangle_801: nodes ( Node_519 Node_520 Node_528 )
material: Thermal
grad phi: x=5.644e+00 y=3.024e+00
flux: x=-5.644e+00 y=-3.024e+00
Triangle_802: nodes ( Node_529 Node_528 Node_520 )
material: Thermal
grad phi: x=5.645e+00 y=3.024e+00
flux: x=-5.645e+00 y=-3.024e+00
Triangle_803: nodes ( Node_521 Node_522 Node_530 )
material: Thermal
grad phi: x=1.274e+01 y=1.036e+01
flux: x=-1.274e+01 y=-1.036e+01
Triangle_804: nodes ( Node_531 Node_530 Node_522 )
material: Thermal
grad phi: x=1.273e+01 y=1.026e+01
flux: x=-1.273e+01 y=-1.026e+01
Triangle_805: nodes ( Node_522 Node_523 Node_531 )
material: Thermal
grad phi: x=1.047e+01 y=8.768e+00
flux: x=-1.047e+01 y=-8.768e+00
Triangle_806: nodes ( Node_532 Node_531 Node_523 )
material: Thermal
grad phi: x=1.039e+01 y=8.375e+00
flux: x=-1.039e+01 y=-8.375e+00
Triangle_807: nodes ( Node_523 Node_524 Node_532 )
material: Thermal
grad phi: x=8.905e+00 y=7.484e+00
flux: x=-8.905e+00 y=-7.484e+00
Triangle_808: nodes ( Node_533 Node_532 Node_524 )
material: Thermal
grad phi: x=8.775e+00 y=7.092e+00
flux: x=-8.775e+00 y=-7.092e+00
Triangle_809: nodes ( Node_524 Node_525 Node_533 )
material: Thermal
grad phi: x=7.730e+00 y=6.479e+00
flux: x=-7.730e+00 y=-6.479e+00
Triangle_810: nodes ( Node_534 Node_533 Node_525 )
material: Thermal
grad phi: x=7.596e+00 y=6.155e+00
flux: x=-7.596e+00 y=-6.155e+00
Triangle_811: nodes ( Node_525 Node_526 Node_534 )
material: Thermal
grad phi: x=6.812e+00 y=5.684e+00
flux: x=-6.812e+00 y=-5.684e+00
Triangle_812: nodes ( Node_535 Node_534 Node_526 )
material: Thermal
grad phi: x=6.697e+00 y=5.438e+00
flux: x=-6.697e+00 y=-5.438e+00
Triangle_813: nodes ( Node_526 Node_527 Node_535 )
material: Thermal
grad phi: x=6.075e+00 y=5.043e+00
flux: x=-6.075e+00 y=-5.043e+00
Triangle_814: nodes ( Node_536 Node_535 Node_527 )
material: Thermal
grad phi: x=5.987e+00 y=4.869e+00
flux: x=-5.987e+00 y=-4.869e+00
Triangle_815: nodes ( Node_527 Node_528 Node_536 )
material: Thermal
grad phi: x=5.472e+00 y=4.517e+00
flux: x=-5.472e+00 y=-4.517e+00
Triangle_816: nodes ( Node_537 Node_536 Node_528 )
material: Thermal
grad phi: x=5.414e+00 y=4.406e+00
flux: x=-5.414e+00 y=-4.406e+00
Triangle_817: nodes ( Node_528 Node_529 Node_537 )
material: Thermal
grad phi: x=4.972e+00 y=4.078e+00
flux: x=-4.972e+00 y=-4.078e+00
Triangle_818: nodes ( Node_538 Node_537 Node_529 )
material: Thermal
grad phi: x=4.940e+00 y=4.017e+00
flux: x=-4.940e+00 y=-4.017e+00
Total running time of the script: (0 minutes 0.377 seconds)