Note
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Heat transfer through a wall
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 within a wall. The inner surface of the wall, \(x=0~m\), is heated to \(200~K\), the outer surface of the wall, \(x=10~m\), to \(300~K\).
The thermal equation for the uni-directional problem can be expressed as
\[\Delta T = \frac{\partial^2 T}{\partial r^2} = 0\]
where \(\Delta\) is the Laplace operator.
The analytic solution follows as
\[T(x) = T_i \left(1-\frac{x}{L}\right) + T_o \left(\frac{x}{L}\right)\]
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 ExampleThermal02(Example):
def problem(self):
# ========== setting mesh parameters ==============
Nx = 4 # number of elements through the wall
Ny = 3 # number of elements parallel to the wall
Lx = 10.00 # wall thickness in m
Ly = 5.00 # wall thickness in m
# ========== 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 = (
( 0, 0), # 0
(Lx, 0), # 1
( 0, Ly), # 2
(Lx, Ly), # 3
(0.67*Lx, 0), # 4
(0.33*Lx, Ly), # 5
)
mesher = PatchMesher(model, pts[0], pts[1], pts[3], pts[2], pts[4], None, pts[5], None)
nodes, elements = mesher.triangleMesh(Nx, Ny, Triangle, Thermal(params))
model.plot(factor=0.0,
title='Uni-directional 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()
if math.isclose(X[0], 0.0):
node.setDOF(['T'],[200.]) # prescribed temperature at x=0.0
if math.isclose(X[0], Lx):
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()
T = node.getDisp('T')
R_list.append(X[0])
T_list.append(T)
# the analytic solution for comparison
x = np.linspace(0, Lx, 21)
T = 200. * (1 - x/Lx) + 300. * x/Lx
fig, axs = plt.subplots()
axs.plot(x,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("X 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 = ExampleThermal02()
ex.run()
System Analysis Report
=======================
Nodes:
---------------------
Node_231:
x: [0. 0.]
u: [0.]
Node_232:
x: [3.775 0. ]
u: [0.]
Node_233:
x: [6.7 0. ]
u: [0.]
Node_234:
x: [8.775 0. ]
u: [0.]
Node_235:
x: [10. 0.]
u: [0.]
Node_236:
x: [0. 1.66666667]
u: [0.]
Node_237:
x: [2.925 1.66666667]
u: [0.]
Node_238:
x: [5.56666667 1.66666667]
u: [0.]
Node_239:
x: [7.925 1.66666667]
u: [0.]
Node_240:
x: [10. 1.66666667]
u: [0.]
Node_241:
x: [0. 3.33333333]
u: [0.]
Node_242:
x: [2.075 3.33333333]
u: [0.]
Node_243:
x: [4.43333333 3.33333333]
u: [0.]
Node_244:
x: [7.075 3.33333333]
u: [0.]
Node_245:
x: [10. 3.33333333]
u: [0.]
Node_246:
x: [0. 5.]
u: [0.]
Node_247:
x: [1.225 5. ]
u: [0.]
Node_248:
x: [3.3 5. ]
u: [0.]
Node_249:
x: [6.225 5. ]
u: [0.]
Node_250:
x: [10. 5.]
u: [0.]
Elements:
---------------------
Triangle_347: nodes ( Node_231 Node_232 Node_236 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_348: nodes ( Node_237 Node_236 Node_232 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_349: nodes ( Node_232 Node_233 Node_237 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_350: nodes ( Node_238 Node_237 Node_233 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_351: nodes ( Node_233 Node_234 Node_238 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_352: nodes ( Node_239 Node_238 Node_234 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_353: nodes ( Node_234 Node_235 Node_239 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_354: nodes ( Node_240 Node_239 Node_235 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_355: nodes ( Node_236 Node_237 Node_241 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_356: nodes ( Node_242 Node_241 Node_237 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_357: nodes ( Node_237 Node_238 Node_242 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_358: nodes ( Node_243 Node_242 Node_238 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_359: nodes ( Node_238 Node_239 Node_243 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_360: nodes ( Node_244 Node_243 Node_239 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_361: nodes ( Node_239 Node_240 Node_244 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_362: nodes ( Node_245 Node_244 Node_240 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_363: nodes ( Node_241 Node_242 Node_246 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_364: nodes ( Node_247 Node_246 Node_242 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_365: nodes ( Node_242 Node_243 Node_247 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_366: nodes ( Node_248 Node_247 Node_243 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_367: nodes ( Node_243 Node_244 Node_248 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_368: nodes ( Node_249 Node_248 Node_244 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_369: nodes ( Node_244 Node_245 Node_249 )
material: Thermal
grad phi: x=0.000e+00 y=0.000e+00
flux: x=0.000e+00 y=0.000e+00
Triangle_370: nodes ( Node_250 Node_249 Node_245 )
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_231:
x: [0. 0.]
fix: ['T']
u: [200.]
Node_232:
x: [3.775 0. ]
u: [237.75]
Node_233:
x: [6.7 0. ]
u: [267.]
Node_234:
x: [8.775 0. ]
u: [287.75]
Node_235:
x: [10. 0.]
fix: ['T']
u: [300.]
Node_236:
x: [0. 1.66666667]
fix: ['T']
u: [200.]
Node_237:
x: [2.925 1.66666667]
u: [229.25]
Node_238:
x: [5.56666667 1.66666667]
u: [255.66666667]
Node_239:
x: [7.925 1.66666667]
u: [279.25]
Node_240:
x: [10. 1.66666667]
fix: ['T']
u: [300.]
Node_241:
x: [0. 3.33333333]
fix: ['T']
u: [200.]
Node_242:
x: [2.075 3.33333333]
u: [220.75]
Node_243:
x: [4.43333333 3.33333333]
u: [244.33333333]
Node_244:
x: [7.075 3.33333333]
u: [270.75]
Node_245:
x: [10. 3.33333333]
fix: ['T']
u: [300.]
Node_246:
x: [0. 5.]
fix: ['T']
u: [200.]
Node_247:
x: [1.225 5. ]
u: [212.25]
Node_248:
x: [3.3 5. ]
u: [233.]
Node_249:
x: [6.225 5. ]
u: [262.25]
Node_250:
x: [10. 5.]
fix: ['T']
u: [300.]
Elements:
---------------------
Triangle_347: nodes ( Node_231 Node_232 Node_236 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_348: nodes ( Node_237 Node_236 Node_232 )
material: Thermal
grad phi: x=1.000e+01 y=5.684e-14
flux: x=-1.000e+01 y=-5.684e-14
Triangle_349: nodes ( Node_232 Node_233 Node_237 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_350: nodes ( Node_238 Node_237 Node_233 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_351: nodes ( Node_233 Node_234 Node_238 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_352: nodes ( Node_239 Node_238 Node_234 )
material: Thermal
grad phi: x=1.000e+01 y=-2.842e-14
flux: x=-1.000e+01 y=2.842e-14
Triangle_353: nodes ( Node_234 Node_235 Node_239 )
material: Thermal
grad phi: x=1.000e+01 y=-2.842e-14
flux: x=-1.000e+01 y=2.842e-14
Triangle_354: nodes ( Node_240 Node_239 Node_235 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_355: nodes ( Node_236 Node_237 Node_241 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_356: nodes ( Node_242 Node_241 Node_237 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_357: nodes ( Node_237 Node_238 Node_242 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_358: nodes ( Node_243 Node_242 Node_238 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_359: nodes ( Node_238 Node_239 Node_243 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_360: nodes ( Node_244 Node_243 Node_239 )
material: Thermal
grad phi: x=1.000e+01 y=-1.137e-13
flux: x=-1.000e+01 y=1.137e-13
Triangle_361: nodes ( Node_239 Node_240 Node_244 )
material: Thermal
grad phi: x=1.000e+01 y=-1.137e-13
flux: x=-1.000e+01 y=1.137e-13
Triangle_362: nodes ( Node_245 Node_244 Node_240 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_363: nodes ( Node_241 Node_242 Node_246 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_364: nodes ( Node_247 Node_246 Node_242 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_365: nodes ( Node_242 Node_243 Node_247 )
material: Thermal
grad phi: x=1.000e+01 y=-5.684e-14
flux: x=-1.000e+01 y=5.684e-14
Triangle_366: nodes ( Node_248 Node_247 Node_243 )
material: Thermal
grad phi: x=1.000e+01 y=-2.842e-14
flux: x=-1.000e+01 y=2.842e-14
Triangle_367: nodes ( Node_243 Node_244 Node_248 )
material: Thermal
grad phi: x=1.000e+01 y=-8.527e-14
flux: x=-1.000e+01 y=8.527e-14
Triangle_368: nodes ( Node_249 Node_248 Node_244 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Triangle_369: nodes ( Node_244 Node_245 Node_249 )
material: Thermal
grad phi: x=1.000e+01 y=2.842e-14
flux: x=-1.000e+01 y=-2.842e-14
Triangle_370: nodes ( Node_250 Node_249 Node_245 )
material: Thermal
grad phi: x=1.000e+01 y=0.000e+00
flux: x=-1.000e+01 y=-0.000e+00
Total running time of the script: (0 minutes 0.302 seconds)