using Finite element analysis>>

@author of the code: Tong Wu

Contact: wu616@purdue.edu

The problem is shown in << Fundamentals of the finite

element method for heat and fluid flow>>

Author: Roland W Lewis, Perumal Nithiarasu

and Kankanhally N. Seetharamu

pp108-pp112 and pp172

|node 1------node 2------node 3

bar1 bar2

ambient temperature 25 C

Initial prescribed temperature, on node 1, 100 C

Other parameters are:

h=120 #convection

k=200 #thermal conductivity [W/m*C]

rho=1 #density

P=10e-3 #perimeter [m]

L=1e-2 #length per element

A=6e-6 #section area [m^2]

C_p=2.42e6 #specific heat [W/m**3*C]

dt=0.1 #time step

"""

def eofhtb():

import sympy

from sympy import Matrix

import numpy as np

import scipy

# H_e, convection; K_e, conduction; C_e, specific heat

h=120 #convection

T_a=25 #ambient temperature [C]

k=200 #thermal conductivity [W/m*C]

rho=1 #density

P=10e-3 #perimeter [m]

L=1e-2 #length per element

A=6e-6 #section area [m^2]

C_p=2.42e6 #specific heat [W/m**3*C]

sympy.var('x')

N=Matrix([[1-x], [x]]) #shape function

B=Matrix([-1,1]) #temperature-gradient function

f_e=Matrix([1,1]) #elementwise heat flux

f_e= (h*P*L*T_a/2)*f_e #elementwise heat flux

#initialize C_e, H_e, K_e

C_ee= N*np.transpose(N)

H_ee= N*np.transpose(N)

K_ee= B*np.transpose(B)

x1,x2=(0,1)

from sympy.utilities import lambdify

from mpmath import quad

C_e=scipy.zeros(C_ee.shape, dtype=float)

H_e=scipy.zeros(H_ee.shape, dtype=float)

K_e=scipy.zeros(K_ee.shape, dtype=float)

#Numerical integration

for (i,j), expr1 in scipy.ndenumerate(H_ee):

H_e[i,j] = quad(lambdify((x),expr1,'math'),(x1,x2))

H_e[i,j] = h*P*L*H_e[i,j]

C_e[i,j] = rho*C_p*L*A*quad(lambdify((x),expr1,'math'),(x1,x2))

for (i,j), expr2 in scipy.ndenumerate(K_ee):

K_e[i,j] = quad(lambdify((x),expr2,'math'),(x1,x2))

K_e[i,j] = ((A*k)/L)*K_e[i,j]

#Compute total stiffness (Conduction+Convection)

K_e_total=H_e+K_e

return (H_e,K_e,C_e,K_e_total,f_e);

def main():

import numpy as np

(H_e,K_e,C_e,K_e_total,f_e)=eofhtb() #import function of stiffness matrix

# initialize

C=np.zeros((3,3)) #specific heat matrix (2 bars)

K=np.zeros((3,3)) #stiffness matrix (2 bars)

F=np.zeros((3,1)) #external heat source

T_0=np.zeros((3,1)) #ambient temperature

T_1=np.zeros((3,1)) #ambient temperature + prescribed temperature

T_2=np.zeros((3,1)) #updagted temperature

F_prescribed=np.zeros((3,1)) #prescribed heat source

#finite element analysis

T_0[:,0]=[25,25,25] #define the ambient temperature

#assemble global stiffness matrix

for i in range(2):

C[i:i+2, i:i+2] += C_e

K[i:i+2, i:i+2] += K_e_total

F[i:i+2, :] += f_e

#set parameters of Back-Euler method

#theta=1 #parameter theta: T^{n+theta}=theta*T^{n+1}+(1-theta)*T^{n}

t=0 #time

dt=0.1 #time step

K_and_C_rhs=(C/dt) #the global matrix of right hand side

K_and_C_lhs=(C/dt)+K #the global matrix of left hand side

F_dt=F #external matrix for discretized delta-T

#1st iteration

loop=0;loopmax=200

T_1[:,0]=[100,25,25] #define the prescribed temperature for node 1.

T_storage=np.zeros((loopmax,4))

#iterative solver of Crank-Nicolson method

while loop<loopmax:

loop=loop+1

t=t+dt

F_dt1=np.matmul(K_and_C_rhs,T_0)+F_dt #

F_prescribed[1:3,0]=-K_and_C_lhs[1:3,0]*T_1[0,0]

F_dt1_update=F_dt1+F_prescribed

F_dt_1_up_redu=F_dt1_update[1:3]

K_and_C_lhs_reduce=K_and_C_lhs[1:3,1:3]

T_2[1:3]=np.linalg.solve(K_and_C_lhs_reduce,F_dt_1_up_redu)

T_2[0]=100

T_storage[loop-1,1:4]=T_2.transpose()

T_storage[loop-1,0]=t

T_0=T_2;T_1=T_2

#print(T_storage)

from tempfile import TemporaryFile

outfile = TemporaryFile()

np.save(outfile,T_storage)

#plot temperature of three nodes

import matplotlib as mpl

import matplotlib.pyplot as plt

fig, ax = plt.subplots()

ax.plot(T_storage[:,0], T_storage[:,1], color="blue", label="T,node 1")

ax.plot(T_storage[:,0], T_storage[:,2], color="red", label="T,node 2")

ax.plot(T_storage[:,0], T_storage[:,3], color="green", label="T,node 3")

ax.set_xlabel("time step (s)")

ax.set_ylabel("temperature (C)")

ax.legend()

plt.show()

#main driver

if __name__ == "__main__":

main()