Transient Heat Conduction problem

dolfin-adjoint
1

Dear all,
I am trying to solve a transient heat equation. I am completely newbie in FEniCS.

For inverse part, i wanted to find the value of the coefficients of thermal conductivity.
However, at first iteration |proj g|= 1.98120D-14 and program stoped.
Please guide me, how can i solve the issue.
My complete code is below:

#“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”“”"

from dolfin import *
from fenics import *
import numpy as np
import matplotlib.pyplot as plt
from fenics_adjoint import *
rho = Constant(1.46555e6) 
C= Constant(20.0)
dt= 1
num_steps=10               

mesh = RectangleMesh(Point(0,0),Point(1,1), 10, 10)  
V = FunctionSpace(mesh, "Lagrange", 1) 
x=SpatialCoordinate(mesh)    
M = FunctionSpace(mesh,"CG",1) 
  
Q_heat = 20000.0
Q = Constant(0.0)                                          
g = Expression('Q_heat', degree=1, Q_heat=Constant(Q_heat))

#Define boudary conditions
u_0 = Constant(0.0)  #Initial temperature over the boundary is zero.
u0=interpolate(u_0, V)

#Define Left Boundary
def left_boundary( x, on_boundary):
    return on_boundary and (abs(x[0])< DOLFIN_EPS)

bc = DirichletBC(V, Constant(0.0), left_boundary)

class right(SubDomain):
    def inside(self, x, on_boundary):
           return on_boundary and ((abs(x[0] - 1.) < DOLFIN_EPS))

vertical = right()
tdim = mesh.topology().dim()    
boundaries = MeshFunction('size_t', mesh, tdim-1)
boundaries.set_all(0)                                        
vertical.mark(boundaries, 1)    
dsB = Measure("ds", subdomain_id=1, subdomain_data=boundaries)                                        

d_target=interpolate(Constant(20.0), M)
d_guess =interpolate(Constant(40.0), M)
e_target=interpolate(Constant(100.0), M)
e_guess=interpolate(Constant(100.0), M)

def forward(d, e):

    u = TrialFunction(V)                       
    v = TestFunction(V)                            
    u0 = Function(V)   
    
    #Define Thermal conductivity
    def k():
        return d + e*x[1]           
  
    a= rho*C*u*v*dx+ dt*inner(grad(v),k()*grad(u))*dx

    L= dt*g*v*dsB+rho*C*inner(u0,v)*dx+dt*Q*v*dx
  
    u = Function(V)                                                   

    #vtkfile = File('Heat_Transfer/solution.pvd')       

    t = 0
    for n in range (num_steps):
        t += dt
        #Compute solution
        solve(a==L, u, bc)  
        #Save solution
        #vtkfile << (u, t)

        plot(u)
    #update solution
    u0.assign(u)
    return u,d,e		

[u_target, d_target_1, e_target_1] = forward(d_target, e_target)
File("forward_modified/targetTempereature_modified.pvd") << u_target
File("forward_modified/target_d_modified.pvd") << d_target_1
File("forward_modified/target_e_up.pvd") << e_target_1

[u,d,e] = forward(d_guess, e_guess)

output = File("inverse_Up/inverse_result_up.pvd")

d_viz = Function(M, name="d")
def eval_cb(j, d):
    d_viz.assign(d)
    output << d_viz


alpha = Constant(0.0)
beta = Constant(1e-2)

#objective function
J = assemble((0.5*inner((u-u_target),u-u_target))*dx+0.5*alpha*sqrt(beta*beta+dot(grad(d),grad(d)))*dx)
#Optimize variables
d_c = Control(d)
J_rf = ReducedFunctional(J, d_c, eval_cb_post=eval_cb)

# scipy optimizer
d_opt = minimize(J_rf,method='L-BFGS-B', bounds=(5.0, 200.0), options={"gtol":1e-15,"ftol":1e-15})
2

Please make sure that you use 3x` syntax to format your code, i.e.

```python 
# add code here
# ....


```
3

edited, please check

4

If you look at your functional, you see that its initial value is of the order 1e-12, and the max difference between u and u_target is 3e-05:

np.max(np.abs(u.vector().get_local() - u_target.vector().get_local()))

so it does not seem like your problem is sensitive to your input parameters.

I would suggest you make sure that the simulation results makes sense with the input parameters before trying to do optimization.

5

Thanks a lot for your suggestion.
It would be better, if you suggest how can i get rid of this problem. As i couldn’t understand properly where to modify.

6

Hi Dokken, i couldn’t solve my problem yet. please tell me how can i understand the order of initial value of functional. please suggest me solving this problem.

7

As I said previously,
it does not seem like your forward model depends a lot on the chosen control value, as not matter what you choose it to, the result of the forward model looks the same.

8

Would it be better to use nonlinear differentiation for transient heat conduction problems instead of linear differentiation.