Heat equation with temperature-dependent source terms

variational formulation
1

I am trying to set up a model to solve the heat equation with temperature dependent terms.
It would look like this:

initial_T = Expression("Tini", Tini=298, degree=3) # initial temperature
T0 = interpolate(initial_T, Space)
T = Function(Space)            # Temperature
V = TestFunction(Space)        # Test Function
dT = TrialFunction(Space)      # Temperature Derivative
q0 = Function(VectorSpace)     # old heat flux 
i = Index()
G = as_tensor(T.dx(i), (i))    #gradient of T
G0 = as_tensor(T0.dx(i), (i))  # old gradient of T

and the equations would be:

Laser = Expression('2*power_Q*(1-eta_s)*eta_l*exp(-pow((x[0] - (-Lx/2+2*r_b)-v_source*dt), 2)/(r_b*r_b)-pow((x[1]-0), 2)/(r_b*r_b))/3.1416/r_b/r_b', degree=3, power_Q=power_Q, eta_s=eta_s, eta_l=eta_l, r_b=r_b, Lx=Lx, Lz=Lz, v_source=source_vel, dt=t)

q = as_tensor(dtime/(dtime + tau_q) * (tau_q/dtime*q0[i] - kappa*(1+tau_T/dtime)*G[i] + kappa*tau_T/dtime*G0[i]),(i)) #heat

S=  as_tensor(exp(-(((T-1100)/20)*((T-1100)/20)))/60000,(i))         #temperature dependent source                

F = (rho*cp/dtime*(T-T0)*V-S[i]*V.dx(i) - q[i]*V.dx(i))* dv - rho*Laser*V*da(3) 

Gain = derivative(F, T, dT)                     # Jacobian

where the dv and da are volume and area elements and the Neuman conditions are not included yet. The other parameters are model parameters.
The equations are solved as:

solve(F==0, T, bcs, J = Gain, solver_parameters={"newton_solver":{"linear_solver": "mumps", "relative_tolerance": 1e-3} })
q_tmp = project(q, VectorSpace)
q0.assign(q_tmp)
T0.assign(T)

However, I get 2 errors.
-The first one in the definition of S:

Traceback (most recent call last):
File “/scratch/snx3000/jfernnde/fenics/fenics-project/HeatFlowModel.py”, line 508, in
model.runModel()
File “/scratch/snx3000/jfernnde/fenics/fenics-project/HeatFlowModel.py”, line 452, in runModel
self._createModel(t)
File “/scratch/snx3000/jfernnde/fenics/fenics-project/HeatFlowModel.py”, line 269, in _createModel
self.S = as_tensor(exp(-(((self.T-1100)/20)*((self.T-1100)/20)))/60000,(self.i))
File “/users/jfernnde/bin/miniconda3/envs/fenics/lib/python3.9/site-packages/ufl/tensors.py”, line 272, in as_tensor
return ComponentTensor(expressions, indices)
File “/users/jfernnde/bin/miniconda3/envs/fenics/lib/python3.9/site-packages/ufl/tensors.py”, line 152, in init
fi, fid, sh = remove_indices(expression.ufl_free_indices,
File “/users/jfernnde/bin/miniconda3/envs/fenics/lib/python3.9/site-packages/ufl/index_combination_utils.py”, line 138, in remove_indices
assert None not in shape
AssertionError

-The second one, in the last 2 lines. It cannot assign the variable:
File “/scratch/snx3000/jfernnde/fenics/fenics-project/HeatFlowModel.py”, line 508, in
model.runModel()
File “/scratch/snx3000/jfernnde/fenics/fenics-project/HeatFlowModel.py”, line 483, in runModel
q_tmp = project(self.q, self.VectorSpace)
File “/users/jfernnde/bin/miniconda3/envs/fenics/lib/python3.9/site-packages/dolfin/fem/projection.py”, line 138, in project
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
RuntimeError:

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at

*** fenics-support@googlegroups.com

*** Remember to include the error message listed below and, if possible,
*** include a minimal running example to reproduce the error.

*** -------------------------------------------------------------------------
*** Error: Unable to successfully call PETSc function ‘KSPSolve’.
*** Reason: PETSc error code is: 76 (Error in external library).
*** Where: This error was encountered inside /home/conda/feedstock_root/build_artifacts/fenics-pkgs_1608660238187/work/dolfin/dolfin/la/PETScKrylovSolver.cpp.
*** Process: 0

*** DOLFIN version: 2019.1.0
*** Git changeset: 640ef70247ae212e16ae9f24fc9bc603b506f78a

Can anyone help?

2

Please use proper formatting of both your code and error messages. You should use ```-encapsulation,

Also, please make the code a minimal working code. Follow the guidelines at: Read before posting: How do I get my question answered?

3

I have already reduced it as much as possible.

The error is that I cannot reassign a variable to another variable with q_tmp = project(q, VectorSpace)

4

The problem is that the code cannot be copy-pasted and ran directly as a script, as you have not defined a mesh or the function space.
Your first error, is related to S not being a tensor. There is no variation in the definition of S, it is simply the shape of T.
Secondly if your space is a Scalar space, The derivative of T is not a tensor, but a vector, and you should use as_vector, and I would write it as:

S=  as_vector([exp(-(((T-1100)/20)*((T-1100)/20)))/60000,]*dimension)

Your second error relates to your computer running out of memory.
I would suggest using an iterative solver for the projection which can be set using the solver_type and preconditioner_type kwarg in project.