Regarding implementation of implicit thermal conduction in Joule miniapp

[Raghavendra0117]

Dear all,

I am new to MFEM and its usage. Presently, I am interested in solving a simple linear thermal diffusion problem. I have gone through some of the existing examples and miniapps that are already available with MFEM and I found the "joule" miniapp which use mixed FE spaces (L2, RT) suits to my requirement.

I was going through the joule mini app coding to understand the actual discrete eqns that are implemented. Presently, I am only interested in the thermal diffusion part of the "miniapp", neglecting E, B and joule heating (W). I got stuck at the thermal flux solver (--I assume it is an implicit solver--). I tried to derive the corresponding equation implemented in the solver with the help of some comments mentioned in joule.cpp & joule_solver.hpp (at line 85) and I found there are some differences. I have attached my derivations here (please refer to the attached file). Please correct me, if my derivations/assumptions are wrong. Also, It will be of great help If someone can suggest any Literature/docs/reports which describes the governing equations and its discrete form.

Thank you in advance
Raghavendra Kollipara

Attachment:
Thermal_conduction_implicit_formulation.pdf

[mlstowell]

Hi, @Raghavendra0117,
Your derivation appears to be correct but we do something a little different in the joule miniapp. Your derivation assumes the solver will compute $F^{n+1}$ which is quite natural. However, our ODE solvers compute $\frac{\partial F}{\partial t}$ and then use this to compute $F^{n+1}=F^n+dt \frac{\partial F}{\partial t}$ as a separate step. This step is hidden from the miniapp inside the ODE integrator's Step method. This miniapp uses one of MFEM's built in ODE integrators but I believe this time integration scheme follows the more sophisticated schemes offered in SUNDIALS' ARKODE library.

Best wishes,
Mark

[Raghavendra0117]

Thanks a lot, @mlstowell.

I went through the code related to solver (BackwardEuler in ode.cpp) and this is my understanding so far (pl correct me if i am wrong).

1. The solver is initialized with oper which is an object of MagneticDiffusionEOperator.
2. odesolver->step calls implicitsolve method of MagnetiDiffusionEOperator that outputs dX_dt = [dP_dt, dE_dt, dB_dt, dF_dt, dT_dt.]
3. This dX_dt is used to calculate the x^{n+1} as you mentioned in your comment.

Assuming my understanding is correct, the "implicit solve" routine will return/update dX_dt.

Inside the implicit solve routine, entries of dX_dt is defined as

dT.MakeRef(&L2FESpace, dX_dt,true_offset[0]);
dF.MakeRef(&HDivFESpace, dX_dt,true_offset[1]);

a2->RecoverFEMSolution(X2,v2,F);

weakDiv->AddMult(F, temp_lf, -1.0);

pcg_m3->Mult(temp_lf, dT);
There are no lines related to calculation of dF. Even if one makes thermal flux=0.0 inside the time loop before "odesolver->step()", there is no change in result of thermal flux^{n+1}. Thus, i came to conclusion that F^{n+1} is calculated and same goes to R.H.S of equation related to dT_dt calculation.

I appreciate your help if you can suggest any literature/publications/docs/reports that describes the discrete form of the implemented equations that help my understanding.

Thank you
Raghavendra Kollipara

[mlstowell]

Hi, @Raghavendra0117,
You've certainly found something odd here. It looks like both E and F are being treated in a non-standard manner. It seems that both of these fields are being solved for directly while dE/dt and dF/dt are both set to zero (so that the X+dt dX/dt step won't alter E and F). My guess is that this is done so that the boundary conditions on E and F can be applied more easily. There will be other ramifications of this choice having to do with the order in which the fields are updated and perhaps other things.

I think we agree that improved documentation is needed here.

Best wishes,
Mark