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I'm trying to implement a phase field solidification model of a ternary alloy using FiPy. I have looked at most of the phase field examples provided on FiPy's website and my model is similar to examples.phase.quaternary.

The evolution equation for the concentrations looks like this:

https://i.sstatic.net/KvM1r.png

Which should be solved for C_1 and C_2 (C_3 is solvent). The concentration equations are coupled with the phase evolution equation and we have D_i = D_i(phi), h = h(phi).

There are nonlinear dependencies of the variable solved for (C_i) in all three terms which makes me unsure on how to define them in FiPy. The first term (red) is a diffusion term with a nonlinear coefficient and that should be fine, but how should I define the counterdiffusion and the phasetransformation terms?

I tried defining them as diffusion and convection terms with nonlinear coefficients but without success. Therefore I am hoping for some advice on how I can define so that FiPy likes it.

Any help is highly appreciated, thanks! /Anders

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This set of equations can be solved in a coupled manner. The counter diffusion term can be defined as a coupled diffusion term and the phase transformation term can be defined as a convection term. The equations will be,

eqn1 = fipy.TransientTerm(var=C_1) == \
       fipy.DiffusionTerm(D_1 - coeff_1 * (D_1 - D_3), var=C_1) \ # coupled
       - fipy.DiffusionTerm(coeff_1 * (D_2 - D_3), var=C_2) \ # coupled
       + fipy.ConvectionTerm(conv_coeff_1, var=C_1)

where

coeff_1 = D_1 * C_1 / ((D_1 - D_3) * C_1 + (D_2 - D_3) * C_2 + D_3)
conv_coeff_1 = Vm / R * D_1 * h.faceGrad * inner_sum_1.faceValue

and inner_sum_1 is the complicated inner sum in the phase transformation term. The $\nabla h$ part has been taken out of the inner sum. You can either use (h.grad * inner_sum_1).faceValue or h.faceGrad * inner_sum_1.faceValue or use face values for the variables that constitute inner_sum_1. I don't know how much difference it makes. After both the C_1 and C_2 equations are defined in a similar manner, then combine them into one equation with

eqn = eqn1 & eqn2
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  • Hi and thanks for the input, much appreciated! However, the variable C_3 is declared as a solvent i.e. : components = [ CellVariable(mesh=mesh, name='C1', hasOld=1), CellVariable(mesh=mesh, name='C2', hasOld=1), ] solvent = Variable(value = 1) for Cj in components: solvent -= Cj solvent.name = 'C3' Such as in examples.phase.quaterny, this means there is no evolution equation for C3, i.e. the constraint between the concentrations is being utilized instead: C1 + C2 + C3 = 1. Can I still write the term for C3 as a DiffusionTerm as specified in your comment?
    – A. Ericsson
    Commented Oct 7, 2016 at 11:03
  • In the case that solving for C3 is redundant, substitute C3 from the equations so there is no need to define it.
    – wd15
    Commented Oct 8, 2016 at 14:18
  • The answer is amended with the C_3 = 1 - C_1 - C_2 substitution.
    – wd15
    Commented Oct 8, 2016 at 15:19
  • Thanks for the comments! A followup question thought, does it make any difference if I define C3 and insert it in the equations or if I just insert it directly? When you use C3.harmonicFaceValue for example?
    – A. Ericsson
    Commented Oct 13, 2016 at 8:28
  • C_3 need not be defined at all anywhere in the script. If you need use it for something (like plotting for example) then define it as C_3 = 1 - C_1 - C_2.
    – wd15
    Commented Oct 14, 2016 at 14:12

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