Hi,
I've adjusted the system a little bit, and I wanted some verification and
more criticism.
I took your suggestions about (Z.grad)**2 and changed it to
numerix.dot(Z.grad, Z.grad) to give me a scalar.
In addition, the terms that could not be easily written as convection terms:
\begin{align}
\frac{1}{n} \frac{\partial T}{\partial x} + \frac{T}{n^2}
\frac{\partial n}{\partial x}
\end{align}
I wrote them by making the coefficients a vector and then dotting with the
gradient:
numerix.dot([(1.0/density),], temperature.grad) +
numerix.dot([(temperature / density**2), ], density.grad)
Are these things a fair / reasonable thing to do? I have looked over my
system in the most original form and it seems there is no way to show them
in a ConvectionTerm or ImplicitSourceTerm.
I apologize if the code is a little difficult to read.
Thanks,
Kevin
On 12 December 2017 at 16:37, Guyer, Jonathan E. Dr. (Fed) <
[email protected]> wrote:
> Kevin -
>
> - Should I model the last term in the temperature (T) equations as a
> convection term or explicit source term?
>
> The best coupling would be obtained by treating this term as a convection
> term on n with a velocity proportional to \partial T/\partial x.
> Unfortunately, (D/n) sits outside the derivatives and so FiPy has no
> implicit way to represent it. I think you're stuck with the source.
>
> - What about the "S_Z" term in the Z equation?
>
> Again, as written, these are just sources.
>
> - Is it required that I declare diffusivity (D) as a cell variable, as it
> is now?
>
> In order to write an equation to solve for D, it must be a CellVariable,
> however, there's no advantage in adding this as a solution equation. I
> would just define D as a function of Z. As such, it's better to define it
> as a FaceVariable for use in the DiffusionTerms.
>
> Note:
>
> The expressions Z*(Z.grad) and (Z.grad)**2 amount to multiplying a scalar
> by a vector and a vector by a vector, but D needs to be scalar. I recommend
> working out what your equations would be using either divergence and
> gradient operators or Einstein notation, as premature introduction of d/dx
> in 1D equations leads to equations that are difficult to debug and render
> properly.
>
> You may find that when you go back to the source equations in general
> vectorial form that the sources in the T and Z equations are amenable to
> being written as convection terms or implicit sources, which would be good.
>
> - Jon
>
>
> > On Dec 11, 2017, at 2:42 PM, Kevin Blondino <[email protected]>
> wrote:
> >
> > Hello,
> >
> > I have a somewhat complicated system of highly nonlinear, 1D PDE's which
> I am having trouble modeling. The issue is that I am not sure how to
> represent certain terms.
> >
> > Here is a the following LaTeX code for the system, as to make sure there
> is no ambiguity:
> > \begin{align}
> > \frac{\partial n}{\partial t} \,&=\, \frac{\partial}{\partial
> x}\left(D \frac{\partial n}{\partial x}\right) \\
> > \frac{\partial T}{\partial t} \,&=\, \frac{\partial}{\partial
> x}\left(D \frac{\partial T}{\partial x}\right) +
> \frac{D}{n}\,\frac{\partial n}{\partial x}\,\frac{\partial T}{\partial x} \\
> > \frac{\partial Z}{\partial t} \,&=\, \frac{\partial}{\partial
> x}\left(D \frac{\partial Z}{\partial x}\right) +
> \frac{T}{n^2}\frac{\partial n}{\partial x} + \frac{1}{n}\frac{\partial
> T}{\partial x} + G(Z) \\
> > G(Z) \,&=\, a + b(Z - Z_S) + c(Z - Z_S)^3 \\
> > D \,&=\, 1 + \frac{1}{1 + Z^2 + Z\cdot\frac{\partial Z}{\partial
> x} + \left(\frac{\partial Z}{\partial x}\right)^2}
> > \end{align}
> >
> > I have purposefully squelched many coefficients as possible to make
> simple.
> >
> > The main code I have been using to model them is as follows:
> >
> > # ----------------- Variable Declarations -----------------
> > density = CellVariable(name=r"$n$", mesh=mesh, hasOld=True)
> >
> > temperature = CellVariable(name=r"$T$", mesh=mesh, hasOld=True)
> >
> > Z = CellVariable(name=r"$Z$", mesh=mesh, hasOld=True)
> >
> > ## Diffusivity
> > D = CellVariable(name=r"$D$", mesh=mesh, hasOld=True)
> >
> > ... (initial and boundary conditions are here) ...
> >
> > # ----------------- PDE Declarations ----------------------
> > # Diffusivity Equation (D)
> > diffusivity_equation = ImplicitSourceTerm(coeff=1.0, var=D) == 1.0
> + 1.0 / (1 + Z**2 + Z*(Z.grad) + (Z.grad)**2)
> >
> >
> > # Density Equation (n)
> > density_equation = TransientTerm(var=density) ==
> DiffusionTerm(coeff=D, var=density)
> >
> >
> > # Temperature Equation (T)
> > S_T = (D/density) * numerix.dot(density.grad,temperature.grad) #
> ???
> > S_T_conv = ConvectionTerm(coeff=(D/density)*density.grad,
> var=temperature) # ??
> >
> > temp_equation = TransientTerm(var=temperature) ==
> DiffusionTerm(coeff=D, var=temperature) + S_T
> >
> >
> > # Z Equation
> > G = a + b*(Z - Z_S) + c*(Z - Z_S)**3
> > S_Z = G + (1 / density)*temperature.grad.mag + (temperature /
> density**2)*density.grad.mag # ???
> > Z_equation = TransientTerm(coeff=1.0, var=Z) ==
> DiffusionTerm(coeff=D, var=Z) + S_Z
> >
> > # Fully-Coupled Equation
> > full_equation = density_equation & temp_equation & Z_equation &
> diffusivity_equation
> >
> > Should I model the last term in the temperature (T) equations as a
> convection term or explicit source term?
> > What about the "S_Z" term in the Z equation?
> > Is it required that I declare diffusivity (D) as a cell variable, as it
> is now?
> >
> > I appreciate any help that can be given.
> >
> > Thank you,
> > Kevin
> > _______________________________________________
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>
>
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