.. _FAQ:
Frequently Asked Questions
==========================
How do I represent an equation in FiPy?
---------------------------------------
As explained in :ref:`chap:Numerics`, the canonical
governing equation that can be solved by :term:`FiPy` for the dependent
:class:`~fipy.variables.cellVariable.CellVariable` :math:`\phi` is
.. math::
\underbrace{
\frac{\partial (\rho \phi)}{\partial t}
}_{\text{transient}}
+
\underbrace{
\vphantom{\frac{\partial (\rho \phi)}{\partial t}}
\nabla \cdot \left( \vec{u} \phi \right)
}_{\text{convection}}
=
\underbrace{
\vphantom{\frac{\partial (\rho \phi)}{\partial t}}
\left[ \nabla \cdot \left( \Gamma_i \nabla \right) \right]^n \phi
}_{\text{diffusion}}
+
\underbrace{
\vphantom{\frac{\partial (\rho \phi)}{\partial t}}
S_{\phi}
}_{\text{source}}
and the individual terms are discussed in :ref:`section:discretization`.
A physical problem can involve many different coupled
governing equations, one for each variable. Numerous specific
examples are presented in Part :ref:`part:Examples`.
Is there a way to model an anisotropic diffusion process or more generally to represent the diffusion coefficient as a tensor so that the diffusion term takes the form :math:`\partial_i \Gamma_{ij}\partial_j \phi`?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Terms of the form :math:`\partial_i \Gamma_{ij}\partial_j \phi` can be
posed in :term:`FiPy` by using a list, tuple rank 1
or rank 2 :class:`~fipy.variables.faceVariable.FaceVariable` to represent a
vector or tensor diffusion coefficient. For example, if we wished to
represent a diffusion term with an anisotropy ratio of 5 aligned along the
x-coordinate axis, we could write the term as,
>>> DiffusionTerm([[[5, 0], [0, 1]]])
which represents :math:`5 \partial_x^2 + \partial_y^2`. Notice that
the tensor, written in the form of a list, is contained within
a list. This is because the first index of the list refers to
the order of the term not the first index of the tensor (see
:ref:`discret-higherOrderDiffusion`). This notation, although succinct can
sometimes be confusing so a number of cases are interpreted below.
>>> DiffusionTerm([[5, 1]])
This represents the same term as the case examined above.
The vector notation is just a short-hand representation
for the diagonal of the tensor. Off-diagonals are assumed
to be zero.
>>> DiffusionTerm([5, 1])
This simply represents a fourth order isotropic diffusion
term of the form :math:`5 \left( \partial_x^2 + \partial_y^2
\right)^2`.
>>> DiffusionTerm([[1, 0], [0, 1]])
Nominally, this should represent a fourth order diffusion
term of the form :math:`\partial_x^2 \partial_y^2`, but :term:`FiPy`
does not currently support anisotropy for higher order
diffusion terms so this may well throw an error or give
anomalous results.
>>> x, y = mesh.cellCenters
>>> DiffusionTerm(CellVariable(mesh=mesh,
... value=[[x**2, x * y], [-x * y, -y**2]])
This represents an anisotropic diffusion coefficient that
varies spatially so that the term has the form
:math:`\partial_x (x^2 \partial_x + x y \partial_y)
+ \partial_y (-x y \partial_x - y^2 \partial_y)
\equiv x \partial_x - y \partial_y + x^2 \partial_x^2 - y^2
\partial_y^2`.
Generally, anisotropy is not conveniently aligned along the coordinate
axes; in these cases, it is necessary to apply a rotation matrix in
order to calculate the correct tensor values, see
:mod:`examples.diffusion.anisotropy` for details.
How do I represent a `...` term that *doesn't* involve the dependent variable?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It is important to realize that, even though an expression may
superficially resemble one of those shown in :ref:`section:discretization`, if the dependent variable
*for that PDE* does not appear in the appropriate place, then that
term should be treated as a source.
How do I represent a diffusive source?
''''''''''''''''''''''''''''''''''''''
If the governing equation for :math:`\phi` is
.. math::
\frac{\partial \phi}{\partial t}
= \nabla\cdot\left( D_1 \nabla \phi\right)
+ \nabla\cdot\left( D_2 \nabla \xi\right)
then the first term is a :class:`~fipy.terms.transientTerm.TransientTerm` and the second term is a
:class:`~fipy.terms.diffusionTerm.DiffusionTerm`, but the third term is simply an explicit source,
which is written in Python as
>>> (D2 * xi.faceGrad).divergence
.. currentmodule:: fipy.variables.cellVariable
Higher order diffusive sources can be obtained by simply nesting the
references to :attr:`~CellVariable.faceGrad` and
:attr:`~fipy.variables.faceVariable.FaceVariable.divergence`.
.. note::
We use :attr:`~CellVariable.faceGrad`, rather than
:attr:`~CellVariable.grad`, in order to obtain a second-order spatial
discretization of the diffusion term in :math:`\xi`, consistent with the
matrix that is formed by
:class:`~fipy.terms.diffusionTerm.DiffusionTerm` for :math:`\phi`.
How do I represent a convective source?
'''''''''''''''''''''''''''''''''''''''
The convection of an independent field :math:`\xi` as in
.. math::
\frac{\partial \phi}{\partial t}
= \nabla\cdot
\left(
\vec{u} \xi
\right)
can be rendered as
>>> (u * xi.arithmeticFaceValue).divergence
when :math:`\vec{u}` is a rank-1 :class:`~fipy.variables.faceVariable.FaceVariable` (preferred) or as
>>> (u * xi).divergence
if :math:`\vec{u}` is a rank-1 :class:`~fipy.variables.cellVariable.CellVariable`.
How do I represent a transient source?
''''''''''''''''''''''''''''''''''''''
The time-rate-of change of an independent variable :math:`\xi`, such as in
.. math::
\frac{\partial (\rho_1 \phi)}{\partial t}
= \frac{\partial (\rho_2 \xi)}{\partial t}
does not have an abstract form in :term:`FiPy` and should be discretized
directly, in the manner of Equation :eq:`num:tra`, as
>>> TransientTerm(coeff=rho1) == rho2 * (xi - xi.old) / timeStep
This technique is used in :mod:`examples.phase.anisotropy`.
What if my term involves the dependent variable, but not where FiPy puts it?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Frequently, viewing the term from a different perspective will allow it to
be cast in one of the canonical forms. For example, the third term in
.. math::
\frac{\partial \phi}{\partial t}
= \nabla\cdot\left( D_1 \nabla \phi\right)
+ \nabla\cdot\left( D_2 \phi \nabla \xi\right)
might be considered as the diffusion of the independent variable :math:`\xi` with
a mobility :math:`D_2\phi` that is a function of the dependent variable :math:`\phi`.
For :term:`FiPy`'s purposes, however, this term represents the convection of
:math:`\phi`, with a velocity :math:`D_2\nabla\xi`, due to the counter-diffusion of
:math:`\xi`, so
>>> eq = TransientTerm() == (DiffusionTerm(coeff=D1)
... + <Specific>ConvectionTerm(coeff=D2 * xi.faceGrad))
.. note::
With the advent of :ref:`CoupledEquations` in FiPy 3.x, it is now
possible to represent both terms with
:class:`~fipy.terms.diffusionTerm.DiffusionTerm`.
What if the coefficient of a term depends on the variable that I'm solving for?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A non-linear coefficient, such as the diffusion coefficient in
:math:`\nabla\cdot[\Gamma_1(\phi) \nabla \phi] = \nabla\cdot[\Gamma_0 \phi (1 -
\phi) \nabla\phi]` is not a problem for :term:`FiPy`. Simply write it as it
appears:
>>> diffTerm = DiffusionTerm(coeff=Gamma0 * phi * (1 - phi))
.. note::
Due to the nonlinearity of the coefficient, it will probably be
necessary to "sweep" the solution to convergence as discussed in
:ref:`FAQ-IterationsTimestepsSweeps`.
How can I see what I'm doing?
-----------------------------
.. currentmodule:: fipy.viewers
How do I export data?
~~~~~~~~~~~~~~~~~~~~~
The way to save your calculations depends on how you plan to make use of
the data. If you want to save it for "restart" (so that you can continue
or redirect a calculation from some intermediate stage), then you'll want
to "pickle" the :term:`Python` data with the :mod:`~fipy.tools.dump` module. This is
illustrated in :mod:`examples.phase.anisotropy`,
:mod:`examples.phase.impingement.mesh40x1`,
:mod:`examples.phase.impingement.mesh20x20`, and
:mod:`examples.levelSet.electroChem.howToWriteAScript`.
On the other hand, pickled :term:`FiPy` data is of little use to anything
besides :term:`Python` and :term:`FiPy`. If you want to import your calculations into
another piece of software, whether to make publication-quality graphs or
movies, or to perform some analysis, or as input to another stage of a
multiscale model, then you can save your data as an :abbr:`ASCII` text
file of tab-separated-values with a :class:`~tsvViewer.TSVViewer`. This is illustrated
in :mod:`examples.diffusion.circle`.
How do I save a plot image?
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Some of the viewers have a button or other mechanism in the user interface
for saving an image file. Also, you can supply an optional keyword
``filename`` when you tell the viewer to
:meth:`~viewer.AbstractViewer.plot`, *e.g.*
>>> viewer.plot(filename="myimage.ext")
which will save a file named :file:`myimage.ext` in your current working
directory. The type of image is determined by the file extension
":file:`.ext`". Different viewers have different capabilities:
:term:`Matplotlib`
accepts ":file:`.eps`," ":file:`.jpg`"
(`Joint Photographic Experts Group <http://www.jpeg.org/>`_),
and ":file:`.png`"
(`Portable Network Graphics <http://www.w3.org/Graphics/PNG/>`_).
.. attention::
Actually, :term:`Matplotlib` supports different extensions, depending
on the chosen `backend
<http://matplotlib.sourceforge.net/backends.html>`_, but our
:class:`MatplotlibViewer
<fipy.viewers.matplotlibViewer.abstractMatplotlibViewer.AbstractMatplotlibViewer>` classes
don't properly support this yet.
What if I only want the saved file, with no display on screen?
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
To our knowledge, this is only supported by :term:`Matplotlib`, as is explained
in the
`Matplotlib FAQ on image backends <http://matplotlib.sourceforge.net/faq/howto_faq.html#generate-images-without-having-a-window-popup>`_.
Basically, you need to tell :term:`Matplotlib` to use an "image
backend," such as "``Agg``" or "``Cairo``." Backends are discussed at
http://matplotlib.sourceforge.net/backends.html.
How do I make a movie?
~~~~~~~~~~~~~~~~~~~~~~
:term:`FiPy` has no facilities for making movies. You will need to save
individual frames (see the previous question) and then stitch them together
into a movie, using one of a variety of different free, shareware, or
commercial software packages. The guidance in the
`Matplotlib FAQ on movies <http://matplotlib.sourceforge.net/faq/howto_faq.html#make-a-movie>`_
should be adaptable to other :class:`Viewer <viewer.AbstractViewer>`\s.
Why doesn't the :class:`Viewer <viewer.AbstractViewer>` look the way I want?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
:term:`FiPy`'s viewers are utilitarian. They're designed to let you see
*something* with a minimum of effort. Because different plotting
packages have different capabilities and some are easier to install on some
platforms than on others, we have tried to support a range of :term:`Python`
plotters with a minimal common set of features. Many of these packages are
capable of much more, however. Often, you can invoke the
:class:`Viewer <viewer.AbstractViewer>` you
want, and then issue supplemental commands for the underlying plotting
package. The better option is to make a "subclass" of the :term:`FiPy`
:class:`Viewer <viewer.AbstractViewer>` that comes closest to producing the image you want. You can
then override just the behavior you wan to change, while letting :term:`FiPy` do
most of the heavy lifting. See :mod:`examples.phase.anisotropy` and
:mod:`examples.phase.polyxtal` for examples of creating a custom
:term:`Matplotlib` :class:`Viewer <viewer.AbstractViewer>` class; see
:mod:`examples.cahnHilliard.sphere` for an example of creating a custom
:term:`Mayavi` :class:`Viewer <viewer.AbstractViewer>` class.
.. _FAQ-IterationsTimestepsSweeps:
Iterations, timesteps, and sweeps? Oh, my!
------------------------------------------
Any non-linear solution of partial differential equations is an
approximation. These approximations benefit from repetitive
solution to achieve the best possible answer. In :term:`FiPy` (and in
many similar PDE solvers), there are three layers of repetition.
iterations
This is the lowest layer of repetition, which you'll generally need to
spend the least time thinking about. :term:`FiPy` solves PDEs by
discretizing them into a set of linear equations in matrix form, as
explained in :ref:`section:discretization` and
:ref:`section:linear-equations`. It is not always practical, or even
possible, to exactly solve these matrix equations on a computer.
:term:`FiPy` thus employs "iterative solvers", which make successive
approximations until the linear equations have been satisfactorily
solved. :term:`FiPy` chooses a default number of iterations and solution
tolerance, which you will not generally need to change. If you do wish to
change these defaults, you'll need to create a new
:class:`~fipy.solvers.solver.Solver` object with the desired number of
iterations and solution tolerance, *e.g.*
>>> mySolver = LinearPCGSolver(iterations=1234, tolerance=5e-6)
:
:
>>> eq.solve(..., solver=mySolver, ...)
.. note::
The older :class:`~fipy.solvers.solver.Solver` ``steps=`` keyword is now deprecated
in favor of ``iterations=`` to make the role clearer.
Solver iterations are changed from their defaults in
:mod:`examples.flow.stokesCavity` and
:mod:`examples.updating.update0_1to1_0`.
sweeps
This middle layer of repetition is important
when a PDE is non-linear (*e.g.*, a diffusivity that
depends on concentration) or when multiple PDEs are coupled
(*e.g.*, if solute diffusivity depends on temperature and
thermal conductivity depends on concentration). Even if the
:class:`~fipy.solvers.solver.Solver` solves the *linear* approximation of the
PDE to absolute perfection by performing an infinite number of
iterations, the solution may still not be a very good
representation of the actual *non-linear* PDE. If we
resolve the same equation *at the same point in elapsed
time*, but use the result of the previous solution instead of
the previous timestep, then we can get a refined solution to
the *non-linear* PDE in a process known as "sweeping."
.. note::
Despite references to the "previous timestep," sweeping is
not limited to time-evolving problems. Nonlinear sets of
quasi-static or steady-state PDEs can require sweeping, too.
We need to distinguish between the value of the variable at
the last timestep and the value of the variable at the last
sweep (the last cycle where we tried to solve the
*current* timestep). This is done by first modifying the
way the variable is created:
>>> myVar = CellVariable(..., hasOld=True)
and then by explicitly moving the current value of the
variable into the "old" value only when we want to:
>>> myVar.updateOld()
Finally, we will need to repeatedly solve the equation until
it gives a stable result. To clearly distinguish that a
single cycle will not truly "solve" the equation, we invoke
a different method ":meth:`~fipy.terms.term.Term.sweep`:
>>> for sweep in range(sweeps):
... eq.sweep(var=myVar, ...)
Even better than sweeping a fixed number of cycles is to do it
until the non-linear PDE has been solved satisfactorily:
>>> while residual > desiredResidual:
... residual = eq.sweep(var=myVar, ...)
Sweeps are used to achieve better solutions in
:mod:`examples.diffusion.mesh1D`,
:mod:`examples.phase.simple`,
:mod:`examples.phase.binaryCoupled`, and :mod:`examples.flow.stokesCavity`.
timesteps
This outermost layer of repetition is of most practical interest to
the user. Understanding the time evolution of a problem is frequently
the goal of studying a particular set of PDEs. Moreover, even when
only an equilibrium or steady-state solution is desired, it may not
be possible to simply solve that directly, due to non-linear coupling
between equations or to boundary conditions or initial conditions.
Some types of PDEs have fundamental limits to how large a timestep
they can take before they become either unstable or inaccurate.
.. note::
Stability and accuracy are distinctly different. An unstable
solution is often said to "blow up", with radically
different values from point to point, often diverging to
infinity. An inaccurate solution may look perfectly
reasonable, but will disagree significantly from an
analytical solution or from a numerical solution obtained by
taking either smaller or larger timesteps.
For all of these reasons, you will frequently need to advance
a problem in time and to choose an appropriate interval
between solutions. This can be simple:
>>> timeStep = 1.234e-5
>>> for step in range(steps):
... eq.solve(var=myVar, dt=timeStep, ...)
or more elaborate:
>>> timeStep = 1.234e-5
>>> elapsedTime = 0
>>> while elapsedTime < totalElapsedTime:
... eq.solve(var=myVar, dt=timeStep, ...)
... elapsedTime += timeStep
... timeStep = SomeFunctionOfVariablesAndTime(myVar1, myVar2, elapsedTime)
A majority of the examples in this manual illustrate time evolving
behavior. Notably, boundary conditions are made a function of elapsed
time in :mod:`examples.diffusion.mesh1D`. The timestep is
chosen based on the expected interfacial velocity in
:mod:`examples.phase.simple`. The timestep is gradually
increased as the kinetics slow down in
:mod:`examples.cahnHilliard.mesh2DCoupled`.
Finally, we can (and often do) combine all three layers of repetition:
>>> myVar = CellVariable(..., hasOld=1)
:
:
>>> mySolver = LinearPCGSolver(iterations=1234, tolerance=5e-6)
:
:
>>> while elapsedTime < totalElapsedTime:
... myVar.updateOld()
... while residual > desiredResidual:
... residual = eq.sweep(var=myVar, dt=timeStep, ...)
... elapsedTime += timeStep
Why the distinction between :class:`~fipy.variables.cellVariable.CellVariable` and :class:`~fipy.variables.faceVariable.FaceVariable` coefficients?
---------------------------------------------------------------------------------------------------------------------------------------------------
:term:`FiPy` solves field variables on the cell centers. Transient and
source terms describe the change in the value of a field at the cell
center, and so they take a
:class:`~fipy.variables.cellVariable.CellVariable` coefficient. Diffusion
and convection terms involve fluxes *between* cell centers, and are
calculated on the face between two cells, and so they take a
:class:`~fipy.variables.faceVariable.FaceVariable` coefficient.
.. note::
If you supply a :class:`~fipy.variables.cellVariable.CellVariable`
``var`` when a :class:`~fipy.variables.faceVariable.FaceVariable`
is expected, :term:`FiPy` will automatically substitute
``var.``:attr:`~fipy.variables.cellVariable.CellVariable.arithmeticFaceValue`.
This can have undesirable consequences, however. For one thing, the
arithmetic face average of a non-linear function is not the same as the
same non-linear function of the average argument, *e.g.*, for
:math:`f(x) = x^2`,
.. math::
f(\frac{1+2}{2}) = \frac{9}{4} \neq
\frac{f(1) + f(2)}{2} = \frac{5}{2}
This distinction is not generally important for smoothly
varying functions, but can dramatically affect the solution
when sharp changes are present. Also, for many problems, such
as a conserved concentration field that cannot be allowed to
drop below zero, a harmonic average is more appropriate than
an arithmetic average.
If you experience problems (unstable or wrong results, or
excessively small timesteps), you may need to explicitly supply the
desired :class:`~fipy.variables.faceVariable.FaceVariable` rather than letting :term:`FiPy`
assume one.
How do I represent boundary conditions?
---------------------------------------
.. currentmodule:: fipy.variables.cellVariable
See the :ref:`BoundaryConditions` section for more details.
What does this error message mean?
----------------------------------
``ValueError: frames are not aligned``
This error most likely means that you have provided a
:class:`~fipy.variables.cellVariable.CellVariable` when :term:`FiPy`
was expecting a :class:`~fipy.variables.faceVariable.FaceVariable`
(or vice versa).
``MA.MA.MAError: Cannot automatically convert masked array to Numeric because data is masked in one or more locations.``
This not-so-helpful error message could mean a number of things, but
the most likely explanation is that the solution has become unstable
and is diverging to :math:`\pm\infty`. This can be caused by taking
too large a timestep or by using explicit terms instead of implicit
ones.
``repairing catalog by removing key``
This message (not really an error, but may cause test failures) can
result when using the :mod:`weave` package via the
:option:`--inline` flag. It is due to a bug in :term:`SciPy` that has been
patched in their source repository:
http://www.scipy.org/mailinglists/mailman?fn=scipy-dev/2005-June/003010.html.
``numerix Numeric 23.6``
This is neither an error nor a warning. It's just a sloppy
message left in :term:`SciPy`:
http://thread.gmane.org/gmane.comp.python.scientific.user/4349.
.. _FAQ-FlagsAndEnvironmentVariables:
How do I change FiPy's default behavior?
-------------------------------------------
:term:`FiPy` tries to make reasonable choices, based on what packages
it finds installed, but there may be times that you wish to override
these behaviors. See the :ref:`FlagsAndEnvironmentVariables` section
for more details.
How can I tell if I'm running in parallel?
------------------------------------------
See :ref:`PARALLEL`.
Why don't my scripts work anymore?
----------------------------------
:term:`FiPy` has experienced three major API changes. The steps
necessary to upgrade older scripts are discussed in
:ref:`sec:UpdateFiPy`.
What if my question isn't answered here?
----------------------------------------
Please post your question to the
mailing list <http://www.ctcms.nist.gov/fipy/mail.html>
or file an issue at <https://github.com/usnistgov/fipy/issues/new>.