This document explains how to use :term:`FiPy` in a practical sense. To see the problems that :term:`FiPy` is capable of solving, you can run any of the scripts in the :ref:`examples <part:examples>`.
Note
We strongly recommend you proceed through the :ref:`examples <part:examples>`, but at the very least work through :mod:`examples.diffusion.mesh1D` to understand the notation and basic concepts of :term:`FiPy`.
We exclusively use either the UNIX command line or :term:`IPython` to interact with :term:`FiPy`. The commands in the :ref:`examples <part:examples>` are written with the assumption that they will be executed from the command line. For instance, from within the main :term:`FiPy` directory, you can type:
$ python examples/diffusion/mesh1D.py
A viewer should appear and you should be prompted through a series of examples.
In order to customize the examples, or to develop your own scripts, some knowledge of Python syntax is required. We recommend you familiarize yourself with the excellent Python tutorial :cite:`PythonTutorial` or with Dive Into Python :cite:`DiveIntoPython`. Deeper insight into Python can be obtained from the :cite:`PythonReference`.
As you gain experience, you may want to browse through the :ref:`FlagsAndEnvironmentVariables` that affect :term:`FiPy`.
Diagnostic information about a :term:`FiPy` run can be obtained using the :mod:`logging` module. For example, at the beginning of your script, you can add:
>>> import logging
>>> log = logging.getLogger("fipy")
>>> console = logging.StreamHandler()
>>> console.setLevel(logging.INFO)
>>> log.addHandler(console)
in order to see informational messages in the terminal. To have more verbose debugging information save to a file:
>>> logfile = logging.FileHandler(filename="fipy.log") >>> logfile.setLevel(logging.DEBUG) >>> log.addHandler(logfile) >>> log.setLevel(logging.DEBUG)
To restrict logging to, e.g., information about the :term:`PETSc` solvers:
>>> petsc = logging.Filter('fipy.solvers.petsc')
>>> logfile.addFilter(petsc)
More complex configurations can be specified by setting the :envvar:`FIPY_LOG_CONFIG` environment variable. In this case, it is not necessary to add any logging instructions to your own script. Example configuration files can be found in :file:`{FiPySource}/fipy/tools/logging/`.
If Solving in Parallel, the :mod:`mpilogging` package (which can be installed via :term:`PyPI`) enables reporting which MPI rank each log entry comes from. For example:
>>> from mpilogging import MPIScatteredFileHandler >>> mpilog = MPIScatteredFileHandler(filepattern="fipy.%(mpirank)d_of_%(mpisize)d.log" >>> mpilog.setLevel(logging.DEBUG) >>> log.addHandler(mpilog)
will generate a unique log file for each MPI rank.
For a general installation, :term:`FiPy` can be tested by running:
$ python -c "import fipy; fipy.test()"
This command runs all the test cases in :ref:`FiPy's modules <part:modules>`, but doesn't include any of the tests in :ref:`FiPy's examples <part:examples>`. To run the test cases in both :ref:`modules <part:modules>` and :ref:`examples <part:examples>`, use:
$ python setup.py test
Note
You may need to first run:
$ python setup.py egg_info
for this to work properly.
in an unpacked :term:`FiPy` archive. The test suite can be run with a number of different configurations depending on which solver suite is available and other factors. See :ref:`FlagsAndEnvironmentVariables` for more details.
:term:`FiPy` will skip tests that depend on :ref:`OPTIONALPACKAGES` that have not been installed. For example, if :term:`Mayavi` and :term:`Gmsh` are not installed, :term:`FiPy` will warn something like:
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Skipped 131 doctest examples because `gmsh` cannot be found on the $PATH Skipped 42 doctest examples because the `tvtk` package cannot be imported !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Although the test suite may show warnings, there should be no other errors. Any errors should be investigated or reported on the issue tracker. Users can see if there are any known problems for the latest :term:`FiPy` distribution by checking :term:`FiPy`'s :ref:`CONTINUOUSINTEGRATION` dashboard.
Below are a number of common Command-line Flags for testing various :term:`FiPy` configurations.
If :term:`FiPy` is configured for :ref:`PARALLEL`, you can run the tests on multiple processor cores with:
$ mpirun -np {# of processors} python setup.py test --trilinos
or:
$ mpirun -np {# of processors} python -c "import fipy; fipy.test('--trilinos')"
:term:`FiPy` chooses a default run time configuration based on the available packages on the system. The Command-line Flags and Environment Variables sections below describe how to override :term:`FiPy`'s default behavior.
You can add any of the following case-insensitive flags after the name of a script you call from the command line, e.g.:
$ python myFiPyScript --someflag
.. cmdoption:: --inline Causes many mathematical operations to be performed in C, rather than Python, for improved performance. Requires the :mod:`weave` package.
.. cmdoption:: --cache Causes lazily evaluated :term:`FiPy` :class:`~fipy.variables.variable.Variable` objects to retain their value.
.. cmdoption:: --no-cache Causes lazily evaluated :term:`FiPy` :class:`~fipy.variables.variable.Variable` objects to always recalculate their value.
The following flags take precedence over the :envvar:`FIPY_SOLVERS` environment variable:
.. cmdoption:: --pysparse Forces the use of the :ref:`PYSPARSE` solvers.
.. cmdoption:: --trilinos Forces the use of the :ref:`TRILINOS` solvers, but uses :ref:`PYSPARSE` to construct the matrices.
.. cmdoption:: --no-pysparse Forces the use of the :ref:`TRILINOS` solvers without any use of :ref:`PYSPARSE`.
.. cmdoption:: --scipy Forces the use of the :ref:`SCIPY` solvers.
.. cmdoption:: --pyamg Forces the use of the :ref:`PYAMG` preconditioners in conjunction with the :ref:`SCIPY` solvers.
.. cmdoption:: --pyamgx Forces the use of the :ref:`PYAMGX` solvers.
.. cmdoption:: --lsmlib Forces the use of the :ref:`LSMLIBDOC` level set solver.
.. cmdoption:: --skfmm Forces the use of the :ref:`SCIKITFMM` level set solver.
You can set any of the following environment variables in the manner appropriate for your shell. If you are not running in a shell (e.g., you are invoking :term:`FiPy` scripts from within :term:`IPython` or IDLE), you can set these variables via the :data:`os.environ` dictionary, but you must do so before importing anything from the :mod:`fipy` package.
.. envvar:: FIPY_DISPLAY_MATRIX .. currentmodule:: fipy.terms.term If present, causes the graphical display of the solution matrix of each equation at each call of :meth:`~Term.solve` or :meth:`~Term.sweep`. Setting the value to "``terms``" causes the display of the matrix for each :class:`Term` that composes the equation. Requires the :term:`Matplotlib` package. Setting the value to "``print``" causes the matrix to be printed to the console.
.. envvar:: FIPY_INLINE If present, causes many mathematical operations to be performed in C, rather than Python. Requires the :mod:`weave` package.
.. envvar:: FIPY_INLINE_COMMENT If present, causes the addition of a comment showing the Python context that produced a particular piece of :mod:`weave` C code. Useful for debugging.
.. envvar:: FIPY_LOG_CONFIG
Specifies a :term:`JSON`-formatted logging configuration file, suitable
for passing to :func:`logging.config.dictConfig`. Example configuration
files can be found in :file:`{FiPySource}/fipy/tools/logging/`.
.. envvar:: FIPY_SOLVERS Forces the use of the specified suite of linear solvers. Valid (case-insensitive) choices are "``petsc``", "``scipy``", "``pysparse``", "``trilinos``", "``no-pysparse``", and "``pyamg``".
.. envvar:: FIPY_VERBOSE_SOLVER If present, causes the :class:`~fipy.solvers.pyAMG.linearGeneralSolver.LinearGeneralSolver` to print a variety of diagnostic information. All other solvers should use `Logging`_ and :envvar:`FIPY_LOG_CONFIG`.
.. envvar:: FIPY_VIEWER
Forces the use of the specified viewer. Valid values are any
:samp:`{<viewer>}` from the
:samp:`fipy.viewers.{<viewer>}Viewer`
modules. The special value of ``dummy`` will allow the script
to run without displaying anything.
.. envvar:: FIPY_INCLUDE_NUMERIX_ALL If present, causes the inclusion of all functions and variables of the :mod:`~fipy.tools.numerix` module in the :mod:`fipy` namespace.
.. envvar:: FIPY_CACHE If present, causes lazily evaluated :term:`FiPy` :class:`~fipy.variables.variable.Variable` objects to retain their value.
.. envvar:: PETSC_OPTIONS `PETSc configuration options`_. Set to "`-help`" and run a script with :ref:`PETSC` solvers in order to see what options are possible. Ignored if solver is not :ref:`PETSC`.
:term:`FiPy` can use :ref:`PETSC` or :ref:`TRILINOS` to solve equations in
parallel. Most mesh classes in :mod:`fipy.meshes` can solve in
parallel. This includes all "...Grid..." and "...Gmsh..."
class meshes. Currently, the only remaining serial-only meshes are
:class:`~fipy.meshes.tri2D.Tri2D` and
:class:`~fipy.meshes.skewedGrid2D.SkewedGrid2D`.
Attention!
:term:`FiPy` requires :term:`mpi4py` to work in parallel.
Tip
You are strongly advised to force the use of only one :term:`OpenMP` thread with :ref:`Trilinos`:
$ export OMP_NUM_THREADS=1
See :ref:`THREADS_VS_RANKS` for more information.
Note
Trilinos 12.12 has support for Python 3, but PyTrilinos on conda-forge presently only provides 12.10, which is limited to Python 2.x. :ref:`PETSC` is available for both :term:`Python 3` and :term:`Python` 2.7.
It should not generally be necessary to change anything in your script. Simply invoke:
$ mpirun -np {# of processors} python myScript.py --petsc
or:
$ mpirun -np {# of processors} python myScript.py --trilinos
instead of:
$ python myScript.py
The following plot shows the scaling behavior for multiple processors. We compare solution time vs number of Slurm tasks (available cores) for a Method of Manufactured Solutions Allen-Cahn problem.
.. plot:: documentation/pyplots/scaling.py Scaling behavior of different solver packages
"Speedup" relative to :ref:`PySparse` (bigger numbers are better) versus number of tasks (processes) on a log-log plot. The number of threads per :term:`MPI` rank is indicated by the line style (see legend). :term:`OpenMP` threads \times :term:`MPI` ranks = Slurm tasks.
A few things can be observed in this plot:
- Both :ref:`PETSc` and :ref:`Trilinos` exhibit power law scaling, but the
power is only about 0.7. At least one source of this poor scaling is
that our "
...Grid..." meshes parallelize by dividing the mesh into slabs, which leads to more communication overhead than more compact partitions. The "...Gmsh..." meshes partition more efficiently, but carry more overhead in other ways. We'll be making efforts to improve the partitioning of the "...Grid..." meshes in a future release. - :ref:`PETSc` and :ref:`Trilinos` have fairly comparable performance, but lag :ref:`PySparse` by a considerable margin. The :ref:`SciPy` solvers are even worse. Some of this discrepancy may be because the different packages are not all doing the same thing. Different solver packages have different default solvers and preconditioners. Moreover, the meaning of the solution tolerance depends on the normalization the solver uses and it is not always obvious which of several possibilities a particular package employs. We will endeavor to normalize the normalizations in a future release.
- :ref:`PETSc` with one thread is faster than with two threads until the number of tasks reaches about 10 and is faster than with four threads until the number of tasks reaches more than 20. :ref:`Trilinos` with one thread is faster than with two threads until the number of tasks is more than 30. We don't fully understand the reasons for this, but there may be a modest benefit, when using a large number of cpus, to allow two to four :term:`OpenMP` threads per :term:`MPI` rank. See :ref:`THREADS_VS_RANKS` for caveats and more information.
These results are likely both problem and architecture dependent. You should develop an understanding of the scaling behavior of your own codes before doing "production" runs.
The easiest way to confirm that :term:`FiPy` is properly configured to solve in parallel is to run one of the examples, e.g.,:
$ mpirun -np 2 examples/diffusion/mesh1D.py
You should see two viewers open with half the simulation running in one of them and half in the other. If this does not look right (e.g., you get two viewers, both showing the entire simulation), or if you just want to be sure, you can run a diagnostic script:
$ mpirun -np 3 python examples/parallel.py
which should print out:
mpi4py PyTrilinos petsc4py FiPy processor 0 of 3 :: processor 0 of 3 :: processor 0 of 3 :: 5 cells on processor 0 of 3 processor 1 of 3 :: processor 1 of 3 :: processor 1 of 3 :: 7 cells on processor 1 of 3 processor 2 of 3 :: processor 2 of 3 :: processor 2 of 3 :: 6 cells on processor 2 of 3
If there is a problem with your parallel environment, it should be clear that there is either a problem importing one of the required packages or that there is some problem with the :term:`MPI` environment. For example:
mpi4py PyTrilinos petsc4py FiPy processor 0 of 3 :: processor 0 of 1 :: processor 0 of 3 :: 10 cells on processor 0 of 1 [my.machine.com:69815] WARNING: There were 4 Windows created but not freed. processor 1 of 3 :: processor 0 of 1 :: processor 1 of 3 :: 10 cells on processor 0 of 1 [my.machine.com:69814] WARNING: There were 4 Windows created but not freed. processor 2 of 3 :: processor 0 of 1 :: processor 2 of 3 :: 10 cells on processor 0 of 1 [my.machine.com:69813] WARNING: There were 4 Windows created but not freed.
indicates :term:`mpi4py` is properly communicating with :term:`MPI` and is running in parallel, but that :ref:`TRILINOS` is not, and is running three separate serial environments. As a result, :term:`FiPy` is limited to three separate serial operations, too. In this instance, the problem is that although :ref:`TRILINOS` was compiled with :term:`MPI` enabled, it was compiled against a different :term:`MPI` library than is currently available (and which :term:`mpi4py` was compiled against). The solution, in this instance, is to solve with :ref:`PETSC` or to rebuild :ref:`TRILINOS` against the active :term:`MPI` libraries.
When solving in parallel, :term:`FiPy` essentially breaks the problem
up into separate sub-domains and solves them (somewhat) independently.
:term:`FiPy` generally "does the right thing", but if you find that
you need to do something with the entire solution, you can use
var.:attr:`~fipy.variables.cellVariable.CellVariable.globalValue`.
Note
One option for debugging in parallel is:
$ mpirun -np {# of processors} xterm -hold -e "python -m ipdb myScript.py"
By default, :ref:`PETSc` and :ref:`Trilinos` spawn as many :term:`OpenMP` threads as there are cores available. This may very well be an intentional optimization, where they are designed to have one :term:`MPI` rank per node of a cluster, so each of the child threads would help with computation but would not compete for I/O resources during ghost cell exchanges and file I/O. However, Python's Global Interpreter Lock (GIL) binds all of the child threads to the same core as their parent! So instead of improving performance, each core suffers a heavy overhead from managing those idling threads.
The solution to this is to force these solvers to use only one :term:`OpenMP` thread:
$ export OMP_NUM_THREADS=1
Because this environment variable affects all processes launched in the current session, you may prefer to restrict its use to :term:`FiPy` runs:
$ OMP_NUM_THREADS=1 mpirun -np {# of processors} python myScript.py --trilinos
The difference can be extreme. We have observed the :term:`FiPy` test
suite to run in just over two minutes when OMP_NUM_THREADS=1,
compared to over an hour and 23 minutes when :term:`OpenMP` threads are
unrestricted. We don't know why, but other platforms do not suffer the
same degree of degradation.
Conceivably, allowing these parallel solvers unfettered access to :term:`OpenMP` threads with no :term:`MPI` communication at all could perform as well or better than purely :term:`MPI` parallelization. The plot below demonstrates this is not the case. We compare solution time vs number of :term:`OpenMP` threads for fixed number of slots for a Method of Manufactured Solutions Allen-Cahn problem. :term:`OpenMP` threading always slows down FiPy performance.
.. plot:: documentation/pyplots/cpus_vs_threads.py Effect of having more :term:`OpenMP` threads for each :term:`MPI` rank
"Speedup" relative to one thread (bigger numbers are better) versus number of threads for 32 Slurm tasks on a log-log plot. :term:`OpenMP` threads \times :term:`MPI` ranks = Slurm tasks.
See https://www.mail-archive.com/fipy@nist.gov/msg03393.html for further analysis.
It may be possible to configure these packages to use only one :term:`OpenMP` thread, but this is not the configuration of the version available from conda-forge and building Trilinos, at least, is NotFun™.
_-_WGA2526.jpg#/media/File:Hieronymus_Bosch_-_Triptych_of_Garden_of_Earthly_Delights_(detail)_-_WGA2526.jpg){kind=link}
:term:`FiPy` works with arbitrary polygonal meshes generated by
:term:`Gmsh`. :term:`FiPy` provides two wrappers classes
(:class:`~fipy.meshes.gmshImport.Gmsh2D` and
:class:`~fipy.meshes.gmshImport.Gmsh3D`) enabling :term:`Gmsh` to be
used directly from python. The classes can be instantiated with a set
of :term:`Gmsh` style commands (see
:mod:`examples.diffusion.circle`). The classes can also be
instantiated with the path to either a :term:`Gmsh` geometry file
(.geo) or a :term:`Gmsh` mesh file (.msh) (see
:mod:`examples.diffusion.anisotropy`).
As well as meshing arbitrary geometries, :term:`Gmsh` partitions meshes for parallel simulations. Mesh partitioning automatically occurs whenever a parallel communicator is passed to the mesh on instantiation. This is the default setting for all meshes that work in parallel including :class:`~fipy.meshes.gmshImport.Gmsh2D` and :class:`~fipy.meshes.gmshImport.Gmsh3D`.
Note
:term:`FiPy` solution accuracy can be compromised with highly non-orthogonal or non-conjunctional meshes.
Equations can now be coupled together so that the contributions from
all the equations appear in a single system matrix. This results in
tighter coupling for equations with spatial and temporal derivatives
in more than one variable. In :term:`FiPy` equations are coupled
together using the & operator:
>>> eqn0 = ... >>> eqn1 = ... >>> coupledEqn = eqn0 & eqn1
The coupledEqn will use a combined system matrix that includes
four quadrants for each of the different variable and equation
combinations. In previous versions of :term:`FiPy` there has been no
need to specify which variable a given term acts on when generating
equations. The variable is simply specified when calling solve or
sweep and this functionality has been maintained in the case of
single equations. However, for coupled equations the variable that a
given term operates on now needs to be specified when the equation is
generated. The syntax for generating coupled equations has the form:
>>> eqn0 = Term00(coeff=..., var=var0) + Term01(coeff..., var=var1) == source0 >>> eqn1 = Term10(coeff=..., var=var0) + Term11(coeff..., var=var1) == source1 >>> coupledEqn = eqn0 & eqn1
and there is now no need to pass any variables when solving:
>>> coupledEqn.solve(dt=..., solver=...)
In this case the matrix system will have the form
\left(
\begin{array}{c|c}
\text{\ttfamily Term00} & \text{\ttfamily Term01} \\ \hline
\text{\ttfamily Term10} & \text{\ttfamily Term11}
\end{array} \right)
\left(
\begin{array}{c}
\text{\ttfamily var0} \\ \hline
\text{\ttfamily var1}
\end{array} \right)
=
\left(
\begin{array}{c}
\text{\ttfamily source0} \\ \hline
\text{\ttfamily source1}
\end{array} \right)
:term:`FiPy` tries to make sensible decisions regarding each term's
location in the matrix and the ordering of the variable column
array. For example, if Term01 is a transient term then Term01
would appear in the upper left diagonal and the ordering of the
variable column array would be reversed.
The use of coupled equations is described in detail in :mod:`examples.diffusion.coupled`. Other examples that demonstrate the use of coupled equations are :mod:`examples.phase.binaryCoupled`, :mod:`examples.phase.polyxtalCoupled` and :mod:`examples.cahnHilliard.mesh2DCoupled`. As well as coupling equations, true vector equations can now be written in :term:`FiPy`.
Attention!
Coupled equations are not compatible with :ref:`discret-higherOrderDiffusion` terms. This is not a practical limitation, as any higher order terms can be decomposed into multiple 2nd-order equations. For example, the pair of coupled Cahn-Hilliard & Allen-Cahn 4th- and 2nd-order equations
\frac{\partial C}{\partial t}
&= \nabla\cdot\left[
M\nabla\left(
\frac{\partial f(c, \phi)}{\partial C}
- \kappa_C\nabla^2 C
\right)
\right]
\\
\frac{\partial \phi}{\partial t}
&= -L\left(
\frac{\partial f(c, \phi)}{\partial \phi}
- \kappa_\phi\nabla^2 \phi
\right)
can be decomposed to three 2nd-order equations
\frac{\partial C}{\partial t}
&= \nabla\cdot\left(
M\nabla\mu
\right)
\\
\mu
&= \frac{\partial f(c, \phi)}{\partial C}
- \kappa_C\nabla^2 C
\\
\frac{\partial \phi}{\partial t}
&= -L\left(
\frac{\partial f(c, \phi)}{\partial \phi}
- \kappa_\phi\nabla^2 \phi
\right)
.. currentmodule:: fipy.variables.cellVariable
If no constraints are applied, solutions are conservative, i.e., all boundaries are zero flux. For the equation
\frac{\partial\phi}{\partial t}
&= \nabla\cdot\left(\vec{a}\phi\right) + \nabla\cdot\left(b\nabla\phi\right)
the condition on the boundary S is
\hat{n}\cdot\left(\vec{a}\phi + b\nabla\phi\right) = 0\qquad\text{on $S$.}
To apply a fixed value boundary condition use the :meth:`~CellVariable.constrain` method. For example, to fix var to have a value of 2 along the upper surface of a domain, use
>>> var.constrain(2., where=mesh.facesTop)Note
The old equivalent :class:`~fipy.boundaryConditions.fixedValue.FixedValue` boundary condition is now deprecated.
To apply a fixed Gradient boundary condition use the :attr:`~.CellVariable.faceGrad`.:meth:`~fipy.variables.variable.Variable.constrain` method. For example, to fix var to have a gradient of (0,2) along the upper surface of a 2D domain, use
>>> var.faceGrad.constrain(((0,),(2,)), where=mesh.facesTop)If the gradient normal to the boundary (e.g., \hat{n}\cdot\nabla\phi) is to be set to a scalar value of 2, use
>>> var.faceGrad.constrain(2 * mesh.faceNormals, where=mesh.exteriorFaces)Generally these can be implemented with a judicious use of :attr:`~.CellVariable.faceGrad`.:meth:`~fipy.variables.variable.Variable.constrain`. Failing that, an exterior flux term can be added to the equation. Firstly, set the terms' coefficients to be zero on the exterior faces,
>>> diffCoeff.constrain(0., mesh.exteriorFaces)
>>> convCoeff.constrain(0., mesh.exteriorFaces)then create an equation with an extra term to account for the exterior flux,
>>> eqn = (TransientTerm() + ConvectionTerm(convCoeff)
... == DiffusionCoeff(diffCoeff)
... + (mesh.exteriorFaces * exteriorFlux).divergence)where exteriorFlux is a rank 1 :class:`~fipy.variables.faceVariable.FaceVariable`.
Note
The old equivalent :class:`~fipy.boundaryConditions.fixedFlux.FixedFlux` boundary condition is now deprecated.
Convection terms default to a no flux boundary condition unless the exterior faces are associated with a constraint, in which case either an inlet or an outlet boundary condition is applied depending on the flow direction.
The use of spatial varying boundary conditions is best demonstrated with an example. Given a 2D equation in the domain 0 < x < 1 and 0 < y < 1 with boundary conditions,
\phi = \left\{
\begin{aligned}
xy &\quad \text{on $x>1/2$ and $y>1/2$} \\
\vec{n} \cdot \vec{F} = 0 &\quad \text{elsewhere}
\end{aligned}
\right.
where \vec{F} represents the flux. The boundary conditions in :term:`FiPy` can be written with the following code,
>>> X, Y = mesh.faceCenters
>>> mask = ((X < 0.5) | (Y < 0.5))
>>> var.faceGrad.constrain(0, where=mesh.exteriorFaces & mask)
>>> var.constrain(X * Y, where=mesh.exteriorFaces & ~mask)then
>>> eqn.solve(...)Further demonstrations of spatially varying boundary condition can be found in :mod:`examples.diffusion.mesh20x20` and :mod:`examples.diffusion.circle`
The Robin condition applied on the portion of the boundary S_R
\hat{n}\cdot\left(\vec{a}\phi + b\nabla\phi\right) = g\qquad\text{on $S_R$}
can often be substituted for the flux in an equation
\frac{\partial\phi}{\partial t}
&= \nabla\cdot\left(\vec{a}\phi\right) + \nabla\cdot\left(b\nabla\phi\right)
\\
\int_V\frac{\partial\phi}{\partial t}\,dV
&= \int_S \hat{n} \cdot \left(\vec{a}\phi + b\nabla\phi\right) \, dS
\\
\int_V\frac{\partial\phi}{\partial t}\,dV
&= \int_{S \notin S_R} \hat{n} \cdot \left(\vec{a}\phi + b\nabla\phi\right) \, dS
+ \int_{S \in S_R} g \, dS
At faces identified by mask,
>>> a = FaceVariable(mesh=mesh, value=..., rank=1)
>>> a.setValue(0., where=mask)
>>> b = FaceVariable(mesh=mesh, value=..., rank=0)
>>> b.setValue(0., where=mask)
>>> g = FaceVariable(mesh=mesh, value=..., rank=0)
>>> eqn = (TransientTerm() == PowerLawConvectionTerm(coeff=a)
... + DiffusionTerm(coeff=b)
... + (g * mask * mesh.faceNormals).divergence)When the Robin condition does not exactly map onto the boundary flux, we can attempt to apply it term by term. The Robin condition relates the gradient at a boundary face to the value on that face, however :term:`FiPy` naturally calculates variable values at cell centers and gradients at intervening faces. Using a first order upwind approximation, the boundary value of the variable at face f can be written in terms of the value at the neighboring cell P and the normal gradient at the boundary:
\phi_f &\approx \phi_P - \left(\vec{d}_{fP}\cdot\nabla\phi\right)_f
\\
&\approx \phi_P - \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f
where \vec{d}_{fP} is the distance vector from the face center to the adjoining cell center. The approximation \left(\vec{d}_{fP}\cdot\nabla\phi\right)_f \approx \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f is most valid when the mesh is orthogonal.
Substituting this expression into the Robin condition:
\hat{n}\cdot\left(\vec{a} \phi + b \nabla\phi\right)_f &= g \\
\hat{n}\cdot\left[\vec{a} \phi_P
- \vec{a} \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f
+ b \nabla\phi\right]_f &\approx g \\
\left(\hat{n}\cdot\nabla\phi\right)_f
&\approx \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
we obtain an expression for the gradient at the boundary face in terms of its neighboring cell. We can, in turn, substitute this back into :eq:`upwind1`
\phi_f &\approx \phi_P
- \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
\left(\vec{d}_{fP}\cdot\hat{n}\right)_f \\
&\approx \frac{-g_f \left(\hat{n}\cdot\vec{d}_{fP}\right)_f + b_f\phi_P}
{- \left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
to obtain the value on the boundary face in terms of the neighboring cell.
Substituting :eq:`Robin_facegrad` into the discretization of the :class:`~fipy.terms.diffusionTerm.DiffusionTerm`:
\int_V \nabla\cdot\left(\Gamma\nabla\phi\right) dV
&= \int_S \Gamma \hat{n}\cdot\nabla\phi\, S \\
&\approx \sum_f \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f \\
&= \sum_{f \notin S_R} \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f
+ \sum_{f \in S_R} \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f \\
&\approx \sum_{f \notin S_R} \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f
+ \sum_{f \in S_R} \Gamma_f \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f} A_f
An equation of the form
>>> eqn = TransientTerm() == DiffusionTerm(coeff=Gamma0)can be constrained to have a Robin condition at faces identified by
mask by making the following modifications
>>> Gamma = FaceVariable(mesh=mesh, value=Gamma0)
>>> Gamma.setValue(0., where=mask)
>>> dPf = FaceVariable(mesh=mesh,
... value=mesh._faceToCellDistanceRatio * mesh.cellDistanceVectors)
>>> n = mesh.faceNormals
>>> a = FaceVariable(mesh=mesh, value=..., rank=1)
>>> b = FaceVariable(mesh=mesh, value=..., rank=0)
>>> g = FaceVariable(mesh=mesh, value=..., rank=0)
>>> RobinCoeff = (mask * Gamma0 * n / (-dPf.dot(a) + b)
>>> eqn = (TransientTerm() == DiffusionTerm(coeff=Gamma) + (RobinCoeff * g).divergence
... - ImplicitSourceTerm(coeff=(RobinCoeff * n.dot(a)).divergence)Similarly, for a :class:`~fipy.terms.convectionTerm.ConvectionTerm`, we can substitute :eq:`upwind2`:
\int_V \nabla\cdot\left(\vec{u}\phi\right) dV
&= \int_S \hat{n}\cdot\vec{u} \phi\,dS \\
&\approx \sum_f \left(\hat{n}\cdot\vec{u}\right)_f \phi_f A_f \\
&= \sum_{f \notin S_R} \left(\hat{n}\cdot\vec{u}\right)_f \phi_f A_f
+ \sum_{f \in S_R} \left(\hat{n}\cdot\vec{u}\right)_f
\frac{-g_f \left(\hat{n}\cdot\vec{d}_{fP}\right)_f + b_f\phi_P}
{- \left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f} A_f
Note
An expression like the heat flux convection boundary condition -k\nabla T\cdot\hat{n} = h(T - T_\infty) can be put in the form of the Robin condition used above by letting \vec{a} \equiv h \hat{n}, b \equiv k, and g \equiv h T_\infty.
Applying internal boundary conditions can be achieved through the use of implicit and explicit sources.
An equation of the form
>>> eqn = TransientTerm() == DiffusionTerm()can be constrained to have a fixed internal value at a position
given by mask with the following alterations
>>> eqn = (TransientTerm() == DiffusionTerm()
... - ImplicitSourceTerm(mask * largeValue)
... + mask * largeValue * value)The parameter largeValue must be chosen to be large enough to
completely dominate the matrix diagonal and the RHS vector in cells
that are masked. The mask variable would typically be a
CellVariable Boolean constructed using the cell center values.
An equation of the form
>>> eqn = TransientTerm() == DiffusionTerm(coeff=Gamma0)can be constrained to have a fixed internal gradient magnitude
at a position given by mask with the following alterations
>>> Gamma = FaceVariable(mesh=mesh, value=Gamma0)
>>> Gamma[mask.value] = 0.
>>> eqn = (TransientTerm() == DiffusionTerm(coeff=Gamma)
... + DiffusionTerm(coeff=largeValue * mask)
... - ImplicitSourceTerm(mask * largeValue * gradient
... * mesh.faceNormals).divergence)The parameter largeValue must be chosen to be large enough to
completely dominate the matrix diagonal and the RHS vector in cells
that are masked. The mask variable would typically be a
FaceVariable Boolean constructed using the face center values.
Nothing different needs to be done when applying Robin boundary conditions at internal faces.
Note
While we believe the derivations for applying Robin boundary conditions are "correct", they often do not seem to produce the intuitive result. At this point, we think this has to do with the pathology of "internal" boundary conditions, but remain open to other explanations. :term:`FiPy` was designed with diffuse interface treatments (phase field and level set) in mind and, as such, internal "boundaries" do not come up in our own work and have not received much attention.
Warning
The constraints mechanism is not designed to constrain internal values
for variables that are being solved by equations. In particular, one must
be careful to distinguish between constraining internal cell values
during the solve step and simply applying arbitrary constraints to a
CellVariable. Applying a constraint,
>>> var.constrain(value, where=mask)simply fixes the returned value of var at mask to be
value. It does not have any effect on the implicit value of var at the
mask location during the linear solve so it is not a substitute
for the source term machinations described above. Future releases of
:term:`FiPy` may implicitly deal with this discrepancy, but the current
release does not.
A simple example can be used to demonstrate this:
>>> m = Grid1D(nx=2, dx=1.) >>> var = CellVariable(mesh=m)
We wish to solve \nabla^2 \phi = 0 subject to \phi\rvert_\text{right} = 1 and \phi\rvert_{x < 1} = 0.25. We apply a constraint to the faces for the right side boundary condition (which works).
>>> var.constrain(1., where=m.facesRight)We create the equation with the source term constraint described above
>>> mask = m.x < 1.
>>> largeValue = 1e+10
>>> value = 0.25
>>> eqn = DiffusionTerm() - ImplicitSourceTerm(largeValue * mask) + largeValue * mask * valueand the expected value is obtained.
>>> eqn.solve(var)
>>> print var
[ 0.25 0.75]However, if a constraint is used without the source term constraint an unexpected solution is obtained
>>> var.constrain(0.25, where=mask)
>>> eqn = DiffusionTerm()
>>> eqn.solve(var)
>>> print var
[ 0.25 1. ]although the left cell has the expected value as it is constrained.
:term:`FiPy` has simply solved \nabla^2 \phi = 0 with \phi\rvert_\text{right} = 1 and (by default) \hat{n}\cdot\nabla\phi\rvert_\text{left} = 0, giving \phi = 1 everywhere, and then subsequently replaced the cells x < 1 with \phi = 0.25.
Thanks to the future package and to the contributions of pya and woodscn, :term:`FiPy` runs under both :term:`Python 3` and :term:`Python` 2.7, without conversion or modification.
Because :term:`Python` itself will drop support for Python 2.7 on January 1, 2020 and many of the prerequisites for :term:`FiPy` have pledged to drop support for Python 2.7 no later than 2020, we have prioritized adding support for better :term:`Python 3` solvers, starting with :term:`petsc4py`.
Because the faster :term:`PySparse` and :term:`Trilinos` solvers are not available under :term:`Python 3`, we will maintain :term:`Python` 2.x support as long as practical. Be aware that the conda-forge packages that :term:`FiPy` depends upon are not well-maintained on :term:`Python` 2.x and our support for that configuration is rapidly becoming impractical, despite the present performance benefits. Hopefully, we will learn how to optimize our use of :ref:`PETSc` and/or :ref:`Trilinos` 12.12 will become available on conda-forge.
You can view the manual online at <http://www.ctcms.nist.gov/fipy> or you can download the latest manual from <http://www.ctcms.nist.gov/fipy/download/>. Alternatively, it may be possible to build a fresh copy by issuing the following command in the base directory:
$ python setup.py build_docs --pdf --html
Note
This mechanism is intended primarily for the developers. At a minimum, you will need at least version 1.7.0 of Sphinx, plus all of its prerequisites. Python 2.7 probably won't work.
We install via conda:
$ conda install --channel conda-forge sphinx
Bibliographic citations require the sphinxcontrib-bibtex package:
$ python -m pip install sphinxcontrib-bibtex
Some documentation uses numpydoc styling:
$ python -m pip install numpydoc
Some embeded figures require matplotlib, pandas, and imagemagick:
$ conda install --channel conda-forge matplotlib pandas imagemagick
The PDF file requires SIunits.sty available, e.g., from texlive-science.
Spelling is checked automatically in the course of :ref:`CONTINUOUSINTEGRATION`. If you wish to check manually, you will need pyspelling, hunspell, and the libreoffice dictionaries:
$ conda install --channel conda-forge hunspell $ python -m pip install pyspelling $ wget -O en_US.aff https://cgit.freedesktop.org/libreoffice/dictionaries/plain/en/en_US.aff?id=a4473e06b56bfe35187e302754f6baaa8d75e54f $ wget -O en_US.dic https://cgit.freedesktop.org/libreoffice/dictionaries/plain/en/en_US.dic?id=a4473e06b56bfe35187e302754f6baaa8d75e54f