Using FiPy

This document explains how to use FiPy in a practical sense. To see the problems that FiPy is capable of solving, you can run any of the scripts in the examples.

Note

We strongly recommend you proceed through the examples, but at the very least work through examples.diffusion.mesh1D to understand the notation and basic concepts of FiPy.

We exclusively use either the unix command line or IPython to interact with FiPy. The commands in the examples are written with the assumption that they will be executed from the command line. For instance, from within the main FiPy directory, you can type:

$ python examples/diffusion/mesh1D.py

A viewer should appear and you should be prompted through a series of examples.

Note

From within IPython, you would type:

>>> run examples/diffusion/mesh1D.py

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 [PythonTutorial] or with Dive Into Python [DiveIntoPython].

As you gain experience, you may want to browse through the Command-line Flags and Environment Variables that affect FiPy.

Testing FiPy

For a general installation, FiPy can be tested by running:

$ python -c "import fipy; fipy.test()"

This command runs all the test cases in FiPy’s modules, but doesn’t include any of the tests in FiPy’s examples. To run the test cases in both modules and examples, use:

$ python setup.py test

in an unpacked 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 Command-line Flags and Environment Variables for more details.

FiPy will skip tests that depend on Optional Packages that have not been installed. For example, if Mayavi and Gmsh are not installed, FiPy will warn:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Skipped 131 doctest examples because `gmsh` cannot be found on the $PATH
Skipped 42 doctest examples because the `tvtk` package cannot be imported
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

We have a few known, intermittent failures:

#425

The test suite can freeze, usually in examples.chemotaxis, when running on multiple processors. This has never affected us in an actual parallel simulation, only in the test suite.

#430

When running in parallel, the tests for _BinaryTerm sometimes return one erroneous result. This is not reliably reproducible and doesn’t seem to have an effect on actual simulations.

Although the test suite may show warnings, there should be no other errors. Any errors should be investigated or reported on the tracking system. Users can see if there are any known problems for the latest FiPy distribution by checking FiPy’s automated test display.

Below are a number of common Command-line Flags for testing various FiPy configurations.

Parallel Tests

If FiPy is configured for Solving in 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')"

Command-line Flags and Environment Variables

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 FiPy‘s default behavior.

Command-line Flags

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

--inline

Causes many mathematical operations to be performed in C, rather than Python, for improved performance. Requires the scipy.weave package.

The following flags take precedence over the FIPY_SOLVERS environment variable:

--pysparse

Forces the use of the PySparse solvers.

--trilinos

Forces the use of the Trilinos solvers, but uses PySparse to construct the matrices.

--no-pysparse

Forces the use of the Trilinos solvers without any use of PySparse.

--scipy

Forces the use of the SciPy solvers.

--pyamg

Forces the use of the PyAMG preconditioners in conjunction with the SciPy solvers.

--lsmlib

Forces the use of the LSMLIB level set solver.

--skfmm

Forces the use of the Scikit-fmm level set solver.

Environment Variables

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 FiPy scripts from within IPython or IDLE), you can set these variables via the os.environ dictionary, but you must do so before importing anything from the fipy package.

FIPY_DISPLAY_MATRIX

If present, causes the graphical display of the solution matrix of each equation at each call of solve() or sweep(). Setting the value to “terms,” causes the display of the matrix for each Term that composes the equation. Requires the Matplotlib package.

FIPY_INLINE

If present, causes many mathematical operations to be performed in C, rather than Python. Requires the scipy.weave package.

FIPY_INLINE_COMMENT

If present, causes the addition of a comment showing the Python context that produced a particular piece of scipy.weave C code. Useful for debugging.

FIPY_SOLVERS

Forces the use of the specified suite of linear solvers. Valid (case-insensitive) choices are “pysparse”, “trilinos”, “no-pysparse”, “scipy” and “pyamg”.

FIPY_VERBOSE_SOLVER

If present, causes the linear solvers to print a variety of diagnostic information.

FIPY_VIEWER

Forces the use of the specified viewer. Valid values are any <viewer> from the fipy.viewers.<viewer>Viewer modules. The special value of dummy will allow the script to run without displaying anything.

FIPY_INCLUDE_NUMERIX_ALL

If present, causes the inclusion of all funcions and variables of the numerix module in the fipy namespace.

Solving in Parallel

FiPy can use Trilinos to solve equations in parallel. Most mesh classes in fipy.meshes can solve in parallel. This includes all “...Grid...” and “...Gmsh...” class meshes. Currently, the only remaining serial-only meshes are Tri2D and SkewedGrid2D.

Attention

Trilinos must be compiled with MPI support.

Attention

FiPy requires mpi4py to work in parallel. See the mpi4py installation guide.

Note

Parallel efficiency is greatly improved by installing PySparse in addition to Trilinos. If PySparse is not installed be sure to use the --no-pysparse flag when running in parallel.

It should not generally be necessary to change anything in your script. Simply invoke:

$ mpirun -np {# of processors} python myScript.py --trilinos

instead of:

$ python myScript.py

To confirm that FiPy and Trilinos are properly configured to solve in parallel, the easiest way to tell 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 simultion), 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: processor 0 of 3 :: PyTrilinos: processor 0 of 3 :: FiPy: 5 cells on processor 0 of 3
mpi4py: processor 1 of 3 :: PyTrilinos: processor 1 of 3 :: FiPy: 7 cells on processor 1 of 3
mpi4py: processor 2 of 3 :: PyTrilinos: processor 2 of 3 :: FiPy: 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 MPI environment. For example:

mpi4py: processor 2 of 3 :: PyTrilinos: processor 0 of 1 :: FiPy: 10 cells on processor 0 of 1
[my.machine.com:69815] WARNING: There were 4 Windows created but not freed.
mpi4py: processor 1 of 3 :: PyTrilinos: processor 0 of 1 :: FiPy: 10 cells on processor 0 of 1
[my.machine.com:69814] WARNING: There were 4 Windows created but not freed.
mpi4py: processor 0 of 3 :: PyTrilinos: processor 0 of 1 :: FiPy: 10 cells on processor 0 of 1
[my.machine.com:69813] WARNING: There were 4 Windows created but not freed.

indicates mpi4py is properly communicating with MPI and is running in parallel, but that Trilinos is not, and is running three separate serial environments. As a result, FiPy is limited to three separate serial operations, too. In this instance, the problem is that although Trilinos was compiled with MPI enabled, it was compiled against a different MPI library than is currently available (and which mpi4py was compiled against). The solution is to rebuild Trilinos against the active MPI libraries.

When solving in parallel, FiPy essentially breaks the problem up into separate sub-domains and solves them (somewhat) independently. FiPy generally “does the right thing”, but if you find that you need to do something with the entire solution, you can use var.globalValue.

Note

Trilinos solvers frequently give intermediate output that FiPy cannot suppress. The most commonly encountered messages are

Gen_Prolongator warning : Max eigen <= 0.0

which is not significant to FiPy.

Aztec status AZ_loss: loss of precision

which indicates that there was some difficulty in solving the problem to the requested tolerance due to precision limitations, but usually does not prevent the solver from finding an adequate solution.

Aztec status AZ_ill_cond: GMRES hessenberg ill-conditioned

which indicates that GMRES is having trouble with the problem, and may indicate that trying a different solver or preconditioner may give more accurate results if GMRES fails.

Aztec status AZ_breakdown: numerical breakdown

which usually indicates serious problems solving the equation which forced the solver to stop before reaching an adequate solution. Different solvers, different preconditioners, or a less restrictive tolerance may help.

Meshing with Gmsh

FiPy works with arbitrary polygonal meshes generated by Gmsh. FiPy provides two wrappers classes (Gmsh2D and Gmsh3D) enabling Gmsh to be used directly from python. The classes can be instantiated with a set of Gmsh style commands (see examples.diffusion.circle). The classes can also be instantiated with the path to either a Gmsh geometry file (.geo) or a Gmsh mesh file (.msh) (see examples.diffusion.anisotropy).

As well as meshing arbitrary geometries, 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 Gmsh2D and Gmsh3D.

Note

FiPy solution accuracy can be compromised with highly non-orthogonal or non-conjunctional meshes.

Coupled and Vector Equations

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 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 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

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 equation is described in detail in examples.diffusion.coupled. Other examples that demonstrate the use of coupled equations are examples.phase.binaryCoupled, examples.phase.polyxtalCoupled and examples.cahnHilliard.mesh2DCoupled. As well as coupling equations, true vector equations can now be written in FiPy (see examples.diffusion.coupled for more details).

Boundary Conditions

Applying fixed value (Dirichlet) boundary conditions

To apply a fixed value boundary condition use the 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 FixedValue boundary condition is now deprecated.

Applying fixed gradient boundary conditions (Neumann)

To apply a fixed Gradient boundary condition use the faceGrad.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)

Applying fixed flux boundary conditions

Generally these can be implemented with a judicious use of faceGrad.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 FaceVariable.

Note

The old equivalent FixedFlux boundary condition is now deprecated.

Applying outlet or inlet boundary conditions

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.

Applying spatially varying boundary conditions

The use of spatial varying boundary conditions is best demonstrated with an example. Given a 2D equation in the domain and with boundary conditions,

where represents the flux. The boundary conditions in 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 examples.diffusion.mesh20x20 and examples.diffusion.circle

Applying internal boundary conditions

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.

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 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)

Apply a constraint to the faces for a right side boundary condition (which works).

>>> var.constrain(1., where=m.facesRight)

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 * value

and 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 value 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.

Running under Python 3

It is possible to run FiPy scripts under Python 3, but there is admittedly little advantage in doing so at this time. We still develop and use FiPy under Python 2.x. To use, you must first convert FiPy‘s code to Python 3 syntax. From within the main FiPy directory:

$ 2to3 --write .
$ 2to3 --write --doctests_only .

You can expect some harmless warnings from this conversion.

The minimal prerequisites are:

• NumPy version 1.5 or greater.
• SciPy version 0.9 or greater.
• Matplotlib version 1.2 or greater (this hasn’t been released yet, and we haven’t been able to successfully test the matplotlibViewer classes with their development code).

Manual

You can view the manual online at <http://www.ctcms.nist.gov/fipy> or you can download the latest manual from <http://matforge.org/fipy/wiki/FiPyManual>. 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 1.1.2 of Sphinx, plus all of its prerequisites.