fipy.variables package

Submodules

fipy.variables.addOverFacesVariable module

fipy.variables.arithmeticCellToFaceVariable module

fipy.variables.betaNoiseVariable module

class fipy.variables.betaNoiseVariable.BetaNoiseVariable(*args, **kwds)

Bases: NoiseVariable

Represents a beta distribution of random numbers with the probability distribution

x^{\alpha - 1}\frac{\beta^\alpha e^{-\beta x}}{\Gamma(\alpha)}

with a shape parameter \alpha, a rate parameter \beta, and \Gamma(z) = \int_0^\infty t^{z - 1}e^{-t}\,dt.

Seed the random module for the sake of deterministic test results.

>>> from fipy import numerix
>>> numerix.random.seed(1)

We generate noise on a uniform Cartesian mesh

>>> from fipy.variables.variable import Variable
>>> alpha = Variable()
>>> beta = Variable()
>>> from fipy.meshes import Grid2D
>>> noise = BetaNoiseVariable(mesh = Grid2D(nx = 100, ny = 100), alpha = alpha, beta = beta)

We histogram the root-volume-weighted noise distribution

>>> from fipy.variables.histogramVariable import HistogramVariable
>>> histogram = HistogramVariable(distribution = noise, dx = 0.01, nx = 100)

and compare to a Gaussian distribution

>>> from fipy.variables.cellVariable import CellVariable
>>> betadist = CellVariable(mesh = histogram.mesh)
>>> x = CellVariable(mesh=histogram.mesh, value=histogram.mesh.cellCenters[0])
>>> from scipy.special import gamma as Gamma 
>>> betadist = ((Gamma(alpha + beta) / (Gamma(alpha) * Gamma(beta)))
...             * x**(alpha - 1) * (1 - x)**(beta - 1)) 
>>> if __name__ == '__main__':
...     from fipy import Viewer
...     viewer = Viewer(vars=noise, datamin=0, datamax=1)
...     histoplot = Viewer(vars=(histogram, betadist),
...                        datamin=0, datamax=1.5)
>>> from fipy.tools.numerix import arange
>>> for a in arange(0.5, 5, 0.5):
...     alpha.value = a
...     for b in arange(0.5, 5, 0.5):
...         beta.value = b
...         if __name__ == '__main__':
...             import sys
...             print("alpha: %g, beta: %g" % (alpha, beta), file=sys.stderr)
...             viewer.plot()
...             histoplot.plot()
>>> print(abs(noise.faceGrad.divergence.cellVolumeAverage) < 5e-15)
1
random values with a beta distribution histogram of random values with a beta distribution
Parameters:
  • mesh (Mesh) – The mesh on which to define the noise.

  • alpha (float) – The parameter \alpha.

  • beta (float) – The parameter \beta.

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__(points=None, order=0, nearestCellIDs=None)

Interpolates the CellVariable to a set of points using a method that has a memory requirement on the order of Ncells by Npoints in general, but uses only Ncells when the CellVariable’s mesh is a UniformGrid object.

Tests

>>> from fipy import *
>>> m = Grid2D(nx=3, ny=2)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.))))
[ 0.5  1.5  1.5]
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.)), order=1))
[ 0.25  1.1   1.2 ]
>>> m0 = Grid2D(nx=2, ny=2, dx=1., dy=1.)
>>> m1 = Grid2D(nx=4, ny=4, dx=.5, dy=.5)
>>> x, y = m0.cellCenters
>>> v0 = CellVariable(mesh=m0, value=x * y)
>>> print(v0(m1.cellCenters.globalValue))
[ 0.25  0.25  0.75  0.75  0.25  0.25  0.75  0.75  0.75  0.75  2.25  2.25
  0.75  0.75  2.25  2.25]
>>> print(v0(m1.cellCenters.globalValue, order=1))
[ 0.125  0.25   0.5    0.625  0.25   0.375  0.875  1.     0.5    0.875
  1.875  2.25   0.625  1.     2.25   2.625]
Parameters:
  • points (tuple or list of tuple) – A point or set of points in the format (X, Y, Z)

  • order ({0, 1}) – The order of interpolation, default is 0

  • nearestCellIDs (array_like) – Optional argument if user can calculate own nearest cell IDs array, shape should be same as points

__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the CellVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new CellVariable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
property arithmeticFaceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
cacheMe(recursive=False)
property cellVolumeAverage

Return the cell-volume-weighted average of the CellVariable:

<\phi>_\text{vol}
= \frac{\sum_\text{cells} \phi_\text{cell} V_\text{cell}}
    {\sum_\text{cells} V_\text{cell}}

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx = 3, ny = 1, dx = .5, dy = .1)
>>> var = CellVariable(value = (1, 2, 6), mesh = mesh)
>>> print(var.cellVolumeAverage)
3.0
constrain(value, where=None)

Constrains the CellVariable to value at a location specified by where.

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> v.constrain(0., where=m.facesLeft)
>>> v.faceGrad.constrain([1.], where=m.facesRight)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]

Changing the constraint changes the dependencies

>>> v.constrain(1., where=m.facesLeft)
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can be Variable

>>> c = Variable(0.)
>>> v.constrain(c, where=m.facesLeft)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> c.value = 1.
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can have a Variable mask.

>>> v = CellVariable(mesh=m)
>>> mask = FaceVariable(mesh=m, value=m.facesLeft)
>>> v.constrain(1., where=mask)
>>> print(v.faceValue)
[ 1.  0.  0.  0.]
>>> mask[:] = mask | m.facesRight
>>> print(v.faceValue)
[ 1.  0.  0.  1.]
property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Copy the value of the NoiseVariable to a static CellVariable.

dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

property faceGrad

Return \nabla \phi as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).

property faceGradAverage

Deprecated since version 3.3: use grad.arithmeticFaceValue() instead

Return \nabla \phi as a rank-1 FaceVariable using averaging for the normal direction(second-order gradient)

property faceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
property gaussGrad

Return \frac{1}{V_P} \sum_f \vec{n} \phi_f A_f as a rank-1 CellVariable (first-order gradient).

getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue

Concatenate and return values from all processors

When running on a single processor, the result is identical to value.

property grad

Return \nabla \phi as a rank-1 CellVariable (first-order gradient).

property harmonicFaceValue

Returns a FaceVariable whose value corresponds to the harmonic interpolation of the adjacent cells:

\phi_f = \frac{\phi_1 \phi_2}{(\phi_2 - \phi_1) \frac{d_{f2}}{d_{12}} + \phi_1}

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (0.5 / 1.) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (1.0 / 3.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (5.0 / 55.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property leastSquaresGrad

Return \nabla \phi, which is determined by solving for \nabla \phi in the following matrix equation,

\nabla \phi \cdot \sum_f d_{AP}^2 \vec{n}_{AP} \otimes \vec{n}_{AP} =
\sum_f d_{AP}^2 \left( \vec{n} \cdot \nabla \phi \right)_{AP}

The matrix equation is derived by minimizing the following least squares sum,

F \left( \phi_x, \phi_y \right) = \sqrt{\sum_f \left( d_{AP}
\vec{n}_{AP} \cdot \nabla \phi - d_{AP} \left( \vec{n}_{AP} \cdot
\nabla \phi \right)_{AP} \right)^2 }

Tests

>>> from fipy import Grid2D
>>> m = Grid2D(nx=2, ny=2, dx=0.1, dy=2.0)
>>> print(numerix.allclose(CellVariable(mesh=m, value=(0, 1, 3, 6)).leastSquaresGrad.globalValue, \
...                                     [[8.0, 8.0, 24.0, 24.0],
...                                      [1.2, 2.0, 1.2, 2.0]]))
True
>>> from fipy import Grid1D
>>> print(numerix.allclose(CellVariable(mesh=Grid1D(dx=(2.0, 1.0, 0.5)),
...                                     value=(0, 1, 2)).leastSquaresGrad.globalValue, [[0.461538461538, 0.8, 1.2]]))
True
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property minmodFaceValue

Returns a FaceVariable with a value that is the minimum of the absolute values of the adjacent cells. If the values are of opposite sign then the result is zero:

\phi_f = \begin{cases}
               \phi_1& \text{when $|\phi_1| \le |\phi_2|$},\\
               \phi_2& \text{when $|\phi_2| < |\phi_1|$},\\
               0 & \text{when $\phi1 \phi2 < 0$}
         \end{cases}

>>> from fipy import *
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(1, 2)).minmodFaceValue)
[1 1 2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, -2)).minmodFaceValue)
[-1 -1 -2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, 2)).minmodFaceValue)
[-1  0  2]
property name
property numericValue
property old

Return the values of the CellVariable from the previous solution sweep.

Combinations of CellVariable’s should also return old values.

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> var1 = CellVariable(mesh = mesh, value = (2, 3), hasOld = 1)
>>> var2 = CellVariable(mesh = mesh, value = (3, 4))
>>> v = var1 * var2
>>> print(v)
[ 6 12]
>>> var1.value = ((3, 2))
>>> print(v)
[9 8]
>>> print(v.old)
[ 6 12]

The following small test is to correct for a bug when the operator does not just use variables.

>>> v1 = var1 * 3
>>> print(v1)
[9 6]
>>> print(v1.old)
[6 9]
parallelRandom()
put(indices, value)
random()
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> c = Constraint(0., where=m.facesLeft)
>>> v.constrain(c)
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> v.release(constraint=c)
>>> print(v.faceValue)
[ 0.5  1.   2.   2.5]
scramble()

Generate a new random distribution.

setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
updateOld()

Set the values of the previous solution sweep to the current values.

>>> from fipy import *
>>> v = CellVariable(mesh=Grid1D(), hasOld=False)
>>> v.updateOld()
Traceback (most recent call last):
   ...
AssertionError: The updateOld method requires the CellVariable to have an old value. Set hasOld to True when instantiating the CellVariable.
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.binaryOperatorVariable module

fipy.variables.cellToFaceVariable module

fipy.variables.cellVariable module

class fipy.variables.cellVariable.CellVariable(*args, **kwds)

Bases: _MeshVariable

Represents the field of values of a variable on a Mesh.

A CellVariable can be pickled to persistent storage (disk) for later use:

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 10, ny = 10)
>>> var = CellVariable(mesh = mesh, value = 1., hasOld = 1, name = 'test')
>>> x, y = mesh.cellCenters
>>> var.value = (x * y)
>>> from fipy.tools import dump
>>> (f, filename) = dump.write(var, extension = '.gz')
>>> unPickledVar = dump.read(filename, f)
>>> print(var.allclose(unPickledVar, atol = 1e-10, rtol = 1e-10))
1
Parameters:
  • mesh (Mesh) – the mesh that defines the geometry of this Variable

  • name (str) – the user-readable name of the Variable

  • value (float or array_like) – the initial value

  • rank (int) – the rank (number of dimensions) of each element of this Variable. Default: 0

  • elementshape (tuple of int) – the shape of each element of this variable Default: rank * (mesh.dim,)

  • unit (str or PhysicalUnit) – The physical units of the variable

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__(points=None, order=0, nearestCellIDs=None)

Interpolates the CellVariable to a set of points using a method that has a memory requirement on the order of Ncells by Npoints in general, but uses only Ncells when the CellVariable’s mesh is a UniformGrid object.

Tests

>>> from fipy import *
>>> m = Grid2D(nx=3, ny=2)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.))))
[ 0.5  1.5  1.5]
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.)), order=1))
[ 0.25  1.1   1.2 ]
>>> m0 = Grid2D(nx=2, ny=2, dx=1., dy=1.)
>>> m1 = Grid2D(nx=4, ny=4, dx=.5, dy=.5)
>>> x, y = m0.cellCenters
>>> v0 = CellVariable(mesh=m0, value=x * y)
>>> print(v0(m1.cellCenters.globalValue))
[ 0.25  0.25  0.75  0.75  0.25  0.25  0.75  0.75  0.75  0.75  2.25  2.25
  0.75  0.75  2.25  2.25]
>>> print(v0(m1.cellCenters.globalValue, order=1))
[ 0.125  0.25   0.5    0.625  0.25   0.375  0.875  1.     0.5    0.875
  1.875  2.25   0.625  1.     2.25   2.625]
Parameters:
  • points (tuple or list of tuple) – A point or set of points in the format (X, Y, Z)

  • order ({0, 1}) – The order of interpolation, default is 0

  • nearestCellIDs (array_like) – Optional argument if user can calculate own nearest cell IDs array, shape should be same as points

__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the CellVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new CellVariable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
property arithmeticFaceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
cacheMe(recursive=False)
property cellVolumeAverage

Return the cell-volume-weighted average of the CellVariable:

<\phi>_\text{vol}
= \frac{\sum_\text{cells} \phi_\text{cell} V_\text{cell}}
    {\sum_\text{cells} V_\text{cell}}

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx = 3, ny = 1, dx = .5, dy = .1)
>>> var = CellVariable(value = (1, 2, 6), mesh = mesh)
>>> print(var.cellVolumeAverage)
3.0
constrain(value, where=None)

Constrains the CellVariable to value at a location specified by where.

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> v.constrain(0., where=m.facesLeft)
>>> v.faceGrad.constrain([1.], where=m.facesRight)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]

Changing the constraint changes the dependencies

>>> v.constrain(1., where=m.facesLeft)
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can be Variable

>>> c = Variable(0.)
>>> v.constrain(c, where=m.facesLeft)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> c.value = 1.
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can have a Variable mask.

>>> v = CellVariable(mesh=m)
>>> mask = FaceVariable(mesh=m, value=m.facesLeft)
>>> v.constrain(1., where=mask)
>>> print(v.faceValue)
[ 1.  0.  0.  0.]
>>> mask[:] = mask | m.facesRight
>>> print(v.faceValue)
[ 1.  0.  0.  1.]
property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Make an duplicate of the Variable

>>> a = Variable(value=3)
>>> b = a.copy()
>>> b
Variable(value=array(3))

The duplicate will not reflect changes made to the original

>>> a.setValue(5)
>>> b
Variable(value=array(3))

Check that this works for arrays.

>>> a = Variable(value=numerix.array((0, 1, 2)))
>>> b = a.copy()
>>> b
Variable(value=array([0, 1, 2]))
>>> a[1] = 3
>>> b
Variable(value=array([0, 1, 2]))
dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

property faceGrad

Return \nabla \phi as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).

property faceGradAverage

Deprecated since version 3.3: use grad.arithmeticFaceValue() instead

Return \nabla \phi as a rank-1 FaceVariable using averaging for the normal direction(second-order gradient)

property faceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
property gaussGrad

Return \frac{1}{V_P} \sum_f \vec{n} \phi_f A_f as a rank-1 CellVariable (first-order gradient).

getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue

Concatenate and return values from all processors

When running on a single processor, the result is identical to value.

property grad

Return \nabla \phi as a rank-1 CellVariable (first-order gradient).

property harmonicFaceValue

Returns a FaceVariable whose value corresponds to the harmonic interpolation of the adjacent cells:

\phi_f = \frac{\phi_1 \phi_2}{(\phi_2 - \phi_1) \frac{d_{f2}}{d_{12}} + \phi_1}

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (0.5 / 1.) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (1.0 / 3.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (5.0 / 55.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property leastSquaresGrad

Return \nabla \phi, which is determined by solving for \nabla \phi in the following matrix equation,

\nabla \phi \cdot \sum_f d_{AP}^2 \vec{n}_{AP} \otimes \vec{n}_{AP} =
\sum_f d_{AP}^2 \left( \vec{n} \cdot \nabla \phi \right)_{AP}

The matrix equation is derived by minimizing the following least squares sum,

F \left( \phi_x, \phi_y \right) = \sqrt{\sum_f \left( d_{AP}
\vec{n}_{AP} \cdot \nabla \phi - d_{AP} \left( \vec{n}_{AP} \cdot
\nabla \phi \right)_{AP} \right)^2 }

Tests

>>> from fipy import Grid2D
>>> m = Grid2D(nx=2, ny=2, dx=0.1, dy=2.0)
>>> print(numerix.allclose(CellVariable(mesh=m, value=(0, 1, 3, 6)).leastSquaresGrad.globalValue, \
...                                     [[8.0, 8.0, 24.0, 24.0],
...                                      [1.2, 2.0, 1.2, 2.0]]))
True
>>> from fipy import Grid1D
>>> print(numerix.allclose(CellVariable(mesh=Grid1D(dx=(2.0, 1.0, 0.5)),
...                                     value=(0, 1, 2)).leastSquaresGrad.globalValue, [[0.461538461538, 0.8, 1.2]]))
True
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property minmodFaceValue

Returns a FaceVariable with a value that is the minimum of the absolute values of the adjacent cells. If the values are of opposite sign then the result is zero:

\phi_f = \begin{cases}
               \phi_1& \text{when $|\phi_1| \le |\phi_2|$},\\
               \phi_2& \text{when $|\phi_2| < |\phi_1|$},\\
               0 & \text{when $\phi1 \phi2 < 0$}
         \end{cases}

>>> from fipy import *
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(1, 2)).minmodFaceValue)
[1 1 2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, -2)).minmodFaceValue)
[-1 -1 -2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, 2)).minmodFaceValue)
[-1  0  2]
property name
property numericValue
property old

Return the values of the CellVariable from the previous solution sweep.

Combinations of CellVariable’s should also return old values.

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> var1 = CellVariable(mesh = mesh, value = (2, 3), hasOld = 1)
>>> var2 = CellVariable(mesh = mesh, value = (3, 4))
>>> v = var1 * var2
>>> print(v)
[ 6 12]
>>> var1.value = ((3, 2))
>>> print(v)
[9 8]
>>> print(v.old)
[ 6 12]

The following small test is to correct for a bug when the operator does not just use variables.

>>> v1 = var1 * 3
>>> print(v1)
[9 6]
>>> print(v1.old)
[6 9]
put(indices, value)
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> c = Constraint(0., where=m.facesLeft)
>>> v.constrain(c)
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> v.release(constraint=c)
>>> print(v.faceValue)
[ 0.5  1.   2.   2.5]
setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
updateOld()

Set the values of the previous solution sweep to the current values.

>>> from fipy import *
>>> v = CellVariable(mesh=Grid1D(), hasOld=False)
>>> v.updateOld()
Traceback (most recent call last):
   ...
AssertionError: The updateOld method requires the CellVariable to have an old value. Set hasOld to True when instantiating the CellVariable.
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.constant module

fipy.variables.constraintMask module

fipy.variables.coupledCellVariable module

fipy.variables.distanceVariable module

class fipy.variables.distanceVariable.DistanceVariable(*args, **kwds)

Bases: CellVariable

A DistanceVariable object calculates \phi so it satisfies,

\abs{\nabla \phi} = 1

using the fast marching method with an initial condition defined by the zero level set. The solution can either be first or second order.

Here we will define a few test cases. Firstly a 1D test case

>>> from fipy.meshes import Grid1D
>>> from fipy.tools import serialComm
>>> mesh = Grid1D(dx = .5, nx = 8, communicator=serialComm)
>>> from .distanceVariable import DistanceVariable
>>> var = DistanceVariable(mesh = mesh, value = (-1., -1., -1., -1., 1., 1., 1., 1.))
>>> var.calcDistanceFunction() 
>>> answer = (-1.75, -1.25, -.75, -0.25, 0.25, 0.75, 1.25, 1.75)
>>> print(var.allclose(answer)) 
1

A 1D test case with very small dimensions.

>>> dx = 1e-10
>>> mesh = Grid1D(dx = dx, nx = 8, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., -1., -1., -1., 1., 1., 1., 1.))
>>> var.calcDistanceFunction() 
>>> answer = numerix.arange(8) * dx - 3.5 * dx
>>> print(var.allclose(answer)) 
1

A 2D test case to test _calcTrialValue for a pathological case.

>>> dx = 1.
>>> dy = 2.
>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 2, ny = 3, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., 1., 1., -1., 1.))
>>> var.calcDistanceFunction() 
>>> vbl = -dx * dy / numerix.sqrt(dx**2 + dy**2) / 2.
>>> vbr = dx / 2
>>> vml = dy / 2.
>>> crossProd = dx * dy
>>> dsq = dx**2 + dy**2
>>> top = vbr * dx**2 + vml * dy**2
>>> sqrt = crossProd**2 *(dsq - (vbr - vml)**2)
>>> sqrt = numerix.sqrt(max(sqrt, 0))
>>> vmr = (top + sqrt) / dsq
>>> answer = (vbl, vbr, vml, vmr, vbl, vbr)
>>> print(var.allclose(answer)) 
1

The extendVariable method solves the following equation for a given extensionVariable.

\nabla u \cdot \nabla \phi = 0

using the fast marching method with an initial condition defined at the zero level set.

>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., 1., 1.))
>>> var.calcDistanceFunction() 
>>> extensionVar = CellVariable(mesh = mesh, value = (-1, .5, 2, -1))
>>> tmp = 1 / numerix.sqrt(2)
>>> print(var.allclose((-tmp / 2, 0.5, 0.5, 0.5 + tmp))) 
1
>>> var.extendVariable(extensionVar, order=1) 
>>> print(extensionVar.allclose((1.25, .5, 2, 1.25))) 
1
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 3, ny = 3, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., 1.,
...                                               1., 1., 1.,
...                                               1., 1., 1.))
>>> var.calcDistanceFunction(order=1) 
>>> extensionVar = CellVariable(mesh = mesh, value = (-1., .5, -1.,
...                                                    2., -1., -1.,
...                                                   -1., -1., -1.))
>>> v1 = 0.5 + tmp
>>> v2 = 1.5
>>> tmp1 = (v1 + v2) / 2 + numerix.sqrt(2. - (v1 - v2)**2) / 2
>>> tmp2 = tmp1 + 1 / numerix.sqrt(2)
>>> print(var.allclose((-tmp / 2, 0.5, 1.5, 0.5, 0.5 + tmp,
...                      tmp1, 1.5, tmp1, tmp2))) 
1
>>> answer = (1.25, .5, .5, 2, 1.25, 0.9544, 2, 1.5456, 1.25)
>>> var.extendVariable(extensionVar, order=1) 
>>> print(extensionVar.allclose(answer, rtol = 1e-4)) 
1

Test case for a bug that occurs when initializing the distance variable at the interface. Currently it is assumed that adjacent cells that are opposite sign neighbors have perpendicular normal vectors. In fact the two closest cells could have opposite normals.

>>> mesh = Grid1D(dx = 1., nx = 3, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., -1.))
>>> var.calcDistanceFunction() 
>>> print(var.allclose((-0.5, 0.5, -0.5))) 
1

Testing second order. This example failed with Scikit-fmm.

>>> mesh = Grid2D(dx = 1., dy = 1., nx = 4, ny = 4, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., -1., 1., 1.,
...                                               -1., -1., 1., 1.,
...                                               1., 1., 1., 1.,
...                                               1, 1, 1, 1))
>>> var.calcDistanceFunction(order=2) 
>>> answer = [-1.30473785, -0.5, 0.5, 1.49923009,
...           -0.5, -0.35355339, 0.5, 1.45118446,
...            0.5, 0.5, 0.97140452, 1.76215286,
...            1.49923009, 1.45118446, 1.76215286, 2.33721352]
>>> print(numerix.allclose(var, answer, rtol=1e-9)) 
True

** A test for a bug in both LSMLIB and Scikit-fmm **

The following test gives different result depending on whether LSMLIB or Scikit-fmm is used. There is a deeper problem that is related to this issue. When a value becomes “known” after previously being a “trial” value it updates its neighbors’ values. In a second order scheme the neighbors one step away also need to be updated (if the in between cell is “known” and the far cell is a “trial” cell), but are not in either package. By luck (due to trial values having the same value), the values calculated in Scikit-fmm for the following example are correct although an example that didn’t work for Scikit-fmm could also be constructed.

>>> mesh = Grid2D(dx = 1., dy = 1., nx = 4, ny = 4, communicator=serialComm)
>>> var = DistanceVariable(mesh = mesh, value = (-1., -1., -1., -1.,
...                                               1.,  1., -1., -1.,
...                                               1.,  1., -1., -1.,
...                                               1.,  1., -1., -1.))
>>> var.calcDistanceFunction(order=2) 
>>> var.calcDistanceFunction(order=2) 
>>> answer = [-0.5,        -0.58578644, -1.08578644, -1.85136395,
...            0.5,         0.29289322, -0.58578644, -1.54389939,
...            1.30473785,  0.5,        -0.5,        -1.5,
...            1.49547948,  0.5,        -0.5,        -1.5]

The 3rd and 7th element are different for LSMLIB. This is because the 15th element is not “known” when the “trial” value for the 7th element is calculated. Scikit-fmm calculates the values in a slightly different order so gets a seemingly better answer, but this is just chance.

>>> print(numerix.allclose(var, answer, rtol=1e-9)) 
True

Creates a distanceVariable object.

Parameters:
  • mesh (Mesh) – The mesh that defines the geometry of this variable.

  • name (str) – The name of the variable.

  • value (float or array_like) – The initial value.

  • unit (str or PhysicalUnit) – The physical units of the variable

  • hasOld (bool) – Whether the variable maintains an old value.

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__(points=None, order=0, nearestCellIDs=None)

Interpolates the CellVariable to a set of points using a method that has a memory requirement on the order of Ncells by Npoints in general, but uses only Ncells when the CellVariable’s mesh is a UniformGrid object.

Tests

>>> from fipy import *
>>> m = Grid2D(nx=3, ny=2)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.))))
[ 0.5  1.5  1.5]
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.)), order=1))
[ 0.25  1.1   1.2 ]
>>> m0 = Grid2D(nx=2, ny=2, dx=1., dy=1.)
>>> m1 = Grid2D(nx=4, ny=4, dx=.5, dy=.5)
>>> x, y = m0.cellCenters
>>> v0 = CellVariable(mesh=m0, value=x * y)
>>> print(v0(m1.cellCenters.globalValue))
[ 0.25  0.25  0.75  0.75  0.25  0.25  0.75  0.75  0.75  0.75  2.25  2.25
  0.75  0.75  2.25  2.25]
>>> print(v0(m1.cellCenters.globalValue, order=1))
[ 0.125  0.25   0.5    0.625  0.25   0.375  0.875  1.     0.5    0.875
  1.875  2.25   0.625  1.     2.25   2.625]
Parameters:
  • points (tuple or list of tuple) – A point or set of points in the format (X, Y, Z)

  • order ({0, 1}) – The order of interpolation, default is 0

  • nearestCellIDs (array_like) – Optional argument if user can calculate own nearest cell IDs array, shape should be same as points

__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the CellVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new CellVariable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
property arithmeticFaceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
cacheMe(recursive=False)
calcDistanceFunction(order=2)

Calculates the distanceVariable as a distance function.

Parameters:

order ({1, 2}) – The order of accuracy for the distance function calculation

property cellInterfaceAreas

Returns the length of the interface that crosses the cell

A simple 1D test:

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(dx = 1., nx = 4)
>>> distanceVariable = DistanceVariable(mesh = mesh,
...                                     value = (-1.5, -0.5, 0.5, 1.5))
>>> answer = CellVariable(mesh=mesh, value=(0, 0., 1., 0))
>>> print(numerix.allclose(distanceVariable.cellInterfaceAreas,
...                        answer))
True

A 2D test case:

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 3, ny = 3)
>>> distanceVariable = DistanceVariable(mesh = mesh,
...                                     value = (1.5, 0.5, 1.5,
...                                              0.5, -0.5, 0.5,
...                                              1.5, 0.5, 1.5))
>>> answer = CellVariable(mesh=mesh,
...                       value=(0, 1, 0, 1, 0, 1, 0, 1, 0))
>>> print(numerix.allclose(distanceVariable.cellInterfaceAreas, answer))
True

Another 2D test case:

>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> distanceVariable = DistanceVariable(mesh = mesh,
...                                     value = (-0.5, 0.5, 0.5, 1.5))
>>> answer = CellVariable(mesh=mesh,
...                       value=(0, numerix.sqrt(2) / 4,  numerix.sqrt(2) / 4, 0))
>>> print(numerix.allclose(distanceVariable.cellInterfaceAreas,
...                        answer))
True

Test to check that the circumference of a circle is, in fact, 2\pi r.

>>> mesh = Grid2D(dx = 0.05, dy = 0.05, nx = 20, ny = 20)
>>> r = 0.25
>>> x, y = mesh.cellCenters
>>> rad = numerix.sqrt((x - .5)**2 + (y - .5)**2) - r
>>> distanceVariable = DistanceVariable(mesh = mesh, value = rad)
>>> print(numerix.allclose(distanceVariable.cellInterfaceAreas.sum(), 1.57984690073))
1
property cellVolumeAverage

Return the cell-volume-weighted average of the CellVariable:

<\phi>_\text{vol}
= \frac{\sum_\text{cells} \phi_\text{cell} V_\text{cell}}
    {\sum_\text{cells} V_\text{cell}}

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx = 3, ny = 1, dx = .5, dy = .1)
>>> var = CellVariable(value = (1, 2, 6), mesh = mesh)
>>> print(var.cellVolumeAverage)
3.0
constrain(value, where=None)

Constrains the CellVariable to value at a location specified by where.

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> v.constrain(0., where=m.facesLeft)
>>> v.faceGrad.constrain([1.], where=m.facesRight)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]

Changing the constraint changes the dependencies

>>> v.constrain(1., where=m.facesLeft)
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can be Variable

>>> c = Variable(0.)
>>> v.constrain(c, where=m.facesLeft)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> c.value = 1.
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can have a Variable mask.

>>> v = CellVariable(mesh=m)
>>> mask = FaceVariable(mesh=m, value=m.facesLeft)
>>> v.constrain(1., where=mask)
>>> print(v.faceValue)
[ 1.  0.  0.  0.]
>>> mask[:] = mask | m.facesRight
>>> print(v.faceValue)
[ 1.  0.  0.  1.]
property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Make an duplicate of the Variable

>>> a = Variable(value=3)
>>> b = a.copy()
>>> b
Variable(value=array(3))

The duplicate will not reflect changes made to the original

>>> a.setValue(5)
>>> b
Variable(value=array(3))

Check that this works for arrays.

>>> a = Variable(value=numerix.array((0, 1, 2)))
>>> b = a.copy()
>>> b
Variable(value=array([0, 1, 2]))
>>> a[1] = 3
>>> b
Variable(value=array([0, 1, 2]))
dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

extendVariable(extensionVariable, order=2)

Calculates the extension of extensionVariable from the zero level set.

Parameters:

extensionVariable (CellVariable) – The variable to extend from the zero level set.

property faceGrad

Return \nabla \phi as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).

property faceGradAverage

Deprecated since version 3.3: use grad.arithmeticFaceValue() instead

Return \nabla \phi as a rank-1 FaceVariable using averaging for the normal direction(second-order gradient)

property faceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
property gaussGrad

Return \frac{1}{V_P} \sum_f \vec{n} \phi_f A_f as a rank-1 CellVariable (first-order gradient).

getLSMshape()
getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue

Concatenate and return values from all processors

When running on a single processor, the result is identical to value.

property grad

Return \nabla \phi as a rank-1 CellVariable (first-order gradient).

property harmonicFaceValue

Returns a FaceVariable whose value corresponds to the harmonic interpolation of the adjacent cells:

\phi_f = \frac{\phi_1 \phi_2}{(\phi_2 - \phi_1) \frac{d_{f2}}{d_{12}} + \phi_1}

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (0.5 / 1.) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (1.0 / 3.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (5.0 / 55.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property leastSquaresGrad

Return \nabla \phi, which is determined by solving for \nabla \phi in the following matrix equation,

\nabla \phi \cdot \sum_f d_{AP}^2 \vec{n}_{AP} \otimes \vec{n}_{AP} =
\sum_f d_{AP}^2 \left( \vec{n} \cdot \nabla \phi \right)_{AP}

The matrix equation is derived by minimizing the following least squares sum,

F \left( \phi_x, \phi_y \right) = \sqrt{\sum_f \left( d_{AP}
\vec{n}_{AP} \cdot \nabla \phi - d_{AP} \left( \vec{n}_{AP} \cdot
\nabla \phi \right)_{AP} \right)^2 }

Tests

>>> from fipy import Grid2D
>>> m = Grid2D(nx=2, ny=2, dx=0.1, dy=2.0)
>>> print(numerix.allclose(CellVariable(mesh=m, value=(0, 1, 3, 6)).leastSquaresGrad.globalValue, \
...                                     [[8.0, 8.0, 24.0, 24.0],
...                                      [1.2, 2.0, 1.2, 2.0]]))
True
>>> from fipy import Grid1D
>>> print(numerix.allclose(CellVariable(mesh=Grid1D(dx=(2.0, 1.0, 0.5)),
...                                     value=(0, 1, 2)).leastSquaresGrad.globalValue, [[0.461538461538, 0.8, 1.2]]))
True
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property minmodFaceValue

Returns a FaceVariable with a value that is the minimum of the absolute values of the adjacent cells. If the values are of opposite sign then the result is zero:

\phi_f = \begin{cases}
               \phi_1& \text{when $|\phi_1| \le |\phi_2|$},\\
               \phi_2& \text{when $|\phi_2| < |\phi_1|$},\\
               0 & \text{when $\phi1 \phi2 < 0$}
         \end{cases}

>>> from fipy import *
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(1, 2)).minmodFaceValue)
[1 1 2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, -2)).minmodFaceValue)
[-1 -1 -2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, 2)).minmodFaceValue)
[-1  0  2]
property name
property numericValue
property old

Return the values of the CellVariable from the previous solution sweep.

Combinations of CellVariable’s should also return old values.

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> var1 = CellVariable(mesh = mesh, value = (2, 3), hasOld = 1)
>>> var2 = CellVariable(mesh = mesh, value = (3, 4))
>>> v = var1 * var2
>>> print(v)
[ 6 12]
>>> var1.value = ((3, 2))
>>> print(v)
[9 8]
>>> print(v.old)
[ 6 12]

The following small test is to correct for a bug when the operator does not just use variables.

>>> v1 = var1 * 3
>>> print(v1)
[9 6]
>>> print(v1.old)
[6 9]
put(indices, value)
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> c = Constraint(0., where=m.facesLeft)
>>> v.constrain(c)
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> v.release(constraint=c)
>>> print(v.faceValue)
[ 0.5  1.   2.   2.5]
setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
updateOld()

Set the values of the previous solution sweep to the current values.

>>> from fipy import *
>>> v = CellVariable(mesh=Grid1D(), hasOld=False)
>>> v.updateOld()
Traceback (most recent call last):
   ...
AssertionError: The updateOld method requires the CellVariable to have an old value. Set hasOld to True when instantiating the CellVariable.
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.exponentialNoiseVariable module

class fipy.variables.exponentialNoiseVariable.ExponentialNoiseVariable(*args, **kwds)

Bases: NoiseVariable

Represents an exponential distribution of random numbers with the probability distribution

\mu^{-1} e^{-\frac{x}{\mu}}

with a mean parameter \mu.

Seed the random module for the sake of deterministic test results.

>>> from fipy import numerix
>>> numerix.random.seed(1)

We generate noise on a uniform Cartesian mesh

>>> from fipy.variables.variable import Variable
>>> mean = Variable()
>>> from fipy.meshes import Grid2D
>>> noise = ExponentialNoiseVariable(mesh = Grid2D(nx = 100, ny = 100), mean = mean)

We histogram the root-volume-weighted noise distribution

>>> from fipy.variables.histogramVariable import HistogramVariable
>>> histogram = HistogramVariable(distribution = noise, dx = 0.1, nx = 100)

and compare to a Gaussian distribution

>>> from fipy.variables.cellVariable import CellVariable
>>> expdist = CellVariable(mesh = histogram.mesh)
>>> x = histogram.mesh.cellCenters[0]
>>> if __name__ == '__main__':
...     from fipy import Viewer
...     viewer = Viewer(vars=noise, datamin=0, datamax=5)
...     histoplot = Viewer(vars=(histogram, expdist),
...                        datamin=0, datamax=1.5)
>>> from fipy.tools.numerix import arange, exp
>>> for mu in arange(0.5, 3, 0.5):
...     mean.value = (mu)
...     expdist.value = ((1/mean)*exp(-x/mean))
...     if __name__ == '__main__':
...         import sys
...         print("mean: %g" % mean, file=sys.stderr)
...         viewer.plot()
...         histoplot.plot()
>>> print(abs(noise.faceGrad.divergence.cellVolumeAverage) < 5e-15)
1
random values with an exponential distribution histogram of random values with an exponential distribution
Parameters:
  • mesh (Mesh) – The mesh on which to define the noise.

  • mean (float) – The mean of the distribution \mu.

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__(points=None, order=0, nearestCellIDs=None)

Interpolates the CellVariable to a set of points using a method that has a memory requirement on the order of Ncells by Npoints in general, but uses only Ncells when the CellVariable’s mesh is a UniformGrid object.

Tests

>>> from fipy import *
>>> m = Grid2D(nx=3, ny=2)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.))))
[ 0.5  1.5  1.5]
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.)), order=1))
[ 0.25  1.1   1.2 ]
>>> m0 = Grid2D(nx=2, ny=2, dx=1., dy=1.)
>>> m1 = Grid2D(nx=4, ny=4, dx=.5, dy=.5)
>>> x, y = m0.cellCenters
>>> v0 = CellVariable(mesh=m0, value=x * y)
>>> print(v0(m1.cellCenters.globalValue))
[ 0.25  0.25  0.75  0.75  0.25  0.25  0.75  0.75  0.75  0.75  2.25  2.25
  0.75  0.75  2.25  2.25]
>>> print(v0(m1.cellCenters.globalValue, order=1))
[ 0.125  0.25   0.5    0.625  0.25   0.375  0.875  1.     0.5    0.875
  1.875  2.25   0.625  1.     2.25   2.625]
Parameters:
  • points (tuple or list of tuple) – A point or set of points in the format (X, Y, Z)

  • order ({0, 1}) – The order of interpolation, default is 0

  • nearestCellIDs (array_like) – Optional argument if user can calculate own nearest cell IDs array, shape should be same as points

__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the CellVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new CellVariable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
property arithmeticFaceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
cacheMe(recursive=False)
property cellVolumeAverage

Return the cell-volume-weighted average of the CellVariable:

<\phi>_\text{vol}
= \frac{\sum_\text{cells} \phi_\text{cell} V_\text{cell}}
    {\sum_\text{cells} V_\text{cell}}

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx = 3, ny = 1, dx = .5, dy = .1)
>>> var = CellVariable(value = (1, 2, 6), mesh = mesh)
>>> print(var.cellVolumeAverage)
3.0
constrain(value, where=None)

Constrains the CellVariable to value at a location specified by where.

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> v.constrain(0., where=m.facesLeft)
>>> v.faceGrad.constrain([1.], where=m.facesRight)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]

Changing the constraint changes the dependencies

>>> v.constrain(1., where=m.facesLeft)
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can be Variable

>>> c = Variable(0.)
>>> v.constrain(c, where=m.facesLeft)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> c.value = 1.
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can have a Variable mask.

>>> v = CellVariable(mesh=m)
>>> mask = FaceVariable(mesh=m, value=m.facesLeft)
>>> v.constrain(1., where=mask)
>>> print(v.faceValue)
[ 1.  0.  0.  0.]
>>> mask[:] = mask | m.facesRight
>>> print(v.faceValue)
[ 1.  0.  0.  1.]
property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Copy the value of the NoiseVariable to a static CellVariable.

dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

property faceGrad

Return \nabla \phi as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).

property faceGradAverage

Deprecated since version 3.3: use grad.arithmeticFaceValue() instead

Return \nabla \phi as a rank-1 FaceVariable using averaging for the normal direction(second-order gradient)

property faceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
property gaussGrad

Return \frac{1}{V_P} \sum_f \vec{n} \phi_f A_f as a rank-1 CellVariable (first-order gradient).

getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue

Concatenate and return values from all processors

When running on a single processor, the result is identical to value.

property grad

Return \nabla \phi as a rank-1 CellVariable (first-order gradient).

property harmonicFaceValue

Returns a FaceVariable whose value corresponds to the harmonic interpolation of the adjacent cells:

\phi_f = \frac{\phi_1 \phi_2}{(\phi_2 - \phi_1) \frac{d_{f2}}{d_{12}} + \phi_1}

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (0.5 / 1.) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (1.0 / 3.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (5.0 / 55.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property leastSquaresGrad

Return \nabla \phi, which is determined by solving for \nabla \phi in the following matrix equation,

\nabla \phi \cdot \sum_f d_{AP}^2 \vec{n}_{AP} \otimes \vec{n}_{AP} =
\sum_f d_{AP}^2 \left( \vec{n} \cdot \nabla \phi \right)_{AP}

The matrix equation is derived by minimizing the following least squares sum,

F \left( \phi_x, \phi_y \right) = \sqrt{\sum_f \left( d_{AP}
\vec{n}_{AP} \cdot \nabla \phi - d_{AP} \left( \vec{n}_{AP} \cdot
\nabla \phi \right)_{AP} \right)^2 }

Tests

>>> from fipy import Grid2D
>>> m = Grid2D(nx=2, ny=2, dx=0.1, dy=2.0)
>>> print(numerix.allclose(CellVariable(mesh=m, value=(0, 1, 3, 6)).leastSquaresGrad.globalValue, \
...                                     [[8.0, 8.0, 24.0, 24.0],
...                                      [1.2, 2.0, 1.2, 2.0]]))
True
>>> from fipy import Grid1D
>>> print(numerix.allclose(CellVariable(mesh=Grid1D(dx=(2.0, 1.0, 0.5)),
...                                     value=(0, 1, 2)).leastSquaresGrad.globalValue, [[0.461538461538, 0.8, 1.2]]))
True
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property minmodFaceValue

Returns a FaceVariable with a value that is the minimum of the absolute values of the adjacent cells. If the values are of opposite sign then the result is zero:

\phi_f = \begin{cases}
               \phi_1& \text{when $|\phi_1| \le |\phi_2|$},\\
               \phi_2& \text{when $|\phi_2| < |\phi_1|$},\\
               0 & \text{when $\phi1 \phi2 < 0$}
         \end{cases}

>>> from fipy import *
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(1, 2)).minmodFaceValue)
[1 1 2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, -2)).minmodFaceValue)
[-1 -1 -2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, 2)).minmodFaceValue)
[-1  0  2]
property name
property numericValue
property old

Return the values of the CellVariable from the previous solution sweep.

Combinations of CellVariable’s should also return old values.

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> var1 = CellVariable(mesh = mesh, value = (2, 3), hasOld = 1)
>>> var2 = CellVariable(mesh = mesh, value = (3, 4))
>>> v = var1 * var2
>>> print(v)
[ 6 12]
>>> var1.value = ((3, 2))
>>> print(v)
[9 8]
>>> print(v.old)
[ 6 12]

The following small test is to correct for a bug when the operator does not just use variables.

>>> v1 = var1 * 3
>>> print(v1)
[9 6]
>>> print(v1.old)
[6 9]
parallelRandom()
put(indices, value)
random()
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> c = Constraint(0., where=m.facesLeft)
>>> v.constrain(c)
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> v.release(constraint=c)
>>> print(v.faceValue)
[ 0.5  1.   2.   2.5]
scramble()

Generate a new random distribution.

setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
updateOld()

Set the values of the previous solution sweep to the current values.

>>> from fipy import *
>>> v = CellVariable(mesh=Grid1D(), hasOld=False)
>>> v.updateOld()
Traceback (most recent call last):
   ...
AssertionError: The updateOld method requires the CellVariable to have an old value. Set hasOld to True when instantiating the CellVariable.
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.faceGradContributionsVariable module

fipy.variables.faceGradVariable module

fipy.variables.faceVariable module

class fipy.variables.faceVariable.FaceVariable(*args, **kwds)

Bases: _MeshVariable

Parameters:
  • mesh (Mesh) – the mesh that defines the geometry of this Variable

  • name (str) – the user-readable name of the Variable

  • value (float or array_like) – the initial value

  • rank (int) – the rank (number of dimensions) of each element of this Variable. Default: 0

  • elementshape (tuple of int) – the shape of each element of this variable Default: rank * (mesh.dim,)

  • unit (str or PhysicalUnit) – The physical units of the variable

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__()

“Evaluate” the Variable and return its value

>>> a = Variable(value=3)
>>> print(a())
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b()
7
__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the _MeshVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new Variable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
cacheMe(recursive=False)
constrain(value, where=None)

Constrain the Variable to have a value at an index or mask location specified by where.

>>> v = Variable((0, 1, 2, 3))
>>> v.constrain(2, numerix.array((True, False, False, False)))
>>> print(v)
[2 1 2 3]
>>> v[:] = 10
>>> print(v)
[ 2 10 10 10]
>>> v.constrain(5, numerix.array((False, False, True, False)))
>>> print(v)
[ 2 10  5 10]
>>> v[:] = 6
>>> print(v)
[2 6 5 6]
>>> v.constrain(8)
>>> print(v)
[8 8 8 8]
>>> v[:] = 10
>>> print(v)
[8 8 8 8]
>>> del v.constraints[2]
>>> print(v)
[ 2 10  5 10]
>>> from fipy.variables.cellVariable import CellVariable
>>> from fipy.meshes import Grid2D
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v = CellVariable(mesh=m, rank=1, value=(x, y))
>>> v.constrain(((0.,), (-1.,)), where=m.facesLeft)
>>> print(v.faceValue)
[[ 0.5  1.5  0.5  1.5  0.5  1.5  0.   1.   1.5  0.   1.   1.5]
 [ 0.5  0.5  1.   1.   1.5  1.5 -1.   0.5  0.5 -1.   1.5  1.5]]
Parameters:
  • value (float or array_like) – The value of the constraint

  • where (array_like of bool) – The constraint mask or index specifying the location of the constraint

property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Make an duplicate of the Variable

>>> a = Variable(value=3)
>>> b = a.copy()
>>> b
Variable(value=array(3))

The duplicate will not reflect changes made to the original

>>> a.setValue(5)
>>> b
Variable(value=array(3))

Check that this works for arrays.

>>> a = Variable(value=numerix.array((0, 1, 2)))
>>> b = a.copy()
>>> b
Variable(value=array([0, 1, 2]))
>>> a[1] = 3
>>> b
Variable(value=array([0, 1, 2]))
property divergence

the divergence of self, \vec{u},

\nabla\cdot\vec{u} \approx \frac{\sum_f (\vec{u}\cdot\hat{n})_f A_f}{V_P}

Returns:

divergence – one rank lower than self

Return type:

fipy.variables.cellVariable.CellVariable

Examples

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=3, ny=2)
>>> from builtins import range
>>> var = CellVariable(mesh=mesh, value=list(range(3*2)))
>>> print(var.faceGrad.divergence)
[ 4.  3.  2. -2. -3. -4.]
dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property name
property numericValue
put(indices, value)
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> v = Variable((0, 1, 2, 3))
>>> v.constrain(2, numerix.array((True, False, False, False)))
>>> v[:] = 10
>>> from fipy.boundaryConditions.constraint import Constraint
>>> c1 = Constraint(5, numerix.array((False, False, True, False)))
>>> v.constrain(c1)
>>> v[:] = 6
>>> v.constrain(8)
>>> v[:] = 10
>>> del v.constraints[2]
>>> v.release(constraint=c1)
>>> print(v)
[ 2 10 10 10]
setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.gammaNoiseVariable module

class fipy.variables.gammaNoiseVariable.GammaNoiseVariable(*args, **kwds)

Bases: NoiseVariable

Represents a gamma distribution of random numbers with the probability distribution

x^{\alpha - 1}\frac{\beta^\alpha e^{-\beta x}}{\Gamma(\alpha)}

with a shape parameter \alpha, a rate parameter \beta, and \Gamma(z) = \int_0^\infty t^{z - 1}e^{-t}\,dt.

Seed the random module for the sake of deterministic test results.

>>> from fipy import numerix
>>> numerix.random.seed(1)

We generate noise on a uniform Cartesian mesh

>>> from fipy.variables.variable import Variable
>>> alpha = Variable()
>>> beta = Variable()
>>> from fipy.meshes import Grid2D
>>> noise = GammaNoiseVariable(mesh = Grid2D(nx = 100, ny = 100), shape = alpha, rate = beta)

We histogram the root-volume-weighted noise distribution

>>> from fipy.variables.histogramVariable import HistogramVariable
>>> histogram = HistogramVariable(distribution = noise, dx = 0.1, nx = 300)

and compare to a Gaussian distribution

>>> from fipy.variables.cellVariable import CellVariable
>>> x = CellVariable(mesh=histogram.mesh, value=histogram.mesh.cellCenters[0])
>>> from scipy.special import gamma as Gamma 
>>> from fipy.tools.numerix import exp
>>> gammadist = (x**(alpha - 1) * (beta**alpha * exp(-beta * x)) / Gamma(alpha)) 
>>> if __name__ == '__main__':
...     from fipy import Viewer
...     viewer = Viewer(vars=noise, datamin=0, datamax=30)
...     histoplot = Viewer(vars=(histogram, gammadist),
...                        datamin=0, datamax=1)
>>> from fipy.tools.numerix import arange
>>> for shape in arange(1, 8, 1):
...     alpha.value = shape
...     for rate in arange(0.5, 2.5, 0.5):
...         beta.value = rate
...         if __name__ == '__main__':
...             import sys
...             print("alpha: %g, beta: %g" % (alpha, beta), file=sys.stderr)
...             viewer.plot()
...             histoplot.plot()
>>> print(abs(noise.faceGrad.divergence.cellVolumeAverage) < 5e-15)
1
random values with a gamma distribution histogram of random values with a gamma distribution
Parameters:
  • mesh (Mesh) – The mesh on which to define the noise.

  • shape (float) – The shape parameter, \alpha.

  • rate (float) – The rate or inverse scale parameter, \beta.

__abs__()

Following test it to fix a bug with C inline string using abs() instead of fabs()

>>> print(abs(Variable(2.3) - Variable(1.2)))
1.1

Check representation works with different versions of numpy

>>> print(repr(abs(Variable(2.3))))
numerix.fabs(Variable(value=array(2.3)))
__add__(other)
__and__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) & (b == 1), [False,  True, False, False]).all())
True
>>> print(a & b)
[0 0 0 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) & (b == 1), [False,  True, False, False]))
True
>>> print(a & b)
[0 0 0 1]
__array__(t=None)

Attempt to convert the Variable to a numerix array object

>>> v = Variable(value=[2, 3])
>>> print(numerix.array(v))
[2 3]

A dimensional Variable will convert to the numeric value in its base units

>>> v = Variable(value=[2, 3], unit="mm")
>>> numerix.array(v)
array([ 0.002,  0.003])
__array_priority__ = 100.0
__array_wrap__(arr, context=None)

Required to prevent numpy not calling the reverse binary operations. Both the following tests are examples ufuncs.

>>> print(type(numerix.array([1.0, 2.0]) * Variable([1.0, 2.0]))) 
<class 'fipy.variables.binaryOperatorVariable...binOp'>
>>> from scipy.special import gamma as Gamma 
>>> print(type(Gamma(Variable([1.0, 2.0])))) 
<class 'fipy.variables.unaryOperatorVariable...unOp'>
__bool__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__call__(points=None, order=0, nearestCellIDs=None)

Interpolates the CellVariable to a set of points using a method that has a memory requirement on the order of Ncells by Npoints in general, but uses only Ncells when the CellVariable’s mesh is a UniformGrid object.

Tests

>>> from fipy import *
>>> m = Grid2D(nx=3, ny=2)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.))))
[ 0.5  1.5  1.5]
>>> print(v(((0., 1.1, 1.2), (0., 1., 1.)), order=1))
[ 0.25  1.1   1.2 ]
>>> m0 = Grid2D(nx=2, ny=2, dx=1., dy=1.)
>>> m1 = Grid2D(nx=4, ny=4, dx=.5, dy=.5)
>>> x, y = m0.cellCenters
>>> v0 = CellVariable(mesh=m0, value=x * y)
>>> print(v0(m1.cellCenters.globalValue))
[ 0.25  0.25  0.75  0.75  0.25  0.25  0.75  0.75  0.75  0.75  2.25  2.25
  0.75  0.75  2.25  2.25]
>>> print(v0(m1.cellCenters.globalValue, order=1))
[ 0.125  0.25   0.5    0.625  0.25   0.375  0.875  1.     0.5    0.875
  1.875  2.25   0.625  1.     2.25   2.625]
Parameters:
  • points (tuple or list of tuple) – A point or set of points in the format (X, Y, Z)

  • order ({0, 1}) – The order of interpolation, default is 0

  • nearestCellIDs (array_like) – Optional argument if user can calculate own nearest cell IDs array, shape should be same as points

__div__(other)
__eq__(other)

Test if a Variable is equal to another quantity

>>> a = Variable(value=3)
>>> b = (a == 4)
>>> b
(Variable(value=array(3)) == 4)
>>> b()
0
__float__()
__ge__(other)

Test if a Variable is greater than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a >= 4)
>>> b
(Variable(value=array(3)) >= 4)
>>> b()
0
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
1
__getitem__(index)

“Evaluate” the Variable and return the specified element

>>> a = Variable(value=((3., 4.), (5., 6.)), unit="m") + "4 m"
>>> print(a[1, 1])
10.0 m

It is an error to slice a Variable whose value is not sliceable

>>> Variable(value=3)[2]
Traceback (most recent call last):
      ...
IndexError: 0-d arrays can't be indexed
__getstate__()

Used internally to collect the necessary information to pickle the CellVariable to persistent storage.

__gt__(other)

Test if a Variable is greater than another quantity

>>> a = Variable(value=3)
>>> b = (a > 4)
>>> b
(Variable(value=array(3)) > 4)
>>> print(b())
0
>>> a.value = 5
>>> print(b())
1
__hash__()

Return hash(self).

__int__()
__invert__()

Returns logical “not” of the Variable

>>> a = Variable(value=True)
>>> print(~a)
False
__iter__()
__le__(other)

Test if a Variable is less than or equal to another quantity

>>> a = Variable(value=3)
>>> b = (a <= 4)
>>> b
(Variable(value=array(3)) <= 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
1
>>> a.value = 5
>>> print(b())
0
__len__()
__lt__(other)

Test if a Variable is less than another quantity

>>> a = Variable(value=3)
>>> b = (a < 4)
>>> b
(Variable(value=array(3)) < 4)
>>> b()
1
>>> a.value = 4
>>> print(b())
0
>>> print(1000000000000000000 * Variable(1) < 1.)
0
>>> print(1000 * Variable(1) < 1.)
0

Python automatically reverses the arguments when necessary

>>> 4 > Variable(value=3)
(Variable(value=array(3)) < 4)
__mod__(other)
__mul__(other)
__ne__(other)

Test if a Variable is not equal to another quantity

>>> a = Variable(value=3)
>>> b = (a != 4)
>>> b
(Variable(value=array(3)) != 4)
>>> b()
1
__neg__()
static __new__(cls, *args, **kwds)
__nonzero__()
>>> print(bool(Variable(value=0)))
0
>>> print(bool(Variable(value=(0, 0, 1, 1))))
Traceback (most recent call last):
    ...
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()
__or__(other)

This test case has been added due to a weird bug that was appearing.

>>> a = Variable(value=(0, 0, 1, 1))
>>> b = Variable(value=(0, 1, 0, 1))
>>> print(numerix.equal((a == 0) | (b == 1), [True,  True, False, True]).all())
True
>>> print(a | b)
[0 1 1 1]
>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx=4)
>>> from fipy.variables.cellVariable import CellVariable
>>> a = CellVariable(value=(0, 0, 1, 1), mesh=mesh)
>>> b = CellVariable(value=(0, 1, 0, 1), mesh=mesh)
>>> print(numerix.allequal((a == 0) | (b == 1), [True,  True, False, True]))
True
>>> print(a | b)
[0 1 1 1]
__pos__()
__pow__(other)

return self**other, or self raised to power other

>>> print(Variable(1, "mol/l")**3)
1.0 mol**3/l**3
>>> print((Variable(1, "mol/l")**3).unit)
<PhysicalUnit mol**3/l**3>
__radd__(other)
__rdiv__(other)
__repr__()

Return repr(self).

__rmul__(other)
__rpow__(other)
__rsub__(other)
__rtruediv__(other)
__setitem__(index, value)
__setstate__(dict)

Used internally to create a new CellVariable from pickled persistent storage.

__str__()

Return str(self).

__sub__(other)
__truediv__(other)
all(axis=None)
>>> print(Variable(value=(0, 0, 1, 1)).all())
0
>>> print(Variable(value=(1, 1, 1, 1)).all())
1
allclose(other, rtol=1e-05, atol=1e-08)
>>> var = Variable((1, 1))
>>> print(var.allclose((1, 1)))
1
>>> print(var.allclose((1,)))
1

The following test is to check that the system does not run out of memory.

>>> from fipy.tools import numerix
>>> var = Variable(numerix.ones(10000))
>>> print(var.allclose(numerix.zeros(10000, 'l')))
False
allequal(other)
any(axis=None)
>>> print(Variable(value=0).any())
0
>>> print(Variable(value=(0, 0, 1, 1)).any())
1
property arithmeticFaceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
cacheMe(recursive=False)
property cellVolumeAverage

Return the cell-volume-weighted average of the CellVariable:

<\phi>_\text{vol}
= \frac{\sum_\text{cells} \phi_\text{cell} V_\text{cell}}
    {\sum_\text{cells} V_\text{cell}}

>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx = 3, ny = 1, dx = .5, dy = .1)
>>> var = CellVariable(value = (1, 2, 6), mesh = mesh)
>>> print(var.cellVolumeAverage)
3.0
constrain(value, where=None)

Constrains the CellVariable to value at a location specified by where.

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> v.constrain(0., where=m.facesLeft)
>>> v.faceGrad.constrain([1.], where=m.facesRight)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]

Changing the constraint changes the dependencies

>>> v.constrain(1., where=m.facesLeft)
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can be Variable

>>> c = Variable(0.)
>>> v.constrain(c, where=m.facesLeft)
>>> print(v.faceGrad)
[[ 1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> c.value = 1.
>>> print(v.faceGrad)
[[-1.  1.  1.  1.]]
>>> print(v.faceValue)
[ 1.   1.   2.   2.5]

Constraints can have a Variable mask.

>>> v = CellVariable(mesh=m)
>>> mask = FaceVariable(mesh=m, value=m.facesLeft)
>>> v.constrain(1., where=mask)
>>> print(v.faceValue)
[ 1.  0.  0.  0.]
>>> mask[:] = mask | m.facesRight
>>> print(v.faceValue)
[ 1.  0.  0.  1.]
property constraintMask

Test that constraintMask returns a Variable that updates itself whenever the constraints change.

>>> from fipy import *
>>> m = Grid2D(nx=2, ny=2)
>>> x, y = m.cellCenters
>>> v0 = CellVariable(mesh=m)
>>> v0.constrain(1., where=m.facesLeft)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False False  True False False]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  1.  0.  0.]
>>> v0.constrain(3., where=m.facesRight)
>>> print(v0.faceValue.constraintMask)
[False False False False False False  True False  True  True False  True]
>>> print(v0.faceValue)
[ 0.  0.  0.  0.  0.  0.  1.  0.  3.  1.  0.  3.]
>>> v1 = CellVariable(mesh=m)
>>> v1.constrain(1., where=(x < 1) & (y < 1))
>>> print(v1.constraintMask)
[ True False False False]
>>> print(v1)
[ 1.  0.  0.  0.]
>>> v1.constrain(3., where=(x > 1) & (y > 1))
>>> print(v1.constraintMask)
[ True False False  True]
>>> print(v1)
[ 1.  0.  0.  3.]
property constraints
copy()

Copy the value of the NoiseVariable to a static CellVariable.

dontCacheMe(recursive=False)
dot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{self} \cdot         ext{other}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

property faceGrad

Return \nabla \phi as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).

property faceGradAverage

Deprecated since version 3.3: use grad.arithmeticFaceValue() instead

Return \nabla \phi as a rank-1 FaceVariable using averaging for the normal direction(second-order gradient)

property faceValue

Returns a FaceVariable whose value corresponds to the arithmetic interpolation of the adjacent cells:

\phi_f = (\phi_1 - \phi_2) \frac{d_{f2}}{d_{12}} + \phi_2

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (0.5 / 1.) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (1.0 / 3.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.arithmeticFaceValue[mesh.interiorFaces.value]
>>> answer = (R - L) * (5.0 / 55.0) + L
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
property gaussGrad

Return \frac{1}{V_P} \sum_f \vec{n} \phi_f A_f as a rank-1 CellVariable (first-order gradient).

getsctype(default=None)

Returns the Numpy sctype of the underlying array.

>>> Variable(1).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1))
True
>>> Variable(1.).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array(1.))
True
>>> Variable((1, 1.)).getsctype() == numerix.NUMERIX.obj2sctype(numerix.array((1., 1.)))
True
property globalValue

Concatenate and return values from all processors

When running on a single processor, the result is identical to value.

property grad

Return \nabla \phi as a rank-1 CellVariable (first-order gradient).

property harmonicFaceValue

Returns a FaceVariable whose value corresponds to the harmonic interpolation of the adjacent cells:

\phi_f = \frac{\phi_1 \phi_2}{(\phi_2 - \phi_1) \frac{d_{f2}}{d_{12}} + \phi_1}

>>> from fipy.meshes import Grid1D
>>> from fipy import numerix
>>> mesh = Grid1D(dx = (1., 1.))
>>> L = 1
>>> R = 2
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (0.5 / 1.) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (2., 4.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (1.0 / 3.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
>>> mesh = Grid1D(dx = (10., 100.))
>>> var = CellVariable(mesh = mesh, value = (L, R))
>>> faceValue = var.harmonicFaceValue[mesh.interiorFaces.value]
>>> answer = L * R / ((R - L) * (5.0 / 55.0) + L)
>>> print(numerix.allclose(faceValue, answer, atol = 1e-10, rtol = 1e-10))
True
inBaseUnits()

Return the value of the Variable with all units reduced to their base SI elements.

>>> e = Variable(value="2.7 Hartree*Nav")
>>> print(e.inBaseUnits().allclose("7088849.01085 kg*m**2/s**2/mol"))
1
inUnitsOf(*units)

Returns one or more Variable objects that express the same physical quantity in different units. The units are specified by strings containing their names. The units must be compatible with the unit of the object. If one unit is specified, the return value is a single Variable.

>>> freeze = Variable('0 degC')
>>> print(freeze.inUnitsOf('degF').allclose("32.0 degF"))
1

If several units are specified, the return value is a tuple of Variable instances with with one element per unit such that the sum of all quantities in the tuple equals the the original quantity and all the values except for the last one are integers. This is used to convert to irregular unit systems like hour/minute/second. The original object will not be changed.

>>> t = Variable(value=314159., unit='s')
>>> from builtins import zip
>>> print(numerix.allclose([e.allclose(v) for (e, v) in zip(t.inUnitsOf('d', 'h', 'min', 's'),
...                                                         ['3.0 d', '15.0 h', '15.0 min', '59.0 s'])],
...                        True))
1
itemset(value)
property itemsize
property leastSquaresGrad

Return \nabla \phi, which is determined by solving for \nabla \phi in the following matrix equation,

\nabla \phi \cdot \sum_f d_{AP}^2 \vec{n}_{AP} \otimes \vec{n}_{AP} =
\sum_f d_{AP}^2 \left( \vec{n} \cdot \nabla \phi \right)_{AP}

The matrix equation is derived by minimizing the following least squares sum,

F \left( \phi_x, \phi_y \right) = \sqrt{\sum_f \left( d_{AP}
\vec{n}_{AP} \cdot \nabla \phi - d_{AP} \left( \vec{n}_{AP} \cdot
\nabla \phi \right)_{AP} \right)^2 }

Tests

>>> from fipy import Grid2D
>>> m = Grid2D(nx=2, ny=2, dx=0.1, dy=2.0)
>>> print(numerix.allclose(CellVariable(mesh=m, value=(0, 1, 3, 6)).leastSquaresGrad.globalValue, \
...                                     [[8.0, 8.0, 24.0, 24.0],
...                                      [1.2, 2.0, 1.2, 2.0]]))
True
>>> from fipy import Grid1D
>>> print(numerix.allclose(CellVariable(mesh=Grid1D(dx=(2.0, 1.0, 0.5)),
...                                     value=(0, 1, 2)).leastSquaresGrad.globalValue, [[0.461538461538, 0.8, 1.2]]))
True
property mag

The magnitude of the Variable, e.g., |\vec{\psi}| = \sqrt{\vec{\psi}\cdot\vec{\psi}}.

max(axis=None)
min(axis=None)
>>> from fipy import Grid2D, CellVariable
>>> mesh = Grid2D(nx=5, ny=5)
>>> x, y = mesh.cellCenters
>>> v = CellVariable(mesh=mesh, value=x*y)
>>> print(v.min())
0.25
property minmodFaceValue

Returns a FaceVariable with a value that is the minimum of the absolute values of the adjacent cells. If the values are of opposite sign then the result is zero:

\phi_f = \begin{cases}
               \phi_1& \text{when $|\phi_1| \le |\phi_2|$},\\
               \phi_2& \text{when $|\phi_2| < |\phi_1|$},\\
               0 & \text{when $\phi1 \phi2 < 0$}
         \end{cases}

>>> from fipy import *
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(1, 2)).minmodFaceValue)
[1 1 2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, -2)).minmodFaceValue)
[-1 -1 -2]
>>> print(CellVariable(mesh=Grid1D(nx=2), value=(-1, 2)).minmodFaceValue)
[-1  0  2]
property name
property numericValue
property old

Return the values of the CellVariable from the previous solution sweep.

Combinations of CellVariable’s should also return old values.

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 2)
>>> from fipy.variables.cellVariable import CellVariable
>>> var1 = CellVariable(mesh = mesh, value = (2, 3), hasOld = 1)
>>> var2 = CellVariable(mesh = mesh, value = (3, 4))
>>> v = var1 * var2
>>> print(v)
[ 6 12]
>>> var1.value = ((3, 2))
>>> print(v)
[9 8]
>>> print(v.old)
[ 6 12]

The following small test is to correct for a bug when the operator does not just use variables.

>>> v1 = var1 * 3
>>> print(v1)
[9 6]
>>> print(v1.old)
[6 9]
parallelRandom()
put(indices, value)
random()
property rank
ravel()
rdot(other, opShape=None, operatorClass=None)

Return the mesh-element–by–mesh-element (cell-by-cell, face-by-face, etc.) scalar product

ext{other} \cdot        ext{self}

Both self and other can be of arbitrary rank, and other does not need to be a _MeshVariable.

release(constraint)

Remove constraint from self

>>> from fipy import *
>>> m = Grid1D(nx=3)
>>> v = CellVariable(mesh=m, value=m.cellCenters[0])
>>> c = Constraint(0., where=m.facesLeft)
>>> v.constrain(c)
>>> print(v.faceValue)
[ 0.   1.   2.   2.5]
>>> v.release(constraint=c)
>>> print(v.faceValue)
[ 0.5  1.   2.   2.5]
scramble()

Generate a new random distribution.

setValue(value, unit=None, where=None)

Set the value of the Variable. Can take a masked array.

>>> a = Variable((1, 2, 3))
>>> a.setValue(5, where=(1, 0, 1))
>>> print(a)
[5 2 5]
>>> b = Variable((4, 5, 6))
>>> a.setValue(b, where=(1, 0, 1))
>>> print(a)
[4 2 6]
>>> print(b)
[4 5 6]
>>> a.value = 3
>>> print(a)
[3 3 3]
>>> b = numerix.array((3, 4, 5))
>>> a.value = b
>>> a[:] = 1
>>> print(b)
[3 4 5]
>>> a.setValue((4, 5, 6), where=(1, 0)) 
Traceback (most recent call last):
    ....
ValueError: shape mismatch: objects cannot be broadcast to a single shape
property shape
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(nx=2, ny=3)
>>> var = CellVariable(mesh=mesh)
>>> print(numerix.allequal(var.shape, (6,))) 
True
>>> print(numerix.allequal(var.arithmeticFaceValue.shape, (17,))) 
True
>>> print(numerix.allequal(var.grad.shape, (2, 6))) 
True
>>> print(numerix.allequal(var.faceGrad.shape, (2, 17))) 
True

Have to account for zero length arrays

>>> from fipy import Grid1D
>>> m = Grid1D(nx=0)
>>> v = CellVariable(mesh=m, elementshape=(2,))
>>> numerix.allequal((v * 1).shape, (2, 0))
True
std(axis=None, **kwargs)

Evaluate standard deviation of all the elements of a MeshVariable.

Adapted from http://mpitutorial.com/tutorials/mpi-reduce-and-allreduce/

>>> import fipy as fp
>>> mesh = fp.Grid2D(nx=2, ny=2, dx=2., dy=5.)
>>> var = fp.CellVariable(value=(1., 2., 3., 4.), mesh=mesh)
>>> print((var.std()**2).allclose(1.25))
True
property subscribedVariables
sum(axis=None)
take(ids, axis=0)
tostring(max_line_width=75, precision=8, suppress_small=False, separator=' ')
property unit

Return the unit object of self.

>>> Variable(value="1 m").unit
<PhysicalUnit m>
updateOld()

Set the values of the previous solution sweep to the current values.

>>> from fipy import *
>>> v = CellVariable(mesh=Grid1D(), hasOld=False)
>>> v.updateOld()
Traceback (most recent call last):
   ...
AssertionError: The updateOld method requires the CellVariable to have an old value. Set hasOld to True when instantiating the CellVariable.
property value

“Evaluate” the Variable and return its value (longhand)

>>> a = Variable(value=3)
>>> print(a.value)
3
>>> b = a + 4
>>> b
(Variable(value=array(3)) + 4)
>>> b.value
7

fipy.variables.gaussCellGradVariable module

fipy.variables.gaussianNoiseVariable module

class fipy.variables.gaussianNoiseVariable.GaussianNoiseVariable(*args, **kwds)

Bases: NoiseVariable

Represents a normal (Gaussian) distribution of random numbers with mean \mu and variance \langle \eta(\vec{r}) \eta(\vec{r}\,') \rangle = \sigma^2, which has the probability distribution

\frac{1}{\sigma\sqrt{2\pi}} \exp -\frac{(x - \mu)^2}{2\sigma^2}

For example, the variance of thermal noise that is uncorrelated in space and time is often expressed as

\left\langle
    \eta(\vec{r}, t) \eta(\vec{r}\,', t')
\right\rangle =
M k_B T \delta(\vec{r} - \vec{r}\,')\delta(t - t')

which can be obtained with:

sigmaSqrd = Mobility * kBoltzmann * Temperature / (mesh.cellVolumes * timeStep)
GaussianNoiseVariable(mesh = mesh, variance = sigmaSqrd)

Note

If the time step will change as the simulation progresses, either through use of an adaptive iterator or by making manual changes at different stages, remember to declare timeStep as a Variable and to change its value with its setValue() method.

>>> import sys
>>> from fipy.tools.numerix import *
>>> mean = 0.
>>> variance = 4.

Seed the random module for the sake of deterministic test results.

>>> from fipy import numerix
>>> numerix.random.seed(3)

We generate noise on a non-uniform Cartesian mesh with cell dimensions of x^2 and y^3.

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = arange(0.1, 5., 0.1)**2, dy = arange(0.1, 3., 0.1)**3)
>>> from fipy.variables.cellVariable import CellVariable
>>> volumes = CellVariable(mesh=mesh, value=mesh.cellVolumes)
>>> noise = GaussianNoiseVariable(mesh = mesh, mean = mean,
...                               variance = variance / volumes)

We histogram the root-volume-weighted noise distribution

>>> from fipy.variables.histogramVariable import HistogramVariable
>>> histogram = HistogramVariable(distribution = noise * sqrt(volumes),
...                               dx = 0.1, nx = 600, offset = -30)

and compare to a Gaussian distribution

>>> gauss = CellVariable(mesh = histogram.mesh)
>>> x = histogram