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distanceVariable.py
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#!/usr/bin/env python
## -*-Pyth-*-
# ###################################################################
# FiPy - Python-based finite volume PDE solver
#
# FILE: "distanceVariable.py"
#
# Author: Jonathan Guyer <guyer@nist.gov>
# Author: Daniel Wheeler <daniel.wheeler@nist.gov>
# Author: James Warren <jwarren@nist.gov>
# mail: NIST
# www: http://www.ctcms.nist.gov/fipy/
#
# ========================================================================
# This software was developed at the National Institute of Standards
# and Technology by employees of the Federal Government in the course
# of their official duties. Pursuant to title 17 Section 105 of the
# United States Code this software is not subject to copyright
# protection and is in the public domain. FiPy is an experimental
# system. NIST assumes no responsibility whatsoever for its use by
# other parties, and makes no guarantees, expressed or implied, about
# its quality, reliability, or any other characteristic. We would
# appreciate acknowledgement if the software is used.
#
# This software can be redistributed and/or modified freely
# provided that any derivative works bear some notice that they are
# derived from it, and any modified versions bear some notice that
# they have been modified.
# ========================================================================
#
# ###################################################################
##
__docformat__ = 'restructuredtext'
from fipy.tools import numerix
from fipy.tools.numerix import MA
from fipy.tools.decorators import getsetDeprecated
from fipy.variables.cellVariable import CellVariable
from fipy.tests.doctestPlus import register_skipper
import sys
import os
def module_exists(module_name):
try:
__import__(module_name)
except ImportError:
return False
else:
return True
def _checkForLSMLIB():
return module_exists('pylsmlib')
def _checkForSKFMM():
return module_exists('skfmm')
def _parseLSMSolver():
args = [s.lower() for s in sys.argv[1:]]
# any command-line specified solver takes precedence over environment variables
if '--lsmlib' in args:
if _checkForLSMLIB():
return "lsmlib"
else:
return None
elif '--skfmm' in args:
if _checkForSKFMM():
return "skfmm"
else:
return None
elif 'FIPY_LSM' in os.environ:
return os.environ['FIPY_LSM'].lower()
elif _checkForLSMLIB():
return 'lsmlib'
elif _checkForSKFMM():
return 'skfmm'
else:
return None
LSM_SOLVER = _parseLSMSolver()
register_skipper(flag="LSM",
test=lambda : LSM_SOLVER is not None,
why="neither `lsmlib` nor `skfmm` can be found on the $PATH")
register_skipper(flag="LSMLIB",
test=lambda : LSM_SOLVER == 'lsmlib',
why="`lsmlib` must be used to run some tests")
register_skipper(flag="SKFMM",
test=lambda : LSM_SOLVER == 'skfmm',
why="`skfmm` must be used to run some tests")
__all__ = ["DistanceVariable"]
class DistanceVariable(CellVariable):
r"""
A `DistanceVariable` object calculates :math:`\phi` so it satisfies,
.. math::
\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() #doctest: +LSM
>>> answer = (-1.75, -1.25, -.75, -0.25, 0.25, 0.75, 1.25, 1.75)
>>> print var.allclose(answer) #doctest: +LSM
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() #doctest: +LSM
>>> answer = numerix.arange(8) * dx - 3.5 * dx
>>> print var.allclose(answer) #doctest: +LSM
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)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., 1., 1., -1., 1.))
>>> var.calcDistanceFunction() #doctest: +LSM
>>> 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) #doctest: +LSM
1
The `extendVariable` method solves the following equation for a given
extensionVariable.
.. math::
\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() #doctest: +LSM
>>> 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)) #doctest: +LSM
1
>>> var.extendVariable(extensionVar, order=1) #doctest: +LSM
>>> print extensionVar.allclose((1.25, .5, 2, 1.25)) #doctest: +LSM
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) #doctest: +LSM
>>> 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)) #doctest: +LSM
1
>>> answer = (1.25, .5, .5, 2, 1.25, 0.9544, 2, 1.5456, 1.25)
>>> var.extendVariable(extensionVar, order=1) #doctest: +LSM
>>> print extensionVar.allclose(answer, rtol = 1e-4) #doctest: +LSM
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)
>>> var = DistanceVariable(mesh = mesh, value = (-1., 1., -1.))
>>> var.calcDistanceFunction() #doctest: +LSM
>>> print var.allclose((-0.5, 0.5, -0.5)) #doctest: +LSM
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) #doctest: +LSM
>>> 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) #doctest: +LSM
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) #doctest: +LSM
>>> var.calcDistanceFunction(order=2) #doctest: +LSM
>>> 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) #doctest: +SKFMM
True
"""
def __init__(self, mesh, name = '', value = 0., unit = None, hasOld = 0):
"""
Creates a `distanceVariable` object.
:Parameters:
- `mesh`: The mesh that defines the geometry of this variable.
- `name`: The name of the variable.
- `value`: The initial value.
- `unit`: the physical units of the variable
- `hasOld`: Whether the variable maintains an old value.
"""
CellVariable.__init__(self, mesh, name = name, value = value, unit = unit, hasOld = hasOld)
self._markStale()
def _calcValue(self):
return self._value
def extendVariable(self, extensionVariable, order=2):
"""
Calculates the extension of `extensionVariable` from the zero
level set.
:Parameters:
- `extensionVariable`: The variable to extend from the zero
level set.
"""
dx, shape = self.getLSMshape()
extensionValue = numerix.reshape(extensionVariable, shape)
phi = numerix.reshape(self._value, shape)
if LSM_SOLVER == 'lsmlib':
from pylsmlib import computeExtensionFields as extension_velocities
elif LSM_SOLVER == 'skfmm':
from skfmm import extension_velocities
else:
raise Exception, "Neither `lsmlib` nor `skfmm` can be found on the $PATH"
tmp, extensionValue = extension_velocities(phi, extensionValue, ext_mask=phi < 0., dx=dx, order=order)
extensionVariable[:] = extensionValue.flatten()
def getLSMshape(self):
mesh = self.mesh
if hasattr(mesh, 'nz'):
raise Exception, "3D meshes not yet implemented"
elif hasattr(mesh, 'ny'):
dx = (mesh.dy, mesh.dx)
shape = (mesh.ny, mesh.nx)
elif hasattr(mesh, 'nx'):
dx = (mesh.dx,)
shape = mesh.shape
else:
raise Exception, "Non grid meshes can not be used for solving the FMM."
return dx, shape
def calcDistanceFunction(self, order=2):
"""
Calculates the `distanceVariable` as a distance function.
:Parameters:
- `order`: The order of accuracy for the distance funtion
calculation, either 1 or 2.
"""
dx, shape = self.getLSMshape()
if LSM_SOLVER == 'lsmlib':
from pylsmlib import distance
elif LSM_SOLVER == 'skfmm':
from skfmm import distance
else:
raise Exception, "Neither `lsmlib` nor `skfmm` can be found on the $PATH"
self._value = distance(numerix.reshape(self._value, shape), dx=dx, order=order).flatten()
self._markFresh()
@getsetDeprecated
def getCellInterfaceAreas(self):
return self.cellInterfaceAreas
@property
def cellInterfaceAreas(self):
"""
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 circumfrence of a circle is, in fact,
:math:`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
"""
from fipy.variables.interfaceAreaVariable import _InterfaceAreaVariable
return _InterfaceAreaVariable(self)
@getsetDeprecated
def _getCellInterfaceNormals(self):
return self._cellInterfaceNormals
@property
def _cellInterfaceNormals(self):
"""
Returns the interface normals over the cells.
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> v = 1 / numerix.sqrt(2)
>>> answer = CellVariable(mesh=mesh,
... value=(((0, 0, v, 0),
... (0, 0, 0, 0),
... (0, 0, 0, 0),
... (0, v, 0, 0)),
... ((0, 0, v, 0),
... (0, 0, 0, 0),
... (0, 0, 0, 0),
... (0, v, 0, 0))))
>>> print numerix.allclose(distanceVariable._cellInterfaceNormals, answer)
True
"""
dim = self.mesh.dim
valueOverFaces = numerix.repeat(self._cellValueOverFaces[numerix.newaxis, ...], dim, axis=0)
cellFaceIDs = self.mesh.cellFaceIDs
if cellFaceIDs.shape[-1] > 0:
interfaceNormals = self._interfaceNormals[...,cellFaceIDs]
else:
interfaceNormals = 0
return MA.where(valueOverFaces < 0, 0, interfaceNormals)
@getsetDeprecated
def _getInterfaceNormals(self):
return self._interfaceNormals
@property
def _interfaceNormals(self):
"""
Returns the normals on the boundary faces only, the other are set to zero.
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.faceVariable import FaceVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> v = 1 / numerix.sqrt(2)
>>> answer = FaceVariable(mesh=mesh,
... value=((0, 0, v, 0, 0, 0, 0, v, 0, 0, 0, 0),
... (0, 0, v, 0, 0, 0, 0, v, 0, 0, 0, 0)))
>>> print numerix.allclose(distanceVariable._interfaceNormals, answer)
True
"""
M = self.mesh.dim
interfaceFlag = numerix.repeat(self._interfaceFlag[numerix.newaxis, ...], M, axis=0)
return numerix.where(interfaceFlag, self._levelSetNormals, 0)
@getsetDeprecated
def _getInterfaceFlag(self):
return self._interfaceFlag
@property
def _interfaceFlag(self):
"""
Returns 1 for faces on boundary and 0 otherwise.
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.faceVariable import FaceVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> answer = FaceVariable(mesh=mesh,
... value=(0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0))
>>> print numerix.allclose(distanceVariable._interfaceFlag, answer)
True
"""
adjacentCellIDs = self.mesh._adjacentCellIDs
val0 = numerix.take(numerix.array(self._value), adjacentCellIDs[0])
val1 = numerix.take(numerix.array(self._value), adjacentCellIDs[1])
return numerix.where(val1 * val0 < 0, 1, 0)
@getsetDeprecated
def _getCellInterfaceFlag(self):
return self._cellInterfaceFlag
@property
def _cellInterfaceFlag(self):
"""
Returns 1 for those cells on the interface:
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> answer = CellVariable(mesh=mesh, value=(0, 1, 1, 0))
>>> print numerix.allclose(distanceVariable._cellInterfaceFlag, answer)
True
"""
from fipy.variables.interfaceFlagVariable import _InterfaceFlagVariable
return _InterfaceFlagVariable(self)
@getsetDeprecated
def _getCellValueOverFaces(self):
return self._cellValueOverFaces
@property
def _cellValueOverFaces(self):
"""
Returns the cells values at the faces.
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.cellVariable import CellVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> answer = CellVariable(mesh=mesh,
... value=((-.5, .5, .5, 1.5),
... (-.5, .5, .5, 1.5),
... (-.5, .5, .5, 1.5),
... (-.5, .5, .5, 1.5)))
>>> print numerix.allclose(distanceVariable._cellValueOverFaces, answer)
True
"""
M = self.mesh._maxFacesPerCell
N = self.mesh.numberOfCells
return numerix.reshape(numerix.repeat(numerix.array(self._value)[numerix.newaxis, ...], M, axis=0), (M, N))
@getsetDeprecated
def _getLevelSetNormals(self):
return self._levelSetNormals
@property
def _levelSetNormals(self):
"""
Return the face level set normals.
>>> from fipy.meshes import Grid2D
>>> from fipy.variables.faceVariable import FaceVariable
>>> mesh = Grid2D(dx = .5, dy = .5, nx = 2, ny = 2)
>>> distanceVariable = DistanceVariable(mesh = mesh,
... value = (-0.5, 0.5, 0.5, 1.5))
>>> v = 1 / numerix.sqrt(2)
>>> answer = FaceVariable(mesh=mesh,
... value=((0, 0, v, v, 0, 0, 0, v, 0, 0, v, 0),
... (0, 0, v, v, 0, 0, 0, v, 0, 0, v, 0)))
>>> print numerix.allclose(distanceVariable._levelSetNormals, answer)
True
"""
faceGrad = self.grad.arithmeticFaceValue
faceGradMag = numerix.array(faceGrad.mag)
faceGradMag = numerix.where(faceGradMag > 1e-10,
faceGradMag,
1e-10)
faceGrad = numerix.array(faceGrad)
## set faceGrad zero on exteriorFaces
exteriorFaces = self.mesh.exteriorFaces
if len(exteriorFaces.value) > 0:
faceGrad[..., exteriorFaces.value] = 0.
return faceGrad / faceGradMag
def _test():
import fipy.tests.doctestPlus
return fipy.tests.doctestPlus.testmod()
if __name__ == "__main__":
_test()