/usr/lib/python2.7/dist-packages/Scientific/Geometry/TensorAnalysis.py is in python-scientific 2.9.4-1.
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#
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 2007-6-12
#
"""
Vector and tensor fields with derivatives
"""
from Scientific import N; Numeric = N
from Scientific.Geometry import Vector, Tensor
from Scientific.indexing import index_expression
from Scientific.Functions import Interpolation
#
# General tensor field base class
#
class TensorField(Interpolation.InterpolatingFunction):
"""Tensor field of arbitrary rank
A tensor field is described by a tensor at each point of
a three-dimensional rectangular grid. The grid spacing must
be uniform. Tensor fields are implemented as a subclass
of InterpolatingFunction from the module
Scientific.Functions.Interpolation and thus share all methods
defined in that class.
Evaluation:
- 'tensorfield(x, y, z)' (three coordinates)
- 'tensorfield(coordinates)' (sequence containing three coordinates)
"""
def __init__(self, rank, axes, values, default = None,
period = (None, None, None), check = True):
"""
@param rank: the tensor rank
@type rank: C{int}
@param axes: three arrays specifying the axis ticks for the three
axes
@type axes: sequence of C{Numeric.array} of rank 1
@param values: an array containing the field values. Its first
three dimensions correspond to the x, y, z
directions and must have lengths compatible with
the axis arrays. The remaining dimensions must
have length 3.
@type values: C{Numeric.array} of rank+3 dimensions
@param default: the value of the field for points outside the grid.
A value of 'None' means that an exception will be
raised for an attempt to evaluate the field outside
the grid. Any other value must a tensor of the
correct rank.
@type default: L{Scientific.Geometry.Tensor} or C{NoneType}
@param period: the period for each of the variables, or C{None} for
variables in which the function is not periodic.
@type period: sequence of length three
@raise ValueError: if the arguments are not consistent
"""
if check:
if len(axes) != 3:
raise ValueError('Field must have three axes')
if len(period) != 3:
raise ValueError('Field requires three period values')
if len(values.shape) != 3 + rank:
raise ValueError('Values must have rank ' + `rank`)
if values.shape[3:] != rank*(3,):
raise ValueError('Values must have dimension 3')
self.rank = rank
self.spacing = []
for axis in axes:
d = axis[1:]-axis[:-1]
self.spacing.append(d[0])
if check:
dmin = Numeric.minimum.reduce(d)
if abs(dmin-Numeric.maximum.reduce(d)) > 0.0001*dmin:
raise ValueError('Grid must be equidistant')
Interpolation.InterpolatingFunction.__init__(self, axes, values,
default, period)
def __call__(self, *points):
if len(points) == 1:
points = tuple(points[0])
value = apply(Interpolation.InterpolatingFunction.__call__,
(self, ) + points)
if self.rank == 0:
return value
elif self.rank == 1:
return Vector(value)
else:
return Tensor(value)
def __getitem__(self, index):
if isinstance(index, int):
index = (index,)
rank = self.rank - len(index)
if rank < 0:
raise ValueError('Number of indices too large')
index = index_expression[...] + index + rank*index_expression[::]
try: default = self.default[index]
except TypeError: default = None
if rank == 0:
return ScalarField(self.axes, self.values[index], default,
self.period, False)
elif rank == 1:
return VectorField(self.axes, self.values[index], default,
self.period, False)
else:
return TensorField(self.axes, rank, self.values[index], default,
self.period, False)
def zero(self):
"""
@returns: a tensor of the same rank as the field values
with all elements equal to zero
@rtype: L{Scientifc.Geometry.Tensor}
"""
if self.rank == 0:
return 0.
else:
return Tensor(Numeric.zeros(self.rank*(3,), Numeric.Float))
def derivative(self, variable):
"""
@param variable: 0 for x, 1 for y, 2 for z
@type variable: C{int}
@returns: the derivative with respect to variable
@rtype: L{TensorField}
"""
period = self.period[variable]
if period is None:
ui = variable*index_expression[::] + \
index_expression[2::] + index_expression[...]
li = variable*index_expression[::] + \
index_expression[:-2:] + index_expression[...]
d_values = 0.5*(self.values[ui]-self.values[li]) \
/self.spacing[variable]
diffaxis = self.axes[variable][1:-1]
d_axes = self.axes[:variable]+[diffaxis]+self.axes[variable+1:]
else:
u = N.take(self.values, range(2, len(self.axes[variable]))+[0, 1],
axis=variable)
l = self.values
d_values = (u-l)/self.spacing[variable]
d_axes = self.axes
d_default = None
if self.default is not None:
d_default = Numeric.zeros(self.rank*(3,), Numeric.Float)
return self._constructor(d_axes, d_values, d_default,
self.period, False)
def allDerivatives(self):
"""
@returns: all three derivatives (x, y, z) on equal-sized grids
@rtype: (L{TensorField}, L{TensorField}, L{TensorField})
"""
x = self.derivative(0)
y = self.derivative(1)
z = self.derivative(2)
if self.period[0] is None:
y._reduceAxis(0)
z._reduceAxis(0)
if self.period[1] is None:
x._reduceAxis(1)
z._reduceAxis(1)
if self.period[2] is None:
x._reduceAxis(2)
y._reduceAxis(2)
return x, y, z
def _reduceAxis(self, variable):
self.axes[variable] = self.axes[variable][1:-1]
i = variable*index_expression[::] + \
index_expression[1:-1:] + index_expression[...]
self.values = self.values[i]
def _checkCompatibility(self, other):
if self.period != other.period:
raise ValueError("Periods don't match")
def __add__(self, other):
self._checkCompatibility(other)
if self.default is None or other.default is None:
default = None
else:
default = self.default + other.default
return self._constructor(self.axes, self.values+other.values,
default, self.period, False)
def __sub__(self, other):
self._checkCompatibility(other)
if self.default is None or other.default is None:
default = None
else:
default = self.default - other.default
return self._constructor(self.axes, self.values-other.values,
default, self.period, False)
#
# Scalar field class definition
#
class ScalarField(TensorField):
"""Scalar field (tensor field of rank 0)
A subclass of TensorField.
"""
def __init__(self, axes, values, default = None,
period = (None, None, None), check = True):
"""
@param axes: three arrays specifying the axis ticks for the three
axes
@type axes: sequence of C{Numeric.array} of rank 1
@param values: an array containing the field values. The
three dimensions correspond to the x, y, z
directions and must have lengths compatible with
the axis arrays.
@type values: C{Numeric.array} of 3 dimensions
@param default: the value of the field for points outside the grid.
A value of 'None' means that an exception will be
raised for an attempt to evaluate the field outside
the grid. Any other value must a tensor of the
correct rank.
@type default: number or C{NoneType}
@param period: the period for each of the variables, or C{None} for
variables in which the function is not periodic.
@type period: sequence of length three
@raise ValueError: if the arguments are not consistent
"""
TensorField.__init__(self, 0, axes, values, default, period, check)
def gradient(self):
"""
@returns: the gradient
@rtype: L{VectorField}
"""
x, y, z = self.allDerivatives()
grad = Numeric.transpose(Numeric.array([x.values, y.values, z.values]),
[1,2,3,0])
if self.default is None:
default = None
else:
default = Numeric.zeros((3,), Numeric.Float)
return VectorField(x.axes, grad, default, self.period, False)
def laplacian(self):
"""
@returns: the laplacian (gradient of divergence)
@rtype L{ScalarField}
"""
return self.gradient().divergence()
ScalarField._constructor = ScalarField
#
# Vector field class definition
#
class VectorField(TensorField):
"""Vector field (tensor field of rank 1)
A subclass of TensorField.
"""
def __init__(self, axes, values, default = None,
period = (None, None, None), check = True):
"""
@param axes: three arrays specifying the axis ticks for the three
axes
@type axes: sequence of C{Numeric.array} of rank 1
@param values: an array containing the field values. Its first
three dimensions correspond to the x, y, z
directions and must have lengths compatible with
the axis arrays. The fourth dimension must
have length 3.
@type values: C{Numeric.array} of four dimensions
@param default: the value of the field for points outside the grid.
A value of 'None' means that an exception will be
raised for an attempt to evaluate the field outside
the grid. Any other value must a vector
@type default: L{Scientific.Geometry.Vector} or C{NoneType}
@param period: the period for each of the variables, or C{None} for
variables in which the function is not periodic.
@type period: sequence of length three
@raise ValueError: if the arguments are not consistent
"""
TensorField.__init__(self, 1, axes, values, default, period, check)
def zero(self):
return Vector(0., 0., 0.)
def _divergence(self, x, y, z):
return x[0] + y[1] + z[2]
def _curl(self, x, y, z):
curl_x = y.values[..., 2] - z.values[..., 1]
curl_y = z.values[..., 0] - x.values[..., 2]
curl_z = x.values[..., 1] - y.values[..., 0]
curl = Numeric.transpose(Numeric.array([curl_x, curl_y, curl_z]),
[1,2,3,0])
if self.default is None:
default = None
else:
default = Numeric.zeros((3,), Numeric.Float)
return VectorField(x.axes, curl, default, self.period, False)
def _strain(self, x, y, z):
strain = Numeric.transpose(Numeric.array([x.values, y.values,
z.values]), [1,2,3,0,4])
strain = 0.5*(strain+Numeric.transpose(strain, [0,1,2,4,3]))
trace = (strain[..., 0,0] + strain[..., 1,1] + strain[..., 2,2])/3.
strain = strain - trace[..., Numeric.NewAxis, Numeric.NewAxis] * \
Numeric.identity(3)[Numeric.NewAxis, Numeric.NewAxis,
Numeric.NewAxis, :, :]
if self.default is None:
default = None
else:
default = Numeric.zeros((3, 3), Numeric.Float)
return TensorField(2, x.axes, strain, default, self.period, False)
def divergence(self):
"""
@returns: the divergence
@rtype L{ScalarField}
"""
x, y, z = self.allDerivatives()
return self._divergence(x, y, z)
def curl(self):
"""
@returns: the curl
@rtype L{VectorField}
"""
x, y, z = self.allDerivatives()
return self._curl(x, y, z)
def strain(self):
"""
@returns: the strain
@rtype L{TensorField} of rank 2
"""
x, y, z = self.allDerivatives()
return self._strain(x, y, z)
def divergenceCurlAndStrain(self):
"""
@returns: all derivative fields: divergence, curl, and strain
@rtype (L{ScalarField}, L{VectorField}, L{TensorField})
"""
x, y, z = self.allDerivatives()
return self._divergence(x, y, z), self._curl(x, y, z), \
self._strain(x, y, z)
def laplacian(self):
"""
@returns: the laplacian
@rtype L{VectorField}
"""
x, y, z = self.allDerivatives()
return self._divergence(x, y, z).gradient()-self._curl(x, y, z).curl()
def length(self):
"""
@returns: a scalar field corresponding to the length (norm) of
the vector field.
@rtype: L{ScalarField}
"""
l = Numeric.sqrt(Numeric.add.reduce(self.values**2, -1))
if self.default is None:
default = None
else:
try:
default = Numeric.sqrt(Numeric.add.reduce(self.default))
except ValueError:
default = None
return ScalarField(self.axes, l, default, self.period, False)
VectorField._constructor = VectorField
#
# Test code
#
if __name__ == '__main__':
from Numeric import *
axis = arange(0., 1., 0.1)
values = zeros((10,10,10,3), Float)
zero = VectorField(3*(axis,), values)
div = zero.divergence()
curl = zero.curl()
strain = zero.strain()
zero = VectorField(3*(axis,), values, period=(1.1, 1.1, 1.1))
div = zero.divergence()
curl = zero.curl()
strain = zero.strain()
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