def MagneticDipoleVectorPotential(srcLoc, obsLoc, component, moment=1.,
orientation=np.r_[0., 0., 1.],
mu=mu_0):
"""
Calculate the vector potential of a set of magnetic dipoles
at given locations 'ref. <http://en.wikipedia.org/wiki/Dipole#Magnetic_vector_potential>'
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray,SimPEG.Mesh obsLoc: Where the potentials will be
calculated (x, y, z) or a
SimPEG Mesh
:param str,list component: The component to calculate - 'x', 'y', or
'z' if an array, or grid type if mesh, can
be a list
:param numpy.ndarray orientation: The vector dipole moment
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
# TODO: break this out!
if isinstance(orientation, str):
orientation = orientationDict[orientation]
assert np.linalg.norm(np.array(orientation), 2) == 1., ("orientation must "
"be a unit vector")
if type(component) in [list, tuple]:
out = range(len(component))
for i, comp in enumerate(component):
out[i] = MagneticDipoleVectorPotential(srcLoc, obsLoc, comp,
orientation=orientation,
mu=mu)
return np.concatenate(out)
if isinstance(obsLoc, Mesh.BaseMesh):
mesh = obsLoc
assert component in ['Ex', 'Ey', 'Ez', 'Fx', 'Fy', 'Fz'], ("Components"
"must be in: ['Ex','Ey','Ez','Fx','Fy','Fz']")
return MagneticDipoleVectorPotential(srcLoc, getattr(mesh, 'grid' +
component),
component[1],
orientation=orientation)
if component == 'x':
dimInd = 0
elif component == 'y':
dimInd = 1
elif component == 'z':
dimInd = 2
else:
raise ValueError('Invalid component')
srcLoc = np.atleast_2d(srcLoc)
obsLoc = np.atleast_2d(obsLoc)
orientation = np.atleast_2d(orientation)
nObs = obsLoc.shape[0]
nSrc = srcLoc.shape[0]
m = moment*np.array(orientation).repeat(nObs, axis=0)
A = np.empty((nObs, nSrc))
for i in range(nSrc):
dR = obsLoc - srcLoc[i, np.newaxis].repeat(nObs, axis=0)
mCr = np.cross(m, dR)
r = np.sqrt((dR**2).sum(axis=1))
A[:, i] = +(mu/(4*np.pi)) * mCr[:, dimInd]/(r**3)
if nSrc == 1:
return A.flatten()
return A
def MagneticDipoleFields(srcLoc, obsLoc, component,
orientation='Z', moment=1., mu=mu_0):
"""
Calculate the vector potential of a set of magnetic dipoles
at given locations 'ref. <http://en.wikipedia.org/wiki/Dipole#Magnetic_vector_potential>'
.. math::
B = \frac{\mu_0}{4 \pi r^3} \left( \frac{3 \vec{r} (\vec{m} \cdot
\vec{r})}{r^2})
- \vec{m}
\right) \cdot{\hat{rx}}
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray obsLoc: Where the potentials will be calculated
(x, y, z)
:param str component: The component to calculate - 'x', 'y', or 'z'
:param numpy.ndarray moment: The vector dipole moment (vertical)
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
if isinstance(orientation, str):
assert orientation.upper() in ['X', 'Y', 'Z'], ("orientation must be 'x', "
"'y', or 'z' or a vector"
"not {}".format(orientation)
)
elif (not np.allclose(np.r_[1., 0., 0.], orientation) or
not np.allclose(np.r_[0., 1., 0.], orientation) or
not np.allclose(np.r_[0., 0., 1.], orientation)):
warnings.warn('Arbitrary trasnmitter orientations ({}) not thouroughly tested '
'Pull request on a test anyone? bueller?').format(orientation)
if isinstance(component, str):
assert component.upper() in ['X', 'Y', 'Z'], ("component must be 'x', "
"'y', or 'z' or a vector"
"not {}".format(component)
)
elif (not np.allclose(np.r_[1., 0., 0.], component) or
not np.allclose(np.r_[0., 1., 0.], component) or
not np.allclose(np.r_[0., 0., 1.], component)):
warnings.warn('Arbitrary receiver orientations ({}) not thouroughly tested '
'Pull request on a test anyone? bueller?').format(component)
if isinstance(orientation, str):
orientation = orientationDict[orientation.upper()]
if isinstance(component, str):
component = orientationDict[component.upper()]
assert np.linalg.norm(orientation, 2) == 1., ('orientation must be a unit '
'vector. Use "moment=X to '
'scale source fields')
if np.linalg.norm(component, 2) != 1.:
warnings.warn('The magnitude of the receiver component vector is > 1, '
' it is {}. The receiver fields will be scaled.'
).format(np.linalg.norm(component, 2))
srcLoc = np.atleast_2d(srcLoc)
component = np.atleast_2d(component)
obsLoc = np.atleast_2d(obsLoc)
orientation = np.atleast_2d(orientation)
nObs = obsLoc.shape[0]
nSrc = int(srcLoc.size / 3.)
# use outer product to construct an array of [x_src, y_src, z_src]
m = moment*orientation.repeat(nObs, axis=0)
B = []
for i in range(nSrc):
srcLoc = srcLoc[i, np.newaxis].repeat(nObs, axis=0)
rx = component.repeat(nObs, axis=0)
dR = obsLoc - srcLoc
r = np.sqrt((dR**2).sum(axis=1))
# mult each element and sum along the axis (vector dot product)
m_dot_dR_div_r2 = (m * dR).sum(axis=1) / (r**2)
# multiply the scalar m_dot_dR by the 3D vector r
rvec_m_dot_dR_div_r2 = np.vstack([m_dot_dR_div_r2 * dR[:, i] for
i in range(3)]).T
inside = (3. * rvec_m_dot_dR_div_r2) - m
# dot product with rx orientation
inside_dot_rx = (inside * rx).sum(axis=1)
front = (mu/(4.* np.pi * r**3))
B.append(Utils.mkvc(front * inside_dot_rx))
return np.vstack(B).T
def MagneticLoopVectorPotential(srcLoc, obsLoc, component, radius, orientation='Z', mu=mu_0):
"""
Calculate the vector potential of horizontal circular loop
at given locations
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray,SimPEG.Mesh obsLoc: Where the potentials will be calculated (x, y, z) or a SimPEG Mesh
:param str,list component: The component to calculate - 'x', 'y', or 'z' if an array, or grid type if mesh, can be a list
:param numpy.ndarray I: Input current of the loop
:param numpy.ndarray radius: radius of the loop
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
if isinstance(orientation, str):
if orientation.upper() != 'Z':
raise NotImplementedError, 'Only Z oriented loops implemented'
elif not np.allclose(orientation, np.r_[0., 0., 1.]):
raise NotImplementedError, 'Only Z oriented loops implemented'
if type(component) in [list, tuple]:
out = range(len(component))
for i, comp in enumerate(component):
out[i] = MagneticLoopVectorPotential(srcLoc, obsLoc, comp, radius,
orientation, mu)
return np.concatenate(out)
if isinstance(obsLoc, Mesh.BaseMesh):
mesh = obsLoc
assert component in ['Ex','Ey','Ez','Fx','Fy','Fz'], "Components must be in: ['Ex','Ey','Ez','Fx','Fy','Fz']"
return MagneticLoopVectorPotential(srcLoc, getattr(mesh,'grid'+component), component[1], radius, mu)
srcLoc = np.atleast_2d(srcLoc)
obsLoc = np.atleast_2d(obsLoc)
n = obsLoc.shape[0]
nSrc = srcLoc.shape[0]
if component=='z':
A = np.zeros((n, nSrc))
if nSrc ==1:
return A.flatten()
return A
else:
A = np.zeros((n, nSrc))
for i in range (nSrc):
x = obsLoc[:, 0] - srcLoc[i, 0]
y = obsLoc[:, 1] - srcLoc[i, 1]
z = obsLoc[:, 2] - srcLoc[i, 2]
r = np.sqrt(x**2 + y**2)
m = (4 * radius * r) / ((radius + r)**2 + z**2)
m[m > 1.] = 1.
# m might be slightly larger than 1 due to rounding errors
# but ellipke requires 0 <= m <= 1
K = ellipk(m)
E = ellipe(m)
ind = (r > 0) & (m < 1)
# % 1/r singular at r = 0 and K(m) singular at m = 1
Aphi = np.zeros(n)
# % Common factor is (mu * I) / pi with I = 1 and mu = 4e-7 * pi.
Aphi[ind] = ((mu / (np.pi * np.sqrt(m[ind])) *
np.sqrt(radius / r[ind]) *((1. - m[ind] / 2.) *
K[ind] - E[ind])))
if component == 'x':
A[ind, i] = Aphi[ind] * (-y[ind] / r[ind] )
elif component == 'y':
A[ind, i] = Aphi[ind] * ( x[ind] / r[ind] )
else:
raise ValueError('Invalid component')
if nSrc == 1:
return A.flatten()
return A
def MagneticDipoleVectorPotential(srcLoc, obsLoc, component, moment=1.,
orientation=np.r_[0., 0., 1.],
mu=mu_0):
"""
Calculate the vector potential of a set of magnetic dipoles
at given locations 'ref. <http://en.wikipedia.org/wiki/Dipole#Magnetic_vector_potential>'
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray,SimPEG.Mesh obsLoc: Where the potentials will be
calculated (x, y, z) or a
SimPEG Mesh
:param str,list component: The component to calculate - 'x', 'y', or
'z' if an array, or grid type if mesh, can
be a list
:param numpy.ndarray orientation: The vector dipole moment
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
# TODO: break this out!
if isinstance(orientation, str):
orientation = orientationDict[orientation]
assert np.linalg.norm(np.array(orientation), 2) == 1., ("orientation must "
"be a unit vector")
if type(component) in [list, tuple]:
out = list(range(len(component)))
for i, comp in enumerate(component):
out[i] = MagneticDipoleVectorPotential(srcLoc, obsLoc, comp,
orientation=orientation,
mu=mu)
return np.concatenate(out)
if isinstance(obsLoc, Mesh.BaseMesh):
mesh = obsLoc
assert component in ['Ex', 'Ey', 'Ez', 'Fx', 'Fy', 'Fz'], ("Components"
"must be in: ['Ex','Ey','Ez','Fx','Fy','Fz']")
return MagneticDipoleVectorPotential(srcLoc, getattr(mesh, 'grid' +
component),
component[1],
orientation=orientation)
if component == 'x':
dimInd = 0
elif component == 'y':
dimInd = 1
elif component == 'z':
dimInd = 2
else:
raise ValueError('Invalid component')
srcLoc = np.atleast_2d(srcLoc)
obsLoc = np.atleast_2d(obsLoc)
orientation = np.atleast_2d(orientation)
nObs = obsLoc.shape[0]
nSrc = srcLoc.shape[0]
m = moment*np.array(orientation).repeat(nObs, axis=0)
A = np.empty((nObs, nSrc))
for i in range(nSrc):
dR = obsLoc - srcLoc[i, np.newaxis].repeat(nObs, axis=0)
mCr = np.cross(m, dR)
r = np.sqrt((dR**2).sum(axis=1))
A[:, i] = +(mu/(4*np.pi)) * mCr[:, dimInd]/(r**3)
if nSrc == 1:
return A.flatten()
return A
def MagneticDipoleFields(srcLoc, obsLoc, component,
orientation='Z', moment=1., mu=mu_0):
"""
Calculate the vector potential of a set of magnetic dipoles
at given locations 'ref. <http://en.wikipedia.org/wiki/Dipole#Magnetic_vector_potential>'
.. math::
B = \frac{\mu_0}{4 \pi r^3} \left( \frac{3 \vec{r} (\vec{m} \cdot
\vec{r})}{r^2})
- \vec{m}
\right) \cdot{\hat{rx}}
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray obsLoc: Where the potentials will be calculated
(x, y, z)
:param str component: The component to calculate - 'x', 'y', or 'z'
:param numpy.ndarray moment: The vector dipole moment (vertical)
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
if isinstance(orientation, str):
assert orientation.upper() in ['X', 'Y', 'Z'], ("orientation must be 'x', "
"'y', or 'z' or a vector"
"not {}".format(orientation)
)
elif (not np.allclose(np.r_[1., 0., 0.], orientation) or
not np.allclose(np.r_[0., 1., 0.], orientation) or
not np.allclose(np.r_[0., 0., 1.], orientation)):
warnings.warn('Arbitrary trasnmitter orientations ({}) not thouroughly tested '
'Pull request on a test anyone? bueller?').format(orientation)
if isinstance(component, str):
assert component.upper() in ['X', 'Y', 'Z'], ("component must be 'x', "
"'y', or 'z' or a vector"
"not {}".format(component)
)
elif (not np.allclose(np.r_[1., 0., 0.], component) or
not np.allclose(np.r_[0., 1., 0.], component) or
not np.allclose(np.r_[0., 0., 1.], component)):
warnings.warn('Arbitrary receiver orientations ({}) not thouroughly tested '
'Pull request on a test anyone? bueller?').format(component)
if isinstance(orientation, str):
orientation = orientationDict[orientation.upper()]
if isinstance(component, str):
component = orientationDict[component.upper()]
assert np.linalg.norm(orientation, 2) == 1., ('orientation must be a unit '
'vector. Use "moment=X to '
'scale source fields')
if np.linalg.norm(component, 2) != 1.:
warnings.warn('The magnitude of the receiver component vector is > 1, '
' it is {}. The receiver fields will be scaled.'
).format(np.linalg.norm(component, 2))
srcLoc = np.atleast_2d(srcLoc)
component = np.atleast_2d(component)
obsLoc = np.atleast_2d(obsLoc)
orientation = np.atleast_2d(orientation)
nObs = obsLoc.shape[0]
nSrc = int(srcLoc.size / 3.)
# use outer product to construct an array of [x_src, y_src, z_src]
m = moment*orientation.repeat(nObs, axis=0)
B = []
for i in range(nSrc):
srcLoc = srcLoc[i, np.newaxis].repeat(nObs, axis=0)
rx = component.repeat(nObs, axis=0)
dR = obsLoc - srcLoc
r = np.sqrt((dR**2).sum(axis=1))
# mult each element and sum along the axis (vector dot product)
m_dot_dR_div_r2 = (m * dR).sum(axis=1) / (r**2)
# multiply the scalar m_dot_dR by the 3D vector r
rvec_m_dot_dR_div_r2 = np.vstack([m_dot_dR_div_r2 * dR[:, i] for
i in range(3)]).T
inside = (3. * rvec_m_dot_dR_div_r2) - m
# dot product with rx orientation
inside_dot_rx = (inside * rx).sum(axis=1)
front = (mu/(4.* np.pi * r**3))
B.append(Utils.mkvc(front * inside_dot_rx))
return np.vstack(B).T
def MagneticLoopVectorPotential(srcLoc, obsLoc, component, radius, orientation='Z', mu=mu_0):
"""
Calculate the vector potential of horizontal circular loop
at given locations
:param numpy.ndarray srcLoc: Location of the source(s) (x, y, z)
:param numpy.ndarray,SimPEG.Mesh obsLoc: Where the potentials will be calculated (x, y, z) or a SimPEG Mesh
:param str,list component: The component to calculate - 'x', 'y', or 'z' if an array, or grid type if mesh, can be a list
:param numpy.ndarray I: Input current of the loop
:param numpy.ndarray radius: radius of the loop
:rtype: numpy.ndarray
:return: The vector potential each dipole at each observation location
"""
if isinstance(orientation, str):
if orientation.upper() != 'Z':
raise NotImplementedError('Only Z oriented loops implemented')
elif not np.allclose(orientation, np.r_[0., 0., 1.]):
raise NotImplementedError('Only Z oriented loops implemented')
if type(component) in [list, tuple]:
out = list(range(len(component)))
for i, comp in enumerate(component):
out[i] = MagneticLoopVectorPotential(srcLoc, obsLoc, comp, radius,
orientation, mu)
return np.concatenate(out)
if isinstance(obsLoc, Mesh.BaseMesh):
mesh = obsLoc
assert component in ['Ex','Ey','Ez','Fx','Fy','Fz'], "Components must be in: ['Ex','Ey','Ez','Fx','Fy','Fz']"
return MagneticLoopVectorPotential(srcLoc, getattr(mesh,'grid'+component), component[1], radius, mu)
srcLoc = np.atleast_2d(srcLoc)
obsLoc = np.atleast_2d(obsLoc)
n = obsLoc.shape[0]
nSrc = srcLoc.shape[0]
if component=='z':
A = np.zeros((n, nSrc))
if nSrc ==1:
return A.flatten()
return A
else:
A = np.zeros((n, nSrc))
for i in range (nSrc):
x = obsLoc[:, 0] - srcLoc[i, 0]
y = obsLoc[:, 1] - srcLoc[i, 1]
z = obsLoc[:, 2] - srcLoc[i, 2]
r = np.sqrt(x**2 + y**2)
m = (4 * radius * r) / ((radius + r)**2 + z**2)
m[m > 1.] = 1.
# m might be slightly larger than 1 due to rounding errors
# but ellipke requires 0 <= m <= 1
K = ellipk(m)
E = ellipe(m)
ind = (r > 0) & (m < 1)
# % 1/r singular at r = 0 and K(m) singular at m = 1
Aphi = np.zeros(n)
# % Common factor is (mu * I) / pi with I = 1 and mu = 4e-7 * pi.
Aphi[ind] = ((mu / (np.pi * np.sqrt(m[ind])) *
np.sqrt(radius / r[ind]) *((1. - m[ind] / 2.) *
K[ind] - E[ind])))
if component == 'x':
A[ind, i] = Aphi[ind] * (-y[ind] / r[ind] )
elif component == 'y':
A[ind, i] = Aphi[ind] * ( x[ind] / r[ind] )
else:
raise ValueError('Invalid component')
if nSrc == 1:
return A.flatten()
return A