def getP(self, mesh, Gloc):
if mesh in self._Ps:
return self._Ps[mesh]
# Find indices for pole receivers
inds_dipole = (np.linalg.norm(self.locs[0] - self.locs[1], axis=1) >
self.threshold)
P0 = mesh.getInterpolationMat(self.locs[0][inds_dipole], Gloc)
P1 = mesh.getInterpolationMat(self.locs[1][inds_dipole], Gloc)
P = P0 - P1
# Generate interpolation matrix for pole receivers
if ~np.alltrue(inds_dipole):
P0_pole = mesh.getInterpolationMat(self.locs[0][~inds_dipole],
Gloc)
P = sp.vstack((P, P0_pole))
if self.data_type == 'apparent_resistivity':
P = sdiag(1. / self.geometric_factor) * P
elif self.data_type == 'apparent_chargeability':
P = sdiag(1. / self.dc_voltage) * P
if self.storeProjections:
self._Ps[mesh] = P
return P
def edgeCurl(self):
"""The edgeCurl property."""
if self.nCy > 1:
raise NotImplementedError(
'Edge curl not yet implemented for nCy > 1')
if getattr(self, '_edgeCurl', None) is None:
#1D Difference matricies
dr = sp.spdiags((np.ones((self.nCx + 1, 1)) * [-1, 1]).T, [-1, 0],
self.nCx,
self.nCx,
format="csr")
dz = sp.spdiags((np.ones((self.nCz + 1, 1)) * [-1, 1]).T, [0, 1],
self.nCz,
self.nCz + 1,
format="csr")
#2D Difference matricies
Dr = sp.kron(sp.identity(self.nNz), dr)
Dz = -sp.kron(dz, sp.identity(self.nCx))
A = self.area
E = self.edge
#Edge curl operator
self._edgeCurl = sdiag(1 / A) * sp.vstack((Dz, Dr)) * sdiag(E)
return self._edgeCurl
def faceDivx(self):
"""Construct divergence operator in the x component (face-stg to cell-centres)."""
if getattr(self, '_faceDivx', None) is None:
D1 = kron3(speye(self.nCz), speye(self.nCy), ddx(self.nCx)[:, 1:])
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
self._faceDivx = sdiag(1 / V) * D1 * sdiag(S)
return self._faceDivx
def faceDivz(self):
"""Construct divergence operator in the z component (face-stg to cell-centres)."""
if getattr(self, '_faceDivz', None) is None:
D3 = kron3(ddx(self.nCz), speye(self.nCy), speye(self.nCx))
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1 / V) * D3 * sdiag(S)
return self._faceDivz
def faceDivx(self):
"""Construct divergence operator in the x component (face-stg to cell-centres)."""
if getattr(self, '_faceDivx', None) is None:
D1 = kron3(speye(self.nCz), speye(self.nCy), ddx(self.nCx)[:,1:])
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
self._faceDivx = sdiag(1/V)*D1*sdiag(S)
return self._faceDivx
def faceDivz(self):
"""Construct divergence operator in the z component (face-stg to cell-centres)."""
if getattr(self, '_faceDivz', None) is None:
D3 = kron3(ddx(self.nCz), speye(self.nCy), speye(self.nCx))
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1/V)*D3*sdiag(S)
return self._faceDivz
def faceDivy(self):
"""Construct divergence operator in the y component (face-stg to cell-centres)."""
raise NotImplementedError('Wrapping the ddx is not yet implemented.')
if getattr(self, '_faceDivy', None) is None:
# TODO: this needs to wrap to join up faces which are connected in the cylinder
D2 = kron3(speye(self.nCz), ddx(self.nCy), speye(self.nCx))
S = self.r(self.area, 'F', 'Fy', 'V')
V = self.vol
self._faceDivy = sdiag(1 / V) * D2 * sdiag(S)
return self._faceDivy
def faceDivy(self):
"""Construct divergence operator in the y component (face-stg to cell-centres)."""
raise NotImplementedError('Wrapping the ddx is not yet implemented.')
if getattr(self, '_faceDivy', None) is None:
# TODO: this needs to wrap to join up faces which are connected in the cylinder
D2 = kron3(speye(self.nCz), ddx(self.nCy), speye(self.nCx))
S = self.r(self.area, 'F', 'Fy', 'V')
V = self.vol
self._faceDivy = sdiag(1/V)*D2*sdiag(S)
return self._faceDivy
def fget(self):
if(self.dim < 3): return None
if(self._faceDivz is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1/V)*D3*sdiag(S)
return self._faceDivz
def fget(self):
if(self.dim < 3): return None
if(self._faceDivz is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1/V)*D3*sdiag(S)
return self._faceDivz
def getP(self, mesh, Gloc):
if mesh in self._Ps:
return self._Ps[mesh]
P = mesh.getInterpolationMat(self.locs, Gloc)
if self.data_type == 'apparent_resistivity':
P = sdiag(1. / self.geometric_factor) * P
elif self.data_type == 'apparent_chargeability':
P = sdiag(1. / self.dc_voltage) * P
if self.storeProjections:
self._Ps[mesh] = P
return P
def test_zero(self):
z = Zero()
assert z == 0
assert not (z < 0)
assert z <= 0
assert not (z > 0)
assert z >= 0
assert +z == z
assert -z == z
assert z + 1 == 1
assert z + 3 +z == 3
assert z - 3 == -3
assert z - 3 -z == -3
assert 3*z == 0
assert z*3 == 0
assert z/3 == 0
a = 1
a += z
assert a == 1
a = 1
a += z
assert a == 1
self.assertRaises(ZeroDivisionError, lambda:3/z)
assert mkvc(z) == 0
assert sdiag(z)*a == 0
assert z.T == 0
assert z.transpose == 0
def faceDivy(self):
if (self.dim < 2):
return None
if getattr(self, '_faceDivy', None) is None:
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if (self.dim == 2):
D2 = sp.kron(ddx(n[1]), speye(n[0]))
elif (self.dim == 3):
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fy', 'V')
V = self.vol
self._faceDivy = sdiag(1 / V) * D2 * sdiag(S)
return self._faceDivy
def makeMassMatrices(self, m):
mu = self.muMap * m
self._MfMui = self.mesh.getFaceInnerProduct(1. / mu) / self.mesh.dim
# self._MfMui = self.mesh.getFaceInnerProduct(1./mu)
# TODO: this will break if tensor mu
self._MfMuI = sdiag(1. / self._MfMui.diagonal())
self._MfMu0 = self.mesh.getFaceInnerProduct(1. / mu_0) / self.mesh.dim
def cellGradBC(self):
"""
The cell centered Gradient boundary condition matrix
"""
if getattr(self, '_cellGradBC', None) is None:
BC = self.setCellGradBC(self._cellGradBC_list)
n = self.vnC
if (self.dim == 1):
G = ddxCellGradBC(n[0], BC[0])
elif (self.dim == 2):
G1 = sp.kron(speye(n[1]), ddxCellGradBC(n[0], BC[0]))
G2 = sp.kron(ddxCellGradBC(n[1], BC[1]), speye(n[0]))
G = sp.block_diag((G1, G2), format="csr")
elif (self.dim == 3):
G1 = kron3(speye(n[2]), speye(n[1]),
ddxCellGradBC(n[0], BC[0]))
G2 = kron3(speye(n[2]), ddxCellGradBC(n[1], BC[1]),
speye(n[0]))
G3 = kron3(ddxCellGradBC(n[2], BC[2]), speye(n[1]),
speye(n[0]))
G = sp.block_diag((G1, G2, G3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.aveCC2F * self.vol # Average volume between adjacent cells
self._cellGradBC = sdiag(S / V) * G
return self._cellGradBC
def getq(self, mesh):
srcind = Utils.closestPoints(mesh, self.loc, gridLoc='CC')
q = sdiag(1 / mesh.vol) * np.zeros(mesh.nC)
q[srcind] = 1.
return q
def getq(self, mesh): srcind = Utils.closestPoints(mesh, self.loc, gridLoc='CC') q = sdiag(1/mesh.vol)*np.zeros(mesh.nC) q[srcind] = 1. return q
def getJtJdiag(self, m, W=None):
"""
Return the diagonal of JtJ
"""
dmudm = self.chiMap.deriv(m)
self._dSdm = None
self._dfdm = None
self.model = m
if (self.gtgdiag is None) and (self.modelType != 'amplitude'):
if W is None:
w = np.ones(self.G.shape[1])
else:
w = W.diagonal()
self.gtgdiag = np.zeros(dmudm.shape[1])
for ii in range(self.G.shape[0]):
self.gtgdiag += (w[ii]*self.G[ii, :]*dmudm)**2.
if self.coordinate_system == 'cartesian':
if self.modelType == 'amplitude':
return np.sum((W * self.dfdm * self.G * dmudm)**2., axis=0)
else:
return self.gtgdiag
else: # spherical
if self.modelType == 'amplitude':
return np.sum(((W * self.dfdm) * self.G * (self.dSdm * dmudm))**2., axis=0)
else:
Japprox = sdiag(mkvc(self.gtgdiag)**0.5*dmudm.T) * (self.dSdm * dmudm)
return mkvc(np.sum(Japprox.power(2), axis=0))
def fget(self):
if(self.dim < 2): return None
if(self._faceDivy is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 2):
D2 = sp.kron(ddx(n[1]), speye(n[0]))
elif(self.dim == 3):
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fy', 'V')
V = self.vol
self._faceDivy = sdiag(1/V)*D2*sdiag(S)
return self._faceDivy
def test_zero(self):
z = Zero()
assert z == 0
assert not (z < 0)
assert z <= 0
assert not (z > 0)
assert z >= 0
assert +z == z
assert -z == z
assert z + 1 == 1
assert z + 3 +z == 3
assert z - 3 == -3
assert z - 3 -z == -3
assert 3*z == 0
assert z*3 == 0
assert z/3 == 0
a = 1
a += z
assert a == 1
a = 1
a += z
assert a == 1
self.assertRaises(ZeroDivisionError, lambda:3/z)
assert mkvc(z) == 0
assert sdiag(z)*a == 0
assert z.T == 0
assert z.transpose == 0
def getInterpolationMatCartMesh(self, Mrect, locType='CC'):
"""
Takes a cartesian mesh and returns a projection to translate onto the cartesian grid.
"""
assert self.isSymmetric, "Currently we have not taken into account other projections for more complicated CylMeshes"
if locType == 'F':
# do this three times for each component
X = self.getInterpolationMatCartMesh(Mrect, locType='Fx')
Y = self.getInterpolationMatCartMesh(Mrect, locType='Fy')
Z = self.getInterpolationMatCartMesh(Mrect, locType='Fz')
return sp.vstack((X, Y, Z))
if locType == 'E':
X = self.getInterpolationMatCartMesh(Mrect, locType='Ex')
Y = self.getInterpolationMatCartMesh(Mrect, locType='Ey')
Z = spzeros(Mrect.nEz, self.nE)
return sp.vstack((X, Y, Z))
grid = getattr(Mrect, 'grid' + locType)
# This is unit circle stuff, 0 to 2*pi, starting at x-axis, rotating counter clockwise in an x-y slice
theta = -np.arctan2(grid[:, 0] - self.cartesianOrigin[0],
grid[:, 1] - self.cartesianOrigin[1]) + np.pi / 2
theta[theta < 0] += np.pi * 2.0
r = ((grid[:, 0] - self.cartesianOrigin[0])**2 +
(grid[:, 1] - self.cartesianOrigin[1])**2)**0.5
if locType in ['CC', 'N', 'Fz', 'Ez']:
G, proj = np.c_[r, theta, grid[:, 2]], np.ones(r.size)
else:
dotMe = {
'Fx': Mrect.normals[:Mrect.nFx, :],
'Fy': Mrect.normals[Mrect.nFx:(Mrect.nFx + Mrect.nFy), :],
'Fz': Mrect.normals[-Mrect.nFz:, :],
'Ex': Mrect.tangents[:Mrect.nEx, :],
'Ey': Mrect.tangents[Mrect.nEx:(Mrect.nEx + Mrect.nEy), :],
'Ez': Mrect.tangents[-Mrect.nEz:, :],
}[locType]
if 'F' in locType:
normals = np.c_[np.cos(theta),
np.sin(theta),
np.zeros(theta.size)]
proj = (normals * dotMe).sum(axis=1)
if 'E' in locType:
tangents = np.c_[-np.sin(theta),
np.cos(theta),
np.zeros(theta.size)]
proj = (tangents * dotMe).sum(axis=1)
G = np.c_[r, theta, grid[:, 2]]
interpType = locType
if interpType == 'Fy':
interpType = 'Fx'
elif interpType == 'Ex':
interpType = 'Ey'
Pc2r = self.getInterpolationMat(G, interpType)
Proj = sdiag(proj)
return Proj * Pc2r
def faceDivz(self):
"""
Construct divergence operator in the z component (face-stg to
cell-centers).
"""
if (self.dim < 3):
return None
if getattr(self, '_faceDivz', None) is None:
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1 / V) * D3 * sdiag(S)
return self._faceDivz
def test_mat_shape(self):
o = Identity()
S = sdiag(np.r_[2,3])[:1,:]
self.assertRaises(ValueError, lambda:S + o)
def check(exp,ans):
assert np.all((exp).todense() == ans)
check(S * o, [[2,0]])
check(S * -o, [[-2,0]])
def test_mat_shape(self):
o = Identity()
S = sdiag(np.r_[2,3])[:1,:]
self.assertRaises(ValueError, lambda:S + o)
def check(exp,ans):
assert np.all((exp).todense() == ans)
check(S * o, [[2,0]])
check(S * -o, [[-2,0]])
def faceDivz(self):
"""
Construct divergence operator in the z component (face-stg to
cell-centres).
"""
if(self.dim < 3):
return None
if getattr(self, '_faceDivz', None) is None:
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
self._faceDivz = sdiag(1/V)*D3*sdiag(S)
return self._faceDivz
def fget(self):
if(self._faceDivx is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 1):
D1 = ddx(n[0])
elif(self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
elif(self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
self._faceDivx = sdiag(1/V)*D1*sdiag(S)
return self._faceDivx
def edgeCurl(self):
"""The edgeCurl property."""
if self.nCy > 1:
raise NotImplementedError('Edge curl not yet implemented for nCy > 1')
if getattr(self, '_edgeCurl', None) is None:
#1D Difference matricies
dr = sp.spdiags((np.ones((self.nCx+1, 1))*[-1, 1]).T, [-1,0], self.nCx, self.nCx, format="csr")
dz = sp.spdiags((np.ones((self.nCz+1, 1))*[-1, 1]).T, [0,1], self.nCz, self.nCz+1, format="csr")
#2D Difference matricies
Dr = sp.kron(sp.identity(self.nNz), dr)
Dz = -sp.kron(dz, sp.identity(self.nCx))
A = self.area
E = self.edge
#Edge curl operator
self._edgeCurl = sdiag(1/A)*sp.vstack((Dz, Dr))*sdiag(E)
return self._edgeCurl
def test_NewtonRoot(self):
fun = lambda x, return_g=True: np.sin(x) if not return_g else ( np.sin(x), sdiag( np.cos(x) ) )
x = np.array([np.pi-0.3, np.pi+0.1, 0])
xopt = Optimization.NewtonRoot(comments=False).root(fun,x)
x_true = np.array([np.pi,np.pi,0])
print('Newton Root Finding')
print('xopt: ', xopt)
print('x_true: ', x_true)
self.assertTrue(np.linalg.norm(xopt-x_true,2) < TOL, True)
def test_NewtonRoot(self):
fun = lambda x, return_g=True: np.sin(x) if not return_g else ( np.sin(x), sdiag( np.cos(x) ) )
x = np.array([np.pi-0.3, np.pi+0.1, 0])
xopt = Optimization.NewtonRoot(comments=False).root(fun,x)
x_true = np.array([np.pi,np.pi,0])
print('Newton Root Finding')
print('xopt: ', xopt)
print('x_true: ', x_true)
self.assertTrue(np.linalg.norm(xopt-x_true,2) < TOL, True)
def __init__(self, mesh, **kwargs):
Problem.BaseProblem.__init__(self, mesh, **kwargs)
Pbc, Pin, self._Pout = \
self.mesh.getBCProjWF('neumann', discretization='CC')
Dface = self.mesh.faceDiv
Mc = sdiag(self.mesh.vol)
self._Div = Mc * Dface * Pin.T * Pin
def getInterpolationMatCartMesh(self, Mrect, locType='CC', locTypeTo=None):
"""
Takes a cartesian mesh and returns a projection to translate onto the cartesian grid.
"""
assert self.isSymmetric, "Currently we have not taken into account other projections for more complicated CylMeshes"
if locTypeTo is None:
locTypeTo = locType
if locType == 'F':
# do this three times for each component
X = self.getInterpolationMatCartMesh(Mrect, locType='Fx', locTypeTo=locTypeTo+'x')
Y = self.getInterpolationMatCartMesh(Mrect, locType='Fy', locTypeTo=locTypeTo+'y')
Z = self.getInterpolationMatCartMesh(Mrect, locType='Fz', locTypeTo=locTypeTo+'z')
return sp.vstack((X,Y,Z))
if locType == 'E':
X = self.getInterpolationMatCartMesh(Mrect, locType='Ex', locTypeTo=locTypeTo+'x')
Y = self.getInterpolationMatCartMesh(Mrect, locType='Ey', locTypeTo=locTypeTo+'y')
Z = spzeros(getattr(Mrect, 'n' + locTypeTo + 'z'), self.nE)
return sp.vstack((X,Y,Z))
grid = getattr(Mrect, 'grid' + locTypeTo)
# This is unit circle stuff, 0 to 2*pi, starting at x-axis, rotating counter clockwise in an x-y slice
theta = - np.arctan2(grid[:,0] - self.cartesianOrigin[0], grid[:,1] - self.cartesianOrigin[1]) + np.pi/2
theta[theta < 0] += np.pi*2.0
r = ((grid[:,0] - self.cartesianOrigin[0])**2 + (grid[:,1] - self.cartesianOrigin[1])**2)**0.5
if locType in ['CC', 'N', 'Fz', 'Ez']:
G, proj = np.c_[r, theta, grid[:,2]], np.ones(r.size)
else:
dotMe = {
'Fx': Mrect.normals[:Mrect.nFx,:],
'Fy': Mrect.normals[Mrect.nFx:(Mrect.nFx+Mrect.nFy),:],
'Fz': Mrect.normals[-Mrect.nFz:,:],
'Ex': Mrect.tangents[:Mrect.nEx,:],
'Ey': Mrect.tangents[Mrect.nEx:(Mrect.nEx+Mrect.nEy),:],
'Ez': Mrect.tangents[-Mrect.nEz:,:],
}[locTypeTo]
if 'F' in locType:
normals = np.c_[np.cos(theta), np.sin(theta), np.zeros(theta.size)]
proj = ( normals * dotMe ).sum(axis=1)
if 'E' in locType:
tangents = np.c_[-np.sin(theta), np.cos(theta), np.zeros(theta.size)]
proj = ( tangents * dotMe ).sum(axis=1)
G = np.c_[r, theta, grid[:,2]]
interpType = locType
if interpType == 'Fy':
interpType = 'Fx'
elif interpType == 'Ex':
interpType = 'Ey'
Pc2r = self.getInterpolationMat(G, interpType)
Proj = sdiag(proj)
return Proj * Pc2r
def cellGrady(self):
if self.dim < 2:
return None
if getattr(self, '_cellGrady', None) is None:
G2 = self._cellGradyStencil()
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F', 'Fy', 'V')
self._cellGrady = sdiag(L)*G2
return self._cellGrady
def cellGrady(self):
if self.dim < 2:
return None
if getattr(self, '_cellGrady', None) is None:
G2 = self._cellGradyStencil()
# Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol
L = self.r(self.area / V, 'F', 'Fy', 'V')
self._cellGrady = sdiag(L) * G2
return self._cellGrady
def cellGrad(self):
"""
The cell centered Gradient, takes you to cell faces.
"""
if getattr(self, '_cellGrad', None) is None:
G = self._cellGradStencil()
S = self.area # Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol # Average volume between adjacent cells
self._cellGrad = sdiag(S / V) * G
return self._cellGrad
def cellGrad(self):
"""
The cell centered Gradient, takes you to cell faces.
"""
if getattr(self, '_cellGrad', None) is None:
G = self._cellGradStencil()
S = self.area # Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol # Average volume between adjacent cells
self._cellGrad = sdiag(S/V)*G
return self._cellGrad
def fget(self):
if self.dim < 3: return None
if getattr(self, '_cellGradz', None) is None:
BC = ['neumann', 'neumann']
n = self.vnC
G3 = kron3(ddxCellGrad(n[2], BC), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F','Fz', 'V')
self._cellGradz = sdiag(L)*G3
return self._cellGradz
def fget(self):
if self.dim < 3: return None
if getattr(self, '_cellGradz', None) is None:
BC = ['neumann', 'neumann']
n = self.vnC
G3 = kron3(ddxCellGrad(n[2], BC), speye(n[1]), speye(n[0]))
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F','Fz', 'V')
self._cellGradz = sdiag(L)*G3
return self._cellGradz
def _h(self, bSolution, srcList):
"""
Magnetic field from bSolution
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: magnetic field
"""
n = int(self._aveF2CCV.shape[0] / self._nC) #number of components
VI = sdiag(np.kron(np.ones(n), 1./self.prob.mesh.vol))
return VI * (self._aveF2CCV * (self._MfMui * self._b(bSolution, srcList)))
def cellGradx(self):
"""
Cell centered Gradient in the x dimension. Has neumann boundary
conditions.
"""
if getattr(self, '_cellGradx', None) is None:
G1 = self._cellGradxStencil()
# Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol
L = self.r(self.area / V, 'F', 'Fx', 'V')
self._cellGradx = sdiag(L) * G1
return self._cellGradx
def cellGradx(self):
"""
Cell centered Gradient in the x dimension. Has neumann boundary
conditions.
"""
if getattr(self, '_cellGradx', None) is None:
G1 = self._cellGradxStencil()
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F','Fx', 'V')
self._cellGradx = sdiag(L)*G1
return self._cellGradx
def _e(self, jSolution, srcList):
"""
Electric field from jSolution
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: electric field
"""
n = int(self._aveF2CCV.shape[0] / self._nC) # number of components
VI = sdiag(np.kron(np.ones(n), 1./self.prob.mesh.vol))
return VI * (self._aveF2CCV * (self._MfRho * self._j(jSolution, srcList)))
def faceDivx(self):
"""
Construct divergence operator in the x component (face-stg to
cell-centres).
"""
if getattr(self, '_faceDivx', None) is None:
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if (self.dim == 1):
D1 = ddx(n[0])
elif (self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
elif (self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
self._faceDivx = sdiag(1 / V) * D1 * sdiag(S)
return self._faceDivx
def faceDivx(self):
"""
Construct divergence operator in the x component (face-stg to
cell-centres).
"""
if getattr(self, '_faceDivx', None) is None:
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 1):
D1 = ddx(n[0])
elif(self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
elif(self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
self._faceDivx = sdiag(1/V)*D1*sdiag(S)
return self._faceDivx
def fget(self):
if(self._edgeCurl is None):
assert self.dim > 1, "Edge Curl only programed for 2 or 3D."
# The number of cell centers in each direction
n = self.vnC
# Compute lengths of cell edges
L = self.edge
# Compute areas of cell faces
S = self.area
# Compute divergence operator on faces
if self.dim == 2:
D21 = sp.kron(ddx(n[1]), speye(n[0]))
D12 = sp.kron(speye(n[1]), ddx(n[0]))
C = sp.hstack((-D21, D12), format="csr")
self._edgeCurl = C*sdiag(1/S)
elif self.dim == 3:
D32 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]+1))
D23 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]+1))
D31 = kron3(ddx(n[2]), speye(n[1]+1), speye(n[0]))
D13 = kron3(speye(n[2]), speye(n[1]+1), ddx(n[0]))
D21 = kron3(speye(n[2]+1), ddx(n[1]), speye(n[0]))
D12 = kron3(speye(n[2]+1), speye(n[1]), ddx(n[0]))
O1 = spzeros(np.shape(D32)[0], np.shape(D31)[1])
O2 = spzeros(np.shape(D31)[0], np.shape(D32)[1])
O3 = spzeros(np.shape(D21)[0], np.shape(D13)[1])
C = sp.vstack((sp.hstack((O1, -D32, D23)),
sp.hstack((D31, O2, -D13)),
sp.hstack((-D21, D12, O3))), format="csr")
self._edgeCurl = sdiag(1/S)*(C*sdiag(L))
return self._edgeCurl
def nodalLaplacian(self):
"""
Construct laplacian operator (nodes to edges).
"""
if getattr(self, '_nodalLaplacian', None) is None:
print('Warning: Laplacian has not been tested rigorously.')
# The number of cell centers in each direction
n = self.vnC
# Compute divergence operator on faces
if (self.dim == 1):
D1 = sdiag(1. / self.hx) * ddx(self.nCx)
L = -D1.T * D1
elif (self.dim == 2):
D1 = sdiag(1. / self.hx) * ddx(n[0])
D2 = sdiag(1. / self.hy) * ddx(n[1])
L1 = sp.kron(speye(n[1] + 1), -D1.T * D1)
L2 = sp.kron(-D2.T * D2, speye(n[0] + 1))
L = L1 + L2
elif (self.dim == 3):
D1 = sdiag(1. / self.hx) * ddx(n[0])
D2 = sdiag(1. / self.hy) * ddx(n[1])
D3 = sdiag(1. / self.hz) * ddx(n[2])
L1 = kron3(speye(n[2] + 1), speye(n[1] + 1), -D1.T * D1)
L2 = kron3(speye(n[2] + 1), -D2.T * D2, speye(n[0] + 1))
L3 = kron3(-D3.T * D3, speye(n[1] + 1), speye(n[0] + 1))
L = L1 + L2 + L3
self._nodalLaplacian = L
return self._nodalLaplacian
def fget(self):
if(self._edgeCurl is None):
assert self.dim > 1, "Edge Curl only programed for 2 or 3D."
# The number of cell centers in each direction
n = self.vnC
# Compute lengths of cell edges
L = self.edge
# Compute areas of cell faces
S = self.area
# Compute divergence operator on faces
if self.dim == 2:
D21 = sp.kron(ddx(n[1]), speye(n[0]))
D12 = sp.kron(speye(n[1]), ddx(n[0]))
C = sp.hstack((-D21, D12), format="csr")
self._edgeCurl = C*sdiag(1/S)
elif self.dim == 3:
D32 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]+1))
D23 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]+1))
D31 = kron3(ddx(n[2]), speye(n[1]+1), speye(n[0]))
D13 = kron3(speye(n[2]), speye(n[1]+1), ddx(n[0]))
D21 = kron3(speye(n[2]+1), ddx(n[1]), speye(n[0]))
D12 = kron3(speye(n[2]+1), speye(n[1]), ddx(n[0]))
O1 = spzeros(np.shape(D32)[0], np.shape(D31)[1])
O2 = spzeros(np.shape(D31)[0], np.shape(D32)[1])
O3 = spzeros(np.shape(D21)[0], np.shape(D13)[1])
C = sp.vstack((sp.hstack((O1, -D32, D23)),
sp.hstack((D31, O2, -D13)),
sp.hstack((-D21, D12, O3))), format="csr")
self._edgeCurl = sdiag(1/S)*(C*sdiag(L))
return self._edgeCurl
def fget(self):
if(self._nodalLaplacian is None):
print 'Warning: Laplacian has not been tested rigorously.'
# The number of cell centers in each direction
n = self.vnC
# Compute divergence operator on faces
if(self.dim == 1):
D1 = sdiag(1./self.hx) * ddx(mesh.nCx)
L = - D1.T*D1
elif(self.dim == 2):
D1 = sdiag(1./self.hx) * ddx(n[0])
D2 = sdiag(1./self.hy) * ddx(n[1])
L1 = sp.kron(speye(n[1]+1), - D1.T * D1)
L2 = sp.kron(- D2.T * D2, speye(n[0]+1))
L = L1 + L2
elif(self.dim == 3):
D1 = sdiag(1./self.hx) * ddx(n[0])
D2 = sdiag(1./self.hy) * ddx(n[1])
D3 = sdiag(1./self.hz) * ddx(n[2])
L1 = kron3(speye(n[2]+1), speye(n[1]+1), - D1.T * D1)
L2 = kron3(speye(n[2]+1), - D2.T * D2, speye(n[0]+1))
L3 = kron3(- D3.T * D3, speye(n[1]+1), speye(n[0]+1))
L = L1 + L2 + L3
self._nodalLaplacian = L
return self._nodalLaplacian
def fget(self):
if(self._nodalLaplacian is None):
print 'Warning: Laplacian has not been tested rigorously.'
# The number of cell centers in each direction
n = self.vnC
# Compute divergence operator on faces
if(self.dim == 1):
D1 = sdiag(1./self.hx) * ddx(mesh.nCx)
L = - D1.T*D1
elif(self.dim == 2):
D1 = sdiag(1./self.hx) * ddx(n[0])
D2 = sdiag(1./self.hy) * ddx(n[1])
L1 = sp.kron(speye(n[1]+1), - D1.T * D1)
L2 = sp.kron(- D2.T * D2, speye(n[0]+1))
L = L1 + L2
elif(self.dim == 3):
D1 = sdiag(1./self.hx) * ddx(n[0])
D2 = sdiag(1./self.hy) * ddx(n[1])
D3 = sdiag(1./self.hz) * ddx(n[2])
L1 = kron3(speye(n[2]+1), speye(n[1]+1), - D1.T * D1)
L2 = kron3(speye(n[2]+1), - D2.T * D2, speye(n[0]+1))
L3 = kron3(- D3.T * D3, speye(n[1]+1), speye(n[0]+1))
L = L1 + L2 + L3
self._nodalLaplacian = L
return self._nodalLaplacian
def _b(self, jSolution, srcList):
"""
Secondary magnetic flux density from jSolution
:param numpy.ndarray hSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: secondary magnetic flux density
"""
n = int(self._aveE2CCV.shape[0] / self._nC) # number of components
VI = sdiag(np.kron(np.ones(n), 1./self.prob.mesh.vol))
return VI * (self._aveE2CCV * ( self._MeMu * self._h(jSolution,srcList)) )
def _j(self, eSolution, srcList):
"""
Current density from eSolution
:param numpy.ndarray eSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: current density
"""
aveE2CCV = self._aveE2CCV
n = int(aveE2CCV.shape[0] / self._nC) # number of components (instead of checking if cyl or not)
VI = sdiag(np.kron(np.ones(n), 1./self.prob.mesh.vol))
return VI * (aveE2CCV * (self._MeSigma * self._e(eSolution,srcList) ) )
def fget(self):
if(self._faceDiv is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 1):
D = ddx(n[0])
elif(self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
D2 = sp.kron(ddx(n[1]), speye(n[0]))
D = sp.hstack((D1, D2), format="csr")
elif(self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
D = sp.hstack((D1, D2, D3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.vol
self._faceDiv = sdiag(1/V)*D*sdiag(S)
return self._faceDiv
def fget(self):
if(self._faceDiv is None):
# The number of cell centers in each direction
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 1):
D = ddx(n[0])
elif(self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
D2 = sp.kron(ddx(n[1]), speye(n[0]))
D = sp.hstack((D1, D2), format="csr")
elif(self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
D = sp.hstack((D1, D2, D3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.vol
self._faceDiv = sdiag(1/V)*D*sdiag(S)
return self._faceDiv
def faceDiv(self):
"""
Construct divergence operator (face-stg to cell-centres).
"""
if getattr(self, '_faceDiv', None) is None:
n = self.vnC
# Compute faceDivergence operator on faces
if (self.dim == 1):
D = ddx(n[0])
elif (self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
D2 = sp.kron(ddx(n[1]), speye(n[0]))
D = sp.hstack((D1, D2), format="csr")
elif (self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
D = sp.hstack((D1, D2, D3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.vol
self._faceDiv = sdiag(1 / V) * D * sdiag(S)
return self._faceDiv
def fget(self):
if self.dim < 2: return None
if getattr(self, '_cellGrady', None) is None:
BC = ['neumann', 'neumann']
n = self.vnC
if(self.dim == 2):
G2 = sp.kron(ddxCellGrad(n[1], BC), speye(n[0]))
elif(self.dim == 3):
G2 = kron3(speye(n[2]), ddxCellGrad(n[1], BC), speye(n[0]))
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F','Fy', 'V')
self._cellGrady = sdiag(L)*G2
return self._cellGrady
def faceDiv(self):
"""
Construct divergence operator (face-stg to cell-centres).
"""
if getattr(self, '_faceDiv', None) is None:
n = self.vnC
# Compute faceDivergence operator on faces
if(self.dim == 1):
D = ddx(n[0])
elif(self.dim == 2):
D1 = sp.kron(speye(n[1]), ddx(n[0]))
D2 = sp.kron(ddx(n[1]), speye(n[0]))
D = sp.hstack((D1, D2), format="csr")
elif(self.dim == 3):
D1 = kron3(speye(n[2]), speye(n[1]), ddx(n[0]))
D2 = kron3(speye(n[2]), ddx(n[1]), speye(n[0]))
D3 = kron3(ddx(n[2]), speye(n[1]), speye(n[0]))
D = sp.hstack((D1, D2, D3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.vol
self._faceDiv = sdiag(1/V)*D*sdiag(S)
return self._faceDiv
def fget(self):
if self.dim < 2: return None
if getattr(self, '_cellGrady', None) is None:
BC = ['neumann', 'neumann']
n = self.vnC
if(self.dim == 2):
G2 = sp.kron(ddxCellGrad(n[1], BC), speye(n[0]))
elif(self.dim == 3):
G2 = kron3(speye(n[2]), ddxCellGrad(n[1], BC), speye(n[0]))
# Compute areas of cell faces & volumes
V = self.aveCC2F*self.vol
L = self.r(self.area/V, 'F','Fy', 'V')
self._cellGrady = sdiag(L)*G2
return self._cellGrady
def _j(self, bSolution, srcList):
"""
Secondary current density from bSolution
:param numpy.ndarray bSolution: field we solved for
:param list srcList: list of sources
:rtype: numpy.ndarray
:return: primary current density
"""
n = int(self._aveE2CCV.shape[0] / self._nC) # number of components
VI = sdiag(np.kron(np.ones(n), 1./self.prob.mesh.vol))
return VI * (self._aveE2CCV * ( self._MeSigma * self._e(bSolution,srcList ) ) )