mirror of
https://github.com/wassname/simpeg.git
synced 2026-06-28 17:35:14 +08:00
GudniRos
472 lines
14 KiB
Python
472 lines
14 KiB
Python
from SimPEG import Survey, Problem, Utils, np, sp, Solver as SimpegSolver
|
|
from scipy.constants import mu_0
|
|
from SurveyFDEM import SurveyFDEM
|
|
from FieldsFDEM import FieldsFDEM, FieldsFDEM_e, FieldsFDEM_b, FieldsFDEM_h, FieldsFDEM_j
|
|
from simpegEM.Base import BaseEMProblem
|
|
from simpegEM.Utils.EMUtils import omega
|
|
|
|
|
|
class BaseFDEMProblem(BaseEMProblem):
|
|
"""
|
|
We start by looking at Maxwell's equations in the electric field \\(\\vec{E}\\) and the magnetic flux density \\(\\vec{B}\\):
|
|
|
|
.. math::
|
|
|
|
\\nabla \\times \\vec{E} + i \\omega \\vec{B} = \\vec{S_m} \\\\
|
|
\\nabla \\times \\mu^{-1} \\vec{B} - \\sigma \\vec{E} = \\vec{S_e}
|
|
|
|
"""
|
|
surveyPair = SurveyFDEM
|
|
fieldsPair = FieldsFDEM
|
|
|
|
def forward(self, m, RHS):
|
|
|
|
F = self.fieldsPair(self.mesh, self.survey)
|
|
|
|
for freq in self.survey.freqs:
|
|
A = self.getA(freq)
|
|
rhs = RHS(freq)
|
|
Ainv = self.Solver(A, **self.solverOpts)
|
|
sol = Ainv * rhs
|
|
Srcs = self.survey.getSrcByFreq(freq)
|
|
F[Srcs, self._fieldType] = sol
|
|
|
|
return F
|
|
|
|
def Jvec(self, m, v, u=None):
|
|
if u is None:
|
|
u = self.fields(m)
|
|
|
|
self.curModel = m
|
|
|
|
Jv = self.dataPair(self.survey)
|
|
|
|
for freq in self.survey.freqs:
|
|
A = self.getA(freq)
|
|
Ainv = self.Solver(A, **self.solverOpts)
|
|
|
|
for src in self.survey.getSource(freq):
|
|
u_src = u[src, self.solType]
|
|
w = self.getADeriv(freq, u_src, v)
|
|
Ainvw = Ainv * w
|
|
for rx in src.rxList:
|
|
fAinvw = self.calcFields(Ainvw, freq, rx.projField)
|
|
P = lambda v: rx.projectFieldsDeriv(src, self.mesh, u, v)
|
|
|
|
Jv[src, rx] = - P(fAinvw)
|
|
|
|
df_dm = self.calcFieldsDeriv(u_src, freq, rx.projField, v)
|
|
if df_dm is not None:
|
|
Jv[src, rx] += P(df_dm)
|
|
|
|
return Utils.mkvc(Jv)
|
|
|
|
def Jtvec(self, m, v, u=None):
|
|
if u is None:
|
|
u = self.fields(m)
|
|
|
|
self.curModel = m
|
|
|
|
# Ensure v is a data object.
|
|
if not isinstance(v, self.dataPair):
|
|
v = self.dataPair(self.survey, v)
|
|
|
|
Jtv = np.zeros(self.mapping.nP)
|
|
|
|
for freq in self.survey.freqs:
|
|
AT = self.getA(freq).T
|
|
ATinv = self.Solver(AT, **self.solverOpts)
|
|
|
|
for src in self.survey.getSource(freq):
|
|
u_src = u[src, self.solType]
|
|
|
|
for rx in src.rxList:
|
|
PTv = rx.projectFieldsDeriv(src, self.mesh, u, v[src, rx], adjoint=True)
|
|
fPTv = self.calcFields(PTv, freq, rx.projField, adjoint=True)
|
|
|
|
w = ATinv * fPTv
|
|
Jtv_rx = - self.getADeriv(freq, u_src, w, adjoint=True)
|
|
|
|
df_dm = self.calcFieldsDeriv(u_src, freq, rx.projField, PTv, adjoint=True)
|
|
|
|
if df_dm is not None:
|
|
Jtv_rx += df_dm
|
|
|
|
real_or_imag = rx.projComp
|
|
if real_or_imag == 'real':
|
|
Jtv += Jtv_rx.real
|
|
elif real_or_imag == 'imag':
|
|
Jtv += - Jtv_rx.real
|
|
else:
|
|
raise Exception('Must be real or imag')
|
|
|
|
return Jtv
|
|
|
|
def getSourceTerm(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE or nF, nSrc)
|
|
:return: RHS
|
|
"""
|
|
Srcs = self.survey.getSrcByFreq(freq)
|
|
if self._eqLocs is 'FE':
|
|
S_m = np.zeros((self.mesh.nF,len(Srcs)), dtype=complex)
|
|
S_e = np.zeros((self.mesh.nE,len(Srcs)), dtype=complex)
|
|
elif self._eqLocs is 'EF':
|
|
S_m = np.zeros((self.mesh.nE,len(Srcs)), dtype=complex)
|
|
S_e = np.zeros((self.mesh.nF,len(Srcs)), dtype=complex)
|
|
|
|
for i, src in enumerate(Srcs):
|
|
smi, sei = src.eval(self)
|
|
if smi is not None:
|
|
S_m[:,i] = smi
|
|
if sei is not None:
|
|
S_e[:,i] = sei
|
|
|
|
return S_m, S_e
|
|
|
|
def getSourceTermDeriv(self,freq,m,v,u=None,adjoint=False):
|
|
raise NotImplementedError('getSourceTermDeriv not implemented yet')
|
|
return None, None
|
|
|
|
|
|
##########################################################################################
|
|
################################ E-B Formulation #########################################
|
|
##########################################################################################
|
|
|
|
class ProblemFDEM_e(BaseFDEMProblem):
|
|
"""
|
|
By eliminating the magnetic flux density using
|
|
|
|
.. math::
|
|
|
|
\\vec{B} = \\frac{-1}{i\\omega}\\nabla\\times\\vec{E},
|
|
|
|
we can write Maxwell's equations as a second order system in \\ \\vec{E} \\ only:
|
|
|
|
.. math::
|
|
|
|
\\nabla \\times \\mu^{-1} \\nabla \\times \\vec{E} + i \\omega \\sigma \\vec{E} = \\vec{J_s}
|
|
|
|
This is the definition of the Forward Problem using the E-formulation of Maxwell's equations.
|
|
|
|
|
|
"""
|
|
|
|
_fieldType = 'e'
|
|
_eqLocs = 'FE'
|
|
fieldsPair = FieldsFDEM_e
|
|
|
|
def __init__(self, mesh, **kwargs):
|
|
BaseFDEMProblem.__init__(self, mesh, **kwargs)
|
|
|
|
def getA(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: scipy.sparse.csr_matrix
|
|
:return: A
|
|
"""
|
|
mui = self.MfMui
|
|
sig = self.MeSigma
|
|
C = self.mesh.edgeCurl
|
|
|
|
return C.T*mui*C + 1j*omega(freq)*sig
|
|
|
|
|
|
def getADeriv(self, freq, u, v, adjoint=False):
|
|
sig = self.curModel.transform
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig)(u)
|
|
|
|
if adjoint:
|
|
return 1j * omega(freq) * ( dsig_dm.T * ( dMe_dsig.T * v ) )
|
|
|
|
return 1j * omega(freq) * ( dMe_dsig * ( dsig_dm * v ) )
|
|
|
|
def getRHS(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE, nSrc)
|
|
:return: RHS
|
|
"""
|
|
|
|
S_m, S_e = self.getSourceTerm(freq)
|
|
C = self.mesh.edgeCurl
|
|
MfMui = self.MfMui
|
|
|
|
RHS = C.T * (MfMui * S_m) -1j*omega(freq)*S_e
|
|
|
|
return RHS
|
|
|
|
def getRHSDeriv(self, freq, u, v, adjoint=False):
|
|
raise NotImplementedError('getRHSDeriv not implemented yet')
|
|
return None
|
|
|
|
|
|
class ProblemFDEM_b(BaseFDEMProblem):
|
|
"""
|
|
Solving for b!
|
|
"""
|
|
_fieldType = 'b'
|
|
_eqLocs = 'FE'
|
|
fieldsPair = FieldsFDEM_b
|
|
|
|
def __init__(self, mesh, **kwargs):
|
|
BaseFDEMProblem.__init__(self, mesh, **kwargs)
|
|
|
|
def getA(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: scipy.sparse.csr_matrix
|
|
:return: A
|
|
"""
|
|
mui = self.MfMui
|
|
sigI = self.MeSigmaI
|
|
C = self.mesh.edgeCurl
|
|
iomega = 1j * omega(freq) * sp.eye(self.mesh.nF)
|
|
|
|
A = C*sigI*C.T*mui + iomega
|
|
|
|
if self._makeASymmetric is True:
|
|
return mui.T*A
|
|
return A
|
|
|
|
def getADeriv(self, freq, u, v, adjoint=False):
|
|
|
|
mui = self.MfMui
|
|
C = self.mesh.edgeCurl
|
|
sig = self.curModel.transform
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dMeSigmaI_dI = self._dMeSigmaI_dI
|
|
|
|
vec = (C.T*(mui*u))
|
|
dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig)(vec)
|
|
|
|
if adjoint:
|
|
if self._makeASymmetric is True:
|
|
v = mui * v
|
|
return dsig_dm.T * ( dMe_dsig.T * ( dMeSigmaI_dI.T * ( C.T * v ) ) )
|
|
|
|
if self._makeASymmetric is True:
|
|
return mui.T * ( C * ( dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) ) ) )
|
|
return C * ( dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) ) )
|
|
|
|
|
|
def getRHS(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE, nSrc)
|
|
:return: RHS
|
|
"""
|
|
|
|
S_m, S_e = self.getSourceTerm(freq)
|
|
C = self.mesh.edgeCurl
|
|
MeSigmaI = self.MeSigmaI
|
|
|
|
RHS = S_m + C * ( MeSigmaI * S_e )
|
|
|
|
if self._makeASymmetric is True:
|
|
mui = self.MfMui
|
|
return mui.T*RHS
|
|
|
|
return RHS
|
|
|
|
def getRHSDeriv(self, freq, u, v, adjoint=False):
|
|
raise NotImplementedError('getRHSDeriv not implemented yet')
|
|
return None
|
|
|
|
|
|
|
|
##########################################################################################
|
|
################################ H-J Formulation #########################################
|
|
##########################################################################################
|
|
|
|
|
|
class ProblemFDEM_j(BaseFDEMProblem):
|
|
"""
|
|
Using the H-J formulation of Maxwell's equations
|
|
|
|
.. math::
|
|
\\nabla \\times \\sigma^{-1} \\vec{J} + i\\omega\\mu\\vec{H} = 0
|
|
\\nabla \\times \\vec{H} - \\vec{J} = \\vec{J_s}
|
|
|
|
Since \(\\vec{J}\) is a flux and \(\\vec{H}\) is a field, we discretize \(\\vec{J}\) on faces and \(\\vec{H}\) on edges.
|
|
|
|
For this implementation, we solve for J using \( \\vec{H} = - (i\\omega\\mu)^{-1} \\nabla \\times \\sigma^{-1} \\vec{J} \) :
|
|
|
|
.. math::
|
|
\\nabla \\times ( \\mu^{-1} \\nabla \\times \\sigma^{-1} \\vec{J} ) + i\\omega \\vec{J} = - i\\omega\\vec{J_s}
|
|
|
|
We discretize this to:
|
|
|
|
.. math::
|
|
(\\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega ) \\mathbf{j} = - i\\omega \\mathbf{j_s}
|
|
|
|
.. note::
|
|
This implementation does not yet work with full anisotropy!!
|
|
|
|
"""
|
|
|
|
_fieldType = 'j'
|
|
_eqLocs = 'EF'
|
|
fieldsPair = FieldsFDEM_j
|
|
|
|
def __init__(self, mesh, **kwargs):
|
|
BaseFDEMProblem.__init__(self, mesh, **kwargs)
|
|
|
|
def getA(self, freq):
|
|
"""
|
|
Here, we form the operator \(\\mathbf{A}\) to solce
|
|
.. math::
|
|
\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
|
|
|
|
:param float freq: Frequency
|
|
:rtype: scipy.sparse.csr_matrix
|
|
:return: A
|
|
"""
|
|
|
|
MeMuI = self.MeMuI
|
|
MfSigi = self.MfSigmai
|
|
C = self.mesh.edgeCurl
|
|
iomega = 1j * omega(freq) * sp.eye(self.mesh.nF)
|
|
|
|
A = C * MeMuI * C.T * MfSigi + iomega
|
|
|
|
if self._makeASymmetric is True:
|
|
return MfSigi.T*A
|
|
return A
|
|
|
|
|
|
def getADeriv(self, freq, u, v, adjoint=False):
|
|
"""
|
|
In this case, we assume that electrical conductivity, \(\\sigma\) is the physical property of interest (i.e. \(\sigma\) = model.transform). Then we want
|
|
.. math::
|
|
\\frac{\mathbf{A(\\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
|
|
&= \\mathbf{C} \\mathbf{M^e_{mu}^{-1}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{\\sigma^{-1}}} \\frac{d \\mathbf{\\sigma^{-1}}}{d \\mathbf{\\sigma}} \\frac{d \\mathbf{\\sigma}}{d \\mathbf{m}}
|
|
"""
|
|
|
|
MeMuI = self.MeMuI
|
|
MfSigi = self.MfSigmai
|
|
C = self.mesh.edgeCurl
|
|
sig = self.curModel.transform
|
|
sigi = 1/sig
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dsigi_dsig = -Utils.sdiag(sigi)**2
|
|
dMf_dsigi = self.mesh.getFaceInnerProductDeriv(sigi)(u)
|
|
|
|
if adjoint:
|
|
if self._makeASymmetric is True:
|
|
v = MfSigi * v
|
|
return dsig_dm.T * ( dsigi_dsig.T *( dMf_dsigi.T * ( C * ( MeMuI.T * ( C.T * v ) ) ) ) )
|
|
|
|
if self._makeASymmetric is True:
|
|
return MfSigi.T * ( C * ( MeMuI * ( C.T * ( dMf_dsigi * ( dsigi_dsig * ( dsig_dm * v ) ) ) ) ) )
|
|
return C * ( MeMuI * ( C.T * ( dMf_dsigi * ( dsigi_dsig * ( dsig_dm * v ) ) ) ) )
|
|
|
|
|
|
def getRHS(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE, nSrc)
|
|
:return: RHS
|
|
"""
|
|
|
|
S_m, S_e = self.getSourceTerm(freq)
|
|
C = self.mesh.edgeCurl
|
|
MeMuI = self.MeMuI
|
|
|
|
|
|
RHS = C * (MeMuI * S_m) - 1j * omega(freq) * S_e
|
|
if self._makeASymmetric is True:
|
|
MfSigi = self.MfSigmai
|
|
return MfSigi.T*RHS
|
|
|
|
return RHS
|
|
|
|
def getRHSDeriv(self, freq, u, v, adjoint=False):
|
|
raise NotImplementedError('getRHSDeriv not implemented yet')
|
|
return None
|
|
|
|
|
|
|
|
class ProblemFDEM_h(BaseFDEMProblem):
|
|
"""
|
|
Using the H-J formulation of Maxwell's equations
|
|
|
|
.. math::
|
|
\\nabla \\times \\sigma^{-1} \\vec{J} + i\\omega\\mu\\vec{H} = 0
|
|
\\nabla \\times \\vec{H} - \\vec{J} = \\vec{J_s}
|
|
|
|
Since \(\\vec{J}\) is a flux and \(\\vec{H}\) is a field, we discretize \(\\vec{J}\) on faces and \(\\vec{H}\) on edges.
|
|
|
|
For this implementation, we solve for J using \( \\vec{J} = \\nabla \\times \\vec{H} - \\vec{J_s} \)
|
|
|
|
.. math::
|
|
\\nabla \\times \\sigma^{-1} \\nabla \\times \\vec{H} + i\\omega\\mu\\vec{H} = \\nabla \\times \\sigma^{-1} \\vec{J_s}
|
|
|
|
We discretize and solve
|
|
|
|
.. math::
|
|
(\\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} \\mathbf{C} + i\\omega \\mathbf{M_{\mu}} ) \\mathbf{h} = \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} \\vec{J_s}
|
|
|
|
.. note::
|
|
This implementation does not yet work with full anisotropy!!
|
|
|
|
"""
|
|
|
|
_fieldType = 'h'
|
|
_eqLocs = 'EF'
|
|
fieldsPair = FieldsFDEM_h
|
|
|
|
def __init__(self, mesh, **kwargs):
|
|
BaseFDEMProblem.__init__(self, mesh, **kwargs)
|
|
|
|
def getA(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: scipy.sparse.csr_matrix
|
|
:return: A
|
|
"""
|
|
|
|
MeMu = self.MeMu
|
|
MfSigi = self.MfSigmai
|
|
C = self.mesh.edgeCurl
|
|
|
|
return C.T * MfSigi * C + 1j*omega(freq)*MeMu
|
|
|
|
def getADeriv(self, freq, u, v, adjoint=False):
|
|
|
|
MeMu = self.MeMu
|
|
C = self.mesh.edgeCurl
|
|
sig = self.curModel.transform
|
|
sigi = 1/sig
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dsigi_dsig = -Utils.sdiag(sigi)**2
|
|
|
|
dMf_dsigi = self.mesh.getFaceInnerProductDeriv(sigi)(C*u)
|
|
|
|
if adjoint:
|
|
return (dsig_dm.T * (dsigi_dsig.T * (dMf_dsigi.T * (C * v))))
|
|
return (C.T * (dMf_dsigi * (dsigi_dsig * (dsig_dm * v))))
|
|
|
|
|
|
def getRHS(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE, nSrc)
|
|
:return: RHS
|
|
"""
|
|
|
|
S_m, S_e = self.getSourceTerm(freq)
|
|
C = self.mesh.edgeCurl
|
|
MfSigmai = self.MfSigmai
|
|
|
|
RHS = S_m + C.T * ( MfSigmai * S_e )
|
|
|
|
return RHS
|
|
|
|
def getRHSDeriv(self, freq, u, v, adjoint=False):
|
|
raise NotImplementedError('getRHSDeriv not implemented yet')
|
|
return None
|
|
|