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https://github.com/wassname/simpeg.git
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rowanc1
297 lines
9.4 KiB
Python
297 lines
9.4 KiB
Python
from SimPEG import Survey, Problem, Utils, np, sp, Solver as SimpegSolver
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from scipy.constants import mu_0
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from SurveyFDEM import SurveyFDEM, FieldsFDEM
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from simpegEM.Utils import Sources
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from simpegEM.Base import BaseEMProblem
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def omega(freq):
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"""Change frequency to angular frequency, omega"""
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return 2.*np.pi*freq
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class BaseFDEMProblem(BaseEMProblem):
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"""
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We start by looking at Maxwell's equations in the electric field \\(\\vec{E}\\) and the magnetic flux density \\(\\vec{B}\\):
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.. math::
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\\nabla \\times \\vec{E} + i \\omega \\vec{B} = 0 \\\\
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\\nabla \\times \\mu^{-1} \\vec{B} - \\sigma \\vec{E} = \\vec{J_s}
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"""
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surveyPair = SurveyFDEM
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def forward(self, m, RHS, CalcFields):
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F = FieldsFDEM(self.mesh, self.survey)
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for freq in self.survey.freqs:
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A = self.getA(freq)
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rhs = RHS(freq)
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solver = self.Solver(A, **self.solverOpts)
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sol = solver.solve(rhs)
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for fieldType in self.storeTheseFields:
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Txs = self.survey.getTransmitters(freq)
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F[Txs, fieldType] = CalcFields(sol, freq, fieldType)
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return F
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def Jvec(self, m, v, u=None):
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if u is None:
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u = self.fields(m)
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self.curModel = m
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Jv = self.dataPair(self.survey)
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for freq in self.survey.freqs:
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A = self.getA(freq)
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solver = self.Solver(A, **self.solverOpts)
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for tx in self.survey.getTransmitters(freq):
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u_tx = u[tx, self.solType]
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w = self.getADeriv(freq, u_tx, v)
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Ainvw = solver.solve(w)
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for rx in tx.rxList:
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fAinvw = self.calcFields(Ainvw, freq, rx.projField)
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P = lambda v: rx.projectFieldsDeriv(tx, self.mesh, u, v)
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df_dm = self.calcFieldsDeriv(u_tx, freq, rx.projField, v)
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if df_dm is None:
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Jv[tx, rx] = - P(fAinvw)
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else:
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Jv[tx, rx] = - P(fAinvw) + P(df_dm)
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return Utils.mkvc(Jv)
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def Jtvec(self, m, v, u=None):
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if u is None:
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u = self.fields(m)
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self.curModel = m
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# Ensure v is a data object.
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if not isinstance(v, self.dataPair):
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v = self.dataPair(self.survey, v)
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Jtv = np.zeros(self.mapping.nP)
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for freq in self.survey.freqs:
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AT = self.getA(freq).T
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solver = self.Solver(AT, **self.solverOpts)
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for tx in self.survey.getTransmitters(freq):
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u_tx = u[tx, self.solType]
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for rx in tx.rxList:
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PTv = rx.projectFieldsDeriv(tx, self.mesh, u, v[tx, rx], adjoint=True)
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fPTv = self.calcFields(PTv, freq, rx.projField, adjoint=True)
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w = solver.solve( fPTv )
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Jtv_rx = - self.getADeriv(freq, u_tx, w, adjoint=True)
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df_dm = self.calcFieldsDeriv(u_tx, freq, rx.projField, PTv, adjoint=True)
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if df_dm is not None:
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Jtv_rx += df_dm
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real_or_imag = rx.projComp
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if real_or_imag == 'real':
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Jtv += Jtv_rx.real
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elif real_or_imag == 'imag':
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Jtv += - Jtv_rx.real
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else:
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raise Exception('Must be real or imag')
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return Jtv
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class ProblemFDEM_e(BaseFDEMProblem):
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"""
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By eliminating the magnetic flux density using
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.. math::
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\\vec{B} = \\frac{-1}{i\\omega}\\nabla\\times\\vec{E},
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we can write Maxwell's equations as a second order system in \\ \\vec{E} \\ only:
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.. math::
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\\nabla \\times \\mu^{-1} \\nabla \\times \\vec{E} + i \\omega \\sigma \\vec{E} = \\vec{J_s}
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This is the definition of the Forward Problem using the E-formulation of Maxwell's equations.
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"""
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solType = 'e'
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def __init__(self, model, **kwargs):
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BaseFDEMProblem.__init__(self, model, **kwargs)
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def getA(self, freq):
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"""
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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mui = self.MfMui
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sig = self.MeSigma
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C = self.mesh.edgeCurl
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return C.T*mui*C + 1j*omega(freq)*sig
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def getADeriv(self, freq, u, v, adjoint=False):
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sig = self.curTModel
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dsig_dm = self.curTModelDeriv
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig, v=u)
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if adjoint:
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return 1j * omega(freq) * ( dsig_dm.T * ( dMe_dsig.T * v ) )
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return 1j * omega(freq) * ( dMe_dsig * ( dsig_dm * v ) )
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def getRHS(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nTx)
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:return: RHS
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"""
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Txs = self.survey.getTransmitters(freq)
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rhs = range(len(Txs))
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for i, tx in enumerate(Txs):
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if tx.txType == 'VMD':
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src = Sources.MagneticDipoleVectorPotential
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else:
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raise NotImplemented('%s txType is not implemented' % tx.txType)
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SRCx = src(tx.loc, self.mesh.gridEx, 'x')
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SRCy = src(tx.loc, self.mesh.gridEy, 'y')
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SRCz = src(tx.loc, self.mesh.gridEz, 'z')
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rhs[i] = np.concatenate((SRCx, SRCy, SRCz))
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a = np.concatenate(rhs).reshape((self.mesh.nE, len(Txs)), order='F')
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mui = self.MfMui
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C = self.mesh.edgeCurl
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j_s = C.T*mui*C*a
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return -1j*omega(freq)*j_s
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def calcFields(self, sol, freq, fieldType, adjoint=False):
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e = sol
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if fieldType == 'e':
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return e
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elif fieldType == 'b':
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if not adjoint:
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b = -(1./(1j*omega(freq))) * ( self.mesh.edgeCurl * e )
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else:
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b = -(1./(1j*omega(freq))) * ( self.mesh.edgeCurl.T * e )
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return b
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
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e = sol
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if fieldType == 'e':
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return None
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elif fieldType == 'b':
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return None
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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class ProblemFDEM_b(BaseFDEMProblem):
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"""
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Solving for b!
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"""
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solType = 'b'
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def __init__(self, model, **kwargs):
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BaseFDEMProblem.__init__(self, model, **kwargs)
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def getA(self, freq):
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"""
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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mui = self.MfMui
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sigI = self.MeSigmaI
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C = self.mesh.edgeCurl
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return mui*C*sigI*C.T*mui + 1j*omega(freq)*mui
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def getADeriv(self, freq, u, v, adjoint=False):
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mui = self.MfMui
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C = self.mesh.edgeCurl
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sig = self.curTModel
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dsig_dm = self.curTModelDeriv
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#TODO: This only works if diagonal (no tensors)...
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dMeSigmaI_dI = - self.MeSigmaI**2
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vec = (C.T*(mui*u))
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig, v=vec)
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if adjoint:
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return dsig_dm.T * ( dMe_dsig.T * ( dMeSigmaI_dI.T * ( C.T * ( mui.T * v ) ) ) )
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return mui * ( C * ( dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) ) ) )
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def getRHS(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nTx)
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:return: RHS
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"""
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Txs = self.survey.getTransmitters(freq)
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rhs = range(len(Txs))
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for i, tx in enumerate(Txs):
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if tx.txType == 'VMD':
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src = Sources.MagneticDipoleVectorPotential
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else:
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raise NotImplemented('%s txType is not implemented' % tx.txType)
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SRCx = src(tx.loc, self.mesh.gridEx, 'x')
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SRCy = src(tx.loc, self.mesh.gridEy, 'y')
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SRCz = src(tx.loc, self.mesh.gridEz, 'z')
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rhs[i] = np.concatenate((SRCx, SRCy, SRCz))
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a = np.concatenate(rhs).reshape((self.mesh.nE, len(Txs)), order='F')
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mui = self.MfMui
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C = self.mesh.edgeCurl
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b_0 = C*a
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return -1j*omega(freq)*mui*b_0
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def calcFields(self, sol, freq, fieldType, adjoint=False):
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b = sol
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if fieldType == 'e':
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if not adjoint:
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e = self.MeSigmaI * ( self.mesh.edgeCurl.T * ( self.MfMui * b ) )
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else:
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e = self.MfMui.T * ( self.mesh.edgeCurl * ( self.MeSigmaI.T * b ) )
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return e
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elif fieldType == 'b':
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return b
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
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b = sol
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if fieldType == 'e':
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sig = self.curTModel
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dsig_dm = self.curTModelDeriv
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C = self.mesh.edgeCurl
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mui = self.MfMui
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#TODO: This only works if diagonal (no tensors)...
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dMeSigmaI_dI = - self.MeSigmaI**2
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vec = C.T * ( mui * b )
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig, v=vec)
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if not adjoint:
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return dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) )
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else:
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return dsig_dm.T * ( dMe_dsig.T * ( dMeSigmaI_dI.T * v ) )
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elif fieldType == 'b':
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return None
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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