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351 lines
12 KiB
Python
351 lines
12 KiB
Python
from BaseTDEM import BaseTDEMProblem
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from SimPEG.Utils import mkvc
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import numpy as np
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from SurveyTDEM import SurveyTDEM, FieldsTDEM
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class FieldsTDEM_e_from_b(FieldsTDEM):
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"""Fancy Field Storage for a TDEM survey."""
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knownFields = {'b': 'F'}
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aliasFields = {'e': ['b','E','e_from_b']}
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def startup(self):
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self.MeSigmaI = self.survey.prob.MeSigmaI
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self.edgeCurlT = self.survey.prob.mesh.edgeCurl.T
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self.MfMui = self.survey.prob.MfMui
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def e_from_b(self, b, ind):
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# TODO: implement non-zero js
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return self.MeSigmaI*(self.edgeCurlT*(self.MfMui*b))
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class ProblemTDEM_b(BaseTDEMProblem):
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"""
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Time-Domain EM problem - B-formulation
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TDEM_b treats the following discretization of Maxwell's equations
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.. math::
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\dcurl \e^{(t+1)} + \\frac{\\b^{(t+1)} - \\b^{(t)}}{\delta t} = 0 \\\\
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\dcurl^\\top \MfMui \\b^{(t+1)} - \MeSig \e^{(t+1)} = \Me \j_s^{(t+1)}
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with \\\(\\b\\\) defined on cell faces and \\\(\e\\\) defined on edges.
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"""
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def __init__(self, mesh, mapping=None, **kwargs):
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BaseTDEMProblem.__init__(self, mesh, mapping=mapping, **kwargs)
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solType = 'b' #: Type of the solution, in this case the 'b' field
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surveyPair = SurveyTDEM
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_FieldsTDEM_pair = FieldsTDEM_e_from_b #: used for the forward calculation only
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####################################################
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# Internal Methods
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####################################################
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def getA(self, tInd):
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"""
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:param int tInd: Time index
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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dt = self.timeSteps[tInd]
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return self.MfMui*self.mesh.edgeCurl*self.MeSigmaI*self.mesh.edgeCurl.T*self.MfMui + (1.0/dt)*self.MfMui
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def getRHS(self, tInd, F):
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dt = self.timeSteps[tInd]
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B_n = np.c_[[F[tx,'b',tInd] for tx in self.survey.txList]].T
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RHS = (1.0/dt)*self.MfMui*B_n
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return RHS
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def calcFields(self, sol, tInd):
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if self.solType == 'b':
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b = sol
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# e = self.MeSigmaI*(self.mesh.edgeCurl.T*(self.MfMui*b))
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else:
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raise NotImplementedError('solType "%s" is not implemented in CalcFields.' % self.solType)
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return {'b':b}
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####################################################
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# Derivatives
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####################################################
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def Gvec(self, m, vec, u=None):
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"""
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:param numpy.array m: Conductivity model
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:param numpy.array vec: vector (like a model)
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:param simpegEM.TDEM.FieldsTDEM u: Fields resulting from m
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: f
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Multiply G by a vector
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"""
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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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# Note: Fields has shape (nF/E, nTx, nT+1)
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# However, p will only really fill (:,:,1:nT+1)
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# meaning the 'initial fields' are zero (:,:,0)
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p = FieldsTDEM(self.mesh, self.survey)
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# 'b' at all times is zero.
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# However, to save memory we will **not** do:
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#
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# p[:, 'b', :] = 0.0
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# fake initial 'e' fields
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p[:, 'e', 0] = 0.0
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c = self.mesh.getEdgeInnerProductDeriv(self.curTModel)*(self.curTModelDeriv*vec)
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for i in range(1,self.nT+1):
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# TODO: G[1] may be dependent on the model
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# for a galvanic source (deriv of the dc problem)
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for tx in self.survey.txList:
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p[tx, 'e', i] = -u[tx,'e',i]*c # i.e.: - diag(e) * MsigDeriv * v
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return p
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def Gtvec(self, m, vec, u=None):
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"""
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:param numpy.array m: Conductivity model
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:param numpy.array vec: vector (like a fields)
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:param simpegEM.TDEM.FieldsTDEM u: Fields resulting from m
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:rtype: np.ndarray (like a model)
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:return: p
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Multiply G.T by a vector
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"""
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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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nTx, nE = self.survey.nTx, self.mesh.nE
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tmp = np.zeros(nE)
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# Here we can do internal multiplications of Gt*v and then multiply by MsigDeriv.T in one go.
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for i in range(1,self.nT+1):
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vu = vec[:,'e',i]*u[:,'e',i]
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if nTx > 1:
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vu = vu.sum(axis=1)
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tmp += vu
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p = -mkvc(self.curTModelDeriv.T*(self.mesh.getEdgeInnerProductDeriv(self.curTModel).T*tmp))
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return p
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def solveAh(self, m, p):
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"""
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:param numpy.array m: Conductivity model
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:param simpegEM.TDEM.FieldsTDEM p: Fields object
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: y
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Solve the block-matrix system \\\(\\\hat{A} \\\hat{y} = \\\hat{p}\\\):
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.. math::
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\mathbf{\hat{A}} = \left[
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\\begin{array}{cccc}
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A & 0 & & \\\\
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B & A & & \\\\
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& \ddots & \ddots & \\\\
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& & B & A
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\end{array}
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\\right] \\\\
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\mathbf{A} =
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\left[
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\\begin{array}{cc}
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\\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\\\
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\dcurl^\\top \MfMui & -\MeSig
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\end{array}
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\\right] \\\\
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\mathbf{B} =
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\left[
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\\begin{array}{cc}
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-\\frac{1}{\delta t} \MfMui & 0 \\\\
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0 & 0
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\end{array}
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\\right] \\\\
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"""
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def AhRHS(tInd, y):
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rhs = self.MfMui*(self.mesh.edgeCurl*(self.MeSigmaI*p[:,'e',tInd+1]))
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if 'b' in p:
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rhs = rhs + p[:,'b',tInd+1]
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if tInd == 0:
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return rhs
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dt = self.timeSteps[tInd]
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return rhs + 1.0/dt*self.MfMui*y[:,'b',tInd]
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def AhCalcFields(sol, tInd):
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y_b = sol
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if self.survey.nTx == 1:
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y_b = mkvc(y_b)
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y_e = self.MeSigmaI*(self.mesh.edgeCurl.T*(self.MfMui*y_b))
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if 'e' in p:
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y_e = y_e - self.MeSigmaI*p[:,'e',tInd+1]
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return {'b':y_b, 'e':y_e}
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return self.forward(m, AhRHS, AhCalcFields)
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def solveAht(self, m, p):
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"""
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:param numpy.array m: Conductivity model
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:param simpegEM.TDEM.FieldsTDEM p: Fields object
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: y
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Solve the block-matrix system \\\(\\\hat{A}^\\\\top \\\hat{y} = \\\hat{p}\\\):
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.. math::
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\mathbf{\hat{A}}^\\top = \left[
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\\begin{array}{cccc}
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A & B & & \\\\
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& \ddots & \ddots & \\\\
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& & A & B \\\\
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& & 0 & A
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\end{array}
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\\right] \\\\
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\mathbf{A} =
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\left[
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\\begin{array}{cc}
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\\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\\\
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\dcurl^\\top \MfMui & -\MeSig
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\end{array}
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\\right] \\\\
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\mathbf{B} =
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\left[
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\\begin{array}{cc}
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-\\frac{1}{\delta t} \MfMui & 0 \\\\
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0 & 0
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\end{array}
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\\right] \\\\
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"""
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# Mini Example:
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#
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# nT = 3, len(times) == 4, fields stored in F[:,:,1:4]
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#
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# 0 is held for initial conditions (this shifts the storage by +1)
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# ^
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# fLoc 0 1 2 3
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# |-----|-----|-----|
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# tInd 0 1 2
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# / ___/
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# 2 (tInd=2 uses fields 3 and would use 4 but it doesn't exist)
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# / ___/
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# 1 (tInd=1 uses fields 2 and 3)
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def AhtRHS(tInd, y):
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nTx, nF = self.survey.nTx, self.mesh.nF
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rhs = np.zeros(nF if nTx == 1 else (nF, nTx))
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if 'e' in p:
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rhs += self.MfMui*(self.mesh.edgeCurl*(self.MeSigmaI*p[:,'e',tInd+1]))
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if 'b' in p:
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rhs += p[:,'b',tInd+1]
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if tInd == self.nT-1:
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return rhs
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dt = self.timeSteps[tInd+1]
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return rhs + 1.0/dt*self.MfMui*y[:,'b',tInd+2]
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def AhtCalcFields(sol, tInd):
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y_b = sol
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if self.survey.nTx == 1:
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y_b = mkvc(y_b)
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y_e = self.MeSigmaI*(self.mesh.edgeCurl.T*(self.MfMui*y_b))
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if 'e' in p:
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y_e += - self.MeSigmaI*p[:,'e',tInd+1]
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return {'b':y_b, 'e':y_e}
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return self.adjoint(m, AhtRHS, AhtCalcFields)
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####################################################
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# Functions for tests
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####################################################
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def _AhVec(self, m, vec):
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"""
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:param numpy.array m: Conductivity model
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:param simpegEM.TDEM.FieldsTDEM vec: Fields object
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: f
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Multiply the matrix \\\(\\\hat{A}\\\) by a fields vector where
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.. math::
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\mathbf{\hat{A}} = \left[
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\\begin{array}{cccc}
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A & 0 & & \\\\
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B & A & & \\\\
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& \ddots & \ddots & \\\\
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& & B & A
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\end{array}
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\\right] \\\\
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\mathbf{A} =
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\left[
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\\begin{array}{cc}
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\\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\\\
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\dcurl^\\top \MfMui & -\MeSig
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\end{array}
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\\right] \\\\
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\mathbf{B} =
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\left[
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\\begin{array}{cc}
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-\\frac{1}{\delta t} \MfMui & 0 \\\\
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0 & 0
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\end{array}
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\\right] \\\\
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"""
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self.curModel = m
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f = FieldsTDEM(self.mesh, self.survey)
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for i in range(1,self.nT+1):
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dt = self.timeSteps[i-1]
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b = 1.0/dt*self.MfMui*vec[:,'b',i] + self.MfMui*(self.mesh.edgeCurl*vec[:,'e',i])
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if i > 1:
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b = b - 1.0/dt*self.MfMui*vec[:,'b',i-1]
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f[:,'b',i] = b
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f[:,'e',i] = self.mesh.edgeCurl.T*(self.MfMui*vec[:,'b',i]) - self.MeSigma*vec[:,'e',i]
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return f
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def _AhtVec(self, m, vec):
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"""
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:param numpy.array m: Conductivity model
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:param simpegEM.TDEM.FieldsTDEM vec: Fields object
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: f
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Multiply the matrix \\\(\\\hat{A}\\\) by a fields vector where
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.. math::
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\mathbf{\hat{A}}^\\top = \left[
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\\begin{array}{cccc}
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A & B & & \\\\
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& \ddots & \ddots & \\\\
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& & A & B \\\\
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& & 0 & A
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\end{array}
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\\right] \\\\
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\mathbf{A} =
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\left[
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\\begin{array}{cc}
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\\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\\\
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\dcurl^\\top \MfMui & -\MeSig
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\end{array}
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\\right] \\\\
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\mathbf{B} =
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\left[
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\\begin{array}{cc}
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-\\frac{1}{\delta t} \MfMui & 0 \\\\
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0 & 0
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\end{array}
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\\right] \\\\
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"""
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self.curModel = m
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f = FieldsTDEM(self.mesh, self.survey)
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for i in range(self.nT):
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b = 1.0/self.timeSteps[i]*self.MfMui*vec[:,'b',i+1] + self.MfMui*(self.mesh.edgeCurl*vec[:,'e',i+1])
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if i < self.nT-1:
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b = b - 1.0/self.timeSteps[i+1]*self.MfMui*vec[:,'b',i+2]
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f[:,'b', i+1] = b
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f[:,'e', i+1] = self.mesh.edgeCurl.T*(self.MfMui*vec[:,'b',i+1]) - self.MeSigma*vec[:,'e',i+1]
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return f
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