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https://github.com/wassname/simpeg.git
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Some documentation.
This commit is contained in:
Dave Marchant
+93
-22
@@ -7,11 +7,13 @@ class ProblemTDEM_b(ProblemBaseTDEM):
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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, model, **kwargs):
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ProblemBaseTDEM.__init__(self, mesh, model, **kwargs)
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@@ -56,11 +58,20 @@ class ProblemTDEM_b(ProblemBaseTDEM):
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w = self.Gtvec(m, y, u)
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return w
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def Gvec(self, m, v, u=None):
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def Gvec(self, sigma, vec, u=None):
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"""
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:param numpy.array sigma: 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 sigma
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:rtype: simpegEM.TDEM.FieldsTDEM
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:return: f
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Multiply G by a vector where
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"""
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if u is None:
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u = self.fields(m)
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u = self.fields(sigma)
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p = FieldsTDEM(self.mesh, 1, self.times.size, 'b')
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c = self.mesh.getEdgeMassDeriv()*self.model.transformDeriv(None)*v
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c = self.mesh.getEdgeMassDeriv()*self.model.transformDeriv(None)*vec
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for i in range(self.times.size):
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ei = u.get_e(i)
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pVal = np.empty_like(ei)
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@@ -80,7 +91,6 @@ class ProblemTDEM_b(ProblemBaseTDEM):
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p = -mkvc(self.model.transformDeriv(None).T*self.mesh.getEdgeMassDeriv().T*tmp)
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return p
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def solveAh(self, m, p):
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def AhRHS(tInd, u):
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rhs = self.MfMui*self.mesh.edgeCurl*self.MeSigmaI*p.get_e(tInd) + p.get_b(tInd)
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@@ -118,37 +128,98 @@ class ProblemTDEM_b(ProblemBaseTDEM):
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# Functions for tests
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####################################################
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def AhVec(self, m, u=None):
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if u is None:
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u = self.fields(m)
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self.makeMassMatrices(m)
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def AhVec(self, sigma, vec):
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"""
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:param numpy.array sigma: 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.makeMassMatrices(sigma)
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dt = self.getDt(0)
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b = 1/dt*self.MfMui*u.get_b(0) + self.MfMui*self.mesh.edgeCurl*u.get_e(0)
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e = self.mesh.edgeCurl.T*self.MfMui*u.get_b(0) - self.MeSigma*u.get_e(0)
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b = 1/dt*self.MfMui*vec.get_b(0) + self.MfMui*self.mesh.edgeCurl*vec.get_e(0)
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e = self.mesh.edgeCurl.T*self.MfMui*vec.get_b(0) - self.MeSigma*vec.get_e(0)
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f = FieldsTDEM(self.mesh, 1, self.times.size, 'b')
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f.set_b(b, 0)
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f.set_e(e, 0)
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for i in range(1,self.nTimes):
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dt = self.getDt(i)
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b = 1/dt*self.MfMui*u.get_b(i) + self.MfMui*self.mesh.edgeCurl*u.get_e(i) - 1/dt*self.MfMui*u.get_b(i-1)
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e = self.mesh.edgeCurl.T*self.MfMui*u.get_b(i) - self.MeSigma*u.get_e(i)
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b = 1/dt*self.MfMui*vec.get_b(i) + self.MfMui*self.mesh.edgeCurl*vec.get_e(i) - 1/dt*self.MfMui*vec.get_b(i-1)
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e = self.mesh.edgeCurl.T*self.MfMui*vec.get_b(i) - self.MeSigma*vec.get_e(i)
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f.set_b(b, i)
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f.set_e(e, i)
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return f
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def AhtVec(self, m, u=None):
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if u is None:
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u = self.fields(m)
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self.makeMassMatrices(m)
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def AhtVec(self, sigma, vec):
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"""
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:param numpy.array sigma: 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.makeMassMatrices(sigma)
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f = FieldsTDEM(self.mesh, 1, self.times.size, 'b')
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for i in range(self.nTimes-1):
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b = 1/self.getDt(i)*self.MfMui*u.get_b(i) + self.MfMui*self.mesh.edgeCurl*u.get_e(i) - 1/self.getDt(i+1)*self.MfMui*u.get_b(i+1)
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e = self.mesh.edgeCurl.T*self.MfMui*u.get_b(i) - self.MeSigma*u.get_e(i)
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b = 1/self.getDt(i)*self.MfMui*vec.get_b(i) + self.MfMui*self.mesh.edgeCurl*vec.get_e(i) - 1/self.getDt(i+1)*self.MfMui*vec.get_b(i+1)
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e = self.mesh.edgeCurl.T*self.MfMui*vec.get_b(i) - self.MeSigma*vec.get_e(i)
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f.set_b(b, i)
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f.set_e(e, i)
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N = self.nTimes - 1
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b = 1/self.getDt(N)*self.MfMui*u.get_b(N) + self.MfMui*self.mesh.edgeCurl*u.get_e(N)
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e = self.mesh.edgeCurl.T*self.MfMui*u.get_b(N) - self.MeSigma*u.get_e(N)
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b = 1/self.getDt(N)*self.MfMui*vec.get_b(N) + self.MfMui*self.mesh.edgeCurl*vec.get_e(N)
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e = self.mesh.edgeCurl.T*self.MfMui*vec.get_b(N) - self.MeSigma*vec.get_e(N)
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f.set_b(b, N)
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f.set_e(e, N)
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return f
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