simpeg.electromagnetics.time_domain.Simulation3DMagneticFluxDensity#

class simpeg.electromagnetics.time_domain.Simulation3DMagneticFluxDensity(mesh, survey=None, dt_threshold=1e-08, store_factors=False, **kwargs)[source]#

Bases: BaseTDEMSimulation

3D TDEM simulation in terms of the magnetic flux density.

This simulation solves for the magnetic flux density at each time-step. In this formulation, the electric fields are defined on mesh edges and the magnetic flux density is defined on mesh faces; i.e. it is an EB formulation. See the Notes section for a comprehensive description of the formulation.

Parameters:

meshdiscretize.base.BaseMesh: The mesh.
surveytime_domain.survey.Survey: The time-domain EM survey.
dt_thresholdfloat: Threshold used when determining the unique time-step lengths.

Attributes

`Adcinv`	Inverse of the factored system matrix for the DC resistivity problem.
`Mcc`	Get cell inner product matrix.
`MccI`	Get inverse of cell inner product matrix.
`MccMu`	Get cell center property inner product matrix.
`MccMuI`	Get cell center property inner product inverse matrix.
`MccMui`	Get cell center property inner product matrix.
`MccMuiI`	Get cell center property inner product inverse matrix.
`MccRho`	Get cell center property inner product matrix.
`MccRhoI`	Get cell center property inner product inverse matrix.
`MccSigma`	Get cell center property inner product matrix.
`MccSigmaI`	Get cell center property inner product inverse matrix.
`Me`	Get edge inner product matrix.
`MeI`	Get inverse of edge inner product matrix.
`MeMu`	Get edge property inner product matrix.
`MeMuI`	Get edge property inner product inverse matrix.
`MeMui`	Get edge property inner product matrix.
`MeMuiI`	Get edge property inner product inverse matrix.
`MeRho`	Get edge property inner product matrix.
`MeRhoI`	Get edge property inner product inverse matrix.
`MeSigma`	Get edge property inner product matrix.
`MeSigmaI`	Get edge property inner product inverse matrix.
`Mf`	Get face inner product matrix.
`MfI`	Get inverse of face inner product matrix.
`MfMu`	Get face property inner product matrix.
`MfMuI`	Get face property inner product inverse matrix.
`MfMui`	Get face property inner product matrix.
`MfMuiI`	Get face property inner product inverse matrix.
`MfRho`	Get face property inner product matrix.
`MfRhoI`	Get face property inner product inverse matrix.
`MfSigma`	Get face property inner product matrix.
`MfSigmaI`	Get face property inner product inverse matrix.
`Mn`	Get node inner product matrix.
`MnI`	Get inverse of node inner product matrix.
`MnMu`	Get node property inner product matrix.
`MnMuI`	Get node property inner product inverse matrix.
`MnMui`	Get node property inner product matrix.
`MnMuiI`	Get node property inner product inverse matrix.
`MnRho`	Get node property inner product matrix.
`MnRhoI`	Get node property inner product inverse matrix.
`MnSigma`	Get node property inner product matrix.
`MnSigmaI`	Get node property inner product inverse matrix.
`Vol`	Get volumes.
`clean_on_model_update`	A list of solver objects to clean when the model is updated
`counter`	SimPEG `Counter` object to store iterations and run-times.
`deleteTheseOnModelUpdate`	HasModel.deleteTheseOnModelUpdate has been deprecated.
`dt_threshold`	Threshold used when determining the unique time-step lengths.
`mesh`	Mesh for the simulation.
`model`	The inversion model.
`mu`	Magnetic permeability (h/m) physical property model.
`muDeriv`	Derivative of Magnetic Permeability (H/m) wrt the model.
`muMap`	Mapping of the inversion model to Magnetic Permeability (H/m).
`mui`	Inverse magnetic permeability (m/h) physical property model.
`muiDeriv`	Derivative of Inverse Magnetic Permeability (m/H) wrt the model.
`muiMap`	Mapping of the inversion model to Inverse Magnetic Permeability (m/H).
`nT`	Total number of time steps.
`needs_model`	True if a model is necessary
`rho`	Electrical resistivity (ohm m) physical property model.
`rhoDeriv`	Derivative of Electrical resistivity (Ohm m) wrt the model.
`rhoMap`	Mapping of the inversion model to Electrical resistivity (Ohm m).
`sensitivity_path`	Path to directory where sensitivity file is stored.
`sigma`	Electrical conductivity (s/m) physical property model.
`sigmaDeriv`	Derivative of Electrical conductivity (S/m) wrt the model.
`sigmaMap`	Mapping of the inversion model to Electrical conductivity (S/m).
`solver`	Numerical solver used in the forward simulation.
`solver_opts`	Set solver-specific parameters.
`storeInnerProduct`	Whether to store inner product matrices
`store_factors`	Whether the factorization for each unique time step is stored.
`survey`	The TDEM survey object.
`t0`	Initial time, in seconds, for the time-dependent forward simulation.
`time_mesh`	Time mesh for easy interpolation to observation times.
`time_steps`	Time step lengths, in seconds, for the time domain simulation.
`times`	Evaluation times.
`verbose`	Verbose progress printout.

Methods

`Fields_Derivs`
`Jtvec`(m, v[, f])	Compute the adjoint sensitivity matrix times a vector.
`Jtvec_approx`(m, v[, f])	Approximation of the Jacobian transpose times a vector for the model provided.
`Jvec`(m, v[, f])	Compute the sensitivity matrix times a vector.
`Jvec_approx`(m, v[, f])	Approximation of the Jacobian times a vector for the model provided.
`MccMuDeriv`(u[, v, adjoint])	Get derivative of MccProperty with respect to the model.
`MccMuIDeriv`(u[, v, adjoint])	Get derivative of MccPropertyI with respect to the model.
`MccMuiDeriv`(u[, v, adjoint])	Get derivative of MccProperty with respect to the model.
`MccMuiIDeriv`(u[, v, adjoint])	Get derivative of MccPropertyI with respect to the model.
`MccRhoDeriv`(u[, v, adjoint])	Get derivative of MccProperty with respect to the model.
`MccRhoIDeriv`(u[, v, adjoint])	Get derivative of MccPropertyI with respect to the model.
`MccSigmaDeriv`(u[, v, adjoint])	Get derivative of MccProperty with respect to the model.
`MccSigmaIDeriv`(u[, v, adjoint])	Get derivative of MccPropertyI with respect to the model.
`MeMuDeriv`(u[, v, adjoint])	Get derivative of MeProperty with respect to the model.
`MeMuIDeriv`(u[, v, adjoint])	Get derivative of MePropertyI with respect to the model.
`MeMuiDeriv`(u[, v, adjoint])	Get derivative of MeProperty with respect to the model.
`MeMuiIDeriv`(u[, v, adjoint])	Get derivative of MePropertyI with respect to the model.
`MeRhoDeriv`(u[, v, adjoint])	Get derivative of MeProperty with respect to the model.
`MeRhoIDeriv`(u[, v, adjoint])	Get derivative of MePropertyI with respect to the model.
`MeSigmaDeriv`(u[, v, adjoint])	Get derivative of MeProperty with respect to the model.
`MeSigmaIDeriv`(u[, v, adjoint])	Get derivative of MePropertyI with respect to the model.
`MfMuDeriv`(u[, v, adjoint])	Get derivative of MfProperty with respect to the model.
`MfMuIDeriv`(u[, v, adjoint])	Get derivative of MfPropertyI with respect to the model.
`MfMuiDeriv`(u[, v, adjoint])	Get derivative of MfProperty with respect to the model.
`MfMuiIDeriv`(u[, v, adjoint])	Get derivative of MfPropertyI with respect to the model.
`MfRhoDeriv`(u[, v, adjoint])	Get derivative of MfProperty with respect to the model.
`MfRhoIDeriv`(u[, v, adjoint])	Get derivative of MfPropertyI with respect to the model.
`MfSigmaDeriv`(u[, v, adjoint])	Get derivative of MfProperty with respect to the model.
`MfSigmaIDeriv`(u[, v, adjoint])	Get derivative of MfPropertyI with respect to the model.
`MnMuDeriv`(u[, v, adjoint])	Get derivative of MnProperty with respect to the model.
`MnMuIDeriv`(u[, v, adjoint])	Get derivative of MnPropertyI with respect to the model.
`MnMuiDeriv`(u[, v, adjoint])	Get derivative of MnProperty with respect to the model.
`MnMuiIDeriv`(u[, v, adjoint])	Get derivative of MnPropertyI with respect to the model.
`MnRhoDeriv`(u[, v, adjoint])	Get derivative of MnProperty with respect to the model.
`MnRhoIDeriv`(u[, v, adjoint])	Get derivative of MnPropertyI with respect to the model.
`MnSigmaDeriv`(u[, v, adjoint])	Get derivative of MnProperty with respect to the model.
`MnSigmaIDeriv`(u[, v, adjoint])	Get derivative of MnPropertyI with respect to the model.
`dpred`([m, f])	Predicted data for the model provided.
`fields`(m)	Compute and return the fields for the model provided.
`fieldsPair`
`getAdiag`(tInd)	Return diagonal system matrix for the given time-step index.
`getAdiagDeriv`(tInd, u, v[, adjoint])	Return derivative operation for the diagonal system matrix times a vector.
`getAsubdiag`(tInd)	Return sub-diagonal system matrix for the time-step index provided.
`getAsubdiagDeriv`(tInd, u, v[, adjoint])	Return derivative operation for the sub-diagonal system matrix times a vector.
`getInitialFields`()	Return the fields for all sources at the initial time.
`getInitialFieldsDeriv`(src, v[, adjoint, f])	Return derivative of initial fields with respect to the model for a given source.
`getRHS`(tInd)	Return right-hand sides for the given time index.
`getRHSDeriv`(tInd, src, v[, adjoint])	Return right-hand side derivative times a vector for a given source and time index.
`getSourceTerm`(tInd)	Return the discrete source terms for the time index provided.
`make_synthetic_data`(m[, relative_error, ...])	Make synthetic data for the model and Gaussian noise provided.
`residual`(m, dobs[, f])	The data residual.

Notes

Here, we start with the quasi-static approximation for Maxwell’s equations by neglecting electric displacement:

\[\begin{split}&\nabla \times \vec{e} + \frac{\partial \vec{b}}{\partial t} = - \frac{\partial \vec{s}_m}{\partial t} \\ &\nabla \times \vec{h} - \vec{j} = \vec{s}_e\end{split}\]

where \(\vec{s}_e\) is an electric source term that defines a source current density, and \(\vec{s}_m\) magnetic source term that defines a source magnetic flux density. We define the constitutive relations for the electrical conductivity \(\sigma\) and magnetic permeability \(\mu\) as:

\[\begin{split}\vec{j} &= \sigma \vec{e} \\ \vec{h} &= \mu^{-1} \vec{b}\end{split}\]

We then take the inner products of all previous expressions with a vector test function \(\vec{u}\). Through vector calculus identities and the divergence theorem, we obtain:

\[\begin{split}& \int_\Omega \vec{u} \cdot (\nabla \times \vec{e}) \, dv + \int_\Omega \vec{u} \cdot \frac{\partial \vec{b}}{\partial t} \, dv = - \int_\Omega \vec{u} \cdot \frac{\partial \vec{s}_m}{\partial t} \, dv \\ & \int_\Omega (\nabla \times \vec{u}) \cdot \vec{h} \, dv - \oint_{\partial \Omega} \vec{u} \cdot (\vec{h} \times \hat{n}) \, da - \int_\Omega \vec{u} \cdot \vec{j} \, dv = \int_\Omega \vec{u} \cdot \vec{s}_e \, dv \\ & \int_\Omega \vec{u} \cdot \vec{j} \, dv = \int_\Omega \vec{u} \cdot \sigma \vec{e} \, dv \\ & \int_\Omega \vec{u} \cdot \vec{h} \, dv = \int_\Omega \vec{u} \cdot \mu^{-1} \vec{b} \, dv\end{split}\]

Assuming natural boundary conditions, the surface integral is zero.

The above expressions are discretized in space according to the finite volume method. The discrete electric fields \(\mathbf{e}\) are defined on mesh edges, and the discrete magnetic flux densities \(\mathbf{b}\) are defined on mesh faces. This implies \(\mathbf{j}\) must be defined on mesh edges and \(\mathbf{h}\) must be defined on mesh faces. Where \(\mathbf{u_e}\) and \(\mathbf{u_f}\) represent test functions discretized to edges and faces, respectively, we obtain the following set of discrete inner-products:

\[\begin{split}&\mathbf{u_f^T C e} + \mathbf{u_f^T } \, \frac{\partial \mathbf{b}}{\partial t} = - \mathbf{u_f^T } \, \frac{\partial \mathbf{s_m}}{\partial t} \\ &\mathbf{u_e^T C^T M_f h} - \mathbf{u_e^T M_e j} = \mathbf{u_e^T s_e} \\ &\mathbf{u_e^T M_e j} = \mathbf{u_e^T M_{e\sigma} e} \\ &\mathbf{u_f^T M_f h} = \mathbf{u_f^T M_{f \frac{1}{\mu}} b}\end{split}\]

where

\(\mathbf{C}\) is the discrete curl operator
\(\mathbf{s_m}\) and \(\mathbf{s_e}\) are the integrated magnetic and electric source terms, respectively
\(\mathbf{M_e}\) is the edge inner-product matrix
\(\mathbf{M_f}\) is the face inner-product matrix
\(\mathbf{M_{e\sigma}}\) is the inner-product matrix for conductivities projected to edges
\(\mathbf{M_{f\frac{1}{\mu}}}\) is the inner-product matrix for inverse permeabilities projected to faces

By cancelling like-terms and combining the discrete expressions in terms of the magnetic flux density, we obtain:

\[\mathbf{C M_{e\sigma}^{-1} C^T M_{f\frac{1}{\mu}} b} + \frac{\partial \mathbf{b}}{\partial t} = \mathbf{C M_{e\sigma}^{-1}} \mathbf{s_e} - \frac{\partial \mathbf{s_m}}{\partial t}\]

Finally, we discretize in time according to backward Euler. The discrete magnetic flux density on mesh faces at time \(t_k > t_0\) is obtained by solving the following at each time-step:

\[\mathbf{A}_k \mathbf{b}_k = \mathbf{q_k} - \mathbf{B}_k \mathbf{b}_{k-1}\]

where \(\Delta t_k = t_k - t_{k-1}\) and

\[\begin{split}&\mathbf{A}_k = \mathbf{C M_{e\sigma}^{-1} C^T M_{f\frac{1}{\mu}}} + \frac{1}{\Delta t_k} \mathbf{I} \\ &\mathbf{B}_k = -\frac{1}{\Delta t_k} \mathbf{I}\\ &\mathbf{q}_k = \mathbf{C M_{e\sigma}^{-1}} \mathbf{s}_{\mathbf{e}, k} \; - \; \frac{1}{\Delta t_k} \big [ \mathbf{s}_{\mathbf{m}, k} - \mathbf{s}_{\mathbf{m}, k-1} \big ]\end{split}\]

Although the following system is never explicitly formed, we can represent the solution at all time-steps as:

\[\begin{split}\begin{bmatrix} \mathbf{A_1} & & & & \\ \mathbf{B_2} & \mathbf{A_2} & & & \\ & & \ddots & & \\ & & & \mathbf{B_n} & \mathbf{A_n} \end{bmatrix} \begin{bmatrix} \mathbf{b_1} \\ \mathbf{b_2} \\ \vdots \\ \mathbf{b_n} \end{bmatrix} = \begin{bmatrix} \mathbf{q_1} \\ \mathbf{q_2} \\ \vdots \\ \mathbf{q_n} \end{bmatrix} - \begin{bmatrix} \mathbf{B_1 b_0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \end{bmatrix}\end{split}\]

where the magnetic flux densities at the initial time \(\mathbf{b_0}\) are computed analytically or numerically depending on whether the source carries non-zero current at the initial time.