SimPEG.electromagnetics.time_domain.Simulation3DMagneticFluxDensity

class SimPEG.electromagnetics.time_domain.Simulation3DMagneticFluxDensity(mesh, survey=None, dt_threshold=1e-08, **kwargs)

Bases: SimPEG.electromagnetics.time_domain.simulation.BaseTDEMSimulation

Starting from the quasi-static E-B formulation of Maxwell’s equations (semi-discretized)

$$\begin{array}{r}\mathbf{C}\mathbf{e}+\frac{\partial \mathbf{b}}{\partial t}={\mathbf{s}}_{\mathbf{m}}\\ {\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}\mathbf{b}-{\mathbf{M}}_{\sigma }^{\mathbf{e}}\mathbf{e}={\mathbf{s}}_{\mathbf{e}}\end{array}$$

where ${\mathbf{s}}_{\mathbf{e}}$ is an integrated quantity, we eliminate $\mathbf{e}$ using

$$\mathbf{e}={{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}\mathbf{b}-{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{s}}_{\mathbf{e}}$$

to obtain a second order semi-discretized system in $\mathbf{b}$

$$\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}\mathbf{b}+\frac{\partial \mathbf{b}}{\partial t}=\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{s}}_{\mathbf{e}}+{\mathbf{s}}_{\mathbf{m}}$$

and moving everything except the time derivative to the rhs gives

$$\frac{\partial \mathbf{b}}{\partial t}=-\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}\mathbf{b}+\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{s}}_{\mathbf{e}}+{\mathbf{s}}_{\mathbf{m}}$$

For the time discretization, we use backward euler. To solve for the $n+1$ th time step, we have

$$\frac{{\mathbf{b}}^{n+1}-{\mathbf{b}}^n}{\mathbf{dt}}=-\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}{\mathbf{b}}^{n+1}+\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{{\mathbf{s}}_{\mathbf{e}}}^{n+1}+{{\mathbf{s}}_{\mathbf{m}}}^{n+1}$$

re-arranging to put ${\mathbf{b}}^{n+1}$ on the left hand side gives

$$(\mathbf{I}+\mathbf{dt}\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{\mathbf{C}}^{\mathrm{\top }}{\mathbf{M}}_{{\mu }^{\mathbf{-}\mathbf{1}}}^{\mathbf{f}}){\mathbf{b}}^{n+1}={\mathbf{b}}^n+\mathbf{dt}(\mathbf{C}{{\mathbf{M}}_{\sigma }^{\mathbf{e}}}^{-1}{{\mathbf{s}}_{\mathbf{e}}}^{n+1}+{{\mathbf{s}}_{\mathbf{m}}}^{n+1})$$

Methods

Fields_Derivs	alias of SimPEG.electromagnetics.time_domain.fields.FieldsDerivativesEB
fieldsPair	alias of SimPEG.electromagnetics.time_domain.fields.Fields3DMagneticFluxDensity
getAdiag(tInd)	System matrix at a given time index
getAdiagDeriv(tInd, u, v[, adjoint])	Derivative of ADiag
getAsubdiag(tInd)	Matrix below the diagonal
getRHS(tInd)	Assemble the RHS
getRHSDeriv(tInd, src, v[, adjoint])	Derivative of the RHS

getAsubdiagDeriv

Galleries and Tutorials using SimPEG.electromagnetics.time_domain.Simulation3DMagneticFluxDensity

EM: TDEM: Permeable Target, Inductive Source

EM: TDEM: Permeable Target, Inductive Source

EM: TDEM: 1D: Inversion with VTEM waveform

EM: TDEM: 1D: Inversion with VTEM waveform

Heagy et al., 2017 1D FDEM and TDEM inversions

Heagy et al., 2017 1D FDEM and TDEM inversions

Forward Simulation Including Inductive Response

Forward Simulation Including Inductive Response

3D Forward Simulation for Transient Response on a Cylindrical Mesh

3D Forward Simulation for Transient Response on a Cylindrical Mesh

3D Forward Simulation with User-Defined Waveforms

3D Forward Simulation with User-Defined Waveforms