SimPEG.electromagnetics.time_domain.Simulation3DMagneticFluxDensity

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

Bases: 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 FieldsDerivativesEB
fieldsPair	alias of 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

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

Forward Simulation Including Inductive Response

Forward Simulation Including Inductive Response