SimPEG.electromagnetics.time_domain.Simulation3DMagneticFluxDensity#

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

Bases: BaseTDEMSimulation

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

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

where \(\mathbf{s_e}\) is an integrated quantity, we eliminate \(\mathbf{e}\) using

\[\mathbf{e} = \mathbf{M_{\sigma}^e}^{-1} \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{M_{\sigma}^e}^{-1} \mathbf{s_e}\]

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

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

and moving everything except the time derivative to the rhs gives

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

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

\[\frac{\mathbf{b}^{n+1} - \mathbf{b}^{n}}{\mathbf{dt}} = -\mathbf{C} \mathbf{M_{\sigma}^e}^{-1} \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f} \mathbf{b}^{n+1} + \mathbf{C} \mathbf{M_{\sigma}^e}^{-1} \mathbf{s_e}^{n+1} + \mathbf{s_m}^{n+1}\]

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

\[(\mathbf{I} + \mathbf{dt} \mathbf{C} \mathbf{M_{\sigma}^e}^{-1} \mathbf{C}^{\top} \mathbf{M_{\mu^{-1}}^f}) \mathbf{b}^{n+1} = \mathbf{b}^{n} + \mathbf{dt}(\mathbf{C} \mathbf{M_{\sigma}^e}^{-1} \mathbf{s_e}^{n+1} + \mathbf{s_m}^{n+1})\]

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