simpeg.electromagnetics.time_domain.Simulation3DCurrentDensity.getAdc

Simulation3DCurrentDensity.getAdc()

The system matrix for the DC resistivity problem.

The solution to the DC resistivity problem is necessary at the initial time for galvanic sources whose currents are non-zero at the initial time. The discrete solution to the 3D DC resistivity problem is expressed as:

\[\mathbf{A_{dc}}\,\boldsymbol{\phi_0} = \mathbf{q_{dc}}\]

where \(\mathbf{A_{dc}}\) is the DC resistivity system matrix, \(\boldsymbol{\phi_0}\) is the discrete solution for the electric potentials at the initial time, and \(\mathbf{q_{dc}}\) is the galvanic source term. This method returns the system matrix for the cell-centered formulation, i.e.:

\[\mathbf{D \, M_{f\rho}^{-1} \, G}\]

where \(\mathbf{D}\) is the face divergence operator, \(\mathbf{G}\) is the cell gradient operator with imposed boundary conditions, and \(\mathbf{M_{f\rho}}\) is the inner product matrix for resistivities projected to faces.

The current density at the initial time \(\mathbf{j_0}\) are obtained by applying:

\[\mathbf{j_0} = \mathbf{M_{f\rho}^{-1} \, G} \, \boldsymbol{\phi_0}\]

See the Notes section of the doc strings for resistivity.Simulation3DCellCentered for a full description of the cell centered DC resistivity formulation.

Returns:

(n_cells, n_cells) sp.sparse.csr_matrix

The system matrix for the DC resistivity problem.