simpeg.electromagnetics.time_domain.Simulation3DCurrentDensity.getAdc#

Simulation3DCurrentDensity.getAdc()[source]#

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:

$${\mathbf{A}}_{\mathbf{dc}}{\mathit{\varphi }}_{\mathbf{0}}={\mathbf{q}}_{\mathbf{dc}}$$

where ${\mathbf{A}}_{\mathbf{dc}}$ is the DC resistivity system matrix, ${\mathit{\varphi }}_{\mathbf{0}}$ is the discrete solution for the electric potentials at the initial time, and ${\mathbf{q}}_{\mathbf{dc}}$ is the galvanic source term. This method returns the system matrix for the cell-centered formulation, i.e.:

$$\mathbf{D}{\mathbf{M}}_{\mathbf{f}\rho }^{\mathbf{-}\mathbf{1}}\mathbf{G}$$

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

The current density at the initial time ${\mathbf{j}}_{\mathbf{0}}$ are obtained by applying:

$${\mathbf{j}}_{\mathbf{0}}={\mathbf{M}}_{\mathbf{f}\rho }^{\mathbf{-}\mathbf{1}}\mathbf{G}{\mathit{\varphi }}_{\mathbf{0}}$$

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.