SimPEG.utils.inverse_property_tensor

SimPEG.utils.inverse_property_tensor(mesh, tensor, return_matrix=False, **kwargs)

Construct the inverse of the physical property tensor.

For a given mesh, the input parameter tensor is a numpy.ndarray defining the constitutive relationship (e.g. Ohm’s law) between two discrete vector quantities $\mathit{j}$ and $\mathit{e}$ living at cell centers. Where $\mathit{M}$ is the physical property tensor, inverse_property_tensor explicitly constructs the inverse of the physical property tensor ${\mathit{M}}^{\mathbf{-}\mathbf{1}}$ for all cells such that:

>>> e = Mi @ j

where the Cartesian components of the discrete vectors are organized according to:

>>> j = np.r_[jx, jy, jz]
>>> e = np.r_[ex, ey, ez]

Parameters

meshdiscretize.base.BaseMesh

A mesh

tensornumpy.ndarray or float

• Scalar: A float is entered.

• Isotropic: A 1D numpy.ndarray with a property value for every cell.

• Anisotropic: A (nCell, dim) numpy.ndarray where each row defines the diagonal-anisotropic property parameters for each cell. nParam = 2 for 2D meshes and nParam = 3 for 3D meshes.

• Tensor: A (nCell, nParam) numpy.ndarray where each row defines the full anisotropic property parameters for each cell. nParam = 3 for 2D meshes and nParam = 6 for 3D meshes.

return_matrixbool, optional

• True: the function returns the inverse of the property tensor.

• False: the function returns the non-zero elements of the inverse of the property tensor in a numpy.ndarray in the same order as the input argument tensor.

Returns

numpy.ndarray or scipy.sparse.coo_matrix

• If return_matrix = False, the function outputs the parameters defining the inverse of the property tensor in a numpy.ndarray with the same dimensions as the input argument tensor

• If return_natrix = True, the function outputs the inverse of the property tensor as a scipy.sparse.coo_matrix.

Notes

The relationship between a quantity and its response to external stimuli (e.g. Ohm’s law) in each cell can be defined by a scalar function $\sigma$ in the isotropic case, or by a tensor $\mathrm{\Sigma }$ in the anisotropic case, i.e.:

$$\overrightarrow{j}=\sigma \overrightarrow{e}\text{or}\overrightarrow{j}=\mathrm{\Sigma }\overrightarrow{e}$$

where

$$\begin{array}{r}\mathrm{\Sigma }=\left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{xy}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{xz}& {\sigma }_{yz}& {\sigma }_{zz}\end{array}\right]\end{array}$$

Examples