simpeg.maps.SelfConsistentEffectiveMedium#

class simpeg.maps.SelfConsistentEffectiveMedium(mesh=None, nP=None, sigma0=None, sigma1=None, alpha0=1.0, alpha1=1.0, orientation0='z', orientation1='z', random=True, rel_tol=0.001, maxIter=50, **kwargs)[source]#

Bases: IdentityMap

Two phase self-consistent effective medium theory mapping for ellipsoidal inclusions. The inversion model is the concentration (volume fraction) of the phase 2 material.

The inversion model is $\phi$. We solve for $\sigma$ given ${\sigma }_0$, ${\sigma }_1$ and $\phi$ . Each of the following are implicit expressions of the effective conductivity. They are solved using a fixed point iteration.

Spherical Inclusions

If the shape of the inclusions are spheres, we use

$$\sum _{j=1}^N({\sigma }^{\ast }-{\sigma }_j)R^j=0$$

where $j=[1,N]$ is the each material phase, and N is the number of phases. Currently, the implementation is only set up for 2 phase materials, so we solve

$$\begin{array}{r}(1-\\ varphi)(\sigma -{\sigma }_0)R^{(0)}+\phi (\sigma -{\sigma }_1)R^{(1)}=0.\end{array}$$

Where $R^{(j)}$ is given by

$$R^{(j)}={[1+\frac{1}{3}\frac{{\sigma }_j-\sigma }{\sigma }]}^{-1}.$$

Ellipsoids

If the inclusions are aligned ellipsoids, we solve

$$\sum _{j=1}^N{\phi }_j({\mathrm{\Sigma }}^{\ast }-{\sigma }_j\mathbf{I}){\mathbf{R}}^{j,\ast }=0$$

where

$$\text{Missing begin{split} or extra end{split}}$$

and the depolarization tensor ${\mathbf{A}}_j$ is given by

$$\begin{array}{r}{\mathbf{A}}^{\ast }=\left[\begin{array}{ccc}Q& 0& 0\\ 0& Q& 0\\ 0& 0& 1-2Q\end{array}\right]\end{array}$$

for a spheroid aligned along the z-axis. For an oblate spheroid ($\alpha <1$, pancake-like)

$$Q=\frac{1}{2}(1+\frac{1}{{\alpha }^2-1}[1-\frac{1}{\chi }{\tan }^{-1}(\chi )])$$

where

$$\chi =\sqrt{\frac{1}{{\alpha }^2}-1}$$

For reference, see Torquato (2002), Random Heterogeneous Materials

Attributes

alpha0	Aspect ratio of the phase-0 ellipsoids.
alpha1	Aspect ratio of the phase-1 ellipsoids.
is_linear	Determine whether or not this mapping is a linear operation.
maxIter	Maximum number of iterations for the fixed point iteration calculation.
mesh	The mesh used for the mapping
nP	Number of parameters the mapping acts on.
orientation0	Orientation of the phase-0 inclusions.
orientation1	Orientation of the phase-0 inclusions.
random	Are the inclusions randomly oriented (True) or preferentially aligned (False)?
rel_tol	relative tolerance for convergence for the fixed-point iteration.
shape	Dimensions of the mapping operator
sigma0	Physical property value for phase-0 material.
sigma1	Physical property value for phase-1 material.
sigstart	first guess for sigma
tol	absolute tolerance for the convergence of the fixed point iteration calc

Methods

deriv(m)	Derivative of the effective conductivity with respect to the volume fraction of phase 2 material
dot(map1)	Multiply two mappings to create a simpeg.maps.ComboMap.
getA(alpha, orientation)	Depolarization tensor
getQ(alpha)	Geometric factor in the depolarization tensor
getR(sj, se, alpha[, orientation])	Electric field concentration tensor
getdR(sj, se, alpha[, orientation])	Derivative of the electric field concentration tensor with respect to the concentration of the second phase material.
hashin_shtrikman_bounds(phi1)	Hashin Shtrikman bounds
hashin_shtrikman_bounds_anisotropic(phi1)	Hashin Shtrikman bounds for anisotropic media
inverse(sige)	Compute the concentration given the effective conductivity
test([m, num, random_seed])	Derivative test for the mapping.
wiener_bounds(phi1)	Define Wenner Conductivity Bounds

Galleries and Tutorials using simpeg.maps.SelfConsistentEffectiveMedium#

Effective Medium Theory Mapping

Effective Medium Theory Mapping

Straight Ray with Volume Data Misfit Term

Straight Ray with Volume Data Misfit Term