Gap Model

The gap model spreads the current of each electrode $e_{\ell }\subset \partial \Omega$ uniformly over the electrode and assumes no current in the gaps between electrodes (see Electrode Models):

$$\nabla \cdot (\sigma \nabla u)=0\text{ in }\Omega ,\sigma {\partial }_{\nu }u=\{\begin{array}{ll}{I}_{\ell }/|e_{\ell }|& \text{on }{e}_{\ell },\\ 0& \text{on }\partial \Omega \setminus {\bigcup }_{\ell }{e}_{\ell },\end{array}\sum _{\ell }{I}_{\ell }=0.$$

It is a Neumann Problem with piecewise constant data, so it is well posed up to an additive constant (see Grounding of the Potential). The voltage of electrode $m$ is modelled as the mean potential on it:

$$V_m=\frac{1}{|e_m|}{\int }_{e_m}u\mathrm{d}s.$$

Reciprocity. If the same electrodes inject and measure, the map $I\mapsto V$ is symmetric: the current weights $1/|e_{\ell }|$ on $e_{\ell }$ and the averaging weights coincide. It is positive semidefinite, and $I^{\top }V={\int }_{\Omega }\sigma |\nabla u{|}^2$ is the dissipated power.

Limitations. A metal electrode is a good conductor, so its surface is (nearly) an equipotential and the current density is not uniform. It concentrates at the electrode edges (see Shunt Model). The gap model ignores this and also ignores the contact impedance. It systematically overestimates the resistivity, and the voltages it predicts on current-carrying electrodes miss the contact voltage drop entirely. Voltages on electrodes without current are much less affected (see Measurement Protocols).

Injection and measurement electrodes may differ. Current can be driven through one set of electrodes while voltages are averaged over another set. Then the input-output map is no longer square.

Discretisation. With the finite element basis ${\phi }_i$, the load vector of electrode $\ell$ is ${\int }_{e_{\ell }}{\phi }_i\mathrm{d}s/|e_{\ell }|$, and the measurement matrix is its transpose (see Discrete Electrode Models).

In ModularEIT.jl: GapModel.

References

1. K.-S. Cheng, D. Isaacson, J. C. Newell, D. G. Gisser (1989). Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36(9), 918–924. doi:10.1109/10.35300
2. E. Somersalo, M. Cheney, D. Isaacson (1992). Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography. SIAM J. Appl. Math. 52(4), 1023–1040. doi:10.1137/0152060