Point Electrode Model

The point electrode model idealises every electrode as a single boundary point $x_{\ell }\in \partial \Omega$ (see Electrode Models). The current $I_{\ell }$ enters as a point source:

$$\nabla \cdot (\sigma \nabla u)=0\text{ in }\Omega ,\sigma {\partial }_{\nu }u=\sum _{\ell =1}^L{I}_{\ell }{\delta }_{x_{\ell }}\text{ on }\partial \Omega ,\sum _{\ell }{I}_{\ell }=0.$$

Weak form. For every test function $v$,

$${\int }_{\Omega }\sigma \nabla u\cdot \nabla v\mathrm{d}x=\sum _{\ell =1}^L{I}_{\ell }v(x_{\ell }).$$

The right-hand side needs point values of $v$, which $H^1(\Omega )$ functions do not have in two or more dimensions. The point source is not in $H^{-1/2}(\partial \Omega )$, and the solution is not in $H^1(\Omega )$: near a source in a half-plane with constant $\sigma$, $u\approx -\frac{I_{\ell }}{\pi \sigma }\log |x-x_{\ell }|$.

Consequence for measurements. The potential is infinite at a current-carrying point electrode. Voltages can only be measured at points $x_m$ that carry no current. In a finite element discretisation the nodal value at an injection node stays finite, but it grows like $\log (1/h)$ under mesh refinement and has no physical meaning.

Justification. For small electrodes, the Complete Electrode Model and the point model give almost the same relative data (voltage differences between non-injecting electrodes). The error vanishes as the electrode size tends to zero. This makes point electrodes a useful model for small electrodes and a convenient setting for theory.

Discretisation. The load vector of electrode $\ell$ is the unit vector of the node closest to $x_{\ell }$, and measuring picks nodal values (see Discrete Electrode Models). The voltage-driven counterpart prescribes $u(x_{\ell })=U_{\ell }$ at the electrode nodes and no current elsewhere. For consistent data it is the exact inverse of the current-driven problem.

In ModularEIT.jl: PointElectrodeModel.

References

1. M. Hanke, B. Harrach, N. Hyvönen (2011). Justification of point electrode models in electrical impedance tomography. Math. Models Methods Appl. Sci. 21(6), 1395–1413. doi:10.1142/S0218202511005362
2. 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