Complete Electrode Model in Electrical Impedance Tomography #906
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Complete Electrode Model
For an electrically conductive region Ω, described by its conductivity σ(x), with boundary$\partial$ Ω partially covered by $L$ electrodes {$e_\ell$}, potential $u$ is governed by Laplace's equation.
The electrodes are assumed to be perfectly conductive with a contact impedance$z_\ell$ between the electrodes and the domain. The voltages {$U_\ell$} on these electrodes are
The currents {$I_\ell$} on each electrode are
And there is no boundary current except at electrodes.
Conservation of charge requires
And a "ground" voltage is established by requiring
Equation (1) and the boundary conditions in (2)-(6) describe the Complete Electrode Model (CEM) for which a solution exists and is unique. [https://doi.org/10.1137/0152060]
Variational Form
The CEM can be expressed as a bilinear form [https://doi.org/10.3934/math.2021431]
Finite Dimensional Approximation
The solution can be approximated with a finite dimensional representation
Similarly the$L$ voltages on the electrodes can be represented by a set of basis vectors spanning $\mathcal{R}^{L-1}_\diamond$ , the space of zero mean vectors of length $L-1$ .
Using these representations, and considering$N$ test functions ($\varphi_i$ , $0$ )
Similarly, considering$L-1$ test functions ($0$ , $\phi_i$ )
Choosing$\phi_1 = [-1,1,0,0,0...,0]$ , $\phi_2 = [-1,0,1,0,0...,0]$ , $\phi_3 = [-1,0,0,1,0...,0]$ , etc, and using
allows to write
which can be solved with the usual methods.
With
skfem
edit1: added missing import
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