3
rsta.royalsocietypublishing.org Phil.Trans.R.Soc.A374: 20150333
.........................................................
to estimate the unknown permittivity and shape parameters from the boundary capacitance
measurements. The boundary and finite-elements coupling method is used to simplify the
numerical implementation of the simultaneous reconstruction approach. The simulation results
show that the proposed method can mitigate the shielding effect of the deposit layer on the ROI,
and significantly improve the permittivity reconstruction results. The remainder of the paper is
organized as follow: in §2a, the measurement principles and governing equations of ECT are
briefly described, in §2b, the target interface and permittivity distribution are represented by a set
of parameters and in §2c, the unknown parameters are estimated by using the block coordinate
descent method; in §3, the boundary and finite-elements coupling method in ECT is described; in
§4, several numerical tests are conducted to evaluate the performance of the proposed method.
2. Methodology
(a) Governing equations
Figure 1 gives a schematic of the capacitance tomography problem considered in the paper. The
entire observed domain Ω consists of the electrode mounting layer Ω
p
, the deposit layer Ω
d
and
the inhomogeneous permittivity section Ω
m
(the ROI of the ECT). A set of electrodes E is mounted
around the periphery of the observed domain. Each time, one of the electrodes is stimulated by a
voltage signal (used as the drive electrode), while the others are at the virtual earth potential (used
as the detecting electrodes). Mutual capacitances between the drive and detecting electrodes are
sequentially measured by a measuring circuit.
Following Maxwell’s theory, the electrical potential u at the entire observed region Ω is
governed by the elliptic partial differential equation
∇·ε(x)∇u(x) = 0, x ∈ Ω, (2.1)
where x ∈ R
k
is the k-dimensional spatial variable, ε ∈ C
k
is the permittivity distribution. In
the paper, we assume the electrode mounting layer and the deposit layer are occupied by
the homogeneous permittivity. As the ε is constant in these regions, the associated governing
equations are simplified to the Laplace equations
u(x) = 0, x ∈ Ω
p
∪ Ω
d
. (2.2)
The boundary conditions at the exterior boundary ∂Ω are determined by the data collection
strategy. During ith voltage stimulation, the electric potentials at the electrodes follow the
Dirichlet condition
u(x) = δ(i, j)V
c
, x ∈ E
j
, (2.3)
where δ(i, j) is Kronecker delta equalling zero except when i = j, V
c
is the drive voltage amplitude,
and E
j
denotes the jth electrode. The electric charge densities at the inter-electrode gaps follow
the Neumann condition
ε(x)q(x) = 0, x ∈ G, (2.4)
where q = ∂u/∂ν, ν is the outward unit normal to the counterclockwise-oriented boundary, and
G = ∂Ω \∪
L
l=1
E
l
is the combination of the inter-electrode gaps.
The boundary conditions at the interior interfaces are drawn from the integral formulation of
Maxwell’s equations
u
p
(x) = u
d
(x)andε
p
q
p
(x) =−ε
d
q
d
(x), x ∈ Γ
p
(2.5)
and
u
d
(x) = u
m
(x)andε
d
q
d
(x) =−ε
m
(x)q
m
(x), x ∈ Γ
d
, (2.6)
where Γ
p
= Ω
p
∩ Ω
d
is the interface between the electrode mounting layer and the deposit layer,
and Γ
d
= Ω
d
∩ Ω
m
is the interface between the deposit layer and the inhomogeneous permittivity
section. The subscripts p, d and f stand for the electric quantities calculated at the side of
on December 8, 2016http://rsta.royalsocietypublishing.org/Downloaded from
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