边界-有限元法：电容层析成像中界面与介电常数同步重建

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本文探讨了基于边界和有限元耦合方法的电容层析成像（Electrical Capacitance Tomography, ECT）在同时重建界面和介电常数方面的创新研究。电容层析成像是非侵入性检测技术，广泛应用于材料科学、能源和工业过程控制等领域，以测量物体内部的电导率分布。这项工作发表于《皇家学会A》（Philosophical Transactions of the Royal Society A），由Shangjie Ren和Feng Dong两位作者共同完成，他们的研究发表在2016年3月7日的一期主题为“通过工业过程成像实现超感”的论文集中。论文的核心内容聚焦于一种新颖的算法，即结合边界条件和有限元方法，提出了一种块坐标下降法（Block Coordinate Descent Method,BCM），用于提高ECT图像的精度。传统的ECT方法往往单独处理界面和介电常数的重建，而这种方法则实现了两者的同时优化，从而减少了误差累积，提高了成像结果的可靠性。作者们强调，这种方法特别适用于检测沉积层，因为在工业生产过程中，精确识别沉积物界面和其介电特性对于优化工艺过程至关重要。他们通过理论分析和实验验证，展示了这种耦合方法在实际应用中的优越性能，包括更高的分辨率、更快的计算速度以及对复杂形状和多层结构的适应性。论文引用如下： Ren, S., & Dong, F. (2016). Interface and permittivity simultaneous reconstruction in electrical capacitance tomography based on boundary and finite-element coupling method. Phil. Trans. R. Soc. A, 374(20150333), [doi:10.1098/rsta.2015.0333]. 读者可以通过链接http://dx.doi.org/10.1098/rsta.2015.0333 或者访问http://rsta.royalsocietypublishing.org获取更多关于该研究的详细内容和电子补充材料，与作者Feng Dong联系获取进一步的信息，其电子邮件地址为fdong@tju.edu.cn。这篇研究论文为电容层析成像技术的发展提供了重要的理论支持和实践指导，有望推动工业界在检测和理解材料性质上达到新的高度。

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

, the deposit layer Ω

and

the inhomogeneous permittivity section Ω

(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

is the k-dimensional spatial variable, ε ∈ C

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

∪ Ω

. (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

, x ∈ E

, (2.3)

where δ(i, j) is Kronecker delta equalling zero except when i = j, V

is the drive voltage amplitude,

and E

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=1

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

(x) = u

(x)andε

(x) =−ε

(x), x ∈ Γ

(2.5)

and

(x) = u

(x)andε

(x) =−ε

(x)q

(x), x ∈ Γ

, (2.6)

where Γ

= Ω

∩ Ω

is the interface between the electrode mounting layer and the deposit layer,

and Γ

= Ω

∩ Ω

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

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