ECT界面与介电常数耦合重建：边界与有限元方法

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"这篇研究论文探讨了一种基于边界和有限元耦合方法的ECT(电气电容层析成像)技术，用于同时重建物体界面和介电常数。该技术在电气工程、能源以及应用数学等领域具有重要意义。关键词包括：界面和介电常数分布的同步重建、边界和有限元耦合法、块坐标下降法、ECT技术以及沉积层检测。通讯作者为Feng Dong，邮箱为fdong@tju.edu.cn，研究主要在中国天津大学的工艺测量与控制国家重点实验室进行。文章指出，在工业和生物系统中，容器内壁的沉积层形成是一个普遍存在的问题。对这些沉积层的监测对于提高生产效率和系统安全性至关重要。ECT作为一种非破坏性的检测技术，能够通过测量电容变化来成像内部的介电常数分布。在传统的ECT方法中，通常分别处理界面和介电常数的重建问题，但这种方法可能会导致重建结果的不准确。因此，研究者提出了一种新的耦合方法，结合了边界方法和有限元方法，以实现界面和介电常数的同时重建。这种方法的优势在于能够更精确地捕获复杂形状物体内部的介电特性，并且可以有效地处理由于边界条件引起的计算挑战。文中详细介绍了该耦合方法的理论基础，包括块坐标下降法的应用，这是一种优化算法，可以分步优化多变量函数。通过这种算法，研究人员能够分别处理各个变量，从而实现界面位置和介电常数的同步优化。此外，文章可能还包含了实验验证和数值模拟结果，以证明所提方法的有效性和准确性。这些结果可能展示了在不同条件下的ECT图像重建，包括各种几何形状、介电常数变化以及实际沉积层情况。通过这些实验，研究者可能评估了新方法在解决ECT重建问题时相对于传统方法的改进之处。这项研究为ECT技术提供了一个创新的理论框架，有望改善工业和生物系统中沉积层检测的精确度，为相关领域的工程师和科学家提供了有价值的工具和理论指导。"

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rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000

..................................................................

Figure 1. Measurement principle of electrical capacitance system for deposit layer detection.

To simultaneously visualize the deposit layer proﬁle and materials distribution within it, an

interface and permittivity simultaneous reconstruction algorithm is proposed. The boundary and

ﬁnite elements coupling method is employed to provide an efﬁcient numerical platform for the

simultaneous inversion approach. The block coordinate decent schedule is presented to mitigate

the negative inﬂuences of the inaccurate prior permittivity knowledge on the reconstruction

results and overcome the blocking effect of deposit layer on the internal permittivity changes.

The remainder of the paper is organized as following. In section 2, the measurement principles

and government equations of the electrical capacitance system are brieﬂy described; and then, the

target interface conﬁguration and permittivity distribution are represented by a set of parameters;

ﬁnally, the unknown parameters are estimated from capacitance data by block coordinate descent

method. In section 3, the boundary and ﬁnite elements coupling method is implemented. In

section 4, several numerical tests are conducted to evaluate the proposed method.

2. Methodology

(a) Government equations

Figure 1 gives a schematic representation of the considered capacitance system. The entire

observed domain Ω is consist by the electrode mounting insulation layer (also called insulation

pipe layer in some papers) Ω

, deposit layer Ω

and materials distribution section Ω

. Several

electrodes E are mounted around the periphery of the observed domain. At each time, one of

the electrodes is stimulated by a voltage signal, while the others are at the virtual earth potential.

Mutual capacitances between the drive and detecting electrodes are sequentially measured by

some kind of measuring circuits. Following the Maxwell’s theory, the electrical potential u at the

entire observed region is governed by 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. Since ε

at the pipe and deposit regions is constant, the associated government equations can be simpliﬁed

to Laplace equations

u(x)=0,x∈ Ω

∪ Ω

. (2.2)

The boundary conditions are determined by the data collection strategy. During i-th voltage

stimulation, electric potentials at the electrodes follows Dirichlet condition

u(x)=δ(i, j)V

,x∈ E

, (2.3)

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