This paper presents an electrical impedance tomography (EIT) method using a partial-differential-equation-constrained optimization approach. The forward problem in the inversion framework is described by a complete electrode model (CEM), which seeks the electric potential within the domain and at surface electrodes considering the contact impedance between them. The finite element solution of the electric potential has been validated using a commercial code. The inverse medium problem for reconstructing the unknown electrical conductivity profile is formulated as an optimization problem constrained by the CEM. The method seeks the optimal solution of the domain’s electrical conductivity to minimize a Lagrangian functional consisting of a least-squares objective functional and a regularization term. Enforcing the stationarity of the Lagrangian leads to state, adjoint, and control problems, which constitute the Karush-Kuhn-Tucker (KKT) first-order optimality conditions. Subsequently, the electrical conductivity profile of the domain is iteratively updated by solving the KKT conditions in the reduced space of the control variable. Numerical results show that the relative error of the measured and calculated electric potentials after the inversion is less than 1%, demonstrating the successful reconstruction of heterogeneous electrical conductivity profiles using the proposed EIT method. This method thus represents an application framework for nondestructive evaluation of structures and geotechnical site characterization.
Electrical impedance tomography (EIT) is a nondestructive evaluation method that infers the electrical conductivity, permittivity, or impedance of structures using the surficial measurement of electrical responses due to currents. The method seeks to construct a tomographic image of an object by estimating the spatial distribution of its impedance or electrical conductivity. EIT has been recognized as a highly applicable technique in medical imaging, industrial process monitoring, and near-surface site characterization due to its ease of field experimentation, economic feasibility, and superior ability to penetrate structures. When a structure experiences electric potential difference, current flows from the point of high potential to the point of low potential. In other words, when the electric current is input to a structure, electric potential distribution is formed within the structure and on its boundary. If a structure is heterogeneous, the path of electric current is different from the homogeneous case. Therefore, the electric potential distribution of a structure depends on its material heterogeneity [
The problem of reconstructing the material properties of a structure using the measured electrical response from the surface can be defined as an inverse medium problem. Many mathematical algorithms and numerical strategies have been proposed to improve the solution to this problem. For example, mathematical and statistical regularization methods have been proposed to relieve the ill-posedness of the inverse problem and improve the solution convergence [
For improving the previous development of the EIT for civil structures, this study proposes a nonlinear inversion method using a complete electrode model (CEM), targeting the reconstruction of the electrical conductivity profile of heterogeneous domains. The CEM comprises the Laplace equation for electric potential and the boundary conditions that express the current input to the structure via surface electrodes [
The remainder of this paper is organized as follows: In
To solve the inverse medium problem using measured electric potential values, it is first necessary to resolve the electrostatic forward problem for calculating the electric potential due to the current input. This study uses the CEM as a mathematical model for the forward problem. The CEM accounts for the current loss that occurs when the current flows to a low impedance material through an electrode and the voltage drops due to the contact impedance between the structural surface and electrode [
The forward problem described by the CEM can be formulated as the following boundary value problem:
Using the boundary conditions
Next, we integrate
Adding
The choice of
In
For validation, the forward solutions are compared to the solutions obtained using Technology Computer-Aided Design (TCAD) software. TCAD is used for modeling electronic design processes and the behavior of electrical devices based on fundamental physics [
The error of the forward solution relative to the TCAD result can be calculated using the following equation:
Position | CEM (V) | TCAD (V) | Error (%) |
---|---|---|---|
1 | 10.00 | 10.00 | 0.00 |
2 | 6.93 | 7.03 | 1.42 |
3 | 6.05 | 6.07 | 0.33 |
4 | 5.48 | 5.48 | 0.00 |
5 | 5.00 | 5.00 | 0.00 |
6 | 4.52 | 4.52 | 0.00 |
7 | 3.95 | 3.93 | 0.51 |
8 | 3.07 | 2.97 | 3.37 |
9 | 0.00 | 0.00 | 0.00 |
10 | 3.07 | 2.97 | 3.37 |
11 | 3.95 | 3.93 | 0.51 |
12 | 4.52 | 4.52 | 0.00 |
13 | 5.00 | 5.00 | 0.00 |
14 | 5.48 | 5.48 | 0.00 |
15 | 6.05 | 6.07 | 0.33 |
16 | 6.93 | 7.03 | 1.42 |
For investigating the dependency of the forward solutions on the finite element mesh, two mesh types are explored: a radial mesh composed of eight-node quadrilateral elements and a mixed mesh with a square section inside.
The problem of reconstructing the electrical conductivity profile of a structure using the electric potential measured at surface electrodes can be formulated as the following PDE-constrained optimization problem:
The objective functional
The PDE-constrained optimization problem can be converted to an unconstrained optimization problem using a Lagrange multiplier method. Specifically, the objective functional
At the optimum of the Lagrangian, its first variation with respect to the adjoint variables
At the optimum of the Lagrangian, its first variation with respect to the state variables
Because
Finally, the first variation of the Lagrangian with respect to the control variable
Because
The control problem is a boundary value problem for the electrical conductivity
The adjoint problem described in
Substituting
Solving
The submatrices in
By solving the state and adjoint problems, the first and second optimality conditions are satisfied. Only the true profile of
1) Assuming the initial profile of
2) Using the state solution
3) Using the state and adjoint solutions, calculate the reduced gradient for
4) Determine the search direction using a line search method and update the electrical conductivity profile
In this study, the search direction for the optimal solution of the control variable
Then, the electrical conductivity vector
The misfit functional is evaluated using the updated electrical conductivity
In this study, an inexact line search method is used to determine a step length
Choose |
repeat |
until |
Terminate with |
The choice of the regularization factor
In
Therefore,
Consider a circular domain with a radius of 15 cm, as shown in
To investigate the accuracy of the inversion result, the errors in the inverted electrical conductivity values are calculated at four sampling points, as shown in
Point | Target |
Reconstructed |
Error (%) |
---|---|---|---|
A | 1.00000 | 1.00002 | 0.002 |
B | 1.00000 | 0.99999 | 0.001 |
C | 1.00000 | 1.00026 | 0.026 |
D | 1.00000 | 1.00116 | 0.116 |
To investigate the effect of the initial guess on the reconstructed profile, four cases of the inversion with different initial-to-target ratios are considered.
Case | Target electrical conductivity |
Assumed electrical conductivity |
|
---|---|---|---|
1 | 1.5 | 0.7 | 0.47 |
2 | 15.0 | 1.5 | 0.10 |
3 | 1.5 | 2.3 | 1.53 |
4 | 1.0 | 10.0 | 10.00 |
RMS errors | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|
Next, the inversion is performed to reconstruct heterogeneous electrical conductivity profiles.
RMS errors | Case 5 | Case 6 | Case 7 |
---|---|---|---|
This study was conducted to develop a nonlinear inversion method for electrical impedance tomography of structures using measured electric potentials at the surface. The proposed method minimizes the misfit between calculated and measured electric potential values at surface electrodes within a PDE-constrained optimization framework. The CEM was used as a forward model to calculate the electric potential due to current input. By solving the KKT conditions of the Lagrangian iteratively, the inversion procedure could successfully recover heterogeneous electrical conductivity profiles. The TN regularization scheme was used to mitigate the ill-posedness of the inverse problem and improve the convergence of the solution. A series of numerical results showed that the homogeneous and heterogeneous electrical conductivity profiles were successfully reconstructed using the inversion method developed in this study.
For all numerical example cases, the misfit was decreased from its original value by a factor on the order of
In a uniform finite-element mesh, as shown in
This study used the backward difference method shown in
For a radial finite-element mesh such as that shown in
The central difference method for the polar coordinate system is shown in
In a non-uniform finite-element mesh, as shown in