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 [14,22]. The current loss and voltage drop influence the measured electric potential values detected by the electrodes. As the CEM considers both the contact impedance and current loss at the electrodes, the error between the calculated electric potential and the experimentally derived electric potential is generally smaller than that computed using other models. Fig. 1 schematically shows a circular structure with electrodes on its surface. Solving the forward problem using the CEM reveals the electric potential inside the structure and that at the electrodes due to the current input through the electrodes.

The forward problem described by the CEM can be formulated as the following boundary value problem:

(1)∇⋅(σ∇u)=0,x∈Ω,

(2a)u+zlσ∂u∂n=Ul,l=1,2,…,L,x∈ΓEl,

(2b)∫ΓElσ∂u∂ndΓ=Il,l=1,2,…,L,x∈ΓEl,

(2c)σ∂u∂n=0,x∈∂Ω∖∪l=1LΓEl,where Ω denotes the structural domain, u is the scalar electric potential, σ is the electrical conductivity, n is the outward unit normal on the boundary ∂Ω, El denotes the lth electrode, ΓEl is the lth electrode boundary, zl is the contact impedance of El, Il is the injected current at El, Ul is the electric potential at El, and L is the number of electrodes. To ensure the existence and uniqueness of solutions, we add

(3)∑l=1LIl=0,to the model. To determine the reference point for the electric potential,

(4)∑l=1LUl=0must be satisfied. We multiply Eq. (1) by a test function v(x) and integrate it over the domain Ω, which results in

(5)∫∂Ωvσ∂u∂nd∂Ω−∫Ω∇v⋅(σ∇u)dΩ=0.

Using the boundary conditions (2a) and (2c), Eq. (5) can be rewritten as

(6)∑l=1L1zl∫ΓElv(Ul−u)dΓ−∫Ω∇v⋅(σ∇u)dΩ=0.

Next, we integrate Eq. (2a) over ΓEl, which results in

(7)∫ΓEludΓ=∫ΓEl(Ul−zlσ∂u∂n)dΓ=Ul|el|−zlIl.

Eq. (7) is multiplied by a test value Vl, divided by zl, and summed for all electrodes, which yields

(8)∑l=1L1zlVl(∫ΓEludΓ−Ul|el|+zlIl)=0.

Adding Eqs. (6) and (8) results in

(9)∑l=1L1zl[∫ΓElv(Ul−u)dΓ+Vl(∫ΓEl(u−Ul)dΓ+zlIl)]−∫Ωσ∇v⋅∇udΩ=0,→∑l=1L1zl[∫ΓElv(Ul−u)dΓ+Vl(∫ΓEl(u−Ul)dΓ)]+∑l=1LIlVl−∫Ωσ∇v⋅∇udΩ=0,→∫Ωσ∇u⋅∇vdΩ+∑l=1L1zl∫ΓEl(u−Ul)(v−Vl)dΓ=∑l=1LIlVl.

Eq. (9) is the weak form of the forward problem. The electric potential u in the domain and the potential U at the electrodes are approximated using Legendre basis functions ϕi(x) and basis vectors nj, respectively:

(10)u(x)≅uh(x)=∑j=1Nαjϕj(x),

(11)U≅Uh=∑j=1L−1βjnj,where N is the number of nodes in the finite element mesh. The coefficient αj is the nodal value of u(x), βj is the unknown parameter for linearly combining vectors nj, and Uh is a vector consisting of the electric potential at each electrode. The basis vectors nj∈RL×1 are defined as

(12)n1=[1−100⋮0],n2=[10−10⋮0],n3=[100−1⋮0],…,nL−1=[1000⋮−1].

The choice of nj ensures that Eq. (4) is satisfied. The test function v and the value of Vl at the electrode can be similarly approximated as

(13)vh(x)=∑i=1Nviϕi(x),

(14)Vh=∑i=1L−1Vini,where Vh is a vector consisting of the value of V at each electrode. By substituting Eqs. (10), (11), (13), and (14) into the weak form represented by Eq. (9), a linear system of equations for unknown vector u=[αTβT]T can be obtained

(15)Ku=F;[BCCTD][αβ]=[0I],where the matrices and vectors are written as

(16a)Bij=∫Ωσ∇ϕi⋅∇ϕjdΩ+∑l=1L1zl∫ΓElϕiϕjdΓ(1≤i≤N)(1≤j≤N),

(16b)Cij=−(1z1∫ΓE1ϕidΓ−1zj+1∫ΓEj+1ϕidΓ)(1≤i≤N)(1≤j≤L−1),

(16c)CijT=Cji=−(1z1∫ΓE1ϕjdΓ−1zi+1∫ΓEi+1ϕjdΓ)(1≤i≤L−1)(1≤j≤N),

(16d)Dij={|e1|z1,i≠j|e1|z1−|ej+1|zj+1,i=j(1≤i≤L−1)(1≤j≤L−1),

(17)0∈RN×1,I=[I1−I2I1−I3I1−I4⋮I1−IL]∈R(L−1)×1,

(18)α=[α1α2α3⋮αN]∈RN×1,β=[β1β2β3⋮βL−1]∈R(L−1)×1.

In Eq. (16d), |el| is the length of the lth electrode. Solving Eq. (15) yields nodal values of the electric potential in the domain as well as the values of β. The potentials Ulh at the electrodes can be obtained as follows:

(19)Ulh=[U1hU2hU3h⋮ULh]=[∑l=1L−1βl−β1−β2⋮−βL−1].

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 [23]. Fig. 2 shows the configuration of a circular domain with a radius of 10 cm. The electrical conductivity of the domain is 1.56×10−1S/cm. A total of 16 electrodes is arranged to cover 50% of the boundary; a current of 0.78 A is input into the fifth electrode, and it flows out through the 13th electrode. The contact impedance of the electrodes is 9.8×10−7Ω⋅cm2. In this case, the potential of the fifth electrode is 10 V. Fig. 3 shows the distribution of the calculated electric potential within the domain due to the current input. The numerical solution calculated using the CEM is very close to the result obtained using the TCAD software.

The error of the forward solution relative to the TCAD result can be calculated using the following equation:

(20)Error (%)=|VCEM−VTCAD|VTCAD×100.

Table 1 shows the electric potential values at each electrode and the corresponding relative errors. The maximum error is 3.37%, which demonstrates the accuracy of the forward solution. Fig. 4 depicts the electric potential, Ul, calculated at each electrode.

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. Fig. 5 illustrates these two types of meshes.

Fig. 6 shows a circular domain with a radius of 15 cm and electrodes attached to its boundary. The domain is assumed to be homogeneous, with an electrical conductivity of 1×10−2S/cm, which is typical of concrete materials. A total of 32 electrodes is arranged on the boundary and are spaced evenly to cover 50% of the boundary (as for the domain described in Section 2.2). A current with a cosine pattern along the perimeter is input to each electrode. The magnitude of the current at each electrode is Il=cos⁡(2πl/L), where L is the total number of electrodes, and l is the index for the electrode number. The contact impedance of the electrodes is 0.22Ω⋅cm2. Fig. 7 presents the forward solutions calculated using the two meshes displayed in Fig. 5. Fig. 7c shows the calculated electric potential values at the electrodes on the boundary. The two results show that the forward solution changes minimally with the mesh type.

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:

(21)minσ(x)J:=12∑l=1L∫ΓEl(Ul−Ulm)2dΓ+γ(σ).

The objective functional J consists of a misfit functional in a least-squares sense and a regularization term γ(σ). The misfit functional is expressed as the sum of the squared differences between the calculated electric potential Ul and the measured electric potential Ulm at electrode El. This optimization problem is constrained by the governing Eq. (1) and boundary conditions (2)–(4) of the forward problem. To relieve the ill-posedness of the inverse problem, the regularization term γ(σ) is included in the objective functional J. In this study, the Tikhonov (TN) regularization scheme is used; thus, the regularization term can be written as

(22)γ(σ)=12Rσ∫Ω∇σ⋅∇σdΩ,where Rσ is a regularization factor that controls the penalty to the gradient of the electrical conductivity σ(x). Although TN regularization cannot accurately capture sharply varying material profiles, it still assists the inversion process in narrowing the feasibility space of solutions and improving the solution convergence.

The PDE-constrained optimization problem can be converted to an unconstrained optimization problem using a Lagrange multiplier method. Specifically, the objective functional J can be augmented by Eqs. (1) and (2b) to construct the Lagrangian functional L,

(23)L(u,Ul,w,Wl,σ)=12∑l=1L∫ΓEl(Ul−Ulm)2dΓ+γ(σ)+∫Ωw∇⋅(σ∇u)dΩ+∑l=1L∫ΓElWl(σ∂u∂n−Il)dΓ,where w and Wl are Lagrange multipliers applied to the left-hand terms of the governing equation and boundary conditions, respectively. The electrical conductivity σ(x) that minimizes the Lagrangian is the solution to the inverse problem. For the optimal solutions, first-order optimality conditions are introduced, which enforce the first variation of the Lagrangian with respect to state variables u and Ul, adjoint variables w and Wl, and the control variable σ to vanish.

The adjoint problem described in Section 3.2.2 can be solved using the Galerkin finite element method. Specifically, Eq. (25) is multiplied by a test function v(x) and integrated over the entire domain Ω. Upon integrating the resulting equation by parts and using a procedure similar to that represented by Eqs. (5) to (9), the weak form of the adjoint problem can be derived as

(30)∫Ωσ∇w⋅∇vdΩ+∑l=1L1zl∫ΓEl(w−Wl)(v−Vl)dΓ=∑l=1L(Ul−Ulm)Vl|el|,where Vl is a test value used to derive the weak form of Eq. (26a). The adjoint solutions w and Wl can be approximated using Legendre basis functions ϕi(x) and basis vectors nj, respectively, as follows:

(31)w(x)≅wh(x)=∑j=1Nαjadjϕj(x),

(32)W≅Wh=∑j=1L−1βjadjnj,where αjadj is the nodal value of w(x), βjadj is the unknown parameter for linearly combining vectors nj, and Wh is a vector consisting of Wl at each electrode. The test functions v and Vl are approximated using Eqs. (13) and (14).

Substituting Eqs. (13), (14), (31) and (32) into (30) results in the following linear system of equations:

(33)Kadjuadj=Fadj;[BadjCadj(−Cadj)TDadj][αadjβadj]=[0Iadj],where Kadj, uadj, and Fadj are the stiffness matrix, solution vector, and right-hand side vector of the adjoint problem, respectively. The superscript “adj” denotes the adjoint problem. The matrices and vectors in Eq. (33) can be written as

(34a)Bijadj=∫Ωσ∇ϕi⋅∇ϕjdΩ+∑l=1L1zl∫ΓElϕiϕjdΓ,(1≤i≤N)(1≤j≤N)

(34b)Cijadj=1z1∫ΓE1ϕidΓ−1zj+1∫ΓEj+1ϕidΓ,(1≤i≤N)(1≤j≤L−1)

(34c)(−Cijadj)T=−Cjiadj=−(1z1∫ΓE1ϕjdΓ−1zi+1∫ΓEi+1ϕjdΓ),(1≤i≤L−1)(1≤j≤N)

(34d)Dijadj={−|e1|z1,i≠j−|e1|z1−|ej+1|zj+1,i=j(1≤i≤L−1)(1≤j≤L−1),

(35)0∈RN×1,Iadj=[(U1−U1m)|e1|−(U2−U2m)|e2|(U1−U1m)|e1|−(U3−U3m)|e3|(U1−U1m)|e1|−(U4−U4m)|e4|⋮(U1−U1m)|e1|−(UL−ULm)|eL|]∈R(L−1)×1,

(36)αadj=[α1adjα2adjα3adj⋮αNadj]∈RN×1,βadj=[β1adjβ2adjβ3adj⋮βL−1adj]∈R(L−1)×1.

Solving Eq. (33) yields nodal values of adjoint variable w in the domain as well as the values of βadj. The adjoint variable Wlh at the electrodes can be derived as

(37)Wlh=[W1hW2hW3h⋮WLh]=[∑l=1L−1βladj−β1adj−β2adj⋮−βL−1adj].

The submatrices in Eq. (33) are related to those of the state problem through

(38a)Badj=B,

(38b)Cadj=−C,

(38c)(−Cadj)T=CT,

(38d)Dadj=−D.

Eq. (38) contributes to reducing the computational cost of constructing the system matrices of the adjoint problem in the inversion process.

By solving the state and adjoint problems, the first and second optimality conditions are satisfied. Only the true profile of σ(x) exactly satisfies the control problem represented by Eqs. (28) and (29). Therefore, the material profile σ(x) must be updated to satisfy the third optimality condition. Updating σ(x) to minimize the Lagrangian L requires a reduced gradient for L. The process of updating the control variable σ(x) using the state and adjoint solutions can be summarized as follows:

1) Assuming the initial profile of σ(x), solve the forward problem (1) and (2) for state variables u(x) and Ul.

2) Using the state solution Ul, solve the adjoint problem (25) and (26) for adjoint solutions w(x) and Wl.

3) Using the state and adjoint solutions, calculate the reduced gradient for L with respect to the control variable σ(x) as

(39)gσ≡∇σL=−RσΔσ−∇w⋅∇u=−Rσ(∂2σ∂x2+∂2σ∂y2)+(−∂w∂x∂u∂x−∂w∂y∂u∂y),where the derivatives are calculated using second-order accurate finite difference methods. When using a radial mesh, such as that shown in Fig. 5a, the reduced gradient (40) expressed in a cylindrical coordinate system can be used

(40)gσ=−Rσ(∂2σ∂r2+1r∂σ∂r+1r2∂2σ∂θ2)+(−∂w∂r∂u∂r−1r2∂w∂θ∂u∂θ).

4) Determine the search direction using a line search method and update the electrical conductivity profile σ(x).

Consider a circular domain with a radius of 15 cm, as shown in Fig. 8. A total of 32 electrodes is placed on the boundary and spaced evenly to cover 50% of the boundary. For finite element implementation, the mixed mesh illustrated in Fig. 5b is used. The contact impedance of the electrodes is 0.22Ω⋅cm2. The initial value of the regularization factor Rσ is 1.0, and the weight factor ε for the regularization factor continuation scheme is 0.5. As an excitation to the domain, a current with a cosine pattern, as expressed in Eq. (49), is input to each electrode along the circular perimeter,

(49)Il=cos⁡(2πl/32),l=1,2,…,32.

Fig. 9 shows the target, initial guess, and reconstructed electrical conductivity profiles. The target profile is homogeneous with σtg=1.0 S/cm, which is typical of graphite materials. The inversion starts with an initial guess of σini=0.5 S/cm and reconstructs the target profile accurately at about 700 iterations. Fig. 9d shows the response misfit as a function of the iteration number during the inversion process. The misfit decreases from 3×102 to 1.49×10−3, which represents a reduction on the order of 10−5. The response misfit, Fm, as part of the objective functional J in Eq. (21), can be written as

(50)Fm=12∑l=1L∫ΓEl(Ul−Ulm)2dΓ.

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 Fig. 10: A(0,0), B(4,0), C(8,0), and D(12,0). Table 3 lists the errors at each sampling point. The maximum error is 0.116%, which demonstrates that the inversion for the homogeneous electrical conductivity profile is highly accurate.

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. Table 4 shows the four cases for the initial guess σini and target σtg. For Cases 1 and 2, the initial guess is less than the target, whereas for Cases 3 and 4, it is greater. The electrode arrangement, input current, and contact impedance are the same as those in the previous example.

Figs. 11–14 display the inversion results for the four electrical conductivity profiles presented in Table 4. In all four cases, the homogeneous target values are accurately reconstructed. In particular, as shown in Figs. 12b and 14b, the target values are recovered well even when the target is 10 times smaller or larger than the initial guess. Figs. 11c, 12c, 13c and 14c show the misfit Fm as a function of the number of iterations. For each case, the misfit decreases by a factor of at least 10−5. Figs. 11d, 12d, 13d and 14d show the electric potential values at the 32 electrodes calculated using the reconstructed electrical conductivity profile for each case. The calculated electric potential values nearly coincide with the measured values, which demonstrates that the target profiles are successfully reconstructed. The root-mean-square (RMS) error, eRMS, and the relative RMS error, er,RMS, of the electric potential at the electrodes can be calculated using Eq. (51). In the equations, Ui and Uim denote the calculated and measured electric potential values at the ith electrode, respectively, and |Um|max2 is the absolute maximum of the measured potentials. Table 5 shows the calculated RMS errors for each case of the inversion process. Even in Case 4 with the largest relative error, the value is less than 0.18%. In all cases, the inversion results are evaluated to be sufficiently accurate,

(51)eRMS=1L∑i=1L(Ui−Uim)2,er,RMS=1L∑i=1L(Ui−Uim)2|Um|max2.

Next, the inversion is performed to reconstruct heterogeneous electrical conductivity profiles. Fig. 15 shows three cases of the target profile. In Case 5, the target values of the electrical conductivity are 0.04S/cm for y≥0 and 0.01S/cm for y<0. In Case 6, the target profile is divided into three layers: 0.04S/cm for y≥5, 0.025S/cm for −5≤y<5, and 0.01S/cm for y<−5. The target profile for Case 7 has four quadrants. In quadrants 1 and 3, the target value is 0.04S/cm. In quadrants 2 and 4, the target value is 0.01S/cm. The target conductivity values of Case 5 to Case 7 are typical of air-dried concrete materials in various conditions [27]. Examples of the profile heterogeneity in Fig. 15 include fiber-reinforced composite sandwich plates and concrete specimens under curing. The electrode arrangement, current input, contact impedance, initial value of Rσ, and weight factor ε are the same as in the homogeneous domain cases.

Figs. 16–18 show the inversion results for the heterogeneous electrical conductivity profiles. The target conductivity values are reconstructed reasonably well in the three heterogeneous cases. Figs. 16c, 17c, and 18c show the response misfit Fm as a function of the iteration number. Figs. 16d, 17d, and 18d show the electric potential values calculated at the electrodes using the reconstructed electrical conductivity profile for each case. The calculated electric potential values nearly coincide with the measured ones, which demonstrates that the reconstructed profiles converged to the targets. For each case, the initial misfit decreases by a factor on the order of 10−3. Table 6 shows the RMS error and the relative RMS error of the electrical potentials calculated at the electrodes for each case. The errors were calculated using Eq. (51). They are relatively large compared with those in the homogeneous cases, due to the difficulty in accurately recovering the sharply varying interfaces between layers. Nevertheless, the relative RMS error of the heterogeneous cases is less than 0.80%, indicating that the inversion results are sufficiently accurate.