## 3 Mathematical Model Combining MoM with FEM  
Pipeline corrosion is mainly caused by the resistive coupling of the grounding currents on the pipeline metal. The computation domain includes the soil area, the pipeline conductor area, as well as the infinite air-earth boundary, the grounding conductor boundary, and the interface boundary between the soil area and the pipeline conductor area. Figure 3  
Figure 2: Typical three-electrode system for measuring polarization curves. shows the boundary value problem of the resistive coupling model between the grounding electrode and pipelines. The conductivities are assumed to be homogeneous. Then, the soil field \(\Omega_{\text{s}}\) and pipe conductor field \(\Omega_{\text{p}}\) belong to the constant current field so the potential satisfies Laplace's equations. The air can be considered as electrical insulating medium so the current flowing out of the air-earth boundary (\(S_{\text{g}}\)) is zero. The total leakage current of the grounding electrode should equal the injected current in the HVDC transmission system based on the charge conservation law. According to the interface conditions for electric field vectors, the current density on the soil side should equal the current density on the pipe conductor side. Besides, the leakage current density also should satisfy the polarization relationship with the pipe-to-soil potential (\(\varphi_{\text{s}}-\varphi_{\text{p}}\)). The theoretical equations for the above three-dimensional boundary value problem can be formulated as follows:  
\[\left\{ \begin{array}{l}\
abla^{2}\varphi_{\text{s}}(\overrightarrow{r})=0, \ \ \overrightarrow{r}\in\Omega_{\text{s}}\\ \
abla^{2}\varphi_{\text{p}}(\overrightarrow{r})=0,\ \ \overrightarrow{r}\in\Omega_{\text{p}}\\ \frac{\partial\varphi_{\text{s}}(\overrightarrow{r})}{\partial n}=0,\ \ \overrightarrow{r}\in S_{\text{g}}\\ \int_{S_{\text{ele}}}\sigma_{\text{s}}\frac{\partial\varphi_{\text{s}}( \overrightarrow{r})}{\partial n}ds=I_{\text{hvd}},\ \ \overrightarrow{r}\in S_{\text{ele}}\\ \sigma_{\text{p}}\frac{\partial\varphi_{\text{p}}(\overrightarrow{r})}{ \partial n}=\sigma_{\text{s}}\frac{\partial\varphi_{\text{s}}(\overrightarrow {r})}{\partial n}\\ =-f_{\text{p}}(\varphi_{\text{p}}(\overrightarrow{r})-\varphi_{\text{s}}( \overrightarrow{r})),\ \ \overrightarrow{r}\in S_{\text{p}}\end{array} \right. \tag{3}\]  
where \(\overrightarrow{r}\) is an arbitrary coordinate in the computation domain, \(I_{\text{hvdc}}\) is the grounding current of the HVDC grounding electrode, and \(\sigma_{\text{p}}\) and \(\sigma_{\text{s}}\) are the conductivity of the pipe conductor and soil medium, respectively. A numerical method combining MoM and FEM is proposed to solve the above equations. The Laplace equation in the soil area is discretized by MoM which is suitable for open-domain problems. The Laplace equation in pipe conductor area is discretized by FEM which is suitable for enclosure domain problems. By applying the nonlinear polarization boundary conditions between the two areas, the two methods can be coupled for solving. The potential of an arbitrary point in the soil area can be expressed as follows:  
\[\varphi_{\text{s}}(\overrightarrow{r})=\iint\limits_{S}G(\overrightarrow{r },\overrightarrow{r}^{\prime})\cdot J_{\text{leak}}(\overrightarrow{r}^{ \prime})ds,s\in S_{\text{ele}},S_{\text{p}} \tag{4}\]  
where \(\overrightarrow{r}^{\prime}\) is the current source coordinate in the soil area, \(J_{\text{leak}}(\overrightarrow{r})\) is the leakage current density for the buried conductor, and \(G(\overrightarrow{r},\overrightarrow{r}^{\prime})\) is the Green's function for the above boundary problem. The electrode conductor and pipe conductor are divided into \(M\) and  
Figure 3: Schematic diagram of the boundary value problem. \(N\) segments, respectively. The basis function for different segment current density can be expressed as follows:  
\[v_{i}= \begin{cases}1&\overset{\rightarrow}{r}^{\prime}\in\Delta_{i}\\ 0&\overset{\rightarrow}{r}^{\prime}\
otin\Delta_{i}\end{cases}