## 2 Establishment of a Simulation Model of the Cable Terminal Sheath Structure  
### Governing Equations Employed for the Simulations  
In the study of electric fields in the outer insulation structures of high-voltage electrical equipment, the electrostatic fields and AC constant fields at low frequencies are the most common types of electric fields. When applying [PERSON]'s equations to the analysis, the \(-\partial B/\partial t\) term in them can be neglected and is called an electrically quasi-static field. Electrically quasi-static fields should be used as much as possible when simulating fouling flash on insulated equipment [19]. According to [PERSON]'s equations, for an isotropic linear medium, [PERSON]'s equation for the relevant potential is Equation (1):  
\[\
abla\cdot\
abla\varphi=\
abla^{2}\varphi=-\rho/\varepsilon \tag{1}\]  
Here, \(\rho\) denotes the charge density at any point, \(\varepsilon\) denotes the dielectric constant of the dielectric, and \(\varphi\) denotes the potential. When \(\rho=0\), i.e., there is no space charge in the solution domain, then the equation becomes [PERSON]'s equation:  
\[\
abla^{2}\varphi=0 \tag{2}\]  
The structure of electrical insulation is generally a combination of multiple dielectrics. At this time, the potential function of each dielectric in the structure can be expressed by [PERSON]'s equation and [PERSON]'s equation and should satisfy Formulas (3) and (4) on the different dielectric boundary surfaces:  
\[\varphi^{+}=\varphi^{-} \tag{3}\]  
\[\left(\varepsilon\frac{\partial\varphi}{\partial n}\right)^{+}-\left( \varepsilon\frac{\partial\varphi}{\partial n}\right)^{-}=\sigma \tag{4}\]  
Here, \(\sigma\) denotes the surface charge density at the interfaces of different dielectric manifolds. An important condition in solving the Poisson or Laplace equations for the potentials derived above is the boundary conditions. Knowing the boundary conditions and the initial conditions, the specific solution of the partial differential equation can be solved. For electrostatic fields and constant electric fields in the dielectric surrounding a conductor, there are usually three types of boundary conditions, as follows [20]. Boundary conditions of the first type involve the known potential \(\Phi(\Gamma)\) of the boundary \(\Gamma\). \[\varphi|_{\Gamma}=\Phi(\Gamma) \tag{5}\]Boundary conditions of the second type are the known potential normal derivatives of the boundary \(\Gamma\). \[\frac{\partial\varphi}{\partial n}\mid_{\Gamma}=\gamma \tag{6}\]  
\(n\) is the outer normal of the boundary. Type III boundary conditions are a combination of Type I boundary conditions and Type II boundaries, i.e., known potentials on the boundary \(\Gamma_{1}\) and normal derivatives of the potentials on another part of the boundary \(\Gamma_{2}\). Using the finite element method, the first thing to do is to transform the above margin problem for the electrostatic field into a variational problem. The field is discretized into \(N\) partition elements, and the function \(I(\varphi)\) of the variational problem can be rewritten by assuming a triangular partitioning of the elements:  
\[I[\varphi]=\sum_{e=1}^{N}I_{1}^{e}[\varphi]-\sum_{e=1}^{N}I_{2}^{e}[\varphi] \tag{7}\]  
\[\left|I_{1}^{e}[\varphi]=\frac{1}{2}\iint_{e}\varepsilon[\
abla\varphi]^{T}[ \
abla\varphi]dxdy\right| \tag{8}\]  
\[I_{2}^{e}[\varphi]=\iint_{e}\rho\varphi dxdy \tag{9}\]  
Let the interpolation function be:  
\[\widetilde{\varphi}=ax+by+c \tag{10}\]  
Set the interpolation potential corresponding to the three nodes of the triangular cell \(e\) to be \(\varphi_{l_{1}}\)\(\varphi_{l_{2}}\)\(\varphi_{l_{3}}\), which leads to the following system of equations:  
\[\
abla^{2}\varphi=0 \tag{2}\]0  
Solving the system of equations yields the values of the coefficients \(a\), \(b\), \(c\). Substituting these values into Equation (10) yields the interpolating function:  
\[\
abla^{2}\varphi=0 \tag{2}\]1  
\[N_{i}^{e}(x,y)=\frac{1}{2\Delta}(a_{i}x+b_{i}y+c)\;\left(i=1,2,3\;\Delta=\frac {1}{2}