## 2 Principle of Axisymmetric Extended Finite Element Method  
The essence of XFEM is to use the enrichment function to modify the shape function of CFEM to achieve the purpose of describing discontinuities. The convergence of XFEM is ensured by the partition of unity method (PUM) [18]. The enrichment function is constructed by the level set function and the shape function of CFEM [19]. The axisymmetric field problems are studied on the (\(\rho\), \(z\)) plane in the cylindrical coordinate system. The axisymmetric field can simplify 3D problems effectively, as shown in Figure 1. It has obvious advantages in improving computational efficiency with accuracy. Figure 1: Axisymmetric analysis model from three-dimensional field domain. Figure 2 represents a sphere located in the middle of a cylinder. The elements containing the interface are called the enrichment elements as \(E\). The nodes related to the enrichment elements are named enrichment nodes. XFEM is a method that equates the boundary value problems of partial differential equations to the conditional variational problem. It establishes corresponding discrete equations to solve them. By dividing the solution domain into discrete finite elements, computations are made based on the node of these elements. The Poison boundary value problem of the axisymmetric electrostatic field is used in this paper. \[\left\{ \begin{array}{l}\Omega:-\frac{1}{\rho}\frac{\partial}{\partial\varphi }(\varepsilon p\frac{\partial\varphi}{\partial\varphi})-\frac{\partial}{ \partial z}(\varepsilon\frac{\partial\varphi}{\partial z})=\sigma\\ \Gamma_{1}:\varphi(\rho,z)=\varphi_{0}\\ \Gamma_{1}:\frac{\partial\varphi(\rho,z)}{\partial n}=0\end{array} \right., \tag{1}\]  
where \(\Omega\) is the solution domain, \(\Gamma_{1},\Gamma_{1}\) is the boundary, \(\varphi\) is the value of electric potential, \(\rho\) and \(z\) are the vertical coordinates of cylindrical coordinates, \(\varepsilon\) is the relative permittivity of a medium, \(\sigma\) is surface charge density, \(\varphi_{0}\) is the electric potential at the boundary. The approximation of the electric potential \(\varphi(\rho,z)\) (axisymmetric model) with triangular element mesh can be written as:  
\[\varphi(\rho,z)=\sum_{i=1}^{3}N_{i}(\rho,z)\varphi_{i}+\sum_{j=1}^{m}N_{j}( \rho,z)\Phi(\rho,z)\beta_{j}, \tag{2}\]  
where \(N_{i}\), \(N_{j}\) are the shape functions of CFEM, \(\Phi\) is the enrichment function (its expression is as the Equation (3)), \(m\) is the number of the enrichment nodes in the enrichment element, \(\varphi_{i}\) and \(\beta_{j}\) are unknowns. \[\Phi(\rho,z)=\sum_{I}N_{I}(\rho,z)\cdot|\phi_{I}|-\left|\sum_{I}N_{I}(\rho,z) \cdot\phi_{I}\right|, \tag{3}\]  
where \(\phi_{I}\) is the level set function. The final equilibrium equation can be expressed in a discrete form as follows:  
\[\left[ \begin{array}{cc}K_{II}&K_{II}\\ K_{II}&K_{JJ}\end{array} \right]\left\{ \begin{array}{l}\mathbf{\varphi}_{I}\\ \mathbf{\varphi}_{I}\end{array} \right\}=\left\{ \begin{array}{l}P_{I}\\ P_{I}\end{array} \right\}, \tag{4}\]  
Figure 2: Discrete field with discontinuous circle interfaces. where \(K_{II}\),\(K_{JJ}\) are the self-stiffness matrixes of FEM and enrichment nodes, respectively; \(K_{IJ}\),\(K_{JJ}\) are the mutual stiffness matrixes between the two types of nodes. The stiffness matrix is calculated by a minimum of energy functional of the axisymmetric field as:  
\[