## 1 Introduction  
Insulation characteristics are one of the key factors that affect the reliability and safety of power equipment. The insulation problems of traditional power equipment and the more complex and diverse internal insulation problems in low-temperature environments need to be considered. The bubble problem in liquid nitrogen is one of internal insulation problems for high-temperature superconductivity equipment. Insulation design of high-temperature superconducting power equipment requires accurate knowledge of the electric potential distribution in the liquid nitrogen environment containing bubbles. Hence, an accurate and effective numerical calculation method is needed. Many researchers have experimentally studied the dynamic characteristics of bubbles in the electric field [1, 2, 3, 4] and found that bubbles have complex motion and deformation. However, few numerical methods are presented to analyze the electric potential. The most common numerical method is the conventional finite element method (CFEM). However, CFEM requires a continuous interpolation function, and the material is not allowed to be changed in an element. Due to the small volume, large amount, and continuous motion of bubbles, it is difficult to calculate the electric potential characteristics of liquid nitrogen quickly and accurately with CFEM. The extended finite element method (XFEM), an emerging numerical analysis method, has been developed to deal with extremely discontinuous problems based on CFEM. Compared with CFEM, it has significant advantages:  
1. The interfaces are described by the level set method. A special enrichment term is constructed to reflect the discontinuity characteristics. Therefore, XFEM can better solve the non-convergence problem caused by the discontinuity. For example, the sharp corner problem may lead to a non-converge in CFEM. However, XFEM can solve this problem well. 2. The mesh reconstruction is effectively avoided when involving the moving boundary problems, because an element is allowed to contain two materials. XFEM can calculate each moment of bubble movement using only one set of mesh. XFEM [5; 6; 7; 8] developed from solid mechanics. XFEM has become the most popular computational tool in analyzing crack problems [9]. [PERSON] et al. [10] presented a combined XFEM phase-field model with sharp and diffuse representations of cracks, which divides the phase field and XFEM into two regions to simplify the calculation. The XFEM method has also been used by many scholars to solve the crack problem of piezoelectric materials. [PERSON] et al. [11] used XFEM to analyze static impermeable cracks at the interface of piezoelectric materials. [PERSON], [PERSON] et al. [12] predicted fatigue-cracking behavior of piezo-electric structures in the existence of multiple geometrical discontinuities under cyclic thermal-electrical-mechanical loads using XFEM. [PERSON], [PERSON] et al. [13] used XFEM to solve the inverse problem of detecting multiple cracks in two-dimensional piezoelectric structures under dynamic electric loads. In the field of electrical engineering, [PERSON] et al. [14] introduced the XFEM into the electric field analysis first. [PERSON] [15] used the XFEM to calculate the electric field of an insulating plate with cracks. [PERSON] et al. [16] proposed an improved XFEM for modeling electromagnetic devices with multiple nearby and applied it to the iron core. XFEM is used to analyze a field-circuit coupled high-temperature superconducting cable considering magnetic hysteresis [17]. As for the use of XFEM in an axisymmetric electric field, it is blank before. In this paper, XFEM is first applied to analyze the discontinuity problems in an axisymmetric electric field. It could be used to analyze the electric field characteristics of transformer oil with bubbles, to calculate the current distribution in the conductive layer of high-temperature superconducting cable.