Electrical treeing is a leading cause to the eventual breakdown of dielectric polymers under high voltages. This paper presents a simulation scheme developed based on the phase-field regularized cohesive zone model (PF-CZM) for electrical tree modelling. By using the electrical analog of the crack propagation, the localized breakdown is modelled by the evolution of surface energy, and the electrical treeing is driven by the competition between the surface energy and the stored energy following the laws of thermodynamics. The microscopic Weibull distribution of the dielectric breakdown strength is the key factor resulting in the fractal structures of the electrical tree. The model developed is mesh independent and length-scale insensitive when the mesh size is no greater than $2.5\mathrm{\mu }\mathrm{m}$. The validity of the model was confirmed through experiments, which strengthens its credibility. Three types of composites are designed and compared. The results indicate that the epoxy resin enhanced with 5 vol% silica and 1 vol% graphene sheet has a 3.5 % longer dielectric breakdown time and a 29.2 % higher thermal conductivity than the pure epoxy resin. Overall, the model provides a valuable tool for understanding the physics of electrical treeing and designing new dielectric materials with high withstand voltages.

电树是介电聚合物在高电压下最终击穿的主要原因。本文提出了一种基于相场正则化内聚区模型（PF-CZM）开发的用于电树建模的仿真方案。通过使用裂纹扩展的电学模拟，通过表面能的演化来模拟局部击穿，并且遵循热力学定律，通过表面能和存储能量之间的竞争来驱动电树。介电击穿强度的微观威布尔分布是产生电树分形结构的关键因素。当网格尺寸不大于时，开发的模型与网格无关并且对长度尺度不敏感$2.5\mathrm{\mu }\mathrm{米}$。通过实验证实了模型的有效性，增强了模型的可信度。设计并比较了三种类型的复合材料。结果表明，用5 vol%二氧化硅和1 vol%石墨烯片增强的环氧树脂比纯环氧树脂的介电击穿时间长3.5%，导热率高29.2%。总的来说，该模型为理解电树的物理原理和设计具有高耐压的新型介电材料提供了一个有价值的工具。