We derive a mixed-dimensional 3D-1D formulation of the electrostatic equation in two domains with different dielectric constants to compute, with an affordable computational cost, the electric field and potential in the relevant case of thin inclusions in a larger 3D domain. The numerical solution is obtained by Mixed Finite Elements for the 3D problem and Finite Elements on the 1D domain. We analyze some test cases with simple geometries to validate the proposed approach against analytical solutions, and perform comparisons with the fully resolved 3D problem. We treat the case where ramifications are present in the one-dimensional domain and show some results on the geometry of an electrical treeing, a ramified structure that propagates in insulators causing their failure.

翻译：本文推导了具有不同介电常数的两个域中静电方程的混合维3D-1D公式，以可承受的计算成本计算大三维域中薄夹杂物这一重要情形下的电场和电势。数值解通过三维问题的混合有限元法和一维域上的有限元法获得。我们分析了几何形状简单的测试案例，通过与解析解对比验证所提方法，并与完全解析的三维问题进行对比。我们处理了一维域存在分岔的情形，并展示了电树枝（一种在绝缘体中传播导致其失效的分岔结构）几何形状的若干计算结果。