A mixed-dimensional model for the electrostatic problem on coupled domains

Abstract

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.

Figures (11)

• Figure 1: Domain $\Omega$, given by a cylinder corresponding to the gas subdomain $\Omega_g$, surrounded by a generic volume $\Omega_s$, representing the dielectric domain.
• Figure 2: Junction among three cylinders $\Omega_1$, $\Omega_2$ and $\Omega_3$. The boundary of the junction volume $\mathcal{J}$ is the union of the bases of the three cylinders and the lateral surface of the 3D domain.
• Figure 3: TC1 - Discretizations of the three-dimensional (blue) and one-dimensional (purple) domains.
• Figure 4: TC1 - Computed potential $\Phi$ (right half) and electric field $\mathbf{D}_s$ (left half) in the dielectric domain, osbserved from the top basis of the domain $\Omega$.
• Figure 5: TC1 - Computed potential $\Phi_s$ and electric field $\mathbf{D}_s$ in the 3D dielectric domain on a longitudinal section of $\Omega$. The magnitude of $\mathbf{D}_s$ is expressed in logarithmic scale and the arrows are tangential to the field streamlines.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof