⚙️ Feature: Leontovich Boundary Condition via Surface Conductivity (ADE)

Summary

Implement the Leontovich (also known as surface impedance) boundary condition (SIBC) in Wakis using an Auxiliary Differential Equation (ADE) formulation based on surface conductivity ( \sigma_s ). This allows modeling resistive walls without meshing into conducting materials.

Assumptions

• Surface impedance is constant (non‑dispersive).
• Boundary edges or faces are tagged as “surface impedance” via a boundary detection method using pyvista.cells_inside_surface and applying a gradient-based masking.
• Surface conductivity is defined as: $\sigma_s = \frac{1}{Z_s}$
• Surface impedance is related to material properties as: $Z_s = (1 + j) \sqrt{\frac{\omega \mu}{2 \sigma}}$
• The computed $\sigma_s$ will modify the permittivity and conductivity tensors ieps and sigma at the boundary-detected cells.

Example: Copper at High Frequency

For copper (conductivity $( \sigma = 5.8 \times 10^7 , \text{S/m} ))$ at frequency $( f = 1 , \text{GHz} )$, assuming $( \mu = \mu_0 )$:

• Angular frequency: $\omega = 2\pi f = 2\pi \times 10^9$
• Then:
  $Z_s^\text{(Cu)} \approx (1 + j) \cdot \sqrt{\frac{\omega \mu_0}{2 \cdot \sigma}} \approx (1 + j) \cdot 8.3 \times 10^{-3} \Omega$
• Surface conductivity:
  $\sigma_s = \frac{1}{Z_s} \approx \frac{1}{(1 + j) \cdot 8.3 \times 10^{-3}} \approx (0.060 - j0.060) \cdot 10^3 \text{S}$

This value can be used to define $( \sigma_s )$ at the wall in the ADE formulation.