Fascinating theoretical challenge: Simulating EM eigenmodes in a tetrahedron

General Discussion
1

Hello FreeFEM community,

I am an independent researcher working on a theoretical physics model where the geometry of space is hypothesized to be tetrahedral at a fundamental scale. A key prediction of the model is that the fundamental electromagnetic eigenmode of a perfectly conducting tetrahedron should have a specific dimensionless frequency.

The Theoretical Prediction: For a regular tetrahedron with edge length a, the fundamental eigenvalue k (wave number) should satisfy: (k * a)^2 / π² ≈ 14.6969 (This value comes from 12 * sqrt(3/2)).

The Request: I would be incredibly grateful if someone with experience in electromagnetic simulations in FreeFEM could help me verify this. The task would involve:

  1. Defining the geometry of a regular tetrahedron.
  2. Setting up the eigenvalue problem for EM waves in a perfect electrical conductor (PEC) cavity.
  3. Solving for the fundamental eigenmode and extracting its eigenvalue.
  4. (Optionally) visualizing the E-field distribution to see if it oscillates between opposite edges.

I believe this is a non-trivial and interesting problem that could be of general interest. My strength is in the theoretical side, and I am less experienced with numerical code. Any help or guidance would be immensely appreciated. I am, of course, happy to provide full theoretical context and acknowledge all help in any resulting work.

Operational Definition for the Simulation:

  1. Please define a regular tetrahedron with edge length a = 1.
  2. Set up the electromagnetic eigenvalue problem for a perfect electrical conductor (PEC) cavity, solving the wave equation \nabla^2 \mathbf{E} + k^2 \mathbf{E} = 0 with boundary condition \mathbf{\hat{n}} \times \mathbf{E} = 0.
  3. Solve for the fundamental eigenmode (the mode with the smallest eigenvalue k^2).
  4. The output I need is this smallest eigenvalue \lambda = k^2 .

From this, I will compute the dimensionless constant C = \lambda / \pi^2 and check if it is approximately 14.6969.

A visualization of the E-field for this mode would also be immensely valuable to see if the energy is indeed concentrated between opposite edges.

Thank you for considering this request.

Best regards, Frodo