Finite element method for high-frequency electromagnetics

Question

I am writing a project about the Finite element method for use in high-frequency solutions of Maxwell's equations. This could be for use in antenna design and similar. I have some trouble understanding how to choose the form of the maxwells equation to solve. It seems that for high-frequency problems people use the second order equations, the one for only one field. $$ \nabla \times \mu^{-1} \nabla \times E + \epsilon \frac{\partial^2 E}{\partial t^2} = 0 $$ Others solve the first-order equations for both the electric and magnetic fields. This seems to often be for use in eddy current problems, like electric motors and transformers.

$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{B} = \mu \left( \mathbf{J} + \epsilon \frac{\partial \mathbf{E}}{\partial t}\right) $$

Is this because it is simpler to solve the second-order equation, but you need more information for low-frequency problems like electric motors?

I have mostly used the books: "The finite element method in electromagnetics" by Jian-ming Jin, and "Computational Electromagnetics" by Thomas Rylander, Par Ingelstrom, and Anders Bondeson.

Does anyone have any insight into this?

Answer 1

Using the typical expansion functions (1-forms/edge-elements for E, and 2-forms/facet-elements for B) the formulations are basically the same after spatial discretization and you'd expect more or less the same accuracy. I do think they express slightly different opinions about time integration.

The mixed E/B formulation nudges you in the direction of leapfrog integration, leading towards something FDTD-like but with unstructured gridding instead of sugarcubing. This formulation is not very popular because it is both conditionally stable (like FDTD's CFL condition, but worse because you have less control over element shape) and yet it still requires an implicit solve by a non-diagonal mass matrix (although it is well conditioned / spectrally equivalent to identity, it's still an annoyance).

The all-E formulation nudges you in the direction of Newmark integration, which is unconditionally stable (not limited by element size/shape). This method also requires an implicit solve and the system is more poorly conditioned (it's a weighted sum of the the stiffness and mass matrices, so it inherits unboundedness from the curl-curl operator).

Of course, there's nothing that says you have you use these particular time integrators. You can always submit these systems to a black-box ODE solver (Runge-Kutta, etc).

Since you mention antenna modeling, another method that should be considered is frequency domain FEM, based on the vector wave equation $\nabla \times \mu_r^{-1} \nabla \times \vec E - k^2 {\epsilon}_r \vec E = 0$. These systems are rather difficult to solve/precondition: oscillatory behavior like Helmholtz, compounded with the complexity of faithfully preconditioning the nullspace of the curl operator. But, they can be readily hybridized with other frequency domain techniques (modal expansions, method of moments, etc). These are much more accurate termination conditions than ABC's/PML's/etc, which can be important because many quantities of engineering interest (gain, RCS) are derived from the far-field.

As an aside, I'd recommend Jian-Ming Jin's "Finite Element Analysis of Antennas and Arrays", it's excellent (though a little more advanced). It has in-depth coverage of the time-domain Newmark method and frequency-domain methods/hybrids.