In-Plane E Field or In-Plane H Field

In the general case, in 2D and 2D axisymmetric, solving for three variables for each field is still required. The “in-plane H” or “in-plane E” assumption simplifies the problem to only three dependent variables.

TM Waves in 2D

For TM waves in 2D, solve for an in-plane electric field vector and one out-of-plane variable for the magnetic field. Maxwell’s equations then read

(4-14)

with the flux terms

(4-15)

and

The divergence on ΓE(H) is applied row-wise. The conductivity and permittivity tensors σ and εr represent in-plane material properties, while the relative permeability μr is an out-of-plane scalar property.

The default Lax-Friedrichs flux parameters are τE = 1/(2Z) for Ampère’s law, and the scalar τH = Z/2 for Faraday’s law, where Z is the impedance of a vacuum.

TE Waves in 2D

For TE waves in 2D, solve for an in-plane magnetic field vector and one out-of-plane variable for the electric field. Maxwell’s equations then read

(4-16)

with the flux terms

(4-17)

and

The divergence of ΓH(E) is applied row-wise. The tensor of relative permeability μr represents in-plane material properties, while the relative permittivity εr and conductivity σ are out-of-plane scalar properties.

The default Lax-Friedrichs flux parameters are τE = 1/(2Z) for Ampère’s law, and two scalar τH = Z/2 for Faraday’s law, where Z is the impedance of a vacuum.