Maxwell's equations (steady and time-harmonic)

Time-harmonic Maxwell's equations can be written under the form

Physical properties of each media are set with field MateriauDielec :

# MateriauDielec = ref ISOTROPE epsilon mu sigma
MateriauDielec = 1 ISOTROPE 10.0 2.5  0.1

You can enter complex values (without space between real and imaginary as for Helmholtz equation) for epsilon and mu. This equation is implemented by classes HarmonicMaxwellEquation_2D, HarmonicMaxwellEquation_3D with edge elements, and HarmonicMaxwellEquation_2D_DG, HarmonicMaxwellEquation_3D_DG with LDG formulation (first order system), HarmonicMaxwellEquation_Sipg3D with SIPG formulation. Static Maxwell's equations are equal to :

This equation is implemented by class StaticMaxwellEquation_3D (and similar classes for 2-D and LDG formulation). Transient Maxwell's equations can be deduced :

This equation is implemented by TimeMaxwellEquation_3D. When using TimeMaxwellEquation_3D_DG, LDG formulation is used, with the following first-order system :

Boundary Conditions

The following boundary conditions are implemented

DIRICHLET : such condition is usually called Perfectly conducting condition, and given as:
for homogeneous Dirichlet condition and
for inhomogeneous Dirichlet condition, F is provided when defining the source terms. F is automatically set when computing the diffracted field.
NEUMANN : such condition is given as:
for homogeneous Neumann condition and
for inhomogeneous Neumann condition, F is provided when defining the source terms. F is automatically set when computing the diffracted field.
ABSORBING : such condition is also called Silver-Müller condition, and given as:
When, the total field is computed, a source-term is present
where is the incident field
IMPEDANCE such condition is also called Robin condition, and given as
The impedance Z is given in the data file in the line related to the keyword ConditionReference. F is provided when defining the source terms. F is automatically set when computing the diffracted field.
PERIODICITY periodic conditions and quasi-periodic conditions are implemented. Therefore periodic domains or cyclic domains (periodicity in theta) are supported.

Modes on a section of a waveguide

In order to compute propagative modes along direction z :

One can eliminate E_z and H_z and obtain the following set of equations

The research of eigenvalues β and eigenvectors (E, H) for this set of equations is achieved by class ModeMaxwellEquation (for the definition of the equation), and VarModeMaxwell (for the associated problem