Representing Antenna Fields with Equivalent Surface Currents

The Huygens Principle tells us that the radiated fields of any source distribution confined within a certain volume $V$ can also be reproduced by equivalent surface currents on the surface $\partial V$ of the source volume.

In AntennaFieldRepresentations.jl, surface current densities are represented by a struct SurfaceCurrentDensity{P, E, B, C} which is a subtype of AntennaFieldRepresentation{P, C}. The type parameters have the following meaning

Constructors for a SurfaceCurrentDensity

To generate a SurfaceCurrentDensity, use one of the following constructors:

SurfaceCurrentDensity{P, E, B, C}(functionspace::B, coefficients::AbstractVector{C}, wavenumber <: Number) where{P <: PropagationType, E <: ElmagType, B <: BEAST.Space{T} where{T <: Real}, C <: Complex}

SurfaceCurrentDensity(::P, ::E, functionspace::B, coefficients::AbstractVector{C}, wavenumber <: Number) where{P <: PropagationType, E <: ElmagType, B <: BEAST.Space{T} where{T <: Real}, C <: Complex}

SurfaceCurrentDensity(::P, ::E, functionspace::B, wavenumber <: Number) where{P <: PropagationType, E <: ElmagType, B <: BEAST.Space{T} where{T <: Real}, C <: Complex}

The input arguments for the costructors are

• P <: PropagationType : Can be Radiated, Absorbed, or Incident
• E <: Elmagtype : Can be Electric or magnetic
• coefficients: Coefficient vector. Can be an AbstractVector{C}. If no coefficients vector is given, a zero-filled Vector{C} of appropriate size will be created for initialization.
• B <: BEAST.Space : An object which defines the (triangulated) geometry of the Huygens surface and the space of basis funtions for expanding the surface currents. Refer to the The BEAST.Space Type-section for more detailed information
• wavenumber : Wavenumber $\omega = 2\pi \, f$

The BEAST.Space Type

We don't have to reinvent the wheel. BEAST.jl has many relevant surface current expansions already implemented. The documentation for BEAST.jl can be found here and it is recommended to have a look at the tutorial and the examples to get used with its general usage.

using BEAST, CompScienceMeshes

Creating a BEAST.Space-object contains two steps: Creating the geometry (a mesh) and defining a Finite Element space on this geometry.

Creating the Geometry (the Mesh)

First, we need to define a mesh. The package CompScienceMeshes provides data structures and algorithms for working with simplical meshes in computational science. We can create a mesh from scratch for very simple meshes, e.g., by

using CompScienceMeshes

meshsize = 0.1
sidelengthA = 1.0
sidelengthB = 1.0

mesh = CompScienceMeshes.meshrectangle(sidelengthA, sidelengthB, meshsize)

or

using CompScienceMeshes

radius = 1.0
meshsize = 0.4

mesh = CompScienceMeshes.meshsphere(radius, meshsize)

More elaborate meshes can be created with a meshing tool like gmsh ^[1]. The meshes created by gmsh (usually stored in a .msh-file) can be loaded for our use with

using CompScienceMeshes

mesh = CompScienceMeshes.read_gmsh_mesh("example_box.msh")

Defining the Finite Element Space

Creating the finite element space is easy in BEAST.jl. Several, easy to use constructor functions exist to create many of the most widely used Finite Element spaces for surface current expansion. To create a set of Rao-Wilton-Glisson basis functions on a previously defined triangular mesh Γ, use ^[2]

using BEAST

β = BEAST.raviartthomas(mesh)

At the moment, Rao-Wilton-Glisson (RWG) basis functions are the only type of Finite Element space which is tested to work with AntennaFieldRepresentations.jl. However, many more Finite Element spaces are defined by BEAST.jl which might just "work out of the box" (go ahead and try if you dare and let us know how it worked out for you):

• For rotated RWG (more technically: $\bm{n} \times$RWG) functions, use (the × symbol can be typed in the REPL by typing \times followed by [TAB])

using BEAST

β = BEAST.n × BEAST.raviartthomas(mesh)

• For Buffa-Christiansen basis functions use

using BEAST

β_buffa = BEAST.buffachristiansen(mesh)

• More Finite Element spaces can be found by consulting the source code of BEAST.jl.

Defining the SurfaceCurrentDensity

currents_el = SurfaceCurrentDensity{Radiated,Electric,typeof(β_buffa),ComplexF64}(β_buffa, ones(ComplexF64, numfunctions(β_buffa)), k0)
currents_mag = SurfaceCurrentDensity{Radiated,Magnetic,typeof(β_buffa),ComplexF64}(β_buffa, ones(ComplexF64, numfunctions(β_buffa)), k0)