 |
| |
|
Celia has two types of boundary conditions, first order one way wave equation and perfectly matched layer (PML).
The first order one way wave equation method is a differential technique designed for low accuracy but high speed. It is provided for quick and dirty calculations where the nature of the calculation is, for example, to test the configuration of the model as quickly as possible or to assess early time behaviour of a system before the boundary comes into play.
The first order boundary is configured as a vacuum only condition at the present time.
For high accuracy calculations the perfectly matched layer (PML) method has been implemented. This method uses an anisotropic absorbing material outside of the computational mesh to absorb outgoing waves and prevent boundary reflections. It is a highly effective boundary method with very low reflection coefficient.
The characteristics of the absorbing layer are determined by its thickness, and this is a user defined property. The thicker the layer, the better its performance, but on the negative side a thicker layer requires more storage and time to compute.
The PML implementation used is a development of that of Berenger. This method uses a split field non-Maxwellian set of field equations. In its classic form the Berenger scheme permits neither the termination of dispersive nor lossy media.
In order to extend the PML method to these material types it is necessary to make changes to the split field equation set that is solved in the PML layer to introduce dispersion. Furthermore it is necessary to re-cast the equations governing the calculation of the conductivities within the layer to ensure proper matching, and so reflectionless absorption of the outgoing wave.
The RBC within Celia has been upgraded to incorporate this modified equation set and is capable of handling dispersive and lossy material regions that abut the grid boundary.
A second issue with the Berenger scheme is that, while the method solves a set of split field equations containing a total of 6 field components for both electric and magnetic fields (a total of 12 field components), for storage efficiency Berenger chose to store only 3 values of conductivity within the PML layer.
This approach is the most efficient one for uniform material properties tangential to the boundary (in other words a single material, usually vacuum at the mesh boundary) but it cannot be used if structures within the mesh are to be terminated at the boundary.
This is the case for example with a microstrip circuit where the output port extends to the boundary and must be terminated with its characteristic impedance, or the introduction of a soil medium in a GPR calculation.
The Celia boundary condition has been developed to automatically recognise the material characteristics at the mesh boundary and to solve the appropriate equation set depending on those materials.
For a simple free space boundary the classic Berenger PML method is used for optimum storage efficiency. If, on the other hand, a GPR model is designed with layered soil media abutting the mesh boundary then the full anisotropic dispersive equations are solved.
The profile used for the conductivity in the PML layer is that preferred by most authors in the literature, Equations (1) and (2).
(1)
(2)
In these expressions, v is the wave speed, e the material permittivity, d the layer depth and r the distance into the layer. R is a theoretical reflection coefficient which can be chosen and m is the order of the spatial polynomial. This profile has been shown to operate well in most circumstances.
Within Celia the values of smax and m are chosen automatically according to the layer thickness (Wu and Fang).