PMLs apply a complex coordinate stretching in one, two, or three directions, depending on how the PML domain connects to the physical domain. In each direction, the same form of stretching is used, defined as a function of a dimensionless coordinate ξ, which varies from 0 to 1 over the PML layer. The function returns a new, complex and stretched, coordinate interpreted as relative to the typical wavelength for each simulation frequency. That is, the complex displacement for stretching in a single direction is Δx = λfi(ξ)−Δwξ, where λ is a typical wavelength and Δw is the original width of the PML (as drawn in the geometry). A separate displacement is computed for each stretching direction and summed to make a total displacement.In the PML nodes, you can choose between predefined polynomial and rational stretching functions, or select your own user defined functions. The polynomial stretching function is defined aswhere λ is a typical wavelength parameter, p is a curvature parameter, and s is a scaling factor. The rational stretching function is defined asFor user defined stretching you specify the real and imaginary part as separate functions of one or two arguments. The first argument is interpreted as the dimensionless distance ξ and the second — optional — argument as the typical wavelength λ.
• The typical wavelength represents the longest wavelength of propagating waves in an infinite medium. It is normally provided by a physics interface. For nondispersive media, it is expected to be inversely proportional to the frequency and serve to make the PML perform similarly for all frequencies.
In eigenfrequency studies, the typical wavelength parameter must not depend on the — unknown — frequency. When the typical wavelength is set to be obtained from a physics interface, it is therefore redefined to be equal to the PML width Δw instead. A user-defined typical wavelength applies as entered, but must not be a function of the frequency. It is often most convenient to draw and mesh the PML as if it had been part of the physical domain. To tune its effective thickness, use the scaling factor.
• The PML scaling factor multiplies the typical wavelength to produce an effective scaled width for the PML. For example, to retain perfect absorption for plane waves incident at an angle θ relative to the boundary normal, it is necessary to compensate for the longer wavelength seen by the PML in the stretching direction. In this case, 1/cos(θ) is a suitable scaling factor.Conversely, if resolving the field inside the PML proves too costly, it is possible to lower the scaling factor below its default value 1, to make better use of the available mesh elements. Note that this has a price in terms of less efficient absorption.
• The PML curvature parameter serves to relocate mesh resolution inside the PML. When there are components present which decay inside the PML much faster than the longest waves, the resolution must be increased in the zone closest to the boundary between PML and physical domain. Increasing the curvature parameter effectively moves available mesh elements toward the inner PML boundary. This is often necessary when the wave field contains a mix of different wavelengths or a mix between propagating and evanescent components.