Infinite Element Implementation
Infinite elements apply a semi-infinite coordinate stretching in one, two, or three directions, depending on how the infinite element 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 infinite element layer. The function returns a new, stretched, coordinate interpreted as a new position in the given direction. That is, the displacement for stretching in a single direction is Δx = fi(ξ)−Δwξ, where Δw is the original width of the infinite element domain (as drawn in the geometry). A separate displacement vector is computed for each stretching direction and summed to make a total displacement.
The default rational stretching function is defined as
(5-1)
where Δp is the, so called, pole distance and γ is a number larger than one, computed as
where Δs is the scaled thickness of the infinite element domain. The scaled thickness Δs and the pole distance Δp are user inputs.
For the user-defined stretching option, you specify f(ξ) directly as a functions of the dimensionless distance ξ. This can, for example, be used for modeling a surrounding domain of finite extent, or for implementing a stretching function that better fits a dependent variable that decays with distance slower or faster than a monopole solution.
There is no check that the geometry of the region is correct, so it is important to draw a proper geometry and select the corresponding region type.
Interpreting Infinite Element Parameters
The default infinite element stretching has two user-defined parameters that let you control the thickness of the quasi-infinite region, as perceived by the physics interfaces, as well as how the stretching is distributed across the domain.
Physical width
The scaled width of the infinite element domain, Δs, is by default set to 1e3*dGeomChar, where the constant dGeomChar is a characteristic geometry dimension. The domain is therefore by default scaled to be very much larger than the original geometry, but not quite infinite in order to avoid numerical difficulties. In particular, the finite distance to the far-away boundary allows prescribing standard boundary conditions effectively at infinity.
Pole distance
The coordinate stretching function, Equation 5-1, used in the infinite element domain contains a singularity when ξ = γ. Since γ > 1, this happens outside the infinite element domain. The pole distance, Δp, controls just how far away this singularity is located. If Δp is small compared to the scaled width, Δs, the coordinate stretching is very nonlinear, progressing from gentle close to the boundary with the physical domain to abrupt toward the quasi-infinite boundary. Conversely, if the pole distance is large compared to the scaled width, the stretching is constant across the domain.
The default pole distance is dGeomChar, which is small compared to the physical width. Therefore, the coordinate stretching by default exhibits a nearly 1/r behavior, which is suitable for making optimal use of mesh resolution when the dependent variable also behaves as 1/r for large r, where r is the distance from any sources or inhomogeneities.