HFSS15: Radiation Boundaries

When solving radiating (i.e. antenna) and scattering (i.e. Radar Cross Section) structures in an unbounded, infinite domain, HFSS truncates the problem into a bounded domain and prescribes the appropriate truncation condition. This is generally known as the “radiation boundary condition”. Theoretically, the radiation boundary condition should be a “transparent” condition. In other words it should not produce any unphysical reflection as a result of the artificial truncation. HFSS offers three types of radiation boundaries: first-order absorbing boundary condition (ABC), perfectly matched layers (PML), and boundary integral equations (IE).

ABC and PML Boundaries

Both the ABC and PML boundaries attempt to minimize reflections by absorbing all outgoing waves at the truncation boundary. Because of this, they can only be prescribed at convex surfaces. This is because for concave surfaces, outgoing waves will re-enter the problem domain and should therefore not be completely eliminated. While PMLs absorb any kind of waves including guided waves, ABC imitates radiation to homogeneous background space.

ABCs only absorbs normal or near normal incident waves. Thus in order to produce accurate results, it must be placed sufficiently far away from structures. The typical recommendation is at least a quarter wavelength from the radiating source, although in some cases the radiation boundary may be located closer than one-quarter wavelength, such as portions of the radiation boundary where little radiated energy is expected. ABC is a local condition and thus preserving the sparse nature of the FEM formulation.

PMLs absorb all outgoing waves by adding artificial material layers that are designed such that all of the incident waves impinging upon them are completely transmitted with minimal reflections. Thus PMLs can be placed closer than ABCs. Furthermore, the PML absorbs a much wider range of waves in terms of frequency and direction whereas ABC absorbs only normally incident waves accurately. However, PMLs in general makes it more difficult for the iterative solver to reach convergence compared to ABCs. PMLs also preserves the sparse nature of the FEM formulation.

Boundary Integral Equations

The boundary integral equations (IE) is an exact transparent condition. The Sommerfeld radiation condition at infinity as required by physics is enforced exactly through the employment of appropriate Green’s functions via an integral equation method. This hybridization is commonly known as the finite element-boundary integral (FEBI) method. Unlike ABC and PML, the IE boundary can be of arbitrary shape, both concave and convex thus in some cases allowing the size of the finite element solution domain to be significantly reduced. It can be placed very close to the structures and produce accurate results and from a performance point of view a minimum distance of 0.125 wavelengths is recommended. Since analytic Green’s functions are known only for a few specific situations, the IE boundary is not allowed to touch any other boundary except an infinite ground plane. In addition the IE boundary does not work with symmetry planes. The IE boundary is not compatible with curvilinear finite element and thus should not be placed on a true surface if it is desired to use curvilinear elements in the finite element domain. The IE condition is a non-local condition, producing a partial sparse and partial dense system matrix. For this reason when simulating electrically large structures, the IE condition typically requires more RAM and CPU than either ABC or PML assuming all are placed on the same geometry. However by taking advantage of conformal radiation volumes to reduce the overall finite element solution domain utilization of the IE boundary will result in more efficient simulation for electrically large open boundary problems.