# EDA Focus January 2010: HFSS' New Integral Equation Solver

The High Frequency Structure Simulator (**HFSS**™) is a powerful 3D full-wave electromagnetic field simulator used in microwave/RF, antenna, signal integrity and electromagnetic compatibility simulation and design. HFSS is widely used by corporations and government labs to model a wide variety of microwave components and systems including antennas and antenna platforms. The latest release of HFSS, 12.1, adds a new integral equation solver option - HFSS-IE. This new solver, capable of handling lossy material properties, uses a method of moments (MoM) technique to solve for the sources or currents on the surfaces of the conducting and dielectric objects. HFSS-IE is primarily effective for radiation and scattering studies of large, mostly conducting structures and is available in the industry standard HFSS interface; sharing geometry, material and certain critical solver technology such as adaptive meshing for automated accuracy. In addition one can link in the fields computed from an HFSS design in to an HFSS-IE model giving the user the best of both technologies. The HFSS-IE solver will complement HFSS as a comprehensive antenna design solution.

# Theory

In integral equation techniques the unknown is the surface current. Surface integral equations are derived from the continuity of the tangential electric and magnetic fields on a surface, and they relate the electric and magnetic fields on the surface of a conductor to the electric current on the surface. Scattering from dielectrics is handled using equivalent magnetic and electric surface currents which results in a similar relationship.

Using a surface mesh, these integral equations are converted to a matrix equation using the "method of moments" (MoM). Unfortunately, the conventional solution procedure yields a dense matrix which requires large amounts of memory and time even when using a modern iterative matrix solution technique. This memory and time constraint would normally limit the use of integral equations to small problems, but fortunately, there is a way around it.

Adaptive cross approximation (ACA) is a method for compressing sub-blocks in the impedance matrix that result from the MoM procedure. ACA in conjunction with an iterative matrix solver reduces the memory and complexity requirements, allowing for integral equations to be applied to very large problems. In addition the solver can take advantage of multi-core computers to reduce the overall solution time. For example, consider the antenna mounted on an aircraft shown in Figure 1, for this project if one uses 4 processors the solve time is reduced by a factor of 2.3 when compared to a single processor solution.

Figure 1 The new integral equation solver option operates within the industry-standard HFSS desktop.

# Interface and Links to HFSS

A strength of the HFSS-IE solver is its implementation as a design type in the familiar HFSS desktop, see Figure 1. To a current HFSS user there is little additional training required. Once the model geometry, material and boundary conditions are entered and excitations defined the solution setup is straightforward; the user enters the frequency and the convergence criteria; no additional solver settings are required for the ACA algorithm, which along with mesh generation is fully automated.

Another advantage of using the same desktop as HFSS is the option to use HFSS designs as sources in an HFSS-IE design though a data link. Since the two solvers use the same desktop a user can place both designs in the same project. One creates the feed in HFSS and then links the fields from that simulation into the target HFSS-IE design with a few mouse clicks. As an additional feature the user can also include the feed structure of the source simulation in the target HFSS-IE design. In this case the HFSS-IE solution will include the scattering from the feed when computing the resulting patterns. This option can be a very powerful feature for applications such as reflector antennas where feed blockage is important.

# Adaptive Meshing

One of the most salient features of HFSS and in fact all of Ansoft's 3D electromagnetic field simulation tools is adaptive refinement. In an adaptive solution the solver computes the fields using an initial mesh and then based on a proprietary refinement procedure determines what parts of the model require a finer mesh and automatically refines the mesh in those areas. It then re-solves and repeats until numerical convergence is achieved.

Using adaptive refinement the user allows the solver to place elements where needed, eliminating the guess work and uncertainty of a hand generated mesh and providing automation and reliability to the simulation process. For such a procedure to work efficiently a stopping criteria is needed and for antenna applications one typically uses the change in S parameters although in HFSS-IE, as in HFSS, the option to include one of the fields quantities (e.g. directivity) in determining the convergence is also available.

To illustrate the power of such a procedure consider the blade antenna shown in Figure 2. This antenna is mounted on a larger body not shown. A typical approach one might use to simulate this antenna is to seed the model and then run 1 pass. As a test the mesh was seeded to an edge size of λ/25 and simulated 1 pass. That mesh is shown in Figure 2a. For comparison an adaptive simulation was also run. The mesh after running 7 passes is shown in Figure 2b. As one can see the meshes are radically different. The plot shown in Figure 3 shows the difference in the S11 as compared to a well converged HFSS simulation for the adaptive simulation plotted vs. pass. Included for comparison is the result produced from the mesh displayed in Figure 2a. As we would expect the mesh for that simulation does not accurately predict the input response of this antenna.

Figure 2. Examples of meshes for blade antenna structure (a) is a seeded mesh (b) is a mesh generated using the adaptive process on the same model resulting in a more refined mesh.

Figure 3. Results for blade antenna simulation. Error presented is the difference in the complex magnitude of S11 between the HFSS-IE result and the result from an HFSS simulation. Inset shows current plot on full blade antenna for converged mesh.

The adaptive procedure not only creates the optimum mesh, which in many cases will be more efficient than a uniformly seeded mesh, but it also gets the correct mesh. This gives the user added confidence in the results. It is important to note that the adapted mesh shown in Figure 2b is very dense in the areas around the slot and it is this dense mesh that is important for correctly characterizing the response. The ACA matrix compression procedure mentioned in the last section can easily handle such a non-uniform mesh.

# Examples

In addition to designing the antenna itself often an antenna engineer is concerned with proximity effects. Specifically how does the system the antenna is mounted on affect the performance and/or is there an optimum location for the antenna on the given structure. The latter, referred to as antenna placement studies, are important in vehicle designs. For example consider the unmanned drone shown in Figure 4. A blade antenna similar to the one described previously is mounted on top of the airframe. The drone itself has a wing span of almost 15 meters, more than 40 wavelengths. Again in simulating this model the adaptive refinement of the blade antenna is critical to an accurate result. In the figure the resulting radiation patterns are overlaid on the vehicle model showing the currents on the conducting surfaces.

Figure 4. Simulation results for a 900 MHz blade antenna on an unmanned drone.

As a second example, this time a linked design, consider the reflector antenna shown in Figure 5. The feed is a circular flared horn antenna. The feed antenna was simulated in HFSS. In this case since the feed blockage is critical to the response the HFSS-IE model was setup to include the feed scattering in the final solution. The resulting currents are shown in the figure.

Figure 5. Currents on a reflector antenna modeled in HFSS-IE with the source simulated in a separate HFSS design. The simulation accurately predicts the induced currents on the feed structure.

# Conclusion

HFSS-IE is a new solver option for HFSS that uses an integral equation formulation to efficiently analyze large open, mostly conducting, models. In combination with HFSS the antenna design engineer can now pick the best solver to use for a given situation and in many cases can take advantage of both solvers in a linked project.