How to Improve Convergence for the MLFMM /FEM

The hybrid multilevel fast multipole method (MLFMM) / finite element method (FEM) is an iterative solution method and under certain conditions, the iterative solution may fail to converge. Several model or solution settings are presented that could improve the model's convergence behaviour.

Sometimes when using the MLFMM, the Solver stops with the error message: 
ERROR 4673: Iterative solution of the system of linear equations failed, maybe try another pre-conditioner (solution settings).
The Solver stores the solution with the lowest residuum during the solution. Results are generated if this residuum is considered adequate, but the results may be less accurate if the stopping criterion for the residuum is not exactly met. In this case, the Solver output contains the warning: 
WARNING 830: Maximum number of iterations reached without convergence, using in the following the solution with the smallest residuum.
Note: Warning 830 indicates possible inaccurate results due to inadequate convergence.
One or a combination of the following changes can be made to the model:
  • Adjust the mesh.
  • Change the default box size.
  • Use the CFIE for metallic structures.
  • Use double precision.
  • Change the FEM to use first order basis functions.

Adjusting the Mesh

Slight adjustments to the mesh size (smaller or larger elements) could lead to improved convergence. If a model is discretised too finely or too coarsely, convergence could be negatively affected.

Tip: Reduce the radii of thick wire segments, or replace them with metallic strips (2D meshes) or cylinders (3D meshes) to improve convergence.

Changing the Default MLFMM Box Size

The MLFMM uses a boxing algorithm that encloses the entire computational space in a single box at the highest level, dividing this box in three dimensions into a maximum of eight child boxes and repeating the process iteratively until the side length of each child box is approximately a quarter wavelength at the finest level. Using a different box size at the finest level can sometimes facilitate convergence, although memory consumption could increase if the box size is increased.

The default box size is 0.23. A lower value decreases memory consumption while a higher value increases the memory consumption.

Tip: Try box size values between 0.2 and 0.35 wavelengths with increments of 0.02.

Using the CFIE on MLFMM Metallic Surfaces

The combined field integral equation (CFIE) uses both the electric field integral equation (EFIE) and magnetic field integral equation (MFIE). This produces a better-conditioned matrix leading to better convergence in general.
Note:
  • Use the default preconditioner with the CFIE.
  • CFIE can only be applied to surfaces bounding closed PEC structures.
  • A mixture of CFIE and EFIE surfaces can be used.
  • Sharp corners on CFIE surfaces can lead to inaccurate results.

Sharp corners should be meshed finer if CFIE is applied. If there is uncertainty, the EFIE should rather be applied around sharp corners. A rule of thumb is to apply the EFIE up to a few meshed triangles away from the sharp corners.

Using Double Precision

The Solver uses single precision by default- a single byte is used to store a complex number.

Use double precision when higher accuracy is required and to help resolve convergence issues. Double precision uses two bytes to store a complex number in the matrix. This increases the number of significant digits and reduces numerical noise.
Note:
  • Double precision requires twice the memory compared to single precision.
  • Double precision does not improve convergence for the stabilised MLFMM.

Changing the FEM to Use First Order Basis Functions

The FEM uses higher order (order two) basis functions by default. For large volumes, the higher order results in a much smaller number of tetrahedra in the mesh.