How to Improve Convergence for the MLFMM

The MLFMM 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:
  • Activate additional stabilisation for the MLFMM.
  • Adjust the mesh.
  • Change the preconditioner.
  • Change the default box size.
  • Use the CFIE for metallic structures.
  • Use double precision.
If these options fail to bring convergence, consider if the model warrants using another solution method, or contact technical support for further assistance.
Tip: Highly lossy media (solved with the SEP) in most cases have poor convergence for the MLFMM. The FEM or VEP would be better suited to solving these materials.

Activating Additional Stabilisation for the MLFMM

Activating additional stabilisation may help to achieve convergence for the MLFMM.

Dielectrics, wires and other elements can be included in the model. If the poor convergence is due to something other than the metallic triangles, activating the stabilised MLFMM is unlikely to improve convergence.
Restriction: The stabilised MLFMM improves convergence only for metallic triangles.

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.

If a model is discretised too finely (smaller than λ 10 ) or mesh elements are too coarse (larger than λ 7 ), 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 Preconditioner

The sparse LU is the default preconditioner. Changing to another preconditioner may help to achieve convergence for the MLFMM. Select one of the following preconditioners:

  • Use the sparse approximate inverse (SPAI) preconditioner.
    • The default (accelerated SPAI) preconditioner is fast.
    • The non-accelerated SPAI preconditioner takes longer than its accelerated counterpart, but often converges better.
  • Use the incomplete LU decomposition (ILU) preconditioner.
    Restriction: The ILU preconditioner is supported only for sequential solutions.

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.
  • 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.
  • Double precision requires twice the memory compared to single precision.
  • Double precision does not improve convergence for the stabilised MLFMM.