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Multilevel fast multipole method

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Computational mathematics

The multilevel fast multipole method (MLFMM) is used along with method of moments (MoM) a numerical computational method of solving linear partial differential equations which have been formulated as integral equations of large objects almost faster without loss in accuracy. This method is an alternative formulation of the technology behind the MoM and is applicable to much larger structures like radar cross-section (RCS) analysis, antenna integration on large structures, reflector antenna design, finite size antenna arrays, etc., making full-wave current-based solutions of such structures a possibility.

Method

The MLFMM is based on the Method of Moments (MoM), but reduces the memory complexity from O ( N 2 ) {\displaystyle {\mathcal {O}}(N^{2})} to O ( N log ⁡ N ) {\displaystyle {\mathcal {O}}(N\log N)} , and the solving complexity from O ( N 3 ) {\displaystyle {\mathcal {O}}(N^{3})} to O ( N iter N log ⁡ N ) {\displaystyle {\mathcal {O}}(N_{\textrm {iter}}N\log N)} , where N {\displaystyle N} represents the number of unknowns and N iter {\displaystyle N_{\textrm {iter}}} the number of iterations in the solver. This method subdivides the Boundary Element mesh into different clusters and if two clusters are in each other's far field, all calculations that would have to be made for every pair of nodes can be reduced to the midpoints of the clusters with almost no loss of accuracy. For clusters not in the far field, the traditional BEM has to be applied. That is MLFMM introduces different levels of clustering (clusters made out of smaller clusters) to additionally enhance computation speed.

References

  1. "Multilevel Fast Multipole Method (MLFMM)". Austrian Academy of Sciences – Acoustics Research Institute. Retrieved 20 April 2014.
  2. "Multilevel Fast Multipole Method (MLFMM)". Feko. Retrieved 20 April 2014.
  3. "Multilevel Fast Multipole Method (MLFMM)". E field. 2013-04-30. Retrieved 20 April 2014.
  4. P.-L. Rui; R.-S. Chen; Z.-W. Liu & Y.-N. Gan (2008). "Schwarz-Krylov Subspace Method for MLFMM Analysis of Electromagnetic Wave Scattering Problems". Progress in Electromagnetics Research. 82. PIER: 51–63. doi:10.2528/PIER08013003.
  5. Bingle, M Burger, E.; Jakobus, U.; van Tonder, J.J. (7–9 Nov 2011). "Theory and application of an MLFMM/FEM hybrid framework in FEKO". 2011 IEEE International Conference on Microwaves, Communications, Antennas and Electronic Systems (COMCAS 2011). pp. 1–3. doi:10.1109/COMCAS.2011.6105819. ISBN 978-1-4577-1694-2. S2CID 39160247.{{cite book}}: CS1 maint: multiple names: authors list (link)
  6. D'Ambrosio, K.; Pirich, R.; Kaufman, A.; Mesecher, D. (11 May 2009). "Parallel computation methods for enhanced MOM and MLFMM performance". 2009 IEEE Long Island Systems, Applications and Technology Conference. pp. 1–4. doi:10.1109/LISAT.2009.5031571. ISBN 978-1-4244-2347-7. S2CID 18786124.
  7. Ulrich Jakobus; Johann van Tonder & Marlize Schoeman. "Advanced EMC Modeling by Means of a Parallel MLFMM and Coupling with Network Theory" (PDF). EMSS. Retrieved 20 April 2014.
  8. "(Electrically) Large Applications & Integral Equation solver" (PDF). CST. Retrieved 20 April 2014.
  9. "Multilevel Fast Multipole Method (MLFMM)". ESI. Archived from the original on 20 April 2014. Retrieved 20 April 2014.
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