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Method of Moments (MoM)

Method of Moments (MoM)
Source and surface discretization
for the MoM

General Applicability of the Technique

The MoM is applicable to problems involving currents on metallic and dielectric structures and radiation in free space. The structures are electrically small and are typically made of metals, although special extensions allow the inclusion of dielectrics, either as layered dielectrics or as finite sized shapes.

Technical Foundation

The MoM is a full wave solution of Maxwell's integral equations in the frequency domain. An advantage of the MoM is that it is a "source method" meaning that only the structure in question is discretised, not free space as with "field methods". Boundary conditions do not have to be set and memory requirements scale proportional to the size of the geometry in question and the required solution frequency. The following special extensions have been included in FEKO's MoM formulation to enable the modelling of magnetic and dielectric media.

Amuba image
Special Greens function for
planar multilayered media

Planar Green Functions for Multilayered Media

Multilayered dielectric media may be modelled with Green's functions, e.g. substrates for microstrip. The special Green's function formulation implements 2D infinite planes with finite thickness to handle each layer of the dielectric.  Conducting surfaces and wires inside the dielectric layers have to be discretised, but not the dielectric planes themselves.  Metallic surfaces and wires can be arbitrarily oriented in the media and are allowed to cross multiple layers.  Arbitrarily shaped dielectric objects may be included in multilayered dielectric media by modelling them as surface equivalence principle objects.  Slot coupled structures are easily simulated with magnetic slot models and calculations are sped up with interpolation tables.

Surface Equivalence Principle image
Surface Equivalence
Principle (Surface Mesh)

Surface Equivalence Principle (SEP)

The SEP introduces equivalent electric and magnetic currents on the surface of a closed dielectric body. The surface of such bodies can be arbitrarily shaped and is discretised using triangles.  This method may also be used to bound planar multilayered media to form finite sized dielectric objects.  In such instances the boundary of the object is meshed as SEP triangles, while the internal layers are modeled with the multilayered dielectric media method, thus avoiding the requirement to mesh the dielectric boundaries between different dielectric layers of the object.

Volume Equivalence
Principle (Tetrahedral Mesh)

Volume Equivalence Principle (VEP)

The VEP allows the creation of arbitrarily shaped dielectric bodies using tetrahedral volume elements. More basis functions are typically required than for the SEP, but neighboring tetrahedra may have differing electric and magnetic properties.

Windscreen method

Windscreen Antenna Method

Windscreen antennas typically consist of a number of dielectric layers of varying dielectric properties.  FEKO includes a MoM based formulation that meshes only the metallic antenna elements in a windscreen antenna, while rigorously taking all dielectric layers into account with special methods. The MLFMM can be enable together with this method.

Thin Dielectric Sheets

Multiple layers of thin dielectric and anisotropic sheets can be analysed as a single layer in FEKO. Typical applications are the analysis of radome covered antennas and windscreens of automobiles.

Dielectrically Coated Wires

FEKO implements two methods for the modelling of dielectric and magnetic coatings on wires:

  • Popovic's formulation modifies the radius of the metallic wire core to change the capacitive loading on the wire, while simultaneously adding a corresponding inductive load. The method is restricted in that the loss factor of the layer has to be identical to the loss factor of the surrounding medium.
  • Pure dielectric layers (i.e. relative permeability of the layer equals that of the surrounding medium) should be modelled with the equivalence theorem where the effect of the dielectric layering is accounted for by a volume polarisation current. The only restriction on the method is that the layering may not be magnetic.

Real Ground

Real ground can be modelled with the reflection coefficient approximation or the exact Sommerfeld formulation.

Periodic Boundary Condition (PBC)

Infinite periodic structures with periodicity in either one or two dimensions may be modelled with the PBC feature. The structures can consist of metallic surfaces and thin dielectric sheets. The PBC method is based upon the periodic version of the free space Green function and can thus be used together with an infinite perfect electric or magnetic conducting surface, but not together with dielectric layers. A typical application is the analysis of frequency selective surfaces.

Low Frequency Analysis

FEKO makes the analysis of very low frequency problems possible by automatically decomposing the problem space with special basis functions.  In test problems FEKO accurately computed the current distribution on the object under test for frequencies as low as 0.001 Hz.

Higher Order Basis Functions


RWG (lower order)
basis function


Edge-based HOBF of
order 3.5

Higher Order Basis Functions (HOBF) use higher order polynomial basis functions to model the currents on any particular mesh element.  Such HOBF allow the user to mesh geometry with larger triangles, while obtaining the same solution accuracy.  These larger (more coarse) mesh elements imply that less mesh elements are used to discretize a model, which in turn implies that less unknowns have to be solved during the computation process and that problems can be solved with less memory.  In very large problems solution time will also decrease.

FEKO uses hierarchical basis functions to increase the order of any triangle as necessary.  Small geometric details of a model will still be meshed with electrically small mesh elements, while larger details may be meshed with coarser mesh elements.  When FEKO automatically performs order selection for the model, higher order basis functions are applied to electrically large mesh elements, while lower order basis functions are applied to electrically smaller mesh elements.  With this adaptive scheme FEKO automatically ensures high fidelity MoM solutions, with as little memory as possible and as fast as possible solution times.

FEKO's implimentation of HOBF is supported for curvilinear mesh elements.  Large flat triangular mesh elements that are well suited to meshing of flat or minimally curved surfaces are poorly suited to meshing geometries with complex curvature.  In such cases mesh triangles would traditionally have to be made smaller in order to represent the geometry accurately and consequently to model phase from scattered or transmitted fields properly.  FEKO's curvilinear HOBF support allow large triangular mesh elements to wrap onto curved geometry as demonstrated here.  Large mesh elements may thus be used to mesh a curved surface very accurately, both from a geometrical and EM perspective.

RCS_comparison_RWG_HOBF_Curvilinear.png mesh_sphere_RWG_coarse.jpg
(0.125 λ)
(0.75 λ)
Curvilinear HOBF
(0.75 λ)
RCS of a sphere comparing RWG with flat HOBF and
curvilinear HOBF models

Acceleration of MoM

The MoM forms a densely populated matrix as part of its solution process.  As a result, the traditional implementation of the MoM scales poorly in both memory and runtime requirements as the size of the problem increases.  This problem is overcome in FEKO with the application of the Adaptive Cross-Approximation (ACA) method.  The ACA is a purely mathematical method that improves the solution of complex MoM problems with significant gains in both runtime and memory requirements.

MoM solutions are also accelerated by efficient hardware utilization in FEKO, including:

  • Parallelisation for both shared and distributed memory computers and clusters to maximally utilize RAM and available CPU cores in such systems.  Special coding techniques ensure highly efficient scaling of parallel processed MoM solutions with increasing unknowns or number of processes.
  • Graphic Processing Unit (GPU) implementation of MoM subroutines to speed up solutions.


Typical Application of the MoM

Typical applications of the MoM include wire antennas, antennas mounted on structures, etc.

Magic T delta port driven 3D horisontal plane (small)

Fields in a delta port driven waveguide magic-T
Additional Information

Additional Information