What Are Method of Moments?

The method of moments (MoM) is a numerical technique used in computational electromagnetics to solve integral equations derived from Maxwell's equations by converting them into a system of linear algebraic equations solvable by matrix methods. Unlike differential equation approaches such as the finite difference time domain method, MoM operates directly on integral formulations that already satisfy radiation conditions at infinity, making it particularly well suited for open-region problems involving antennas, scattering, and radiation. The technique was systematically formulated for electromagnetic problems by Roger Harrington in his 1968 textbook, which established the theoretical foundations still in use today.

MoM draws from the broader mathematical framework of weighted residual methods, in which an unknown function is expanded as a finite sum of known basis functions with unknown coefficients. Testing functions are then applied to drive the residual to zero in a weighted average sense, yielding a deterministic set of equations relating the unknown coefficients. In electromagnetics, the unknown is typically a surface current distribution on a conductor or dielectric body, and the integral equation relates that current to observed fields through a Green's function kernel.

Integral Equation Formulation

The two most common integral equation forms in MoM electromagnetics are the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE). The EFIE enforces the boundary condition that the total tangential electric field vanishes on a perfect electric conductor surface, while the MFIE uses the equivalent magnetic current relationship. A combined-field integral equation (CFIE) is formed as a linear combination of both, resolving the internal resonance problems that afflict EFIE or MFIE applied separately at certain frequencies. Basis functions commonly used include the Rao-Wilton-Glisson (RWG) triangular patch functions, which represent current flowing between adjacent triangular surface facets and satisfy current continuity across shared edges. According to an IEEE Xplore publication on the method of moments, these formulations underpin a broad family of full-wave solvers across frequency ranges from HF to millimeter wave.

Matrix Assembly and Solution

Once basis and testing functions are selected, MoM reduces the integral equation to a dense matrix equation of the form Z·I = V, where Z is the impedance matrix, I is the vector of unknown current coefficients, and V is the excitation vector. Each element of Z represents the mutual electromagnetic coupling between a pair of basis functions and is computed by evaluating a double surface integral over source and observation patches. For a geometry discretized into N unknowns, this produces an N-by-N dense complex matrix whose direct factorization requires O(N^2) memory and O(N^3) computation, which limits classical MoM to structures of moderate electrical size. Acceleration techniques including the fast multipole method (FMM) and its multilevel variant (MLFMM) reduce these costs to approximately O(N log N), enabling full-wave analysis of electrically large antennas and scattering bodies that would be intractable with direct factorization alone.

Antenna and Scattering Analysis

MoM is the preferred full-wave method for antenna impedance, gain, and radiation pattern analysis when the geometry can be represented as metallic or dielectric surfaces in an unbounded medium. It computes near-field and far-field quantities directly without the absorbing boundary layers required by volumetric methods. For radar cross-section prediction, MoM solves the scattering problem by illuminating the object with a known incident wave and solving for the induced surface currents, then computing the bistatic or monostatic scattered fields. Boundary element methods in related engineering fields share the same mathematical heritage as MoM but are more commonly applied in acoustics, structural mechanics, and heat transfer, illustrating the breadth of integral equation techniques.

Applications

The method of moments has applications in a range of fields, including:

• Antenna design and impedance matching for wireless communication systems
• Radar cross-section prediction for aerospace and defense platforms
• Electromagnetic compatibility (EMC) and interference analysis
• Microwave circuit and interconnect modeling
• Biomedical electromagnetic dosimetry for implantable devices