The method of moments (MoM), also known as the moment method and method of weighted residuals, is a numerical method in computational electromagnetics. It is used in computer programs that simulate the interaction of electromagnetic fields such as radio waves with matter, for example antenna simulation programs like NEC that calculate the radiation pattern of an antenna. Generally being a frequency-domain method, it involves the projection of an integral equation into a system of linear equations by the application of appropriate boundary conditions. This is done by using discrete meshes as in finite difference and finite element methods, often for the surface. The solutions are represented with the linear combination of pre-defined basis functions; generally, the coefficients of these basis functions are the sought unknowns. Green's functions and Galerkin method play a central role in the method of moments. For many applications, the method of moments is identical to the boundary element method. It is one of the most common methods in microwave and antenna engineering.
History Development of boundary element method and other similar methods for different engineering applications is associated with the advent of digital computing in the 1960s. Prior to this, variational methods were applied to engineering problems at microwave frequencies by the time of World War II. While Julian Schwinger and Nathan Marcuvitz have respectively compiled these works into lecture notes and textbooks, Victor Rumsey has formulated these methods into the "reaction concept" in 1954. The concept was later shown to be equivalent to the Galerkin method. In the late 1950s, an early version of the method of moments was introduced by Yuen Lo at a course on mathematical methods in electromagnetic theory at University of Illinois.
In the 1960s, early research work on the method was published by Kenneth Mei, Jean van Bladel and Jack Richmond. In the same decade, the systematic theory for the method of moments in electromagnetics was largely formalized by Roger Harrington. While the term "the method of moments" was coined earlier by Leonid Kantorovich and Gleb Akilov for analogous numerical applications, Harrington has adapted the term for the electromagnetic formulation. Harrington published the seminal textbook Field Computation by Moment Methods on the moment method in 1968. The development of the method and its indications in radar and antenna engineering attracted interest; MoM research was subsequently supported by the United States government. The method was further popularized by the introduction of generalized antenna modeling codes such as Numerical Electromagnetics Code, which was released into public domain by the United States government in the late 1980s. In the 1990s, introduction of fast multipole and multilevel fast multipole methods enabled efficient MoM solutions to problems with millions of unknowns. Being one of the most common simulation techniques in RF and microwave engineering, the method of moments forms the basis of many commercial design software such as FEKO. Many non-commercial and public domain codes of different sophistications are also available. In addition to its use in electrical engineering, the method of moments has been applied to light scattering and plasmonic problems.
Background
Basic concepts
An inhomogeneous integral equation can be expressed as:
L ( f ) = g {\displaystyle L(f)=g}
where L denotes a linear operator, g denotes the known forcing function and f denotes the unknown function. f can be approximated by a finite number of basis functions ( f n {\displaystyle f_{n}} ):
f ≈ ∑ n N a n f n . {\displaystyle f\approx \sum _{n}^{N}a_{n}f_{n}.}
By linearity, substitution of this expression into the equation yields:
∑ n N a n L ( f n ) ≈ g . {\displaystyle \sum _{n}^{N}a_{n}L(f_{n})\approx g.}
We can also define a residual for this expression, which denotes the difference between the actual and the approximate solution:
R = ∑ n N a n L ( f n ) − g {\displaystyle R=\sum _{n}^{N}a_{n}L(f_{n})-g}
The aim of the method of moments is to minimize this residual, which can be done by using appropriate weighting or testing functions, hence the name method of weighted residuals. After the determination of a suitable inner product for the problem, the expression then becomes:
∑ n N a n ⟨ w m , L ( f n ) ⟩ ≈ ⟨ w m , g ⟩ {\displaystyle \sum _{n}^{N}a_{n}\langle w_{m},L(f_{n})\rangle \approx \langle w_{m},g\rangle }
Thus, the expression can be represented in the matrix form:
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