Method of moments (electromagnetics)
[6] Prior to this, variational methods were applied to engineering problems at microwave frequencies by the time of World War II.
[7] While Julian Schwinger and Nathan Marcuvitz have respectively compiled these works into lecture notes and textbooks,[8][9] Victor Rumsey has formulated these methods into the "reaction concept" in 1954.
[11] In the 1960s, early research work on the method was published by Kenneth Mei, Jean van Bladel[12] and Jack Richmond.
[13] In the same decade, the systematic theory for the method of moments in electromagnetics was largely formalized by Roger Harrington.
[14] The development of the method and its indications in radar and antenna engineering attracted interest; MoM research was subsequently supported United States government.
[18][19][20] 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.
[22] In addition to its use in electrical engineering, the method of moments has been applied to light scattering[23] and plasmonic problems.
We can also define a residual for this expression, which denotes the difference between the actual and the approximate solution:
discrete points and is often used to obtain approximate solutions when the inner product operation is cumbersome to perform.
[33] Triangular patches, introduced by S. Rao, D. Wilton and A. Glisson in 1982,[36] are known as RWG basis functions and are widely used in MoM.
[37] Characteristic basis functions were also introduced to accelerate computation and reduce the matrix equation.
[29] Depending on the application and studied structure, the testing and basis functions should be chosen appropriately to ensure convergence and accuracy, as well as to prevent possible high order algebraic singularities.
[40] Depending on the application and sought variables, different integral or integro-differential equations are used in MoM.
Radiation and scattering by thin wire structures, such as many types of antennas, can be modeled by specialized equations.
[45] For a linear wire that is centered on the origin and aligned with the z-axis, the equation can be written as:
The equation can be generalized to different excitation schemes, including magnetic frills.
[46] The general form of electric field integral equation (EFIE) can be written as:
MFIE is often formulated to be a Fredholm integral equation of the second kind and is generally well-posed.
In many cases, EFIEs are converted to mixed potential integral equations (MFIE) through the use of Lorenz gauge condition; this aims to reduce the orders of singularities through the use of magnetic vector and scalar electric potentials.
[49][50] In order to bypass the internal resonance problem in dielectric scattering calculations, combined-field integral equation (CFIE) and Poggio—Miller—Chang—Harrington—Wu—Tsai (PMCHWT) formulations are also used.
[51] Another approach, the volumetric integral equation, necessitates the discretization of the volume elements and is often computationally expensive.
[52] MoM can also be integrated with physical optics theory[53] and finite element method.
[54] Appropriate Green's function for the studied structure must be known to formulate MoM matrices: automatic incorporation of the radiation condition into the Green's function makes MoM particularly useful for radiation and scattering problems.
[55] Full wave analysis of planarly-stratified structures in particular, such as microstrips or patch antennas, necessitate the derivation of Green's functions that are peculiar to these geometries.
This involves the inverse Hankel transform of the spectral-domain Green's function, which is defined on the Sommerfeld integration path.
Following the extraction of quasi-static and surface pole components, these integrals can be approximated as closed-form complex exponentials through Prony's method or generalized pencil-of-function method; thus, the spatial Green's functions can be derived through the use of appropriate identities such as Sommerfeld identity.
[62][63][64] This method is known in the computational electromagnetics literature as the discrete complex image method (DCIM), since the Green's function is effectively approximated with a discrete number of image dipoles that are located within a complex distance from the origin.
[66] Rational-function fitting method,[67][68] as well as its combinations with DCIM,[64] can also be used to approximate closed-form Green's functions.
Alternatively, the closed-form Green's function can be evaluated through method of steepest descent.
[69] For the periodic structures such as phased arrays and frequency selective surfaces, series acceleration methods such as Kummer's transformation and Ewald summation is often used to accelerate the computation of the periodic Green's function.
