## The Method of MomentsIn the 1960s, R.F. Harrington [5] and others applied a
technique called the (1) where is a linear operator, (2) where Now instead of one equation with a continuous unknown quantity, (3) To solve for the values of weighting or testing
functions, w,
are applied. This yields the following system of N equations in N unknowns:_{n}(4) Or, expressed in matrix form, (5) This linear system of equations has the form, (6) where the elements of [ φ is determined using Equation (2).The Method of Moments (MoM) can be used to solve a wide range of equations involving linear operations including integral and differential equations. This numerical technique has many applications other than electromagnetic modeling; however the MoM is widely used to solve equations derived from Maxwell's equations. In general, moment method codes generate and solve large, dense matrix equations and most of the computational resources required are devoted to filling and solving this matrix equation. The particular form of the equations that is solved and the choice of basis and weighting functions have a great impact on the size of this matrix and ultimately the suitability of a given moment method code to model a given geometry. ## Equation OptionsThe most common equation form solved by CEM modeling codes
based on the Method of Moments is the (7) where Another equation solved by Moment Method codes is the (8) where Moment Method codes based on the EFIE or MFIE alone, may
exhibit unstable behavior when the modeling surfaces form a resonant cavity at
a particular frequency. To avoid this, many moment method codes solve a linear
combination of the EFIE and MFIE known as a Some CEM modeling codes employ the Method of Moments to solve other equations. For example, static modeling codes often solve a form of Laplace's equation relating electric field strengths to charge densities or magnetic field strengths to current densities. The Generalized Multiple Technique (GMT), which is described in another section of this report, employs a moment method to solve equations for the electric field generated by multipole sources. ## Basis and Weighting FunctionsAn appropriate choice of basis and weighting functions can make a tremendous difference in the number of elements, N, required to obtain an accurate solution. Since the solution is represented as a summation of basis functions [see Eq. (2)], it is important to choose basis functions that accurately represent the solution with a small number of terms. For example, when solving for the current distribution on a surface, the basis functions should be current distribution elements that can be summed together in a way that is able to efficiently approximate any overall current distribution that might result from the analysis. Weighting functions should be chosen that maximize the
linear independence of the various weighted forms of the equation. Often, the
best choices of weighting functions are functions that are identical to the
basis functions. Moment method techniques that employ identical basis and
weighting functions are called ## References[1] R.
F. Harrington, [2] R.
F. Harrington, [3] R.
E. Collin, |