TABLE 45.7 Sampling Operations Involved in MoM Modeling DE Model L(sf(s=8(s) Sampling of: Unknown via basis. Subdomain bases usually of low order are Can use either subdomain or entire-c functions b(s) used. Known as FD procedure when Use of latter is generally confined to bodi using f(s=Σab pulse basis is used, and as FE approach rotation. Former is usually of low order, when bases are linear piecewise linear or sinusoidal being the maximum Equation via weight Pointwise matching is commonly employed, using employed, using a delta function. Pulse a delta function. For wires, pulse, linear, and <w(s), L(s scab()>= and linear matching are also used. nusoidal testing is also used. Linear and <w(s),8(s)>to get zi a=8 nusoidal testing is also used for surfaces. Operator Operator sampling for DE models is he nature of the ntwined with sampling the unknown in perator I(s, s). An impo terms of the difference operators used Solution of Z, a =g for th Interaction matrix is sparse. Time-domain teraction matrix is full Solution via factorization approach may be explicit or implicit In or iteration frequency domain, banded-matrix technique usually used a given number of samples. This suggests the benefits that might be derived from using unequal sample sizes in MoM modeling should a systematic way of determining the best nonuniform sampling scheme be developed. The method of moments Numerical implementation of the moment method is a relatively straightforward, and an intuitively logical, extension of these basic elements of numerical analysis, as described in the book by Harrington[1968]and discussed and used extensively in CEM [see, for example, Mittra, 1973, 1975; Strait, 1980; Poggio and Miller, 88]. Whether it is an integral equation, a differential equation, or another approach that is being used for the numerical model, three essential sampling operations are involved in reducing the analytical formulation via the moment method to a computer algorithm as outlined in Table 45.7. We note that operator sampling can ultimately determine the sampling density needed to achieve a desired accuracy in the source-field rela tionships involving integral operators, especially at and near the "self term, " where the observation and source points become coincident or nearly so and the integral becomes nearly singular. Whatever the method used for these sampling operations, they lead to a linear system of equations or matrix approximation of the original integral or differential operators. Because the operations and choices involved in developing this matrix descrip tion are common to all moment-method models, we shall discuss them in somewhat more detail. When using IE techniques, the coefficient matrix in the linear system of equations that results is most often referred to as an impedance matrix because in the case of the E-field form, its multiplication of the vector of unknown currents equals a vector of electric fields or voltages. The inverse matrix similarly is often called an admittance matrix because its multiplication of the electric-field or voltage vector yields the unknown-current vector. In this discussion we instead use the terms direct matrix and solution matrix because they are more generic descriptions whatever the forms of the originating integral or differential equations. As illustrated in the following, development of the direct matrix and solution matrix dominates both the computer time and storage requirements of numerical modeling In the particular case of an IE model, the coefficients of the direct or original matrix are the mutual impedances of the multiport representation which approximates the problem being modeled, and the coeffi cients of its solution matrix (or equivalent thereof)are the mutual admittances. Depending on whether a subdomain or entire-domain basis has been used(see Basic Function Selection), these impedances and admittances represent either spatial or modal interactions among the N ports of the numerical model. In either case, these c 2000 by CRC Press LLC© 2000 by CRC Press LLC a given number of samples. This suggests the benefits that might be derived from using unequal sample sizes in MoM modeling should a systematic way of determining the best nonuniform sampling scheme be developed. The Method of Moments Numerical implementation of the moment method is a relatively straightforward, and an intuitively logical, extension of these basic elements of numerical analysis, as described in the book by Harrington [1968] and discussed and used extensively in CEM [see, for example, Mittra, 1973, 1975; Strait, 1980; Poggio and Miller, 1988]. Whether it is an integral equation, a differential equation, or another approach that is being used for the numerical model, three essential sampling operations are involved in reducing the analytical formulation via the moment method to a computer algorithm as outlined in Table 45.7. We note that operator sampling can ultimately determine the sampling density needed to achieve a desired accuracy in the source–field rela￾tionships involving integral operators, especially at and near the “self term,” where the observation and source points become coincident or nearly so and the integral becomes nearly singular. Whatever the method used for these sampling operations, they lead to a linear system of equations or matrix approximation of the original integral or differential operators. Because the operations and choices involved in developing this matrix descrip￾tion are common to all moment-method models, we shall discuss them in somewhat more detail. When using IE techniques, the coefficient matrix in the linear system of equations that results is most often referred to as an impedance matrix because in the case of the E-field form, its multiplication of the vector of unknown currents equals a vector of electric fields or voltages. The inverse matrix similarly is often called an admittance matrix because its multiplication of the electric-field or voltage vector yields the unknown-current vector. In this discussion we instead use the terms direct matrix and solution matrix because they are more generic descriptions whatever the forms of the originating integral or differential equations. As illustrated in the following, development of the direct matrix and solution matrix dominates both the computer time and storage requirements of numerical modeling. In the particular case of an IE model, the coefficients of the direct or original matrix are the mutual impedances of the multiport representation which approximates the problem being modeled, and the coeffi- cients of its solution matrix (or equivalent thereof) are the mutual admittances. Depending on whether a subdomain or entire-domain basis has been used (see Basic Function Selection), these impedances and admittances represent either spatial or modal interactions among the N ports of the numerical model. In either case, these TABLE 45.7 Sampling Operations Involved in MoM Modeling DE Model IE Model Equation L(s¢)f(s¢) = g(s¢) L(s,s¢)f(s¢) = g(s) Sampling of: Unknown via basis￾functions bj (s¢) using f(s¢) ª Âajbj (s¢) Subdomain bases usually of low order are used. Known as FD procedure when pulse basis is used, and as FE approach when bases are linear. Can use either subdomain or entire-domain bases. Use of latter is generally confined to bodies of rotation. Former is usually of low order, with piecewise linear or sinusoidal being the maximum variation used. Equation via weight functions wi (s) <wi (s),L(s,s¢)Âaj bj (s¢)> = <wi (s),g(s) > to get Zijaj = gi Pointwise matching is commonly employed, using a delta function. Pulse and linear matching are also used. Pointwise matching is commonly employed, using a delta function. For wires, pulse, linear, and sinusoidal testing is also used. Linear and sinusoidal testing is also used for surfaces. Operator Operator sampling for DE models is entwined with sampling the unknown in terms of the difference operators used. Sampling needed depends on the nature of the integral operator L(s,s¢). An important consideration whenever the field integrals cannot be evaluated in closed form. Solution of: Zijaj = gi for the aj Interaction matrix is sparse.Time-domain approach may be explicit or implicit. In frequency domain, banded-matrix technique usually used. Interaction matrix is full. Solution via factorization or iteration