TABLE 45.3 Desirable Attributes in a Computer Mode Attribute Description Accuracy The quantitative degree to which the computed results conform to the mathematical and physical reality being modeled Accuracy, preferably of known and, better yet, selectable value, is the single most important model attribute It is determined by the physical modeling error(Ep)and numerical modeling error Efficiency The relative cost of obtaining the needed results. It is determined by the human effort required to develop the computer input and interpret the output and by the associated computer cost of running the model. Utility The applicability of the computer model in terms of problem size and complexity. Utility also relates to ease of use, reliability of results obtained, etc. TABLE 45.4 Representative Approximations that Arise in Model Development Implementation/Implications Conceptualization Physical optics Surface sources given by tangential components of incident field, with fields subsequently propagated via a Greens function. Best for backscatter and main-lobe region of reflector ntennas, from resonance region(ka> 1) and up in frequ Physical theory of diffraction Combines aspects of physical optics and geometrical theory of diffraction, primarily via use of edge-current corrections to utilize best features of each. Geometrical theory diffraction Fields propagated via a divergence factor with amplitude obtained from diffraction coefficient enerally applicable for ka >2-5 Can involve complicated ray tracing Geometrical optics Ray tracing without diffraction Improves with increasing frequency. Compensation theorem Solution obtained in terms of perturbation from a reference, known solution. Approach used for low-contrast, penetrable objects where sources are estimated from incident Rayleigh Fields at surface of object represented in terms of only outward propagating components in a Formulation Surface impedance Reduces number of field quantities by assuming an impedance relation between tangential E and H at surface of penetrable object. May be use tion with physical optics. Reduces surface integral on thin, wirelike object to a line integral by ignoring circumferential current and circumferential variation of longitudinal current, which is represented as a filament Generally limited to ka< I where a is the wire radius. Numerical Implementation ofox→(,-f∥(x-x) Differentiation and integration of continuous functions represented in terms of analytic Jf(x)dx→∑f(x)△x ons on sampled approximations, for which polynomial or trigonometric functions are often used Inherently a discretizing operation, for which typically Ax< N2r for acceptable Computatio Deviation of numerical model Affects solution accuracy and relatability to physical problem in ways that are difficult to predict from physical reality and quantify. Discretized solutions usually converge globally in proportion to exp(-AN ) with A determined by the problem. At least two solutions using different numbers of unknowns N, are needed to 45.3 Analytical Issues in Developing a Computer Model Attention here is limited primarily to propagators that use either the Maxwell curl equations or source integrals which employ a Greens function, although for completeness we briefly discuss modal and optical techniques as well. Selection of solution domain Either the integral equation(IE)or differential equation(DE) propagator can be formulated in the time domain, where time is treated as an independent variable, or in the frequency domain, where the harmonic c 2000 by CRC Press LLC© 2000 by CRC Press LLC 45.3 Analytical Issues in Developing a Computer Model Attention here is limited primarily to propagators that use either the Maxwell curl equations or source integrals which employ a Green’s function, although for completeness we briefly discuss modal and optical techniques as well. Selection of Solution Domain Either the integral equation (IE) or differential equation (DE) propagator can be formulated in the time domain, where time is treated as an independent variable, or in the frequency domain, where the harmonic TABLE 45.3 Desirable Attributes in a Computer Model Attribute Description Accuracy The quantitative degree to which the computed results conform to the mathematical and physical reality being modeled.Accuracy, preferably of known and, better yet, selectable value, is the single most important model attribute. It is determined by the physical modeling error (eP) and numerical modeling error (eN). Efficiency The relative cost of obtaining the needed results.It is determined by the human effort required to develop the computer input and interpret the output and by the associated computer cost of running the model. Utility The applicability of the computer model in terms of problem size and complexity. Utility also relates to ease of use, reliability of results obtained, etc. TABLE 45.4 Representative Approximations that Arise in Model Development Approximation Implementation/Implications Conceptualization Physical optics Surface sources given by tangential components of incident field, with fields subsequently propagated via a Green’s function. Best for backscatter and main-lobe region of reflector antennas, from resonance region (ka > 1) and up in frequency. Physical theory of diffraction Combines aspects of physical optics and geometrical theory of diffraction, primarily via use of edge-current corrections to utilize best features of each. Geometrical theory diffraction Fields propagated via a divergence factor with amplitude obtained from diffraction coefficient. Generally applicable for ka > 2–5. Can involve complicated ray tracing. Geometrical optics Ray tracing without diffraction. Improves with increasing frequency. Compensation theorem Solution obtained in terms of perturbation from a reference, known solution. Born–Rytov Approach used for low-contrast, penetrable objects where sources are estimated from incident field. Rayleigh Fields at surface of object represented in terms of only outward propagating components in a modal expansion. Formulation Surface impedance Reduces number of field quantities by assuming an impedance relation between tangential E and H at surface of penetrable object. May be used in connection with physical optics. Thin-wire Reduces surface integral on thin, wirelike object to a line integral by ignoring circumferential current and circumferential variation of longitudinal current, which is represented as a filament. Generally limited to ka < 1 where a is the wire radius. Numerical Implementation ¶f /¶x Æ (f+ – f–)/(x+ – x–) Úf(x)dx Æ Âf(xi )Dxi Differentiation and integration of continuous functions represented in terms of analytic operations on sampled approximations, for which polynomial or trigonometric functions are often used. Inherently a discretizing operation, for which typically Dx < l/2p for acceptable accuracy. Computation Deviation of numerical model from physical reality Affects solution accuracy and relatability to physical problem in ways that are difficult to predict and quantify. Nonconverged solution Discretized solutions usually converge globally in proportion to exp(–ANx) with A determined by the problem. At least two solutions using different numbers of unknowns Nx are needed to estimate A