《电子工程师手册》学习资料（英文版）Chapter 45 Computational Electromagnetics

Miller, E.K. "Computational Electromagnetics The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000

Miller, E.K. “Computational Electromagnetics” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

45 Computational Electromagnetics Introduction Background Discussion Modeling as a Transfer Function.Some Issues Involved in Developing a Computer Model 45.3 Analytical Issues in Developing a Computer Selection of Solution Domain. Selection of Field P 45.4 Numerical Issues in Developing a Compute Sampling Functions. The Method of Moments 45.5 Some Practical Consideratio tegral Equation Modeling. Differential Modeling· Discussion· Sampling Requirement 45.6 Ways of Decreasing Computer Time 45.7 Validation, Error Checking, and Error Analysi EK. Miller Modeling Uncertainties. Validation and Error Checking Los Alamos National Laboratory 45.8 Concluding Remark 45.1 Introduction The continuing growth of computing resources is changing how we think about, formulate, solve, and interpret problems In electromagnetics as elsewhere, computational techniques are complementing the more traditional approaches of measurement and analysis to vastly broaden the breadth and depth of problems that are now quantifiable Computational electromagnetics( CEM)may be broadly defined as that branch of electromagnetics that intrinsically and routinely involves using a digital computer to obtain numerical results. with the evolu- tionary development of CEM during the past 20-plus years, the third tool of computational methods has been added to the two classical tools of experimental observation and mathematical analysis. This discussion reviews some of the basic issues involved in CEM and includes only the detail needed illustrate the central ideas involved. The underlying principles that unify the various modeling approaches used in electromagnetics are emphasized while avoiding most of the specifics that make them different. Listed roughout are representative, but not exhaustive, numbers of references that deal with various specialty aspect of CEM. For readers interested in broader, more general expositions, the well-known book on the moment method by Harrington [1968 ]; the books edited by Mittra [1973, 1975), Uslenghi [ 1978], and Strait[1980] the monographs by Stutzman and Thiele [1981], Popovic, et al. [1982], Moore and Pizer [1984], and Wang [1991]; and an IEEE Press reprint volume on the topic edited by Miller et al. [1991] are recommended, as i the article by Miller[ 1988] from which this material is excerpted This chapter is excerpted from E. K. Miller, "A selective survey of computational electromagnetics, "IEEE Trans. Antennas Propagat, voL. AP-36, Pp. 1281-1305, @1988 IEEE c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 45 Computational Electromagnetics 45.1 Introduction 45.2 Background Discussion Modeling as a Transfer Function • Some Issues Involved in Developing a Computer Model 45.3 Analytical Issues in Developing a Computer Model Selection of Solution Domain • Selection of Field Propagator 45.4 Numerical Issues in Developing a Computer Model Sampling Functions • The Method of Moments 45.5 Some Practical Considerations Integral Equation Modeling • Differential Equation Modeling • Discussion • Sampling Requirements 45.6 Ways of Decreasing Computer Time 45.7 Validation, Error Checking, and Error Analysis Modeling Uncertainties • Validation and Error Checking 45.8 Concluding Remarks 45.1 Introduction The continuing growth of computing resources is changing how we think about, formulate, solve, and interpret problems. In electromagnetics as elsewhere, computational techniques are complementing the more traditional approaches of measurement and analysis to vastly broaden the breadth and depth of problems that are now quantifiable. Computational electromagnetics (CEM) may be broadly defined as that branch of electromagnetics that intrinsically and routinely involves using a digital computer to obtain numerical results. With the evolu￾tionary development of CEM during the past 20-plus years, the third tool of computational methods has been added to the two classical tools of experimental observation and mathematical analysis. This discussion reviews some of the basic issues involved in CEM and includes only the detail needed to illustrate the central ideas involved. The underlying principles that unify the various modeling approaches used in electromagnetics are emphasized while avoiding most of the specifics that make them different. Listed throughout are representative, but not exhaustive, numbers of references that deal with various specialty aspects of CEM. For readers interested in broader, more general expositions, the well-known book on the moment method by Harrington [1968]; the books edited by Mittra [1973, 1975], Uslenghi [1978], and Strait [1980]; the monographs by Stutzman and Thiele [1981], Popovic, et al. [1982], Moore and Pizer [1984], and Wang [1991]; and an IEEE Press reprint volume on the topic edited by Miller et al. [1991] are recommended, as is the article by Miller [1988] from which this material is excerpted. This chapter is excerpted from E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propagat., vol. AP-36, pp. 1281–1305, ©1988 IEEE. E.K. Miller Los Alamos National Laboratory

45.2 Background discussion Electromagnetics is the scientific discipline that deals with electric and magnetic sources and the fields these sources produce in specified environments. Maxwell's equations provide the starting point for the study of electromagnetic problems, together with certain principles and theorems such as osition, reciprocity, equivalence, induction, duality, linearity, and uniqueness, derived therefrom [Stratton, 1941; Harrington, 1961] While a variety of specialized problems can be identified, a common ingredient of essentially all of them is that of establishing a quantitative relationship between a cause(forcing function or input) and its effect(the response or output), a relationship which we refer to as a field propagator, the computational characteristics of which are determined by the mathematical form used to describe it. Modeling as a Transfer Function The foregoing relationship may be viewed as a gener PROBLEM DESCRIPTION alized transfer function(see Fig. 45. 1)in which two (ELECTRICAL GEOMETRICAL) ysis or the direct problem, the input is known and the transfer function is derivable from the problem sp mined. For the case of the synthesis or inverse INPUT/ TRANSFER FUNCTION OUTPUT ification, with the output or response to be deter DERIVED FROM problem, two problem classes may be identified. The MAXWELL asier synthesis problem involves finding the input, EQUATIONS NEAR AND FAR given the output and transfer function, an example of which is that of determining the source voltages FIGURE 45. 1 The electromagnetic transfer function relates that produce an observed pattern for a known the input, output, and problem. antenna array. The more difficult synthesis problem itself separates into two problems. One is that of finding the transfer function, given the input and output,an cample of which is that of finding a source distribution that produces a given far field. The other and still more difficult is that of finding the object geometry that produces an observed scattered field from a known ting field. The latter problem is the most difficult of the three synthesis problems to solve because it is intrinsically transcendental and nonlinear Electromagnetic propagators are derived from a particular solution of Maxwells equations, as the cause mentioned above normally involves some specified or known excitation whose effect is to induce some to-be- determined response(e.g, a radar cross section, antenna radiation pattern). It therefore follows that the essence of electromagnetics is the study and determination of field propagators to thereby obtain an input-output transfer function for the problem of interest, and it follows that this is also the goal of CEM. Some Issues Involved in Developing a Computer Model We briefly consider here a classification of model types, the steps involved in developing a computer model, the desirable attributes of a computer model, and finally the role of approximation throughout the modeling process. Classification of Model Types It is convenient to classify solution techniques for electromagnetic modeling in terms of the field propagator that might be used, the anticipated application, and the problem type for which the model is intended to be ed, as is outlined in Table 45. 1. Selection of a field propagator in the form, for example, of the Maxwell curl equations, a Greens function, modal or spectral expansions, or an optical description is a necessary first step developing a solution to any electromagnetic problem. Development of a Computer Model Development of a computer model in electromagnetics or literally any other disciplinary activity can be decomposed into a small number of basic, generic steps. These steps might be described by different names but c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 45.2 Background Discussion Electromagnetics is the scientific discipline that deals with electric and magnetic sources and the fields these sources produce in specified environments. Maxwell’s equations provide the starting point for the study of electromagnetic problems, together with certain principles and theorems such as superposition, reciprocity, equivalence, induction, duality, linearity, and uniqueness, derived therefrom [Stratton, 1941; Harrington, 1961]. While a variety of specialized problems can be identified, a common ingredient of essentially all of them is that of establishing a quantitative relationship between a cause (forcing function or input) and its effect (the response or output), a relationship which we refer to as a field propagator, the computational characteristics of which are determined by the mathematical form used to describe it. Modeling as a Transfer Function The foregoing relationship may be viewed as a gener￾alized transfer function (see Fig. 45.1) in which two basic problem types become apparent. For the anal￾ysis or the direct problem, the input is known and the transfer function is derivable from the problem spec￾ification, with the output or response to be deter￾mined. For the case of the synthesis or inverse problem, two problem classes may be identified. The easier synthesis problem involves finding the input, given the output and transfer function, an example of which is that of determining the source voltages that produce an observed pattern for a known antenna array. The more difficult synthesis problem itself separates into two problems. One is that of finding the transfer function, given the input and output, an example of which is that of finding a source distribution that produces a given far field. The other and still more difficult is that of finding the object geometry that produces an observed scattered field from a known exciting field. The latter problem is the most difficult of the three synthesis problems to solve because it is intrinsically transcendental and nonlinear. Electromagnetic propagators are derived from a particular solution of Maxwell’s equations, as the cause mentioned above normally involves some specified or known excitation whose effect is to induce some to-be￾determined response (e.g., a radar cross section, antenna radiation pattern). It therefore follows that the essence of electromagnetics is the study and determination of field propagators to thereby obtain an input–output transfer function for the problem of interest, and it follows that this is also the goal of CEM. Some Issues Involved in Developing a Computer Model We briefly consider here a classification of model types, the steps involved in developing a computer model, the desirable attributes of a computer model, and finally the role of approximation throughout the modeling process. Classification of Model Types It is convenient to classify solution techniques for electromagnetic modeling in terms of the field propagator that might be used, the anticipated application, and the problem type for which the model is intended to be used, as is outlined in Table 45.1. Selection of a field propagator in the form, for example, of the Maxwell curl equations, a Green’s function, modal or spectral expansions, or an optical description is a necessary first step in developing a solution to any electromagnetic problem. Development of a Computer Model Development of a computer model in electromagnetics or literally any other disciplinary activity can be decomposed into a small number of basic, generic steps. These steps might be described by different names but FIGURE 45.1 The electromagnetic transfer function relates the input, output, and problem

TABLE 45.1 Classification of Model Types in CEM Field Propagator escription Based on Integral operator Green's function for infinite medium or special boundaries Differential operator Maxwell curl equations or their integral counterparts Modal expansions Solutions of MaxwellI's equations in a particular coordinate system and expansion Rays and diffraction coefficients Application Determining the originating sources of a field and patte terns they produc Obtaining the fields distant from a known source Determining the perturbing effects of medium inhomogeneities onfiguration or wave number 1D, 2D, 3D Electrical properties of medium Dielectric, lossy, perfectly conducting, anisotropic, inhomogeneous, nonlinear, bianisotropic Linear, curved, segmented, compound, arbitrary TABLE 45.2 Steps in Developing a Computer Model Encapsulating observation and analysis in terms of elementary physical principles and their mathematical descriptions leshing out of the elementary description into a more complete, formally solved, mathematical representation umerical implementation Transforming into a computer algorithm using various numerical techniques Computation Obtaining quantitative results validation Determining the numerical and physical credibility of the computed results would include at a minimum those outlined in Table 45. 2. Note that by its nature, validation is an open-ended process that cumulatively can absorb more effort than all the other steps together. The primary focus of the following discussion is on the issue of numerical implementation. Desirable Attributes of a Computer Model A computer model must have some minimum set of basic properties to be useful From the long list of attributes rtant a summarized in Table 45.3. Accuracy is put foremost because results of insufficient or unknown accuracy have uncertain value and may even be harmful. On the other hand, a code that produces accurate results but at unacceptable cost will have hardly any more value. Finally, a code's applicability in terms of the depth and breadth of the problems for which it can be used determines its utility. The Role of Approximation As approximation is an intrinsic part of each step involved in developing a computer model, we summarize some of the more commonly used approximations in Table 45. 4. We note that the distinction between ar approximation at the conceptualization step and during the formulation is somewhat arbitrary, but choose to use the former category for those approximations that occur before the formulation itself. c 2000 by CRC Press LLC

© 2000 by CRC Press LLC would include at a minimum those outlined in Table 45.2. Note that by its nature, validation is an open-ended process that cumulatively can absorb more effort than all the other steps together. The primary focus of the following discussion is on the issue of numerical implementation. Desirable Attributes of a Computer Model A computer model must have some minimum set of basic properties to be useful. From the long list of attributes that might be desired, we consider: (1) accuracy, (2) efficiency, and (3) utility the three most important as summarized in Table 45.3. Accuracy is put foremost because results of insufficient or unknown accuracy have uncertain value and may even be harmful. On the other hand, a code that produces accurate results but at unacceptable cost will have hardly any more value. Finally, a code’s applicability in terms of the depth and breadth of the problems for which it can be used determines its utility. The Role of Approximation As approximation is an intrinsic part of each step involved in developing a computer model, we summarize some of the more commonly used approximations in Table 45.4. We note that the distinction between an approximation at the conceptualization step and during the formulation is somewhat arbitrary, but choose to use the former category for those approximations that occur before the formulation itself. TABLE 45.1 Classification of Model Types in CEM Field Propagator Description Based on Integral operator Green’s function for infinite medium or special boundaries Differential operator Maxwell curl equations or their integral counterparts Modal expansions Solutions of Maxwell’s equations in a particular coordinate system and expansion Optical description Rays and diffraction coefficients Application Requires Radiation Determining the originating sources of a field and patterns they produce Propagation Obtaining the fields distant from a known source Scattering Determining the perturbing effects of medium inhomogeneities Problem type Characterized by Solution domain Time or frequency Solution space Configuration or wave number Dimensionality 1D, 2D, 3D Electrical properties of medium and/or boundary Dielectric, lossy, perfectly conducting, anisotropic, inhomogeneous, nonlinear, bianisotropic Boundary geometry Linear, curved, segmented, compound, arbitrary TABLE 45.2 Steps in Developing a Computer Model Step Activity Conceptualization Encapsulating observation and analysis in terms of elementary physical principles and their mathematical descriptions Formulation Fleshing out of the elementary description into a more complete, formally solved, mathematical representation Numerical implementation Transforming into a computer algorithm using various numerical techniques Computation Obtaining quantitative results Validation Determining the numerical and physical credibility of the computed results

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