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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
A comprehensive, step-by-step reference to the Nyström Method for solving Electromagnetic problems using integral equations Computational electromagnetics studies the numerical methods or techniques that solve electromagnetic problems by computer programming. Currently, there are mainly three numerical methods for electromagnetic problems: the finite-difference time-domain (FDTD), finite element method (FEM), and integral equation methods (IEMs). In the IEMs, the method of moments (MoM) is the most widely used method, but much attention is being paid to the Nyström method as another IEM, because it possesses some unique merits which the MoM lacks. This book focuses on that method--providing information on everything that students and professionals working in the field need to know. Written by the top researchers in electromagnetics, this complete reference book is a consolidation of advances made in the use of the Nyström method for solving electromagnetic integral equations. It begins by introducing the fundamentals of the electromagnetic theory and computational electromagnetics, before proceeding to illustrate the advantages unique to the Nyström method through rigorous worked out examples and equations. Key topics include quadrature rules, singularity treatment techniques, applications to conducting and penetrable media, multiphysics electromagnetic problems, time-domain integral equations, inverse scattering problems and incorporation with multilevel fast multiple algorithm. * Systematically introduces the fundamental principles, equations, and advantages of the Nyström method for solving electromagnetic problems * Features the unique benefits of using the Nyström method through numerical comparisons with other numerical and analytical methods * Covers a broad range of application examples that will point the way for future research The Nystrom Method in Electromagnetics is ideal for graduate students, senior undergraduates, and researchers studying engineering electromagnetics, computational methods, and applied mathematics. Practicing engineers and other industry professionals working in engineering electromagnetics and engineering mathematics will also find it to be incredibly helpful.
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Seitenzahl: 807
Veröffentlichungsjahr: 2020
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Table of Contents
Cover
About the Authors
Preface
Acknowledgment
1 Electromagnetics, Physics, and Mathematics
1.1 A Brief History of Electromagnetics
1.2 Enduring Legacy of Electromagnetic Theory–Why?
1.3 The Rise of Quantum Optics and Electromagnetics
1.4 The Early Days – Descendent from Fluid Physics
1.5 The Complete Development of Maxwell's Equations
1.6 Circuit Physics, Wave Physics, Ray Physics, and Plasmonic Resonances
1.7 The Age of Closed Form Solutions
1.8 The Age of Approximations
1.9 The Age of Computations
1.10 Fast Algorithms
1.11 High Frequency Solutions
1.12 Inverse Problems
1.13 Metamaterials
1.14 Small Antennas
1.15 Conclusions
Bibliography
Notes
2 Computational Electromagnetics
2.1 Introduction
2.2 Analytical Methods
2.3 Numerical Methods
2.4 Electromagnetic Integral Equations
2.5 Summary
Bibliography
3 The Nyström Method
3.1 Introduction
3.2 Basic Principle
3.3 Singularity Treatment
3.4 Higher‐Order Scheme
3.5 Comparison to the Method of Moments
3.6 Comparison to the Point‐Matching Method
3.7 Summary
Bibliography
4 Numerical Quadrature Rules
4.1 Introduction
4.2 Definition and Design
4.3 Quadrature Rules for a Segmental Mesh
4.4 Quadrature Rules for a Surface Mesh
4.5 Quadrature Rules for a Volumetric Mesh
4.6 Summary
Bibliography
5 Singularity Treatment
5.1 Introduction
5.2 Singularity Subtraction
5.3 Singularity Cancellation
5.4 Evaluation of Hypersingular and Weakly‐Singular Integrals over Triangular Patches
5.5 Different Scheme for Evaluating Strongly‐Singular and Hypersingular Integrals Over Triangular Patches
5.6 Evaluation of Singular Integrals Over Volume Domains
5.7 Evaluation of Near‐Singular Integrals
5.8 Summary
Bibliography
6 Application to Conducting Media
6.1 Introduction
6.2 Solution for 2D Structures
6.3 Solution for Body‐of‐Revolution (BOR) Structures
6.4 Solutions of the Electric Field Integral Equation
6.5 Solutions of the Magnetic Field Integral Equation
6.6 Solutions of the Combined Field Integral Equation
6.7 Summary
Bibliography
7 Application to Penetrable Media
7.1 Introduction
7.2 Surface Integral Equations for Homogeneous and Isotropic Media
7.3 Volume Integral Equations for Homogeneous and Isotropic Media
†
7.4 Volume Integral Equations for Inhomogeneous or/and Anisotropic Media
7.5 Volume Integral Equations for Conductive Media
7.6 Volume‐Surface Integral Equations for Mixed Media
7.7 Summary
Bibliography
Note
8 Incorporation with Multilevel Fast Multipole Algorithm
8.1 Introduction
8.2 Multilevel Fast Multipole Algorithm
8.3 Surface Integral Equations for Conducting Objects
8.4 Surface Integral Equations for Penetrable Objects
8.5 Volume Integral Equations for Conductive Media
8.6 Volume‐Surface Integral Equations for Conducting‐Anisotropic Media
8.7 Summary
Bibliography
9 Application to Solve Multiphysics Problems
9.1 Introduction
9.2 Solution of Elastic Wave Problems
9.3 MLFMA Acceleration for Solve Large Elastic Wave Problems
9.4 Solution of Acoustic Wave Problems with MLFMA Acceleration
9.5 Unified Boundary Integral Equations for Elastic Wave and Acoustic Wave
9.6 Coupled Integral Equations for Electromagnetic Wave and Elastic Wave
9.7 Summary
Appendix 9.1 Analytical Solutions for the Strongly-Singular Integrals in (9.35)–(9.37)
Appendix 9.2 Asymptotic Solutions of Elastic Wave Scattering by Spherical Obstacles with
Bibliography
10 Application to Solve Time Domain Integral Equations
10.1 Introduction
10.2 Time Domain Surface Integral Equations for Conducting Media
10.3 Time Domain Surface Integral Equations for Penetrable Media
10.4 Time Domain Volume Integral Equations for Penetrable Media
10.5 Time Domain Combined Field Integral Equations for Mixed Media
10.6 Summary
Bibliography
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Gaussian quadrature rules with different points and orders.
Table 4.2 Typical quadrature rules defined over a triangular patch.
Table 4.3 Typical quadrature rules defined over a square patch.
Table 4.4 Typical quadrature rules defined over a tetrahedral element.
Table 4.5 Typical quadrature rules defined over a cuboid element.
Chapter 5
Table 5.1 Comparison between the results of new formula and those of old form...
Table 5.2 Comparison between the results of new formula and those of old form...
Table 5.3 Comparison between the results of new formula and those of old form...
Table 5.4 Comparison of numerical integration values with exact values for
....
Table 5.5 Comparison of numerical integration values with exact values for
....
Table 5.6 Comparison of numerical integration values with exact values for
....
Table 5.7 Comparison of numerical integration values with exact values for
....
Table 5.8 Comparison of numerical integration values with exact values for
....
Chapter 6
Table 6.1 Comparison of condition numbers (CNs) of impedance matrices for thr...
Table 6.2 Accuracy comparison between numerical integration, new formula and ...
Chapter 7
Table 7.1 A ten‐point quadrature rule for tetrahedral element.
Table 7.2 A comparison of computational costs between the Nys...
Table 7.3 A comparison of computational costs between the VSI...
Chapter 8
Table 8.1 Summary of CPU time
(Second) and memory usage
(GB) for the three...
Table 8.2 Comparison of computational costs between the Nystr...
Chapter 9
Table 9.1 CPU time T (Second) and memory usage M (GB) for scattering by a lar...
Table 9.2 CPU time T (Second) and memory usage M (GB) for scattering by trunc...
Table 9.3 Summary of CPU time
(Second) and memory usage
(MB) for solving a...
List of Illustrations
Chapter 1
Figure 1.1 The relationship of classical electromagnetic theory with other k...
Figure 1.2 A negative resistor is needed in order for the total voltage drop...
Figure 1.3 Two pieces of wire constituting a transmission line can be approx...
Figure 1.4 In an inhomogeneous profile with decreasing index, the ray will b...
Figure 1.5 Matched asymptotics can be used to find the solution of the capac...
Figure 1.6 The occurrence of a thin double layer outside a charged particle ...
Figure 1.7 The essential physics of high‐frequency scattering is controlled ...
Figure 1.8 The factorization of the matrix in (1.143) allows the matrix‐ve...
Figure 1.9 A multiregion problem can be solved semi‐analytically using numer...
Figure 1.10 The use of multiple‐scattering information in multiple scatterin...
Figure 1.11 Since super‐resolution information is buried in the evanescent f...
Chapter 2
Figure 2.1 Scattering by an homogeneous dielectric spherical core coated wit...
Figure 2.2 Scattering by an homogeneous dielectric spherical core coated wit...
Figure 2.3 A pair of triangles that define the RWG basis function.
Figure 2.4 A pair of tetrahedrons that define the SWG basis function.
Figure 2.5 Scattering by an impenetrable PEC object.
Figure 2.6 Scattering by a penetrable object that is homogeneous and istropi...
Figure 2.7 Scattering by an inhomogeneous anisotropic object embedded in fre...
Figure 2.8 Scattering by an object with mixed PEC and penetrable media which...
Chapter 4
Figure 4.1 Illustration of quadrature rules defined over a segmental mesh.
Figure 4.2 Distribution of quadrature points over a triangular patch for dif...
Figure 4.3 Geometric mapping for transforming a quadrature rule defined over...
Figure 4.4 Geometric mapping for transforming a quadrature rule defined over...
Figure 4.5 Distribution of quadrature points over a tetrahedral element for ...
Figure 4.6 Geometric mapping for transforming a quadrature rule defined over...
Figure 4.7 Geometric mapping for transforming a quadrature rule defined over...
Chapter 5
Figure 5.1 Illustration of singularity subtraction for a singular kernel, wh...
Figure 5.2 Geometry of a triangular patch in a global coordinate system
.
Figure 5.3 Geometry of a triangular patch in a local coordinate system
. Th...
Figure 5.4 A local Cartesian coordinate system
and polar coordinate system...
Figure 5.5 Geometries of conducting scatterers: (a) A tetrahedron; (b) A dia...
Figure 5.6 Bistatic RCS solutions for a PEC sphere with a radius
.
Figure 5.7 Bistatic RCS solutions for a PEC tetrahedron defined by four vert...
Figure 5.8 Bistatic RCS solutions for a PEC diamond defined with
and
.
Figure 5.9 Bistatic RCS solutions for a PEC circular thin disk with
and
....
Figure 5.10 Bistatic RCS solutions for a PEC triangular cylinder (prism) tha...
Figure 5.11 The convergence of the integral
(real part) with a comparison ...
Figure 5.12 The convergence of the integral
(real part) with a comparison ...
Figure 5.13 Subtriangles when the projection of observation point on the tri...
Figure 5.14 A triangular patch is divided into two parts,
and
for derivi...
Figure 5.15 Geometries of scatterers. (a) A conducting sphere. (b) A conduct...
Figure 5.16 Current density distribution along the principal cut for EM scat...
Figure 5.17 Bistatic RCS solutions for EM scattering by a conducting thin di...
Figure 5.18 Bistatic RCS solutions for EM scattering by a conducting cube wi...
Figure 5.19 Bistatic RCS solutions for EM scattering by a conducting thin di...
Figure 5.20 A local coordinate system
whose
plane coincides with the fla...
Figure 5.21 Bistatic RCS solutions for a PEC sphere with a radius
.
Figure 5.22 Bistatic RCS solutions for a dielectric sphere with a radius
a...
Figure 5.23 Bistatic RCS solutions for a PEC sphere with a one‐layer full di...
Figure 5.24 Derivation of hypersingular integrals in a local coordinate syst...
Figure 5.25 A cylindrical element for evaluating hypersingular integrals.
Figure 5.26 Bistatic RCS solutions for a dielectric sphere with a radius
a...
Figure 5.27 Bistatic RCS solutions for a dielectric cylinder with a radius o...
Figure 5.28 Bistatic RCS solutions for a dielectric cube with a side length
Figure 5.29 Bistatic RCS solutions for EM scattering by a dielectric sphere ...
Figure 5.30 Definition of NF.
is the source triangular patch and
is the ...
Figure 5.31 Dependence of relative error on
for
.
Figure 5.32 Dependence of relative error on
for
.
Figure 5.33 Dependence of relative error on
for
.
Figure 5.34 Dependence of relative error on
for
Figure 5.35 Dependence of relative error on NF for
.
Figure 5.36 Geometries of thin objects. (a) Diamond. (b) Wire. (c) Disk.
Figure 5.37 Bistatic RCS solutions of a thin diamond scatterer in vertical p...
Figure 5.38 Bistatic RCS solutions of a thin wire scatterer in horizontal po...
Figure 5.39 Bistatic RCS solutions of a thin disk scatterer in horizontal po...
Chapter 6
Figure 6.1 TM
z
wave scattering by a 2D conducting scatterer.
Figure 6.2 Geometry of scattering by a 2D conducting strip with a finite wid...
Figure 6.3 Geometry of a 2D conducting L‐shape cylinder.
Figure 6.4 Nyström solution with
unknowns for the ...
Figure 6.5 MoM solution with
unknowns for the current density on the L‐sha...
Figure 6.6 Current density distribution on the strip surface for TM polariza...
Figure 6.7 Current density distribution on the strip surface for TM polariza...
Figure 6.8 Current density distribution on the strip surface for TE polariza...
Figure 6.9 Current density distribution on the strip surface for TE polariza...
Figure 6.10 Interpolated current density distribution using very coarse mesh...
Figure 6.11 Interpolated current density distribution using very coarse mesh...
Figure 6.12 RMS error of the current density for TM polarization,
.
Figure 6.13 RMS error of the current density for TM polarization,
.
Figure 6.14 RMS error of the current density for TE polarization,
.
Figure 6.15 RMS error of the current density for TE polarization,
.
Figure 6.16 RMS error of the current density for TM polarization without the...
Figure 6.17 Geometry of a conducting body of revolution (BOR) where
and ...
Figure 6.18 Discretization of a conducting BOR's generating arc where
and
Figure 6.19 Geometries of three conducting BOR scatterers: (a) sphere; (b) c...
Figure 6.20 Current density distribution along the principal cut at the surf...
Figure 6.21 Bistatic RCS solutions for EM scattering by a conducting cone in...
Figure 6.22 Bistatic RCS solutions for EM scattering by a conducting step‐ra...
Figure 6.23 A canonical geometrical mapping for a curvilinear triangular pat...
Figure 6.24 Geometries of two scatterers. (a) A conducting ogive scatterer. ...
Figure 6.25 Monostatic RCS solutions of an ogive scatterer in HH and VV pola...
Figure 6.26 RMS Error versus density of unknowns for the monostatic RCS solu...
Figure 6.27 Bistatic RCS solution for a pencil‐like target in HH polarizatio...
Figure 6.28 RMS error versus density of unknowns for the bistatic RCS soluti...
Figure 6.29 A surface integral over a triangular patch can be changed into a...
Figure 6.30 Geometries of three super‐thin conducting scatterers and a sampl...
Figure 6.31 Bistatic RCS solutions for a super‐thin conducting hexagonal pla...
Figure 6.32 Bistatic RCS solutions for a super‐thin conducting annulus plate...
Figure 6.33 Bistatic RCS solutions for a super‐thin conducting dumbbell‐like...
Figure 6.34 Comparison of current density solutions from the MFIE and EFIE f...
Figure 6.35 Comparison of error convergence of current density solutions fro...
Figure 6.36 Bistatic RCS solutions for a square plate with different thickne...
Figure 6.37 Bistatic RCS solutions for a square plate with different thickne...
Figure 6.38 Quality of triangular meshes which is measured by the quality fa...
Figure 6.39 Geometries of scatterers. (a) A thin rectangular diamond with a ...
Figure 6.40 Bistatic RCS solutions for a thin conducting rectangular diamond...
Figure 6.41 Bistatic RCS solutions (horizontal polarization) for a thin cond...
Figure 6.42 Bistatic RCS solutions (vertical polarization) for a thin conduc...
Chapter 7
Figure 7.1 Scattering by a single penetrable body. (a) Original problem. (b)...
Figure 7.2 Scattering by a dielectric object with
‐layer full dielectric co...
Figure 7.3 Scattering by a fully coated conductor. (a) Original problem. (b)...
Figure 7.4 Scattering by a partially coated conductor. (a) Original problem....
Figure 7.5 Typical non‐product quadrature rules used in the...
Figure 7.6 Defective meshes in the MoM. (a) One vertex A of a triangle is lo...
Figure 7.7 Geometries of four objects. (a) Fully coated PEC sphere. (b) Full...
Figure 7.8 Normalized magnitude of scattered electric field by a fully coate...
Figure 7.9 Normalized magnitude of scattered electric field by a fully coate...
Figure 7.10 Normalized magnitude of scattered electric field by a dielectric...
Figure 7.11 Normalized magnitude of scattered electric field by a magnetic s...
Figure 7.12 Normalized magnitude of scattered electric field by a lossy diel...
Figure 7.13 Bistatic RCS solutions for a partially coated PEC sphere with
,...
Figure 7.14 Bistatic RCS solutions for a partially coated PEC sphere with
,...
Figure 7.15 A comparison of the Nyström method (NM) solut...
Figure 7.16 The electric shielding effectiveness
of a rectangular PEC encl...
Figure 7.17 Quadrature rules for a tetrahedron. (a) One point. (b) Four poin...
Figure 7.18 Typical defective tetrahedral elements in the MoM. The shadowed ...
Figure 7.19 Derivation of hypersingular integrals in a local coordinate syst...
Figure 7.20 Bistatic RCS solutions for a dielectric sphere with
and
.
Figure 7.21 Bistatic RCS solutions for a dielectric sphere with
and
. The...
Figure 7.22 Bistatic RCS solutions for a dielectric cylinder with
,
, and
Figure 7.23 Bistatic RCS solutions for a dielectric cube with
and
.
Figure 7.24 Bistatic RCS solutions for a sphere with
,
, and
.
Figure 7.25 RMS error of the bistatic RCS solutions for a dielectric cube wi...
Figure 7.26 RMS error of the bistatic RCS solutions for a dielectric cube wi...
Figure 7.27 Scattering by a 3D inhomogeneous anisotropic object embedded in ...
Figure 7.28 Geometries of scatterers. (a) A dielectric sphere with two‐layer...
Figure 7.29 Solution of scattered near electric field at
for the scatterer...
Figure 7.30 Bistatic RCS solutions (vertical polarization) for the scatterer...
Figure 7.31 Bistatic RCS solutions (vertical polarization) for the scatterer...
Figure 7.32 A comparison of convergence for the bistatic RCS solutions betwe...
Figure 7.33 Geometries of lossy conductor structures. (a) A conductive spher...
Figure 7.34 Bistatic RCS solutions of a conductive sphere with a radius
m ...
Figure 7.35 Bistatic RCS solutions of a conductive sphere with a radius
m ...
Figure 7.36 Scattered electric field (
‐component) by a conductive sphere wi...
Figure 7.37 The dissipated power of a rectangular conductive plate versus it...
Figure 7.38 Bistatic RCS solutions (VV) of a conductive sphere coated with a...
Figure 7.39 Convergence of radar cross section solutions based on the relati...
Figure 7.40 A typical interconnect and packaging structure. The signal lines...
Figure 7.41 Geometries of typical interconnect and packaging structures. (a)...
Figure 7.42 S parameters for the interconnect and packaging structure includ...
Figure 7.43 S parameters for the interconnect and packaging structure includ...
Figure 7.44 S parameters for the interconnect and packaging structure includ...
Figure 7.45 S parameters at low frequencies for the interconnect and packagi...
Figure 7.46
parameters for the interconnect and packaging structure includ...
Figure 7.47
parameters for the interconnect and packaging structure with o...
Figure 7.48
parameters for the interconnect and packaging structure with t...
Chapter 8
Figure 8.1 Inverted tree structure for calculating the matrix‐vector multipl...
Figure 8.2 Geometry of a conducting sphere.
Figure 8.3 Bistatic RCS solutions of a conducting sphere with a radius
. Th...
Figure 8.4 Bistatic RCS solutions of a conducting sphere with a radius
. Th...
Figure 8.5 Geometry of a conducting airplane wing.
Figure 8.6 Bistatic RCS solutions of a conducting airplane wing with a lengt...
Figure 8.7 Bistatic RCS solutions of a conducting airplane wing with a lengt...
Figure 8.8 Scattering by a single penetrable body. (a) Original problem. (b)...
Figure 8.9 Geometries of radially stratified spherical scatterers. (a) A PEC...
Figure 8.10 Bistatic RCS solutions for a PEC sphere with one‐layer full diel...
Figure 8.11 Bistatic RCS solutions for a PEC sphere with one‐layer full diel...
Figure 8.12 Bistatic RCS solutions for a dielectric sphere with one‐layer fu...
Figure 8.13 Bistatic RCS solutions for a dielectric sphere with one‐layer fu...
Figure 8.14 Bistatic RCS solutions for a dielectric sphere with two‐layer fu...
Figure 8.15 Bistatic RCS solutions for a dielectric sphere with two‐layer fu...
Figure 8.16 Geometry of a conductive sphere.
Figure 8.17 Bistatic RCS solutions for a conductive sphere with a radius
m...
Figure 8.18 Bistatic RCS solutions for a conductive sphere with a radius
m...
Figure 8.19 Bistatic RCS solutions (VV) for a conductive sphere with a radiu...
Figure 8.20 Bistatic RCS solutions (HH) for a conductive sphere with a radiu...
Figure 8.21 Bistatic RCS solutions with a vertical polarization for a lossy ...
Figure 8.22 Bistatic RCS solutions with a horizontal polarization for a loss...
Figure 8.23 EM Scattering by a 3D conducting object coated with an anisotrop...
Figure 8.24 Geometries of anisotropic or isotropic scatterers. (a) An anisot...
Figure 8.25 Bistatic RCS of a uniaxially anisotropic sphere in vertical pola...
Figure 8.26 Bistatic RCS of a uniaxially anisotropic sphere in horizonal pol...
Figure 8.27 Bistatic RCS of a PEC sphere coated with a uniaxially anisotropi...
Figure 8.28 Bistatic RCS of a PEC sphere coated with a uniaxially anisotropi...
Figure 8.29 Convergence of RCS solutions based on the relative RMS error for...
Figure 8.30 Bistatic RCS of an isotropic dielectric triangular cylinder with...
Chapter 9
Figure 9.1 Elastic wave scattering by an obstacle embedded in an infinite el...
Figure 9.2 Local coordinate system over a singular element. The field point ...
Figure 9.3 Total radial and tangential components of traction along the elev...
Figure 9.4 Total radial and tangential components of traction along the elev...
Figure 9.5 Total radial and tangential components of traction along the elev...
Figure 9.6 Scattered radial and tangential displacement of a spherical cavit...
Figure 9.7 Scattered radial and tangential displacement of a spherical cavit...
Figure 9.8 Total radial and tangential components of displacement along the ...
Figure 9.9 Total radial and tangential components of displacement along the ...
Figure 9.10 Total radial components of traction and displacement along the e...
Figure 9.11 Scattered radial and tangential displacement of an elastic spher...
Figure 9.12 Elastic wave scattering by a 3D elastic object embedded in an el...
Figure 9.13 Radial and tangential (elevated) components of total traction fi...
Figure 9.14 Radial and tangential (elevated) components of scattered displac...
Figure 9.15 Radial and tangential (elevated) components of scattered displac...
Figure 9.16 Radial and tangential (elevated) components of scattered displac...
Figure 9.17 Radial and tangential (elevated) components of total displacemen...
Figure 9.18 Local coordinate system on a triangular patch plane.
Figure 9.19 A meshed sphere surface.
Figure 9.20 Magnitude of the scattered pressure field at the principal cut o...
Figure 9.21 Magnitude of the scattered pressure field at the principal cut o...
Figure 9.22 Magnitude of the scattered pressure field at the principal cut o...
Figure 9.23 Magnitude of the scattered pressure field at the principal cut o...
Figure 9.24 Zoom‐in plot for the magnitude of the scattered pressure field a...
Figure 9.25 A meshed trench structure embedded in truncated ground (top view...
Figure 9.26 Geometrical parameters of the trench structure, in which
is th...
Figure 9.27 Radiation pattern for the truncated ground with 1 trench.
Figure 9.28 Radiation pattern for the truncated ground with 5 trenches.
Figure 9.29 Radiation pattern for the truncated ground with
trenches.
Figure 9.30 Radiation pattern for the truncated ground with
trenches.
Figure 9.31 Comparison of radiation patterns for different truncated sizes o...
Figure 9.32 Elastic wave scattering by an arbitrarily shaped homogeneous obs...
Figure 9.33 RWG basis function
defined in two neighboring triangles
and
Figure 9.34 Radial and tangential components of total traction along the pri...
Figure 9.35 Radial and tangential components of total traction along the pri...
Figure 9.36 Radial and tangential components of total traction along the pri...
Figure 9.37 Radial and tangential components of total displacement along the...
Figure 9.38 Radial and tangential components of total displacement along the...
Figure 9.39 Tangential components of total displacement and traction along t...
Figure 9.40 Radial and tangential components of total displacement along the...
Figure 9.41 Radial and tangential components of scattered displacement along...
Figure 9.42 Radial and tangential components of total traction along the pri...
Figure 9.43 Tangential components of total displacement and traction along t...
Figure 9.44 Radial components of total traction along the principal cut at t...
Figure 9.45 Radial components of total traction along the principal cut at t...
Figure 9.46 Radial components of total displacement along the principal cut ...
Figure 9.47 Radial and tangential components of total displacement and tract...
Figure 9.48 Scattering by a dielectric object. (a) Original problem. (b) Equ...
Figure 9.49 Bistatic RCS solutions for a dielectric cube with a side length
Figure 9.50 Bistatic RCS solutions for a dielectric sphere with a radius
....
Figure 9.51 Normalized displacement field scattered by an elastic sphere wit...
Figure 9.52 Normalized displacement field inside an elastic cube illuminated...
Figure 9.53 Normalized displacement field from a traction‐free cubic cavity ...
Figure 9.54 Normalized displacement field from an elastic cube illuminated b...
Chapter 10
Figure 10.1 Geometry of two scatterers sitting in a rectangular coordinate s...
Figure 10.2
‐directed transient current density at
on the surface of a co...
Figure 10.3
‐component of normalized far‐zone scattered electric field obse...
Figure 10.4 Monostatic radar cross section of a conducting sphere.
Figure 10.5
‐directed transient current density at
on the surface of a co...
Figure 10.6
‐component of normalized far‐zone scattered electric field obse...
Figure 10.7 Monostatic radar cross section of a conducting cylinder.
Figure 10.8 Geometry of the scatterers sited in a rectangular coordinate sys...
Figure 10.9
‐directed transient current density at
on the surface of a co...
Figure 10.10
‐component of normalized far‐zone scattered electric field obs...
Figure 10.11 Root‐mean‐square (RMS) error of surface current density compare...
Figure 10.12
‐directed transient current density at
on the surface of a c...
Figure 10.13
‐component of a normalized far‐zone scattered electric field o...
Figure 10.14
‐directed transient current density at
on the surface of a c...
Figure 10.15
‐component of a normalized far‐zone scattered electric field o...
Figure 10.16 Geometries of dielectric scatterers centered at the origin of a...
Figure 10.17
‐directed electric current density at the centroid of a triang...
Figure 10.18
‐directed magnetic current density at the centroid of a triang...
Figure 10.19
‐component of normalized far‐zone scattered electric field obs...
Figure 10.20
‐directed electric current density at the centroid of a triang...
Figure 10.21
‐directed magnetic current density at the centroid of a triang...
Figure 10.22
‐component of normalized far‐zone scattered electric field obs...
Figure 10.23 RMS errors versus number of meshes for electric current density...
Figure 10.24 RMS errors versus number of Laguerre basis functions for electr...
Figure 10.25
‐directed electric current density at the centroid of a triang...
Figure 10.26
‐directed magnetic current density at the centroid of a triang...
Figure 10.27 Geometries of dielectric scatterers centered at the origin of a...
Figure 10.28
‐component of normalized far‐zone scattered electric field obs...
Figure 10.29
‐component of normalized far‐zone scattered electric field obs...
Figure 10.30
‐component of normalized far‐zone scattered electric field obs...
Figure 10.31
‐component of near‐zone scattered electric field observed at
Figure 10.32
‐component of far‐zone scattered electric field at
and
m f...
Figure 10.33 Bistatic RCS solutions at the frequency
MHz for a dielectric ...
Figure 10.34 Convergence rates of bistatic RCS solutions at the frequency
...
Figure 10.35 Illustration of transient EM scattering by a composite scattere...
Figure 10.36 Geometry of composite scatterers. (a) A dielectric cone backed ...
Figure 10.37
‐component of transient current density sampled at
and
m o...
Figure 10.38
‐component of transient current density sampled at
m and
o...
Figure 10.39
‐component of transient current density sampled at the north p...
Figure 10.40 RMS error of surface current density versus the number of tempo...
Guide
Cover
Table of Contents
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The Nyström Method in Electromagnetics
Mei Song Tong
Tongji University, Shanghai, China
Weng Cho Chew
Purdue University, West Lafayette, USA
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Library of Congress Cataloging-in-Publication Data
Names: Tong, Mei Song, author. | Chew,Weng Cho, author.Title: The NystrÖm method in electromagnetics / Mei Song Tong (Tongji University Shanghai, China), Weng Cho Chew (Purdue University, West Lafayette, USA).Description: Hoboken, NJ, USA :Wiley-IEEE Press, 2020. | Includes bibliographical references and index. |Identifiers: LCCN 2018060139 (print) | LCCN 2019002836 (ebook) | ISBN 9781119284888 (Adobe PDF) | ISBN 9781119284871 (ePub) | ISBN 9781119284840 | ISBN 9781119284840(hardcover) | ISBN 1119284848(hardcover)Subjects: LCSH: Electromagnetism–Mathematics. | Integral equations–Numerical solutions.Classification: LCC QC760 (ebook) | LCC QC760 .T59 2019 (print) | DDC 537.01/51–dc23LC record available at https://lccn.loc.gov/2018060139
Cover Design:WileyCover Image: © Mark Garlick / Science Photo Library/Getty Images
To Mei Song Tong's FamilyWei Liu and Zikang Tong
To Weng Cho Chew's FamilyChew Chin, Huibin Amelia, Shinen Ethan, and Sharon Ling
About the Authors
Professor Mei Song Tong received the Ph. D. degree in electrical engineering from Arizona State University, Tempe, Arizona, USA, in 2004. He is currently the Distinguished Professor, Department Head of Electronic Science and Technology, and Vice Dean of College of Microelectronics, Tongji University, Shanghai, China. Before he joined Tongji University, he was a Research Scientist at the Center for Computational Electromagnetics and Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana‐Champaign, Urbana, Illinois, USA. He is currently an adjunct professor at the University of Illinois at Urbana‐Champaign and an honorary professor at the University of Hong Kong, China. He has published more than 400 papers in refereed journals and conference proceedings, and co‐authored six books. His research interests include electromagnetic field theory, antenna theory and design, simulation and design of RF/microwave circuits and devices, interconnect and packaging analysis, inverse electromagnetic scattering for imaging, and computational electromagnetics.
Professor Tong is a Fellow of Electromagnetics Academy, Fellow of Japan Society for the Promotion of Science (JSPS), Full Member (Commission B) of the U.S. National Committee for the International Union of Radio Science, and Member of the Applied Computational Electromagnetics Society and the Sigma Xi Honor Society. He has been the chair of Shanghai Chapter and the chair of SIGHT committee of IEEE Antennas and Propagation Society since 2014 and in 2018, respectively. He has served as an associate editor or guest editor for several well‐known international journals, including IEEE Antennas and Propagation Magazine, IEEE Transactions on Antennas and Propagation, IEEE Transactions on Components, Packaging and Manufacturing Technology, International Journal of Numerical Modeling: Electronic Networks, Devices and Fields, Progress in Electromagnetics Research, and Journal of Electromagnetic Waves and Applications, etc. He also served as a session organizer, session chair, technical program committee chair or member, and general chair for some prestigious conferences in the electromagnetics community such as IEEE International Symposium on Antennas and Propagation (IEEE AP‐S) and USNC/URSI National Radio Science Meeting, Progress in Electromagnetics Research Symposium (PIERS), and IEEE International Conference on Computational Electromagnetics (ICCEM). He was the recipient of a Visiting Professorship Award from Kyoto University, Japan, in 2012. He advised and coauthored ten papers that received the Best Student Paper Award from 2014 International Workshop on Finite Elements for Microwave Engineering, PIERS'2014, PIERS'2015, PIERS'2016, ICCEM'2018, ICCEM'2019, PIERS'2019, and 2019 International Applied Computational Electromagnetics Society Symposium, respectively. He was the recipient of the Travel Fellowship Award of USNC/URSI for the 31th General Assembly and Scientific Symposium (GASS) of URSI in 2014, Advance Award of Science and Technology of Shanghai Municipal Government in 2015, Fellowship Award of JSPS in 2016, Innovation Award of Universities' Achievements of Ministry of Education of China in 2017, and many teaching‐type awards from Tongji University and Shanghai Municipal Government, respectively. In 2018, Professor Tong was selected as the distinguished lecturer by IEEE Antennas and Propagation Society for the 2019‐2021 term.
Professor Weng Cho Chew received all his degrees from Massachusetts Institute of Technology (MIT). His research interests are in wave physics, specializing in fast algorithms for multiple scattering imaging and computational electromagnetics in the last 30 years. His recent research interest is in combining quantum theory with electromagnetics, and differential geometry with computational electromagnetics. After MIT, he joined Schlumberger‐Doll Research in 1981. In 1985, he joined University of Illinois at Urbana‐Champaign, and was then the director of the Electromagnetics Lab from 1995 to 2007. During 2000–2005, he was the Founder Professor, 2005–2009 the Y. T. Lo Chair Professor, and 2013–2017 the Fisher Distinguished Professor. During 2007–2011, he was the Dean of Engineering at the University of Hong Kong. He joined Purdue University in August 2017 as a Distinguished Professor. He has co‐authored three books, many lecture notes, over 400 journal papers, and over 600 conference papers. He is a fellow of various societies, and an ISI's highly‐cited author. In 2000, he received the IEEE Graduate Teaching Award, in 2008, he received the C. T. Tai Distinguished Educator Award of IEEE Antennas and Propagation Society, in 2013, he was elected to the National Academy of Engineering, and in 2015, he received the ACES Computational Electromagnetics Award. Also, he received the IEEE Electromagnetics Award in 2017 and served as the president of IEEE Antennas and Propagation Society in 2018.
Preface
When I was invited with multiple times by the representative of Wiley to write this book in 2015, I was very hesitant because I worried about whether or not there will be enough readers who want to read the book. This is because the method of moments (MoM) as the primary numerical method for solving electromagnetic (EM) integral equations has been so widely used and accepted, it is very difficult to attract readers' interest by presenting another similar numerical method without demonstrating its obvious advantages. Fortunately, the Nyström method as one of nontraditional numerical methods has received more and more attention and has been fast‐increasingly applied to solve various EM problems since it was introduced to EM community in 1990. The Nyström method has shown certain merits which the MoM does not have and has become a strong competitor and alternative in many applications. Also, as a nontraditional and relatively‐new approach, it is natural to be lately accepted since people need some time to recognize its value, but this should not become a reason to stop studying and promoting it. No new exploration and competition means no advance! The rule should be applicable to all situations, including the investigation on different numerical methods for solving EM problems. Therefore, we can confidently write this book now.
This book is a monograph instead of a textbook which is mainly used for research reference. Many authors of such type of books said that the reason why they wrote their books was that there were not related books on relevant topics or the existent books were too old, so they should present new books. This is of course a strong reason, but we think that more books on similar topics in some research areas are also needed and are very worthy to be written even though there have been other books which have not been very old, because a diversity of books should be presented to satisfy different readers' diverse needs. Also, different books can provide different views or emphases even on similar topics. More importantly, keeping a moderate competition can certainly improve the quality of books just like papers, tons of which can simultaneously address the same or similar topics.
The technical contents of books usually come from published papers which have gone through a peer review, but writing books is still necessary and is actually a very important thing even if readers can find similar contents in papers. This is because books and papers serve different groups of readers who could have very different needs. The papers mainly aim at those professional researchers who have enough background and knowledge on related topics and can easily understand what papers present. In the contrast, the books primarily serve the newbies or inexperienced researchers whose representatives could be graduate students who have not had enough background and knowledge on relevant topics and could feel hard to directly read papers. An investigation by our graduate students association shows that more than 80% graduate students strongly rely on reading books instead of papers in their research work. The books should re‐narrate the sophisticated technical contents of papers by an easier‐understanding style so that readers can more easily understand and accept them. This requires that the authors of books creatively compose, explain, clarify, or illustrate the technical contents of papers according to their understanding and experience, instead of just a simple and mechanical collection or repeat of papers' contents. Due to the significant difference of functions between the books and papers, we should not make the books become a collective copy of some papers as some books' authors did. Those authors often claimed that their books mainly wanted to provide a convenience for readers to read related papers. To avoid such kind of style, we have provided an extensive introduction to the background, basic principle or mechanism, implementation method, physical applications, etc. for the book's theme, i.e. the Nyström method, with a more easily‐accepted manner. We believe that these non‐paper contents are very necessary for inexperienced researchers and students although experienced researchers may not need them.
This book is a summary for the work that we have done on the Nyström method in the past sixteen years. Thanks to Professor Weng Cho Chew who initialized the study on the interesting method and gave a continuous and huge instruction and advice on later investigations, resulting in making a significant progress on this method. Compared with other similar books, we specially strengthen the introduction to the singularity treatment techniques which could be viewed as the life and core of the Nyström method and its great expansion on practical applications, including the incorporation with the multilevel fast multipole algorithm (MLFMA), solution of multiphysics problems, and solution of time‐domain integral equations. These features have never or seldom been dealt with by other authors and could be the distinct highlights of this book.
Also, the Nyström method is an integral equation solver or belongs to an integral equation approach, so it is intimately related to integral equations and knowing the EM integral equations is a prerequisite for reading this book. We already wrote a book entitled “Integral Equation Methods for Electromagnetic and Elastic Waves”, which was published by Morgan & Claypool, San Rafael, CA, in 2008. That book mainly focuses on addressing the mathematics and physics behind the EM integral equations although it also includes some numerical solution methods and numerical examples. As a comparison, this book can be thought of as an engineering book which emphasizes the implementations and applications of the Nyström method for solving EM integral equations for practical engineering problems. Therefore, it may be a good idea to jointly read or refer to these two books.
This book has ten chapters in total and is divided into two parts. The first part includes the first five chapters which mainly address the background of EM or computational EM (CEM), and the basic principle and necessary elements of implementation such as quadrature rules and singularity treatment in the Nyström method. The second part includes other five chapters which mainly deal with the applications and physical implementations of the Nyström method for solving various EM problems as we have emphasized in the above. The first chapter was written by Professor Chew and the other nine chapters were written by myself. The primary contents of each chapter are summarized as follows.
Chapter 1 can be thought of as Professor Chew's recent observation and thinking about the EM and CEM. He first recalls the history of classical EM theory and draws a picture describing its connection with other well‐known physical theories like quantum electrodynamics and Yang‐Mills theory to emphasize the mathematics and physics behind the EM theory. He then clarifies the relationship of the rising quantum optics and quantum EM to the classical EM which can be viewed as the descendent of the fluid physics. Immediately followed is his narration about the complete development of the Maxwell's equations and derivation of wave equations, which allows him to divide the wave into three regimes in terms of frequency, i.e. the circuit physics, wave physics, and ray physics, respectively, and the mechanism of plasmonic resonance is also explained by the way. With the physics, he then starts to address the solution methods for the wave problems. He first narrates the closed‐form solutions and their asymptotic approximations at high frequencies. Next, he addresses the CEM or numerical methods and shows a procedure of converting an operator equation into a matrix equation which is the underpinning method behind the finite element method (FEM) and the MoM. After that, he further addresses fast algorithms and particularly introduces the MLFMA and domain decomposition method (DDM). Finally, he reviews some key components of high‐frequency solutions, inverse problems, metamaterials, and small antennas. With those reviews, he predicts that EM will still remain an important area of study and a fundamental status impacting many other technologies, even if the onset of quantum mechanics could bring new possibilities to make a big change.
Chapter 2 also addresses the CEM as the background of applying the Nyström method, but it is in a more microscopic manner compared with the more macroscopic picture and wider range of Chapter 1. The CEM, which is the basis of solving EM problems by computer modeling and simulation, is first defined and categorized. The analytical solutions of spherical objects, which are Mie‐series solutions and are usually used as exact solutions, are then presented. After that, a brief introduction to the three mainstream numerical methods, i.e. the finite‐difference time‐domain (FDTD) method, FEM, and MoM, is given. Finally, all EM integral equations for different cases are summarily presented and they serve the basis of applying the Nyström method to solve EM problems.
Chapter 3 presents a careful look on the theme of this book, i.e. the Nyström method. It introduces the history, basic principle, and implementation scheme for the Nyström method, but there is no demonstration and application which will be shown in later chapters. It also deals with the singularity treatment which is the key of the Nyström method and describes a higher‐order scheme in principle. Finally, it makes a comparison between the Nyström method and the MoM or point‐matching method, with which many readers could be concerned.
Chapter 4 is devoted to the introduction for numerical quadrature rules which also play a vital role in the Nyström method. The definition of quadrature rules is introduced first, and then different types of quadrature rule are presented in details. These rules are designed for segmental, surface, and volumetric meshes, respectively. For the segmental meshes, the Gaussian quadrature rules in different orders with different points are addressed. For the surface and volumetric meshes, the widely used quadrature rules defined over triangular patches and tetrahedral elements are presented, but the quadrature rules defined over square patches and cuboid elements are also considered because they could be needed in certain cases although they are less employed.
Chapter 5 deals with the singularity treatment techniques which are very essential to the Nyström method. The singularity subtraction method is presented first, which is indispensable for strongly singular and hypersingular integrals. It is based on the Tayor's series expansion of the scalar Green's function to distinguish the singular cores from singular integrands and then subtract them to regularize the original singular integrals. Next, the singularity cancellation method is addressed, which is preferable but can only be applied to weakly‐singular integrals. Much effort is then paid to the derivation of closed‐form formulations for evaluating strongly‐singular and hypersingular integrals which are from the subtraction based on the Cauchy‐principal‐value (CPV) sense. Various cases including surface domain with triangular patches and volume domain with tetrahedral or cylindrical elements are considered. Finally, the near‐singular integrals which are seldom considered are also addressed and a nearness factor (NF) is proposed to determine if the near singularity should be handled or not.
In Chapter 6, we narrate the application of the Nyström method to solve EM problems with conducting media. The two‐dimensional (2D) structures with or without incorporating edge conditions are considered and the treatment of logarithm singularity is described as a supplementary. We then apply the Nyström method to solve the body‐of‐revolution (BOR) problems by using Fourier series expansion and the formulations of evaluating singular modal Green's function and its derivative are derived. Regarding regular three‐dimensional (3D) conducting objects, we first present a higher‐order Nyström scheme to solve the electric field integral equation (EFIE) and then particularly consider very‐thin or super‐thin conducting structures by using the magnetic field integral equation (MFIE) and combined field integral equation (CFIE) to formulate, respectively. Robust formulations for evaluating those near‐singular integrals based on the Green's lemma or the Stokes' theorem are developed and they are particularly suitable for being used in thin structures.
Chapter 7 depicts the application of the Nyström method or Nyström‐based mixed scheme to solve EM problems with penetrable media. The surface integral equations (SIEs), including the EFIEs, MFIEs and CFIEs, with an assumption of homogeneous and isotropic media are solved first and a comparison with the MoM is illustrated. The Nyström method is then applied to solve volume integral equations (VIEs) which allow the inclusion of inhomogeneous and anisotropic media by developing an efficient local correction scheme for singular or near singular integrals over tetrahedral elements. The method is also applied to the case with conductive media which are treated as penetrable media when the skin depth is large. Finally, the Nyström‐based mixed scheme is developed for analyzing interconnect and packaging problems with both penetrable dielectrics and impenetrable conductors based on the volume‐surface integral equations (VSIEs) and the comparison with the conventional MoM is also presented.
In Chapter 8, we recount the incorporation of the Nyström method with the MLFMA for solving electrically large problems. The basic principle and characteristics of MLFMA are introduced first and the Nyström‐based MLFMA is developed for solving various problems. We consider to solve the SIEs with conducting objects, SIEs with penetrable media, VIEs with conductive media, and VSIEs with conducting‐anisotropic media, respectively. Finally, a comparison between the Nyström‐based MLFMA and the conventional MoM‐based MLFMA is made and its distinct features are clarified.
Chapter 9 discusses the application of the Nyström method to solve multiphysics problems which are of interdisciplinary feature. Although there are many types of multiphysics problems, we select the coupled EM–elastodynamic problem as an example to illustrate the solution by the Nyström method. We begin from the introduction of elastic wave equations and acoustic wave equations, with which EM researchers may not be familiar, and then present their numerical solutions by the Nyström method. We also give a unified form of these two wave equations because the acoustic wave is actually the special case of elastic wave. Moreover, we develop the Nyström‐based MLFMA for solving both elastic wave equations and acoustic wave equations of large problems, in particular, we develop a multiple‐tree structure of MLFMA which does not exist in EM problems. Based on those prerequisite investigations, we then develop the coupled integral equations for the multiphysical coupling of EM wave and elastic wave and their Nyström solutions based on an alternative iterative scheme are illustrated.
In the last chapter, Chapter 10, we focus on the solutions of time‐domain integral equations (TDIEs), including the time‐domain version of EFIEs (TDEFIEs), MFIEs (TDMFIEs), CFIEs (TDCFIEs), and VIEs (TDVIEs), for transient EM problems. A corresponding hybrid scheme is developed by combining the Nyström method in space domain and Galerkin's method with Laguerre basis and testing functions in time domain, and the TDIEs are solved with a march‐on in degree (MOD) manner. Based on the hybrid scheme, we solve the time‐domain surface integral equations (TDSIEs) for conducting objects, TDSIEs for penetrable media, TDVIEs for penetrable media, and TDCFIEs for mixed media which include both conducting and penetrable media, respectively. A comparison with the conventional scheme, which uses the MoM to discretize the space domain and employs a march‐on in time (MOT) scheme to discretize the time domain, is performed and its advantages or disadvantages are clarified.
The Nyström method as a nontraditional method is still growing rapidly and many relevant topics can be further studied or explored. For instance, the singularity treatment techniques still have a big space to be investigated as it is tightly related to the Green's function which is full of wave physics. Also, the applications of the Nyström method can be significantly expanded since it has not been applied to many areas such as those mentioned in Chapter 1. We hope that this book can stimulate the readers' strong interest on the interesting method and attract more researchers to study and innovate it.
Mei Song TongTongji UniversityShanghai, China
Acknowledgment
The book reports what we have done on the Nyström method in electromagnetics in the past sixteen years. It was impossible to do such an extensive study in such a long time without enough financial supports. Therefore, we are highly grateful to our research sponsors for their generous fund supports on the research work. Particularly, we would like to strongly appreciate U.S. Air Force Office of Scientific Research (AFOSR), Construction Engineering Research Laboratory (CERL) of U.S. Army, Semiconductor Research Corporation (SRC), Intel Corporation, IBM Corporation, National Natural Science Foundation of China (NSFC), Ministry of Science and Technology (MOST) of China, Ministry of Education (MOE) of China, Science and Technology Committee of Shanghai Municipality (STCSM) of China, and Shanghai Institute of Intelligent Science and Technology of Tongji University. Also, we would like to thank all colleagues and students who have been involved in the related research work. Finally, our gratitude would go to the related staffs of Wiley who have made the publication of this book become possible. Their hard work and tremendous support and help are highly appreciated.
1Electromagnetics, Physics, and Mathematics
Mathematics is the mother of science, science is the mother of technology and father of innovation, and technology is a gift of God.
1.1 A Brief History of Electromagnetics
Electromagnetics is a subject that entails the study of electromagnetic theory, its physical interaction with environments and objects, and its use for design and engineering applications. Electromagnetic theory is governed by a set of equations that originated from many great scientists such as Gauss (1835), Amperes (1823), Faraday (1838), and Coulomb (1785) [1–4]. However, this set of equations is now commonly known as Maxwell's equations due to the work of James Clerk Maxwell who put on them in mathematical form in 1864 and added a term to Ampere's law that included displacement current density [5, 6, 33]1.
This set of equations is used to describe static electromagnetic fields, radio waves, microwaves, optical fields, as well as x‐rays. It is valid from subatomic length scale to inter‐galactic length scale. The Coulomb potential due to a point charge of a proton or an electron inside an atom can be derived from Maxwell's equations. Meanwhile, the propagation of radio waves and optical signals from outer galaxies can be described by the same set of equations [7].
