电磁与弹性波的积分方程方法

5星 · 超过95%的资源需积分: 9 69 浏览量更新于2024-07-25 收藏 9.31MB PDF 举报

"Integral equation methods是解决电磁波和弹性波问题的一种强大工具，由Weng Cho Chew、Mei Song Tong和Bin Hu共同探讨。本书详细介绍了如何利用积分方程方法来处理这两种物理现象中的波动问题。它涵盖了Green's函数、积分方程等相关概念，并在电磁学和固体力学的背景下进行了深入讨论。该书适用于科研人员和学生，旨在提供计算电磁学和计算力学领域的理论基础和应用实例。" 积分方程方法是分析和求解复杂边界条件下的电磁和弹性波问题的有效途径。这些方法基于场量之间的关系，通常将偏微分方程转化为边界上的积分方程，从而简化问题的求解。在电磁学中，这种方法常用于研究天线、雷达散射、微波器件以及光子晶体等领域的现象。在弹性波传播中，它们则被用来解决地震波、声波在复杂介质中的传播问题，或者在结构工程中分析材料的振动和应力分布。 Green's函数是积分方程方法的核心组成部分，它是描述在特定边界条件下任意点处的场响应与源位置之间关系的函数。在电磁学中，Green's函数可以表示为一个点电荷或磁偶极子在空间中产生的场；在弹性波问题中，它对应于一个点源在介质中引起的位移场。通过解析或数值计算得到Green's函数，然后可以求解整个区域内的场分布。积分方程通常以边界值问题的形式出现，例如Lippmann-Schwinger方程或Sommerfeld积分方程。这些方程可以直接对边界上的场或其导数进行操作，而无需考虑边界内部的细节。这使得积分方程方法特别适合处理具有复杂几何形状或不规则边界的场景，因为它们只需要在边界上定义问题，而不需要在整个域内进行离散化。该书详细阐述了积分方程方法的理论基础，包括Fredholm第一型和第二型积分方程的求解策略，以及相关的数值方法，如矩量法（Method of Moments, MoM）和边界元方法（Boundary Element Method, BEM）。这些数值技术允许将积分方程转化为线性代数系统，进而用计算机求解。此外，书中可能还涵盖了迭代解法，如迭代MoM，以及加速收敛的技术，如预条件器和多极展开。作者可能会讨论各种物理现象的实际应用，比如射线光学、波导模式分析、声学散射、结构动力学等，以展示积分方程方法的实用性。《Integral Equation Methods for Electromagnetic and Elastic Waves》是一本深入介绍积分方程在电磁学和弹性波领域应用的著作，对于希望理解和应用这些方法的科研人员和学生来说，是一份宝贵的资源。

Acknowledgements

Many in our research group gave feedback and helped proofread the manuscript at various

stages. Thanks are due Mao-Kun LI, Yuan LIU, Andrew Hesford, Lin SUN, Zhiguo QIAN, Jie

XIONG, Clayton Davis, William Tucker, and Bo HE. During my stay as a visiting professor

at Nanyang Technological University, I received useful feedback from Eng Leong TAN on

some of the chapters. Peter Monk gave useful feedback on part of Chapter 2.

We are grateful to our research sponsors for their supports in the last decade, notably,

AFOSR, SAIC-DEMACO, Army-CERL, Raytheon-UTD, GM, SRC, Intel, IBM, Mentor-

Graphics, and Schlumberger. WCC also thanks supports from the Founder Professorship and

Y.T. Lo Endowed Chair Professorship, as well as supports from the IBM Visiting Professorship

at Brown University, the Distinguished Visitor Program at the Center for Computational

Sciences, AFRL, Dayton, the Cheng Tsang Man Visiting Professorship at NTU, Singapore,

and the MIT Visiting Scientist Program. Supports from Arje Nachman and Mike White of the

government agencies are gratefully acknowledged. We appreciate interactions with partners

and colleagues in the industry, notably, Steve Cotten, Ben Dolgin, Val Badrov, HP Hsu,

Jae Song, Hennning Braunisch, Kemal Aygun, Alaeddin Aydiner, Kaladhar Radhakrishnan,

Alina Deutsch, Barry Rubin, Jason Morsey, Li-Jun Jiang, Roberto Suaya, Tarek Habashy,

and John Bruning. Also, discussions with university partners on various projects such as Ben

Steinberg, Steve Dvorak, Eric Michielsen, and John Volakis have been useful. Last but not

least, constant synergism with members of the EM Laboratory at Illinois, Shun-Lien Chuang,

Andreas Cangellaris, Jianming Jin, Jose Schutt-Aine, and Jennifer Bernhard are enlightening.

This book is also dedicated to the memory of Y.T. Lo, Don Dudley, and Jin Au Kong, who

have been WCC’s mentors and friends.

MC1483_Chew_Lecture.qxd 8/18/08 12:29 PM Page xv

1.1 Mathematical Modeling—A Historical Perspective

In the beginning, mathematics played an important role in human science and technology

advancements by providing arithmetic, geometry, and algebra to enable scientists and tech-

nologists to solve complex problems. For example, precalculus was used by Kepler to arrive

at laws for predicting celestial events in the 1600s. In the late 1600s and early 1700s, Leibniz

and Newton developed early calculus for a better description of physical laws of mechanics [1].

Later, the development of diﬀerential equations enabled the better description of many physi-

cal phenomena via the use of mathematics. The most notable of these were the Navier-Stokes

equations for understanding the physics of ﬂuids in the 1800s [2–4].

Computational ﬂuid dynamics (CFD) [5] has a longer history than computational electro-

magnetics (CEM) (Richardson 1910 [6], Courant, Friedrichs, and Lewy 1928 [7], Lax 1954 [8],

Lax and Wendroﬀ 1960 [9], MacCormack 1969 [10]). The early works are mainly those of

mathematicians looking for numerical means to solve partial diﬀerential equation (PDE), with

the knowledge that many such equations have no closed form solution. Hence, the reason for

the early development of CFD is simple: many ﬂuids problems are analytically intractable.

This stems from the nonlinearity inherent in the Navier-Stokes equations. Early ﬂuids cal-

culations were done numerically, albeit laboriously, with the help of human “computers”. In

fact, the design of the ﬁrst electronic computers was driven by the need to p erform laborious

ﬂuids calculations (Eniac 1945, Illiac I 1952 [11]).

In contrast, most electromagnetic media are linear. Hence, the governing equations for

electromagnetics are, for the most part, linear. The linearity of electromagnetic equations

allows for the application of many analytic techniques for solving these equations.

CHAPTER 1

Introduction to Computational

Electromagnetics

MC1483_Chew_Lecture.qxd 8/18/08 12:29 PM Page 1

2 INTEGRAL EQUATION METHODS FOR ELECTROMAGNETIC AND ELASTIC WAVES

The electromagnetic equations are:

∇×E = −

∂B

∂

(1.1)

∇×H =

∂D

∂

+ J (1.2)

∇·D = ρ (1.3)

∇·B = 0 (1.4)

These equations are attributable to the works of Faraday (Eq. (1.1), Faraday’s law 1843 [12]),

Ampere (Eq. (1.2), Ampere’s law 1823 [13]), Coulomb (Eq. (1.3), Coulomb’s law 1785 [14]; it

was enunciated in a divergence form via Gauss’ law), and Gauss (Eq. (1.4), Gauss’ law 1841

[15]). In 1864, Maxwell [16] completed the electromagnetic theory by adding the concept of

displacement current (∂D/∂t) in Eq. (1.2). Hence, this equation is also known as generalized

Ampere’s law. The set of equations above is also known as Maxwell’s equations. The original

Maxwell’s equations were expressed in some 20 equations. It was Heaviside [17] who distilled

Maxwell’s equations into the above four equations. Hence, they are sometimes known as the

Maxwell-Heaviside equations. These equations have been proven to be valid from subatomic

length scale to intergalactic length scale.

Because of their linearity, one useful technique for solving electromagnetic equations is the

principle of linear superposition. The principle of linear superposition further allows for the

use of the Fourier transform technique. Hence, one can Fourier transform the above equations

in the time variable and arrive at

∇×E = iωB (1.5)

∇×H = −iωD + J (1.6)

∇·D = ρ (1.7)

∇·B = 0 (1.8)

The preceding equations are the frequency-domain version of the electromagnetic equations.

When the region is source free, that is, ρ = 0 and J = 0, as in vacuum, B = µ

H and

D = ²

E, the simplest homogeneous solution to the above equations is a plane wave of the

form a exp(ik · r) where a is orthogonal to k and |k| = ω

√

Hence, Maxwell’s work

bridged the gap between optics and electromagnetics because scientists once thought that the

electromagnetic phenomenon was diﬀerent from the optical phenomenon.

Also, for electrodynamic problems, the last two of the above equations can be derived

from the ﬁrst two with appropriate initial conditions. Hence, the last two equations can be

ignored for dynamic problems [18].

The use of Fourier transform technique brings about the concept of convolution. The

concept of convolution leads to the development of the Green’s function technique [19]. From

the concept of Green’s functions, integral equation can be derived and solved [18].

As a result, electromagneticists have a whole gamut of analytic tools to play with. Many

engineering problems and theoretical modeling can be solved by the application of these tools,

oﬀering physical insight and shedding light on many design problems. The use of intensive

computations to arrive at critical data can be postponed until recently. Hence, CEM concepts

MC1483_Chew_Lecture.qxd 8/18/08 12:29 PM Page 2

INTRODUCTION TO COMPUTATIONAL ELECTROMAGNETICS 3

did not develop until 1960s [20, 21]. Although laborious calculations were done in the early

days, they were not key to the development of many technologies.

1.2 Some Things Do Not Happen in CEM Frequently—

Nonlinearity

Nonlinearity is not completely absent in electromagnetics. Though most nonlinear phenomena

are weak, they are usually ampliﬁed by short wavelengths, or high frequencies. When the

wavelength is short, any anomalies will be ampliﬁed and manifested as a cumulative phase

phenomenon. For instance, the Kerr eﬀect, which is a 10

−

eﬀect, is responsible for the

formation of solitons at optical frequencies [22].

Nonlinear eﬀects do not occur frequently in electromagnetics. A nonlinear phenomenon

has the ability to generate high frequency ﬁelds from low frequency ones (Just imagine f(x)=

.Ifx(t) = sin ωt, f[x(t)] = sin

ωt has a frequency component twice as high as that of

x(t)). Nonlinearity can give rise to shock front and chaos. These phenomena do not occur in

linear electromagnetics.

Nonlinearity in electromagnetics can be a blessing as well as a curse. The successful

propagation of a soliton in an optical ﬁber was heralded as a breakthrough in optics for

optical communications whereby a pulse can propagate distortion-free in an optical ﬁber

for miles needing no rejuvenation. However, the nonlinear phenomenon in an optical ﬁber

is besieged with other woes that make the propagation of a pure soliton diﬃcult. After

two decades of research, no commercially viable communication system using solitons has

been developed. On the contrary, optical ﬁbers are engineered to work in the linear regime

where dispersion management is used to control pulse shapes for long distance propagation.

Dispersion management uses alternating sections of optical ﬁbers with positive and negative

dispersion where pulse expansion is followed by pulse compression to maintain the integrity

of the signal [23].

1.3 The Morphing of Electromagnetic Physics

Electromagnetic ﬁeld is like a chameleon—its physics changes as the frequency changes from

static to optical frequencies [24]. Electromagnetics is described by ﬁelds—a ﬁeld is a function

of 3D space and time. These ﬁelds satisfy their respective PDEs that govern their physics.

Hence, electromagnetic physics changes depending on the regime. The morphing of the

physics in electromagnetics can be easily understood from a simple concept: the comparison

of derivatives. PDEs that govern electromagnetics have many derivative operators in them—

depending on the regime of the physics, diﬀerent parts of the partial derivative operators

become important.

Another important concept is that of the boundary condition. The boundary condition

dictates the manner in which a ﬁeld behaves around an object. In other words, a ﬁeld has to

contort around an object to satisfy the boundary condition. Hence, the variation of a ﬁeld

around an object is primarily determined by the geometrical shape of the object.

MC1483_Chew_Lecture.qxd 8/18/08 12:29 PM Page 3

剩余258页未读，继续阅读

kalaxia

粉丝: 0
资源: 3

电磁与弹性波的积分方程方法

Helmholtz Decomposition Based On Integral Equation Method For Electromagnetic Analysis

Higher Order Electromagnetic Modeling in Current and Charge Integral Equation Method

Marching-on-in-Degree Time-Domain Integral Equation Solver with High-Level Basis Functions for Transient Electromagnetic Scattering

Numerical solutions of Green's integral equation for the diffraction of femtosecond laser pulses through a subwavelength aperture

The L-integral description for a brittle elastic plane with two holes before, during, and after coalescence

A new collocation method for solving certain Hadamard finite-part integral equation

Regularized solution to the Fredholm integral equation of the first kind with noisy data

The application of Continuity Condition in Volume-Surface Integral Equation Combined with the Multilevel Fast Multipole Algorithm

Differential-and-Integral-Methods:微分和积分方法，理学士特拉维夫大学电气和电子工程

Numerical_integral_methods_used_in_integral_equati_complex integ

最新资源