|
|
马上注册，结交更多好友，享用更多功能，让你轻松玩转社区。
您需要 登录 才可以下载或查看，没有账号？注册
x
Preface
Numerical solutions of electromagnetic field problems constitute an area of paramount interest in academia, industry, and government. Many numerical techniques exist based on the solutions of both the differential and integral forms of Maxwell's equations. It is not recognized very often that for electromagnetic analysis of both conducting and practical linear piecewise homogeneous isotropic bodies, integral equations are still one of the most versatile techniques that one can use in both efficiency and accuracy to solve challenging problems including analysis of electrically large structures.
In this book, our emphasis is to deal primarily with the solution of time domain integral equations. Time domain integral equations are generally not as popular as frequency domain integral equations and there are very good reasons for it. Frequency domain integral equations arise from the solution of the unconditionally stable elliptic partial differential equations of boundary-value problems. The time domain integral equations originate from the solution of the hyperbolic partial differential equations, which are initial value problems. Hence, they are conditionally stable. Even though many ways to fix this problem have been proposed over the years, an unconditionally stable scheme still has been a dream. However, the use of the Laguerre polynomials in the solution of time domain integral equations makes it possible to numerically solve time domain problems without the time variable, as it can be eliminated from the computations associated with the final equations analytically. Of course, another way to eliminate the time variable is to take the Fourier transform of the temporal equations, which gives rise to the classical frequency domain methodology. The computational problem with a frequency domain methodology is that at each frequency, the computational complexity scales as 6{í*) [C(·) denotes of "the order of, where N is the number of spatial discretizations]. Time domain methods are often preferred for solving broadband large complex problems over frequency domain methods as they do not involve the repeated solution of a large complex matrix equation. The frequency domain solution requires the solution of a large matrix equation at every frequency step, whereas in the solution of a time domain integral equation using an implicit method, one only needs to solve a real large dense matrix once and then use its inverse in the subsequent computations for each
timestep. So that, a time domain method requires only a vector-matrix product of large dimension at each time step. Thus, in this case, at each timestep, the computational complexity is 0{N2). The use of the orthogonal associated Laguerre functions for the approximation of the transient responses not only has the advantage of an implicit transient solution technique as the computation time is quite small, but also separates the space and the time variables, making it possible to eliminate the time variable from the final computations analytically. Hence, the numerical dispersion due to time discretizations does not exist for this methodology. This methodology has been presented in this book. The xv
xvi PREFACE conventional marching-on-in-time method is also presented along with its various modifications.
The frequency domain solution methodology is also included, so that one can verify the time domain solutions by taking an inverse Fourier transform of the frequency domain data. This will provide verification for the accuracy and stability of the new Laguerre-based time domain methodology, which is called the marching-on-in-degree/order technique.
Finally, a hybrid method based on the time and frequency domain techniques is presented to illustrate that it is possible to go beyond the limitations of the computing resources through intelligent processing. The goal of this hybrid approach is to generate early-time information using a time domain code and lowfrequency information using a frequency domain code, which is not computationally demanding. Then an orthogonal series is used to fit the data both in the early-time and low-frequency domains, and then it is shown that one can use the same coefficients to extrapolate the response imultaneously both for the latetime and high-frequency regions. In this approach, one is not creating any new information but is using the existing information to extrapolate the responses simultaneously in the time and frequency domains. Three different orthogonal polynomials have been presented, and it is illustrated how they can be utilized to achieve the stated goal of combining low-frequency and early-time data to generate a broadband solution. This forms the core of the parametric approach. The fact that Fourier transforms of the presented polynomials are analytic functions allows us to work simultaneously with time and frequency domain data. A priori error bounds are also provided and a processing-based methodology is outlined to know when the data record is sufficient for extrapolation. A nonparametric approach is also possible using either a Neumann series approximation or through the use of the Hubert transform implemented by the fast Fourier transform. This hybrid methodology may make it possible to combine measured vector network analyzer data with a time domain reflectometer measurements in a seamless and accurate fashion without using any of the approximate inaccurate methodologies to process the measured data that currently are implemented in most commercial time domain reflectometer (TDR) and network analyzer measurement systems. |
|
/1 
eetop公众号
创芯大讲堂
创芯人才网
发表于 2013-10-13 15:58:36
楼主