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摘要随着信息科技的不断发展，尤其是计算机技术的不断完善，复杂电磁问题的求解也取得了长足进步。为了探求多个复杂源点在多个场点产生的响应，人们提出了格林函数法。当应用到分层介质时，因为空域格林函数形式复杂，难以求得，难以计算，所以人们通过求解谱域格林函数的Sommerfeld积分来得到空域格林函数。由于Sommerfeld积分的快震荡和慢衰减特性，计算量非常大，耗时非常长，对此人们依然没有很好的办法。27486
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本文主要研究在格林函数求解中的多文插值技术。推导分层介质的空域与谱域格林函数；将多项式插值、Hermite插值与样条插值应用到Sommerfeld积分的计算。本文核心内容概括如下：
第一部分，研究分层介质的空域与谱域格林函数，并将其应用到微带线上。
第二部分，将插值法应用到Sommerfeld积分的计算。由于Sommerfeld积分的快震荡和慢衰减特性，直接计算非常困难，效率极低。为了解决这个问题，我们可以采用插值法来逼近格林函数。插值法较直接计算效率有所提高，受到了比较多的关注。
第三部分，我们对直接计算格林函数与使用插值法计算举例演算，并进行数值分析。然而插值法也会存在一些问题，比如某些奇异点会无法处理。我们可以通过原函数减去奇异项等方法进行插值，限于篇幅，本文只做了粗浅的讨论。
毕业论文关键词 格林函数、Sommerfeld积分、多项式插值、Hermite插值、样条插值、奇异值
Title The multi-dimensional interpolation in solving Green’s functions
Abstract
With the continuous development of information technology, especially the constant improvement of computer technology, the complicated electromagnetic problem solving also made great progress. In order to explore the response in multiple sites from complex source points, green's function method are put forward. When used in layered medium, because the spatial green's function form is complex and difficult to get, so people get the spatial green's function by solving Sommerfeld integrals of spectral domain green's function. Due to fast volatility and slow attenuation characteristics of Sommerfeld integrals, the amount of calculation is very large, very time consuming, people still do not have a good idea about this.
This paper mainly research multi-dimensional interpolation technique in solving the green's functions . Layered medium is derived spatial and spectral domain green's function; The Polynomial interpolation, Hermite interpolation and spline interpolation are applied to calculate Sommerfeld integrals. In this paper, the core content is summarized as follows:
The first part, the research of layered medium’s spatial and spectral domain green's function, and its application to microstrip line.
The second part, the application of interpolation techniques to calculate Sommerfeld integrals. Due to fast volatility and slow attenuation characteristics of Sommerfeld integrals, direct calculation is very difficult and has very low efficiency. In order to solve this problem, we can use interpolation approximation of spectral domain green's function. Interpolation method is more efficient than direct calculation and receives more attention.
The third part, we directly calculate the Green's function and use interpolation method to calculate the calculus, we take examples and compare numerical analysis. However interpolation method also have some problems, such as some singular point cannot be handled through the original function .We can use interpolation minus a singularity to solve it. Due to limited space, this paper just gives easy discussions.
Keywords green’s function、Sommerfeld integrals、Polynomial interpolation、Hermite interpolation、spline interpolation、singular point