离散奇异卷积及其在计算电磁学中的应用
Discrete singular convolution and its application to computational electromagnetics
AI总结：
本文提出离散奇异卷积（DSC）算法，用于计算电磁学，通过统一框架推导多种实现，控制带宽与精度，算例验证其有效性与潜力。
摘要：
本文针对计算电磁学问题，提出了一种新的计算算法——离散奇异卷积（DSC）。研究了DSC算法用于逼近函数及其导数的基本思想。通过构造带限再生核或近似再生核来实现对delta分布的逼近。提出了一种系统化的方法来处理实际应用中出现的多种边界条件。从加权残量法的角度，探讨了DSC算法求解微分方程的统一特性。证明了该算法的不同实现方法，如全局法、局部法、Galerkin法、配点法和有限差分法，均可从同一出发点推导出来。DSC算法的计算带宽和精度均可控。通过三个算例来展示算法的实用性、检验其精度并探索其局限性。采用Galerkin诱导的配点法，对规则和不规则域中的波导分析以及通过势函数进行静电场估计进行了研究。利用广义有限差分法对三维空间中的电磁波传播进行积分，在DSC参数的特定极限下，该方法可转化为全局有限差分格式。数值实验表明，所提算法是解决电磁学问题的一种有前景的方法。
英文摘要：
A new computational algorithm, the discrete singular convolution (DSC), is introduced for computational electromagnetics. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. A systematic procedure is proposed to handle a number of boundary conditions which occur in practical applications. The unified features of the DSC algorithm for solving differential equations are explored from the point of view of the method of weighted residuals. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Both the computational bandwidth and the accuracy of the DSC algorithm are shown to be controllable. Three example problems are employed to illustrate the usefulness, test the accuracy and explore the limitation of the DSC algorithm. A Galerkin-induced collocation approach is used for a waveguide analysis in both regular and irregular domains and for electrostatic field estimation via potential functions. Electromagnetic wave propagation in three spatial dimensions is integrated by using a generalized finite difference approach, which becomes a global-finite difference scheme at certain limit of DSC parameters. Numerical experiments indicate that the proposed algorithm is a promising approach for solving problems in electromagnetics.
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