二维电磁散射问题的一种高精度算法研究
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摘要
由R. F. Harington于1968年提出的矩量法(MoM),是在解决电磁散射的诸多方法中运用最普遍的方法,如可用于各种复杂目标体的电磁散射问题、确定性目标散射、分析天线问题、随机粗糙面散射等。它是通过引入基函数与权函数的方法,离散积分方程为矩阵方程的方式来进行求解的。实现其高精度计算的核心问题是阻抗矩阵中对角元素的计算,这些计算涉及弱/超奇异积分的数值处理。
机械求积法直接使用数值积分公式离散化,不用计算大量的积分,只需要直接赋值即可,这样可以节省大量的计算时间。该方法基于推导出的公式,用计算机求积,没用任何变换,给出一求奇异积分的公式,这样计算出的结果精度很高,但如何构造恰当求积公式及阐述相应求积方法的可靠性是计算数学的一大难题。本文所采用的机械求积公式是关于奇异积分高精度计算的最新公式。而矩量法采用的是汉克函数的小自变量公式,这就使得机械求积技术的精度要高于矩量法。径向基函数方法作为一个本质上用一元函数描述多元函数的强有力工具,是在处理大规模散乱数据时经常用到的方法。近十几年来,无网格径向基函数方法受到了人们越来越多的关注。作为一种新型的数值计算方法,无网格法仅仅需要在域内分布一些相互独立的点,而不是相互连接的单元,所以可以减少大量的数据准备,避免了普通有限元法和边界元法在计算中需要的网格生成或重生成,以及在大变形(如金属成型,高速碰撞等)计算中可能遇到的单元自锁、扭曲、畸变、移动等问题。本文主要做了以下工作：首先利用关于奇异积分高精度计算的最新公式结合积分方程的配置法处理2维散射问题,利用算子分裂方法将光滑边界推广到分段光滑边界问题,本文具体是将光滑边界推广到三角形边界,还可以用类似方法推广到多边形边界；另外考虑到脉冲匹配的精度问题,利用径向基函数逼近表面电流,并采用高精度机械求积公式(MQM)计算涉及到的奇异积分,数值结果表明了这种方法的计算效率。
The Method of moments (MoM) is the one of the most common method used tosolve electromagnetic scattering.the core issue to achieve its high-precision calculation is the calculation of the diagonal elements in its impedance matrix, which involves numerical treatment of the weak/hypersingular integralshese. The MoM may solve some problems related with complex boundary that analytic method is inoperative. As a result, the MoM is broadly used in various electromagnetic researches, such as EMC, microwave net, micro-strip analysis, antenna design, radiation effort and antenna issues, etc.This essay will improve the method of mechanical quadrature (MQM) related with two-dimensional scattering issue as described below:With the latest formula in high-precision calculation of singular integral, and combining the method of dealing with 2-dimensional scattering problem, this paper extends smooth boundary to piecewise smooth boundary problem by using operator splitting method. Adopts the singular integral related to high-precision mechanical quadrature formula (MQM) calculations related to. The numerical results show the computational efficiency of this method. We could obtain improved MQM with better accuracy than it of MQM and MoM with calculation time saving advantages.Taking into account of the accuracy of pulse matching, it approximates surface currentthe by using radial basis function. The interpolation theory of RBF may help to reduce large amount of workload in grid subdivision, and therefore highly improve the efficiency and result accuracy of algorithm.
机械求积法直接使用数值积分公式离散化,不用计算大量的积分,只需要直接赋值即可,这样可以节省大量的计算时间。该方法基于推导出的公式,用计算机求积,没用任何变换,给出一求奇异积分的公式,这样计算出的结果精度很高,但如何构造恰当求积公式及阐述相应求积方法的可靠性是计算数学的一大难题。本文所采用的机械求积公式是关于奇异积分高精度计算的最新公式。而矩量法采用的是汉克函数的小自变量公式,这就使得机械求积技术的精度要高于矩量法。径向基函数方法作为一个本质上用一元函数描述多元函数的强有力工具,是在处理大规模散乱数据时经常用到的方法。近十几年来,无网格径向基函数方法受到了人们越来越多的关注。作为一种新型的数值计算方法,无网格法仅仅需要在域内分布一些相互独立的点,而不是相互连接的单元,所以可以减少大量的数据准备,避免了普通有限元法和边界元法在计算中需要的网格生成或重生成,以及在大变形(如金属成型,高速碰撞等)计算中可能遇到的单元自锁、扭曲、畸变、移动等问题。本文主要做了以下工作：首先利用关于奇异积分高精度计算的最新公式结合积分方程的配置法处理2维散射问题,利用算子分裂方法将光滑边界推广到分段光滑边界问题,本文具体是将光滑边界推广到三角形边界,还可以用类似方法推广到多边形边界；另外考虑到脉冲匹配的精度问题,利用径向基函数逼近表面电流,并采用高精度机械求积公式(MQM)计算涉及到的奇异积分,数值结果表明了这种方法的计算效率。
The Method of moments (MoM) is the one of the most common method used tosolve electromagnetic scattering.the core issue to achieve its high-precision calculation is the calculation of the diagonal elements in its impedance matrix, which involves numerical treatment of the weak/hypersingular integralshese. The MoM may solve some problems related with complex boundary that analytic method is inoperative. As a result, the MoM is broadly used in various electromagnetic researches, such as EMC, microwave net, micro-strip analysis, antenna design, radiation effort and antenna issues, etc.This essay will improve the method of mechanical quadrature (MQM) related with two-dimensional scattering issue as described below:With the latest formula in high-precision calculation of singular integral, and combining the method of dealing with 2-dimensional scattering problem, this paper extends smooth boundary to piecewise smooth boundary problem by using operator splitting method. Adopts the singular integral related to high-precision mechanical quadrature formula (MQM) calculations related to. The numerical results show the computational efficiency of this method. We could obtain improved MQM with better accuracy than it of MQM and MoM with calculation time saving advantages.Taking into account of the accuracy of pulse matching, it approximates surface currentthe by using radial basis function. The interpolation theory of RBF may help to reduce large amount of workload in grid subdivision, and therefore highly improve the efficiency and result accuracy of algorithm.
引文
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