任意光滑凸曲面电磁波绕射建模方法研究
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摘要
随着雷达、通信、电子等工程技术的飞速发展,金属曲面的电磁波绕射在很多工程设计中逐渐成为人们研究的热点,如载体天线的辐射问题,电磁兼容,和雷达探测等。本文基于实际工程背景的需求,对任意光滑凸曲面电磁波绕射建模方法进行了深入的研究。
本文的研究工作主要包括：
1)研究了任意曲面网格剖分、曲面单元参数化拟合描述的方法。考虑到NURBS曲面基函数便于几何建模,而有理Bezier曲面基函数便于数值计算。作者首先运用NURBS曲面对目标进行几何建模,然后利用Cox-De Boor方法将NURBS曲面转换成一系列有理Bezier曲面的组合,并通过Berstein基函数对目标表面进行参数表示。
2)通过差分法求解测地线微分方程,实现了任意曲面单元内部爬行波轨迹(短程线或者测地线)的快速寻迹。通过数值仿真结果的验证,发现了差分法对于任意形状曲面上的爬行波寻迹存在离散误差,并分析了误差存在的原因,为后面的精确寻迹奠定基础。
3)通过龙格-库塔方法求解测地线微分方程,精确实现了任意曲面单元内部爬行波轨迹的寻迹。为提高爬行波寻迹的精度,文中引入4阶龙格-库塔法代替差分法求解测地线微分方程,从而得到高阶精度的测地线微分方程的数值解。
4)研究了基于测地线性质的任意曲面单元内部爬行波轨迹的快速精确寻迹方法。微分几何中规定,曲面上测地线的主法向量处处是曲面的法向量,根据这一性质,结合泰勒级数,实现了任意曲面单元内部的爬行波寻迹。
5)研究了曲面单元之间爬行波轨迹平滑过渡的方法。由于曲面单元之间参数的独立性,造成了多面片曲面目标中曲面单元间参数定义域的不连续。为克服这种不连续给爬行波寻迹造成的影响,作者在两曲面单元的公共边处,引入一参数直线函数处理公共边的爬行波平滑过渡；而在多个曲面单元共有的公共点处,则通过修改Bernstein基函数使得曲面单元间的参数定义域连续,从而实现公共点处爬行波平滑过渡。
6)研究了任意PEC凸曲面目标电磁波绕射建模方法。在任意凸曲面爬行波精确寻迹的基础之上,利用一致性几何绕射理论(UTD)实现任意PEC凸曲面目标电磁波绕射建模。
7)研究了任意阻抗凸曲面目标的爬行波寻迹方法。爬行波寻迹方面,各向同性阻抗曲面与PEC曲面寻迹方法无异。然而对于各向异性阻抗曲面,在进行爬行波寻迹时则需要综合考虑介质材料的光轴与曲面表面外形(曲率半径)两方面的作用。本文借助泛函变分原理得到了各向异性阻抗表面爬行波射线所满足的微分方程。
With the development of engineering technology in radar, communication and electronic technique fields, the diffraction of electromagnetic wave by surfaces become the interest to many researchers, such as radiation problem of carrier antennas, electromagnetic compatibility, radar detection and so on. Based on the requirement from engineering, the modeling method for electromagnetic waves diffraction on arbitrarily shaped smooth surfaces is investigated in this paper.
The main content of this paper includes the following parts.
1) The parametric surfaces is employed to describe the geometry of target, and complex targets are described by a combination of several surface patches. Considering that NURBS format is more efficient for the storage and representation of a model, while Bezier is more stable for the numerical computations, the NURBS patches are subsequently subdivided into a combination of rational Bezier patches using Cox-De Boor algorithm. Then targets are described using Bernstein basis function.
2) The geodesic differential equations are solved using difference method to realize the fast creeping-ray tracing (or geodesic computation) on one single parametric surface patch. According to the numerical results, the creeping-ray tracing method based on difference method may produce discretization error on arbitrarily shaped surface patches. The reason for the discretization error is analyzed in the paper to lay the foundation for the accurate creeping-ray tracing in the further reaserch.
3) In order to realize the accurate creeping-ray tracing on arbitrarily parametric surface patch, the geodesic equations are solved by using Runge-Kutta method. With the aim to raise the accuracy of creeping-ray tracing, the paper employed the4th-order Runge-Kutta method instead of difference method to solve the geodesic equations. In this way, the high-order accurate numerical solution of the geodesic equations can be obtained.
4) A new method is developed based on the property of geodesic in this paper to realize the accurate and efficient creeping-ray tracing on one parametric surface patch. In differential geometry, the principle normal vector of every point on a geodesic curve coin side with the normal vector to the surface which it lays. According to the principle, and together with Taylor series, the accurate and efficient creeping-ray tracing on one arbitrarily shaped parametric surface patch can be performed.
5) The transition method is presented in the paper to deal with the transition of creeping-ray tracing between adjacent parametric surface patches. Since the independence of the parameters between surface patches, the parameters are discontinuous between the surface patches. In order to solve the problem, the author employed a parametric line function to deal with the transition across the common sides between two adjacent surface patches, and improving the format of Bernstein basic function to remove the parameter discontinuity at the common point among several adjacent surface patches, in this way the transition of creeping-ray tracing across the common point can be performed.
6) The simulation method is proposed in this paper for the surface diffraction by arbitrarily shaped PEC convex surface targets. On the base of the accurate creeping-ray tracing for the surface targets with arbitrary shapes, the diffraction by the PEC targets can be analyzed by using the uniform geometric theory of diffraction (UTD).
7) The creeping-ray tracing method for arbitrarily shaped anisotropic convex surface targets is preliminarily studied in this paper. On the isotropical surfaces, the creeping-ray tracing is the same as that on PEC surfaces, but on the anisotropic surfaces, not only the affect from surface shapes (curvature radius) but also that from the optical axis should be considered in the creeping-ray tracing. This paper employed the knowledge of variation calculus to develop the differential equations for the creeping-ray on the anisotropic surfaces.
本文的研究工作主要包括：
1)研究了任意曲面网格剖分、曲面单元参数化拟合描述的方法。考虑到NURBS曲面基函数便于几何建模,而有理Bezier曲面基函数便于数值计算。作者首先运用NURBS曲面对目标进行几何建模,然后利用Cox-De Boor方法将NURBS曲面转换成一系列有理Bezier曲面的组合,并通过Berstein基函数对目标表面进行参数表示。
2)通过差分法求解测地线微分方程,实现了任意曲面单元内部爬行波轨迹(短程线或者测地线)的快速寻迹。通过数值仿真结果的验证,发现了差分法对于任意形状曲面上的爬行波寻迹存在离散误差,并分析了误差存在的原因,为后面的精确寻迹奠定基础。
3)通过龙格-库塔方法求解测地线微分方程,精确实现了任意曲面单元内部爬行波轨迹的寻迹。为提高爬行波寻迹的精度,文中引入4阶龙格-库塔法代替差分法求解测地线微分方程,从而得到高阶精度的测地线微分方程的数值解。
4)研究了基于测地线性质的任意曲面单元内部爬行波轨迹的快速精确寻迹方法。微分几何中规定,曲面上测地线的主法向量处处是曲面的法向量,根据这一性质,结合泰勒级数,实现了任意曲面单元内部的爬行波寻迹。
5)研究了曲面单元之间爬行波轨迹平滑过渡的方法。由于曲面单元之间参数的独立性,造成了多面片曲面目标中曲面单元间参数定义域的不连续。为克服这种不连续给爬行波寻迹造成的影响,作者在两曲面单元的公共边处,引入一参数直线函数处理公共边的爬行波平滑过渡；而在多个曲面单元共有的公共点处,则通过修改Bernstein基函数使得曲面单元间的参数定义域连续,从而实现公共点处爬行波平滑过渡。
6)研究了任意PEC凸曲面目标电磁波绕射建模方法。在任意凸曲面爬行波精确寻迹的基础之上,利用一致性几何绕射理论(UTD)实现任意PEC凸曲面目标电磁波绕射建模。
7)研究了任意阻抗凸曲面目标的爬行波寻迹方法。爬行波寻迹方面,各向同性阻抗曲面与PEC曲面寻迹方法无异。然而对于各向异性阻抗曲面,在进行爬行波寻迹时则需要综合考虑介质材料的光轴与曲面表面外形(曲率半径)两方面的作用。本文借助泛函变分原理得到了各向异性阻抗表面爬行波射线所满足的微分方程。
With the development of engineering technology in radar, communication and electronic technique fields, the diffraction of electromagnetic wave by surfaces become the interest to many researchers, such as radiation problem of carrier antennas, electromagnetic compatibility, radar detection and so on. Based on the requirement from engineering, the modeling method for electromagnetic waves diffraction on arbitrarily shaped smooth surfaces is investigated in this paper.
The main content of this paper includes the following parts.
1) The parametric surfaces is employed to describe the geometry of target, and complex targets are described by a combination of several surface patches. Considering that NURBS format is more efficient for the storage and representation of a model, while Bezier is more stable for the numerical computations, the NURBS patches are subsequently subdivided into a combination of rational Bezier patches using Cox-De Boor algorithm. Then targets are described using Bernstein basis function.
2) The geodesic differential equations are solved using difference method to realize the fast creeping-ray tracing (or geodesic computation) on one single parametric surface patch. According to the numerical results, the creeping-ray tracing method based on difference method may produce discretization error on arbitrarily shaped surface patches. The reason for the discretization error is analyzed in the paper to lay the foundation for the accurate creeping-ray tracing in the further reaserch.
3) In order to realize the accurate creeping-ray tracing on arbitrarily parametric surface patch, the geodesic equations are solved by using Runge-Kutta method. With the aim to raise the accuracy of creeping-ray tracing, the paper employed the4th-order Runge-Kutta method instead of difference method to solve the geodesic equations. In this way, the high-order accurate numerical solution of the geodesic equations can be obtained.
4) A new method is developed based on the property of geodesic in this paper to realize the accurate and efficient creeping-ray tracing on one parametric surface patch. In differential geometry, the principle normal vector of every point on a geodesic curve coin side with the normal vector to the surface which it lays. According to the principle, and together with Taylor series, the accurate and efficient creeping-ray tracing on one arbitrarily shaped parametric surface patch can be performed.
5) The transition method is presented in the paper to deal with the transition of creeping-ray tracing between adjacent parametric surface patches. Since the independence of the parameters between surface patches, the parameters are discontinuous between the surface patches. In order to solve the problem, the author employed a parametric line function to deal with the transition across the common sides between two adjacent surface patches, and improving the format of Bernstein basic function to remove the parameter discontinuity at the common point among several adjacent surface patches, in this way the transition of creeping-ray tracing across the common point can be performed.
6) The simulation method is proposed in this paper for the surface diffraction by arbitrarily shaped PEC convex surface targets. On the base of the accurate creeping-ray tracing for the surface targets with arbitrary shapes, the diffraction by the PEC targets can be analyzed by using the uniform geometric theory of diffraction (UTD).
7) The creeping-ray tracing method for arbitrarily shaped anisotropic convex surface targets is preliminarily studied in this paper. On the isotropical surfaces, the creeping-ray tracing is the same as that on PEC surfaces, but on the anisotropic surfaces, not only the affect from surface shapes (curvature radius) but also that from the optical axis should be considered in the creeping-ray tracing. This paper employed the knowledge of variation calculus to develop the differential equations for the creeping-ray on the anisotropic surfaces.
引文
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