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Convex surface ray tracing based on adaptive cutting surface adjustment under exact normal vector
LI Yaoyao , SU Donglin , LIU Yan , YANG Zhao
School of Electronic and Information Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-05-23; Accepted: 2016-07-07; Published online: 2016-07-27 12:42
Foundation item: National Natural Science Foundation of China (61427803)
Corresponding author. SU Donglin, Tel.:010-82317224, E-mail:sdl@buaa.edu.cn
Abstract: Electrical large targets are difficult to be accurately and analytically expressed and thus it is difficult to use uniform geometrical theory of diffraction (UTD) method for field computation. Aimed at this problem, a novel creeping ray triangular mesh tracing (TM-tracing) algorithm for arbitrary convex surface was proposed. Based on practical engineering triangular mesh and its protocol, a net-like triangular mesh data storage list which meets the rapid multilateral search criteria was designed. A high accuracy normal vector algorithm was proposed to satisfy the tracing requirement. Then a dynamic adjustment of the arc cutting surface fitting tracing method was proposed to realize creeping wave tracing algorithm. Finally, combined with UTD, shadow field value solving algorithm was realized. Aircraft-based ray tracing results show that TM-tracing algorithm can be applied to arbitrary smooth convex surfaces including sphere, cylinder and cone. Tracing speed is 2.8 seconds and deviation is less than 1.61%. It shows that the proposed algorithm has an application value in engineering.
Key words: triangular mesh     uniform geometrical theory of diffraction (UTD)     ray tracing     geodesic     creeping wave

1 相关定义与模型 1.1 射线寻迹问题描述

 图 1 射线寻迹问题描述 Fig. 1 Description of ray tracing problem

UTD理论应用到工程问题中一般分为两步：第1步，射线寻迹；第2步，计算各类射线的并矢绕射系数。其中射线寻迹问题是最为关键的一步。射线寻迹问题的描述，如图 1所示，射线类型包括了直射线、爬行波、反射线、尖端绕射线和边缘绕射线，其中传统UTD方法对于直射线、反射线、边缘绕射线、尖端绕射线研究较多，因为这些射线都是考虑的直线与面或点之间的几何关系，利用费马原理可相对容易地求解。而对于爬行波的求解，则由于曲面的表达式未知而寻迹困难。射爬行波求解问题的核心是求短程线，如图 1所示：从源点RS点到观察点R0点，两点之间只有一条曲线能保证其路径长度为最小值，这条特定的曲线称为短程线，需要在机身一定范围内寻找一组切点[Q1, Q2, …]，之后寻找各个切点沿着掠入射方向到观察点的短程线QiR0，射线RSR0=RSQi+QiR0，因此Qi点及曲线QiR0的求解即射线寻迹的核心问题。

1.2 网格模型的链表设计

 (1)
 (2)
 (3)
 (4)
 图 2 三角网格模型的数学几何表达 Fig. 2 Mathematic and geometrical expression of triangle mesh model
1.3 爬行波射线场模型

 sd1，sd2-入射采样点和出射点；ni-顶点连接的第i个三角片的单位法矢；t1，t2-初始传播方向和出射方向；ρ-射线焦散距离。 图 3 凸曲面爬行波示意图 Fig. 3 Schematic diagram of convex surface creeping wave

 (5)
 (6)

 (7)
 (8)
 (9)

 (10)

 (11)

 (12)

2 射线寻迹算法设计与实现 2.1 爬行波寻迹算法流程

1) 采用式(1)~式(4)，生成求解目标的网格链表矩阵。

2) 计算入射方向下，可能的入射采样点。

 (13)
 (14)

3)  筛选入射采样点。对采样点逐一判断是否存在遮挡，如果存在遮挡，则剔除该点作为采样点的资格。

4)  对入射采样点采用适用于任意凸曲面射线寻迹算法求短程线。

5) 依据法向量和曲率拟合短程线弧长。

2.2 离散点曲面高精度法矢求解

 (15)

 参量 x/m y/m z/m 模长/m 理论 -0.0074 0.1002 0.9877 1 本文算法 -0.0016 0.1222 0.9925 0.99999165

 图 4 复杂结构法矢计算 Fig. 4 Normal vector computation for complex structure
2.3 采样点遮挡判断

 图 5 点源入射时遮挡示意图 Fig. 5 Schematic diagram of shelter from a point source incidence

 (16)

 (17)

 (18)

det[A-C  B-C  R-S]=0时，方程无解，即射线未被遮挡，否则依次求解αβ、1-α-βλ，有任意一个参数不在区间[0，1]内时，则线段未被该面片遮挡，依次循环，完成该采样点的存在性判别。

2.4 切割面自适应寻迹方法

1)  记入射点为S1，加入测地线S′向量，采用(RS-sd1)/|RS-sd1|计算切向传播方向d1

2)  由式(15)求S1点的法向量nS1

3) 计算由d1nS1构成的平面与前向三角形的交点记为S2

4) 采用步骤2)和步骤3)的方法，逐渐调整传播方向和切割面，依次求解测地线上的离散点S3, …, Si, Si+1

 S′2-未修正的传播交点；n′S2-未修正传播交点的法向量。 图 6 切割面调整寻迹方法示意图 Fig. 6 Schematic diagram of cutting surface adjustment tracing method
 (19)
 (20)

3 算法验证及应用

3.1 不可展凸曲面爬行波寻迹精度验证

 图 7 球体上短程线寻迹 Fig. 7 Geodesic tracing on a sphere

 方法 长度/m 偏差/m 误差/% 时间开销/s 理论方法 3.1237 0.0496 1.61 2.8 最小夹角法[5] 3.6925 0.6184 20.11 1.2 本文算法 3.0741

3.2 不可展凸曲面爬行波寻迹收敛性验证

 图 8 寻迹结果收敛性对比 Fig. 8 Tracing result convergence comparison

 寻迹结果 数值 起点 (-1.9904, 0.0581, 0.0791) 初始方向 (1, 2, 2) 结束点 理论方法 (-1.9904, 0.0581, 0.0791) 本文算法 (-1.9672, 0.0381, 0.0238) 最直测地线方法[16] (-1.8932, 0.1692, -0.1424) 长度/m 理论方法 5.141 6 本文算法 5.230 8 最直测地线方法[16] 5.420 1 寻迹时间/s 本文算法 2.85 最直测地线方法[16] 2.75

 图 9 柱体表面、球体表面、锥体表面及复杂结构爬行波寻迹 Fig. 9 Creeping wave tracing on cylinder surface, sphere surface, cone surface and complex structure
3.3 凸曲面寻迹方法在UTD中的应用

 图 10 理想导体圆柱面绕射场求解配置图 Fig. 10 Configuration diagram of diffraction field solve on a perfect electric conductor cylinder surface
 图 11 圆柱阴影区场强值对比 Fig. 11 Comparison of shadow field strength on a cylinder

4 结论

1)  本文所提出的爬行波寻迹方法具有较高的精度和速度，寻迹偏差小于1.61%，1万网格的寻迹速度约为2.8 s。

2)  本文算法可适用于任意数值凸曲面的射线寻迹问题。

 [1] 苏东林, 谢树果, 戴飞, 等. 系统级电磁兼容性量化设计理论与方法[M]. 北京: 国防工业出版社, 2015 : 115 . SU D L, XIE S G, DAI F, et al. The theory and method of quantification design on system-level electromagnetic compatibility[M]. Beijing: National Defense Industry Press, 2015 : 115 . (in Chinese) [2] FU S, ZHANG Y H, HE S Y, et al. Creeping ray tracing algorithm for arbitrary NURBS surfaces based on adaptive variable step euler method[J]. International Journal of Antennas and Propagation, 2015, 2015 : 604861 . [3] JONATHAN R P, ERIC L S. Exact geodesics and shortest paths on polyhedral surfaces[J]. IEEE Transactions on Pathtern Analysis and Machine Intelligence, 2009, 31 (6) : 1006 –1015. DOI:10.1109/TPAMI.2008.213 [4] 王冰切, 苏东林, 张晓雷. 飞机表面绕射射线的寻迹方法[J]. 北京航空航天大学学报, 2007, 33 (7) : 785 –788. WANG B Q, SU D L, ZHANG X L. Discrete ray path tracing on aircraft[J]. Journal of Beijing University of Aeronautics and Astronautics, 2007, 33 (7) : 785 –788. (in Chinese) [5] 陈志贤, 苏东林, 刘焱, 等. 任意曲面上射线的寻迹方法[J]. 北京航空航天大学学报, 2013, 39 (5) : 665 –669. CHEN Z X, SU D L, LIU Y, et al. Ray path tracing on discrete surface[J]. Journal of Beijing University of Aeronautics and Astronautics, 2013, 39 (5) : 665 –669. (in Chinese) [6] WANG N, DU X X, WANG Y, et al. Double diffraction and double reflection in NURBS-UTD method[J]. Microwave and Optical Technology Letters, 2013, 55 (7) : 1549 –1553. DOI:10.1002/mop.27652 [7] CHEN X, HE S Y, YU D F, et al. Geodesic computation on nurbs surfaces for UTD analysis[J]. IEEE Antennas and Wireless Propagation Letters, 2013, 12 : 194 –197. DOI:10.1109/LAWP.2013.2245291 [8] WANG N, LIANG C H, YUAN H B. Calculation of pattern in UTD method based on NURBS modeling with the source on surface[J]. Microwave and Optical Technology Letters, 2007, 49 (10) : 2492 –2498. DOI:10.1002/mop.22727 [9] CHEN X, HE S Y, YU D F, et al. Ray-tracing method for creeping waves on arbitrarily shaped nonuniform rational B-splines surfaces[J]. Journal of the Optical Society of America A, 2013, 30 (4) : 663 –670. DOI:10.1364/JOSAA.30.000663 [10] RUAN Y C, ZHOU X Y, JESSIE Y C, et al.The UTD analysis to EM scattering by arbitrarily convex objects using ray tracing of creeping waves on numerical meshes[C]//IEEE Antennas and Propagation Society International Symposium.Piscataway, NJ:IEEE Press, 2008:1-4. [11] SURAZHSKY V, SURAZHSKY T, KIRSANOV D, et al. Fast exact and approximate geodesics on meshes[J]. ACM Transactions on Graphics, 2005, 24 (3) : 553 –560. DOI:10.1145/1073204 [12] FRANK W. Ray tracing with PO/PTD for RCS modeling of large complex objects[J]. IEEE Transactions on Antennas and Propagation, 2006, 54 (6) : 1797 –1806. DOI:10.1109/TAP.2006.875910 [13] PATHAK P H, BURNSIDE W D, MARHEKA R J. A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface[J]. IEEE Transactions on Antennas and Propagation, 1980, 28 (5) : 631 –642. DOI:10.1109/TAP.1980.1142396 [14] 汪茂光. 几何绕射理论[M]. 西安: 西安电子科技大学出版社, 1994 : 198 -210. WANG M G. Geometrical theory of diffraction[M]. Xi'an: Xidian University Press, 1994 : 198 -210. (in Chinese) [15] 神会存, 周来水. 基于离散曲率计算的三角网格模型优化调整[J]. 航空学报, 2006, 27 (2) : 318 –324. SHEN H C, ZHOU L S. Triangular mesh regularization based on discrete curvature estimation[J]. Acta Aeronautica et Astronautica Sinica, 2006, 27 (2) : 318 –324. (in Chinese) [16] MARTINEZ D, VELHO L, CARVALHO P C. Computing geodesics on triangular meshes[J]. Computers & Graphics, 2005, 29 (5) : 667 –675.

#### 文章信息

LI Yaoyao, SU Donglin, LIU Yan, YANG Zhao

Convex surface ray tracing based on adaptive cutting surface adjustment under exact normal vector

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(12): 2632-2639
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0442