An optimal preconditioning scheme for the discontinuous Galerkin solution of multi-region target scattering problems
-
摘要: 在电磁散射问题中，由均匀介质和金属组合而成的多区域结构目标在天线仿真、雷达成像等工程问题中有着广泛应用. 针对多区域目标的散射问题，研究了不连续伽辽金（discontinuous Galerkin, GD）方法在多区域面积分（surface integral equation, SIE）矩量法中的使用，同时提出了一种优化的距离稀疏预处理（optimized distance sparse preconditioner, O-DSP）方法。该方法根据阻抗矩阵中不同积分算子随距离变化的特性来个性化选择预处理矩阵，进一步增加了预处理矩阵的稀疏性. 数值计算表明，相比之前的距离稀疏预处理方法，优化的预处理矩阵非零元素仅为以前的一半，而且具有相同加速迭代效果.
-
关键词:
- 多区域目标 /
- 面积分(SIE)方程 /
- 预处理技术 /
- 不连续伽辽金(DG)方法 /
- 距离稀疏预处理器(DSP)
Abstract: The electromagnetic scattering from multi-region targets with homogeneous dielectrics and metals are of significant importance in various microwave applications from antenna design to radar imaging. Aiming at the multi-region target scattering problems, the discontinuous Galerkin(GD) solution in the surface integral equation(SIE) method of moment(MoM) is studied, and an optimized distance sparse preconditioner(O-DSP) is proposed. Based on the different matrix characteristics associated with different integrodifferential operators, the proposed preconditioner is constructed by adopting customized sparse strategies, and the sparsity of the preconditioner is further improved. Numerical studies demonstrate that, compared with the conventional DSP, the number of nonzero elements in the proposed precondition matrix is nearly halved, while maintaining similar convergence rate. -
图 1 任意交界表面与其两侧体积区域示意图
Fig. 1 An arbitrary surface with its internal and external regions
图 3 介质球体不同算子矩阵块内元素模值随距离约束的变化
Fig. 3 The variation of modulus value with distance constraint of a dielectric sphere
图 6 K算子矩阵迭代次数和元素数量随距离约束的变化
Fig. 6 The iteration numbers and element numbers of K operator matrix block, as a function of distance constraint
图 8 介质球内嵌不同个数的导体块模型
Fig. 8 Different numbers of conducting spheres embedded in one dielectric sphere
图 9 介质球内嵌不同个数导体块模型的双栈VV-RCS
Fig. 9 The bistatic VV-RCS for the model of multiple conducting spheres embedded in one dielectric sphere
图 10 射频芯片模型的区域分解
Fig. 10 Illustration of subdomain partition scheme of a radio frequency chip model
图 12 使用不同预处理器求解射频芯片模型的迭代收敛情况
Fig. 12 Convergence histories for the radio frequency chip model under different preconditioners
图 13 四旋翼无人机模型的区域分解示意图
Fig. 13 Illustration of subdomain partition scheme of a four-rotor aircraft model
图 15 使用不同预处理器求解四旋翼无人机模型的迭代收敛情况
Fig. 15 Convergence histories for the four-rotor aircraft model under different preconditioners
表 1 不同预处理器的DG方法计算数值效果对比
Tab. 1 Comparison of the n umerical performance of the DG solution with different preconditioner
预处理策略 迭代次数 矩阵元素数 百分比/% $ \mathcal{L} $：ST4，$ \mathcal{K} $：ST4 （NP） 542 - - $ \mathcal{L} $：ST4，$ \mathcal{K} $： ST4 （NFP） 15 19 541 312 100.0 $ \mathcal{L} $：ST2，$ \mathcal{K} $： ST2 （DSP） 14 4 354 944 22.3 $ \mathcal{L} $：ST3，$ \mathcal{K} $：ST3 142 179 136 0.9 $ \mathcal{L} $：ST2，$ \mathcal{K} $： ST4（DS-2BDP） 30 2 177 472 11.1 $ \mathcal{L} $：ST2，$ \mathcal{K} $： ST3（O-DSP） 14 2 213 664 11.3 注：NP为无预处理器（No preconditioner）；DS-2BDP为距离稀疏二对角块预处理器（distance sparse two-partition block-diagonal preconditioner） 
下载: 导出CSV
表 2 不同分解区域数对DSP和O-DSP的迭代次数和元素比率的影响
Tab. 2 Comparison of the iteration numbers and element ratios of the DG solution with DSP and O-DSP for different decomposed spheres
M DSP 迭代次数 O-DSP 迭代次数 元素比/% 4 14 14 50.831 8 17 16 50.815 16 17 17 50.807 32 44 43 50.757
下载: 导出CSV
表 3 导体块数量不同情况下三种预处理器迭代次数和元素个数对比
Tab. 3 Comparison of the iteration numbers and element numbers by 3 different preconditioners with different numbers of conducting spheres
预处理器 导体块数量 矩阵元素数 迭代次数 DSP 2 5 055 045 27 4 5 779 960 23 8 7 287 724 22 DS-2BDP 2 2 615 816 266 4 3 077 832 187 8 4 065 434 221 O-DSP 2 2 651 908 35 4 3 114 024 29 8 4 106 265 38
下载: 导出CSV
表 4 DSP和O-DSP预处理器计算四旋翼无人机迭代次数和矩阵元素数量对比
Tab. 4 Comparison of the iteration numbers and element numbers of the DG solution with DSP and O-DSP for the four-rotor aircraft model
预处理器 迭代次数 矩阵元素数 百分比/% DSP 140 172 320 815 100.0 O-DSP 164 98 643 743 57.2
下载: 导出CSV
-
[1] GLISSON A W, WILTON D R. Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces[J]. IEEE transactions on antennas and propagation,1980,28(5):593-603. DOI: 10.1109/TAP.1980.1142390 [2] HARRINGTON R F. Field computation by moment methods[M]. New York: Macmillan, 1968. [3] RAO S, WILTON D, GLISSON A. Electromagnetic scattering by surfaces of arbitrary shape[J]. IEEE transactions on antennas and propagation,1982,30(1):409-418. [4] YLA-OIJALA P, TASKINEN M, SARVAS J. Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions[J]. Progress in electromagnetics research,2005,52:81-108. DOI: 10.2528/PIER04071301 [5] CHU Y, CHEW W C, ZHAO J, et al. A surface integral equation formulation for low-frequency scattering from a composite object[J]. IEEE transactions on antennas and propagation,2003,51(10):2837-2844. DOI: 10.1109/TAP.2003.817999 [6] LI M K, CHEW W C. Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme[J]. IEEE transactions on antennas propagation,2008,56(8):2389-2397. DOI: 10.1109/TAP.2008.926785 [7] VIPIANA F, FRANCAVILLA M A, VECCHI G. EFIE modeling of high-definition multiscale structures[J]. IEEE transactions on antennas and propagation,2010,58(7):2362-2374. DOI: 10.1109/TAP.2010.2048855 [8] 杨明林, 吴比翼, 黄晓伟, 等. 三维电磁散射问题区域分解技术研究进展[J]. 电波科学学报, 2020, 35(1): 34-42.YANG M L, WU B Y, HUANG X W, et al. The progress of domain decomposition methods for 3D electromagnetic scattering problems, 2020, 35(1): 34-42. (in Chinese) [9] “电磁计算”专刊编委会. 电磁计算方法研究进展综述[J]. 电波科学学报,2020,35(1):13-25.The Editorial Board of Special Issue for “Computational Electromagnetics” . Progress in computational electromagnetic methods[J]. Chinese journal of radio science,2020,35(1):13-25. (in Chinese) [10] GUO L W, CHEN Y P, HU J, et al. A novel JMCFIE-DDM for analysis of EM scattering and radiation by composite objects[J]. IEEE antennas and wireless propagation letters,2017,16:389-392. DOI: 10.1109/LAWP.2016.2580567 [11] ZHAO R, HU J, ZHAO H, et al. EFIE-PMCHWT-based domain decomposition method for solving electromagnetic scattering from complex dielectric/metallic composite objects[J]. IEEE antennas wireless propagation letters,2017,16:1293-1296. DOI: 10.1109/LAWP.2016.2633061 [12] PENG Z, LIM K H, LEE J F. A discontinuous Galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets[J]. IEEE transactions on antennas propagation,2013,61(7):3617-3628. DOI: 10.1109/TAP.2013.2258394 [13] PENG Z, HIPTMAIR R, SHAO Y, et al. Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems[J]. IEEE transactions on antennas and propagation,2016,64(1):210-223. DOI: 10.1109/TAP.2015.2500908 [14] HAN K, CHEN Y, QUE X, et al. A domain decomposition scheme with curvilinear discretizations for solving large and complex PEC scattering problems[J]. IEEE antennas wireless propagation letters,2018,17(2):242-246. DOI: 10.1109/LAWP.2017.2782734 [15] HOU Y, XIAO G, TIAN X. A discontinuous Galerkin augmented electric field integral equation for multiscale electromagnetic scattering problems[J]. IEEE transactions on antennas and propagation,2017,65(7):3615-3622. DOI: 10.1109/TAP.2017.2702716 [16] KONG B B, SHENG X Q. A discontinuous Galerkin surface integral equation method for scattering from multiscale homogeneous objects[J]. IEEE transactions on antennas and propagation,2018,66(4):1937-1946. DOI: 10.1109/TAP.2018.2803133 [17] HUANG X W, SHENG X Q. A discontinuous Galerkin self-dual integral equation method for scattering from IBC objects[J]. IEEE transactions on antennas and propagation,2019,67(7):4708-4717. DOI: 10.1109/TAP.2019.2905670 [18] GAO H W, YANG M L, SHENG X Q. A new SDIE based on CFIE for electromagnetic scattering from IBC objects[J]. IEEE transactions on antennas and propagation,2020,68(1):388-399. DOI: 10.1109/TAP.2019.2940609 [19] CARPENTIERI B, DUFF I S, GIRAUD L, et al. Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations[J]. SIAM journal on scientific computing,2005,27(3):774-792. DOI: 10.1137/040603917 [20] MALAS T, GUREL L. Incomplete LU preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering[J]. SIAM journal on scientific computing,2007,29(4):1476-1494. DOI: 10.1137/060659107 [21] BERTOLO J M, TABOADA J M, LANDESA L, et al. . Near field preconditioner for the fast multipole method[C]// Proceedings of the Fourth European Conference on Antennas and Propagation, Barcelona, 2010: 1-4. [22] KONG B B, HUANG X W, SHENG X Q. A discontinuous Galerkin surface integral solution for scattering from homogeneous objects with high dielectric constant[J]. IEEE transactions on antennas and propagation,2020,68(1):598-603. DOI: 10.1109/TAP.2019.2947139 [23] SONG J M, CHEW W C. Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering[J]. Microwave optical technology letters,1995,10(1):14-19. DOI: 10.1002/mop.4650100107 -
点击查看大图
计量
- 文章访问数: 170
- HTML全文浏览量: 38
- PDF下载量: 20
- 被引次数: 0
