An optimal preconditioning scheme for the discontinuous Galerkin solution of multi-region target scattering problems
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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.
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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. -
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) 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 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 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 -
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