Stability analysis and improvement of conformal leapfrog alternating direction implicit finite-difference time-domain method
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摘要: 提出了一种基于共形网格技术的共形单步交替方向隐式时域有限差分(CLeapfrog ADI-FDTD)方法。与常规FDTD方法相比，此方法能够减小由于目标边界不契合网格划分而引入的阶梯近似误差，提高算法计算不规则目标时的精度；同时算法稳定性更强，计算效率更高。由于引入共形技术后显著降低了原差分法的无条件稳定性，本文利用增长矩阵本征值方法理论分析了算法的稳定性，然后采用了一种改进的共形面积计算方法，在此基础上提出了一种稳定性更高的改进的共形单步交替方向隐式时域有限差分(ICLeapfrog ADI-FDTD)方法。数值算例验证了ICLeapfrog ADI-FDTD是一种具有高稳定性和高精度的高效算法。
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关键词:
- 共形网格 /
- 时域有限差分 /
- Leapfrog ADI-FDTD /
- 无条件稳定 /
- 稳定性改进
Abstract: A conformal leapfrog alternating direction implicit finite-difference time-domain (CLeapfrog ADI-FDTD) method based on conformal technique was proposed in the article. Compared with the conventional FDTD method, the proposed method decreased the step approximation error, it was used to simulate the irregular object whose boundary couldn't match the orthogonal grid; at the same time, this method could have a high efficiency because leapfrog alternating direction implicit finite-difference time-domain (Leapfrog ADI-FDTD) is a method with unconditional stability. However, CLeapfrog ADI-FDTD method may lose the stability expected with the Leapfrog ADI-FDTD schemes, and instability factor in CLeapfrog ADI-FDTD was analyzed through eigenvalue of the growth matrix, then a new method named improved conformal leapfrog alternating direction implicit finite-difference time-domain (ICLeapfrog ADI-FDTD) with a modified conformal technique was proposed, which could improve the stability without losing the calculation accuracy. The accuracy and efficiency of the proposed ICLeapfrog ADI-FDTD method were verified by numerical results. -
图 1 计算目标边界上的两类共形网格
Figure 1. Two kinds of conformal grids in the boundary of computational target
图 3 CLeapfrog ADI-FDTD方法中p与 |λ|max的关系
Figure 3. Relationship between p and |λ|max in CLeapfrog ADI-FDTD method
图 5 CLeapfrog ADI-FDTD和ICLeapfrog ADI-FDTD中a与p的关系
Figure 5. Relationship between a and p in CLeapfrog ADI-FDTD and ICLeapfrog ADI-FDTD methods
图 6 无限长圆柱划分网格中的非稳定网格
Figure 6. Instability grids in division of infinite- long perfectly conducting cylinder
图 7 CLeapfrog ADI-FDTD方法和ICLeapfrog ADI-FDTD方法的稳定性对比
Figure 7. Stability comparison of CLeapfrog ADI-FDTD and ICLeapfrog ADI-FDTD methods
图 8 无限长金属圆柱双站RCS
Figure 8. Bistatic radar cross section(RCS) of infinite-long perfectly conducting cylinder
表 1 不同时间步长下共形方法的增长矩阵模的最大值
Table 1. Maximum eigenvalue in the growth matrix at different time step
C CLeapfrog ADI-FDTD ICLeapfrog ADI-FDTD 0.1 1.025 1.065 0.5 1.129 1.371 1.0 1.276 1.909 1.5 1.444 2.717 2.0 1.637 3.903 3.0 2.114 7.551 4.0 2.740 12.700 
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