A Collaborative Hybrid Algorithm of WHIE-FDTD with CPML and Non-Uniform Grid for Multi-Scale Electromagnetic Structures
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摘要: 时域有限差分法(FDTD)是计算电磁学领域最具代表性的时域求解算法，但由于受到时间稳定性条件(CFL)的限制，在求解多尺度问题时往往面临巨大的时间开销。为了充分削弱CFL的影响，提高仿真效率，该文提出了一种基于弱混合隐式-显式时域有限差分法(WHIE-FDTD)的混合算法——NG-WHIE。该方法使用非均匀网格(NG)代替传统的均匀Yee网格剖分，仅在几何尺寸较小的区域设置细网格剖分，并且在细网格和粗网格之间的区域设置线性变化的平滑过渡网格，以减小网格剖分差异引起的误差；并为其引入卷积完美匹配层(CPML)边界条件，有效截断电磁波传播至边界后产生的反射，降低因电磁波反射引入的数值震荡。该文加工测试了一款多输入多输出(MIMO)贴片天线，与FDTD方法、高频结构仿真软件(HFSS)计算结果以及测试结果相比，所提出方法的计算数据表现出高度一致性。结果表明，该文方法具有极高的精确性、鲁棒性，在不显著消耗额外内存的情况下节省了大量时间开销，为多尺度结构的电磁场仿真问题提供了新的解决方案。Abstract:
The finite-difference time-domain (FDTD) method is the most representative time-domain solution algorithm in the field of computational electromagnetics. However, due to the limitations imposed by the Courant-Friedrichs-Lewy (CFL) condition, it often incurs a high computational cost when solving multi-scale problems. To fully mitigate the impact of the CFL and improve simulation efficiency, this paper proposes a hybrid algorithm solution named NG-WHIE based on the weakly hybrid implicit-explicit FDTD (WHIE-FDTD). By replacing the traditional uniform Yee grid partition with non-uniform grids (NG), only fine grid partitions are set in regions with smaller geometric dimensions. Linearly varying smooth transition grids are set in the regions between fine and coarse grids to reduce the error caused by the grid partition differences, and the convolutional perfect matching layer (CPML) boundary condition is introduced for it. It effectively truncates the reflection of electromagnetic waves after they reach the boundary, reducing the numerical oscillations caused by the reflection of electromagnetic waves. This paper processed and tested a multi-input multi-output (MIMO) patch antenna. Compared with the FDTD method, the calculation results of the high-frequency structure simulation software (HFSS) and the test results, the calculated data of the proposed method show high consistency. The results indicate that this method has extremely high accuracy and robustness, and saves a significant amount of time consumption without significantly consuming additional memory. This provides a new solution for electromagnetic field simulation problems of multi-scale structures. Objective The FDTD method is one of the most widely used time-domain numerical techniques in electromagnetic calculations. It is favored because of its direct time-domain solution and its ability to naturally handle broadband and complex electromagnetic problems. However, due to its explicit nature, the conventional FDTD algorithm is subjected to the CFL stability condition, under which the time step is coupled to the smallest spatial grid size in the computational domain. In multi-scale problems, where electrically large regions coexist with fine geometrical details requiring extremely small cells, a prohibitively small global time step is imposed by this constraint. Consequently, excessive computational time is incurred, and simulation efficiency is severely limited. To effectively mitigate the CFL bottleneck, a hybrid algorithm based on the WHIE-FDTD method is proposed in this paper. Methods The WHIE-FDTD is based on Maxwell's equations and employs an alternating explicit-implicit scheme, which overcomes the limitations imposed by CFL. Subsequently, by using different perturbation terms, different directions can be obtained as the fine direction of the electromagnetic field iteration equation. This article takes the y direction as an example. Using NG instead of the traditional uniform Yee grid partitioning enables a more scientific spatial discretization. Different-sized grids are adopted for different simulation areas. Only fine grids are used in the structurally detailed areas, while coarse grids are used in the remaining areas. Linearly varying smooth transition grids are set in the transitional areas to reduce the error introduced by the difference in grid partitioning. By introducing the CPML as the absorbing boundary condition, the problem of infinite propagation of electromagnetic waves in the open domain is truncated within a limited simulation space. At the truncation boundary, the absorbing boundary is used instead of the traditional reflective boundary to effectively suppress boundary reflection and reduce the numerical oscillations caused by electromagnetic wave reflection. Results and Discussions Taking the MIMO patch antenna as the implementation example, the numerical results show that when CFLN = 1, the S-parameter curve and radiation pattern of the WHIE-FDTD method are highly consistent with the simulation results of the FDTD method and the HFSS software, ensuring the accuracy of the WHIE-FDTD method; after introducing the NG technology, the echo loss depth measured by the NG-WHIE method is approximately 3 dB lower than that of the FDTD method, fully verifying that the NG-WHIE method has higher accuracy than the FDTD method; when CFLN = 9, the results measured by the NG-WHIE method show no divergence trend and exhibit extremely high stability. At the same time, the calculation time was reduced by approximately 61.8%, which strongly demonstrates that this method not only ensures calculation accuracy but also offers extremely high computational efficiency and robustness. Conclusions This paper presents a hybrid algorithm, NG-WHIE, based on WHIE-FDTD. By replacing the traditional uniform Yee grid with the NG technology, a more scientific spatial discretization is achieved. By introducing the Convolutionally Perfect Matching Layer (CPML) as the absorbing boundary condition, it effectively suppresses boundary reflections and reduces the numerical oscillations caused by electromagnetic wave reflections. The numerical results show that when CFLN = 1, the S-parameter curve and radiation pattern of the NG-WHIE method are highly consistent with the simulation results from the FDTD method and HFSS software, as well as actual measurements, thereby fully verifying the accuracy of the NG-WHIE method. When CFLN = 9, the results obtained by the NG-WHIE method show no divergence trend and exhibit extremely high stability. At the same time, the computation time has been reduced by approximately 61.8%, demonstrating that this method not only ensures calculation accuracy but also offers extremely high computational efficiency and good robustness. -
表 1 四端口贴片天线详细参数
参数 值(mm) 参数 值(mm) 参数 值(mm) 参数 值(mm) 参数 值(mm) lp1 4.5 wp1 22 lg1 17 wg1 10 W 50 lp2 2.5 wp2 14.5 lg2 3 wg2 4 xp 13 lp3 1.4 wp3 1.6 lg3 1 wg3 4 xg 5 
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表 2 不同方法的计算时间和内存占用
方法 CFLN CPU时间(s) 内存(MB) FDTD - 377.7 3766.1 WHIE-FDTD 1 1324.2 4598.5 NG-WHIE 1 1235.7 4161.3 5 253.7 4161.3 9 144.3 4161.3
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