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电磁场B样条间断有限元方法

华沁怡 李林 齐红新

华沁怡, 李林, 齐红新. 电磁场B样条间断有限元方法[J]. 强激光与粒子束, 2024, 36: 073004. doi: 10.11884/HPLPB202436.240076
引用本文: 华沁怡, 李林, 齐红新. 电磁场B样条间断有限元方法[J]. 强激光与粒子束, 2024, 36: 073004. doi: 10.11884/HPLPB202436.240076
Hua Qinyi, Li Lin, Qi Hongxin. B-spline discontinuous Galerkin method for Maxwell’s equations[J]. High Power Laser and Particle Beams, 2024, 36: 073004. doi: 10.11884/HPLPB202436.240076
Citation: Hua Qinyi, Li Lin, Qi Hongxin. B-spline discontinuous Galerkin method for Maxwell’s equations[J]. High Power Laser and Particle Beams, 2024, 36: 073004. doi: 10.11884/HPLPB202436.240076

电磁场B样条间断有限元方法

doi: 10.11884/HPLPB202436.240076
基金项目: 国家重点研发计划(2020YFA0709800);国家自然科学基金项目(12192251、12274134);上海市教委基金项目(2023ZKZD35)
详细信息
    作者简介:

    华沁怡,1098600977@qq.com

    通讯作者:

    齐红新,hxqi@phy.ecnu.edu.cn

  • 中图分类号: O441.4

B-spline discontinuous Galerkin method for Maxwell’s equations

  • 摘要: 在计算电磁学领域,时域间断有限元算法(DGTD)一般基于模型空间的不规则网格划分和单元上高阶多项式插值计算。同样的插值阶数,二维空间四边形网格划分比三角形网格划分具有更少的自由度和更高的计算效率。然而,传统基于等参变换和多项式张量积插值的基函数空间在四边形单元上仅具有低阶完备性,且稳定性和精度受网格畸变影响较大。为此,提出了一种基于不规则四边形网格的高阶B样条插值DGTD方法,用于Maxwell方程的求解。文章采用的B样条基不仅具有高阶多项式空间的插值完备性,而且完全消除了单元内部自由度。此外,Maxwell方程离散系统的各系数矩阵还具有精确的解析形式。使用该方法分析腔体的本征模和楔形体的电磁散射,结果表明,相较于COMSOL软件最大允许时间步长提高2.5倍,计算所需未知量减少25%,证实了本文算法的高稳定性和高精度特点。
  • 图  1  凸四边形单元三次样条域点及插值节点分布

    Figure  1.  Distribution of points and interpolation nodes in cubic spline field of convex quadrilateral element

    图  2  方形谐振腔四边形网格划分

    Figure  2.  Quadrilateral meshing of square resonator

    图  3  不同模式下的本征谱

    Figure  3.  Eigen frequency of different modes

    图  4  模式(3,1)相对误差随网格精度的变化

    Figure  4.  Variation of relative error with mesh size

    图  5  (4,1)模式下$ {E}_{y} $相对误差随网格尺寸的变化

    Figure  5.  Relative error of (4,1) mode with mesh size

    图  6  六种共振模式下$ {E}_{y} $分布

    Figure  6.  Square vacuum cavity: component $ {E}_{y} $ of six resonant modes

    图  7  测量点处数值结果与参考值的误差比较

    Figure  7.  Error comparison of numerical results with reference values at measuring point

    图  8  不同监视点数值结果和参考结果的波形对比

    Figure  8.  Comparison of numerical results and reference results of different monitor points

    图  9  电磁波传播至右侧总场边界时的场分布

    Figure  9.  Distribution of electromagnetic wave propagating to the right boundary of total field

    图  10  不同网格划分对比

    Figure  10.  Comparison of mesh distribution

    表  1  腔体的谐振频率

    Table  1.   Analytical and numerical results for resonant frequencies of different modes

    mode analytical/MHz B-DGTD/MHz error (B-DGTD)/% COMSOL/MHz error (COMSOL)/%
    (1,0) 150.000 150.002 0.0013 149.540 0.3067
    (3,2) 540.833 540.800 0.0061 541.598 0.1414
    (4,1) 618.466 618.400 0.0107 622.269 0.6149
    (4,4) 848.528 848.399 0.0152 847.794 0.0865
    (5,3) 874.643 874.480 0.0186 878.505 0.4416
    (6,2) 948.528 948.800 0.0287 950.451 0.2027
    (7,1) 1060.660 1060.974 0.0296 1062.114 0.1371
    (10,0) 1500.000 1500.481 0.0321 1501.771 0.1181
    (11,0) 1650.000 1650.589 0.0357 1645.462 0.2750
    (11,1) 1656.804 1657.584 0.0471 1665.451 0.5219
    下载: 导出CSV
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出版历程
  • 收稿日期:  2024-03-06
  • 修回日期:  2024-04-07
  • 录用日期:  2024-04-07
  • 网络出版日期:  2024-04-19
  • 刊出日期:  2024-05-31

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