电磁场B样条间断有限元方法  

B-spline discontinuous Galerkin method for Maxwell’s equations

作　　者：华沁怡 李林 齐红新 Hua Qinyi;Li Lin;Qi Hongxin(State Key Laboratory of Precision Spectroscopy,East China Normal University,Shanghai 200241,China;School of Physics and Electronic Science,East China Normal University,Shanghai 200241,China)

机构地区：[1]华东师范大学精密光谱科学与技术国家重点实验室,上海200241 [2]华东师范大学物理与电子科学学院,上海200241

出　　处：《强激光与粒子束》2024年第7期35-42,共8页High Power Laser and Particle Beams

基　　金：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)。

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。In the field of computational electromagnetics,the discontinuous Galerkin time domain(DGTD)method typically relies on irregular grid partitioning in model space and high-order polynomial interpolation calculations on elements.When comparing two-dimensional spatial quadrilateral mesh partitioning to triangular mesh partitioning at the same interpolation order,quadrilateral meshing offers fewer degrees of freedom and higher computational efficiency.However,traditional basis function spaces,relying on isoparametric transformations and polynomial tensor product interpolation,only possess low-order completeness on quadrilateral elements.Consequently,their stability and accuracy are significantly influenced by grid distortion.To address this challenge,this thesis proposes a high-order B-spline interpolation DGTD method based on irregular quadrilateral meshes for solving Maxwell's equations.The advantage of B-spline interpolation lies in its high-order completeness on irregular elements,effectively eliminating internal degrees of freedom within the elements.Furthermore,the coefficient matrices of the discrete system for Maxwell's equations also possess exact analytical forms.Analyzing the eigenmodes of cavities and the electromagnetic scattering of wedge structures,thus the maximum allowable time step increasing by 2.5 times and reducing the required unknowns by 25%compared to COMSOL software,the proposed algorithm exhibits notable advantages in terms of higher stability and precision.

关 键 词：B样条 间断有限元 瞬态电磁学 高阶精度 畸变网格 

分 类 号：O441.4[理学—电磁学]