电场积分方程中基于几何自适应聚类树的宽带方向性 $\mathcal{H}^2$-矩阵压缩
Wideband Directional $\mathcal{H}^2$-Matrix Compression for the Electric Field Integral Equation with Geometry-Adaptive Cluster Trees
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中文总结 AI 辅助
本文提出一种支持几何自适应聚类树的宽带方向性H²-矩阵构造方法，通过电尺寸确定方向数、球面斐波那契点构造方向及鲁棒化主元选择，实现低存储和O(N log N)复杂度。
中文摘要 AI 辅助
我们提出了一种用于电场积分方程的高效宽带方向性 $\mathcal{H}^2$-矩阵构造方法，与现有构造不同，该方法不仅支持盒树，还支持几何自适应聚类树。为适应几何自适应聚类树，我们根据每个簇的电尺寸（而非盒树的层级）确定方向数量，使用球面斐波那契点构造方向，并通过角度最近邻映射在簇及其子簇的方向集之间建立层次结构。我们使用不完全自适应交叉近似构造嵌套方向表示，为此引入了一种鲁棒化的树模拟主元选择策略，该策略可防止由特定几何和网格产生的块结构矩阵出现过早收敛。数值结果表明，所提方法达到了预期精度，存储需求不超过基于八叉树的构造，在几何或离散化与八叉树聚类匹配不佳时存储需求显著减少，并且在高频问题中展现出预期的 $\mathcal{O}(N\log N)$ 缩放复杂度。
英文摘要
We present an efficient wideband construction of directional $\mathcal{H}^2$-matrices for the electrical field integral equation that, in contrast to existing constructions, supports not only box trees but also geometry-adaptive cluster trees. To accommodate geometry-adaptive cluster trees, we determine the number of directions from the electrical size of each cluster (instead of the level of a box tree), construct the directions using Spherical-Fibonacci points, and establish a hierarchy between the direction sets of a cluster and its children through an angular nearest-neighbor mapping. We construct the nested directional representation using the incomplete adaptive cross approximation, for which we introduce a robustified tree-mimicry pivoting strategy that prevents premature convergence for block-structured matrices arising from certain geometries and meshes. Numerical results demonstrate that the proposed approach achieves the desired accuracy, requires no more storage than the octree-based construction and substantially less when the geometry or discretization is poorly matched to octree clustering, and exhibits the expected $\mathcal{O}(N\log N)$ scaling for high-frequency problems.
