论文标题
在倍数连接的细分表面上的完整helmholtz分解及其在积分方程中的应用
A Complete Helmholtz Decomposition on Multiply Connected Subdivision Surfaces and Its Application to Integral Equations
论文作者
论文摘要
基于回路细分的等几何分析（IGA）框架中电磁散射的分析长期以来仅限于简单连接的几何形状。无法分析乘数的对象是一个明显的遗漏。在本文中，我们解决了这一挑战。 IGA通过使用相同的基础集来表示两者，从而提供几何和分析之间的无缝集成。特别是，使用细分基集的IgA方法利用了以下事实，即用于表面描述的基础函数几乎到处都是平滑的（具有连续的第二个衍生物）。在简单连接的表面上，这允许定义无差异和无卷发的基集。此套件中缺少的是一个基集，该集合既是无差异和无卷曲，这是对多重连接结构上电流完全分解的必要成分。在本文中，我们使用随机多项式矢量字段对这种缺失的成分进行数值影响。我们表明，此基集在分析上是无差异和无卷曲的。此外，我们表明这些基础恢复了无卷曲，无差，无卷曲和无差异的场。最后，我们使用此基础设置来离散一个条件良好的积分方程，以分析完美传导对象并与其他方法表现出极好的一致性。
The analysis of electromagnetic scattering in the isogeometric analysis (IGA) framework based on Loop subdivision has long been restricted to simply-connected geometries. The inability to analyze multiply-connected objects is a glaring omission. In this paper, we address this challenge. IGA provides seamless integration between the geometry and analysis by using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply-connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free and curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply-connected structures. In this paper, we effect this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these basis recovers curl-free, divergence-free, and curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.
