标题： 关于三角曲面的$H^1(Γ, \partial Γ^c)$和$\partial Γ^c \subset \partial Γ$的生成器：全局环路的构造与分类$Γ$ 显示英文标题

标题： Generators of $H^1(Γ, \partial Γ^c)$ with $\partial Γ^c \subset \partial Γ$ for Triangulated Surfaces $Γ$: Construction and Classification of Global Loops

摘要： 给定一个嵌入到流形$\mathbb R^3$中的紧致曲面$\Gamma$并带有边界$\partial \Gamma$，我们的目标是为相对上同调群$H^1(\Gamma, \partial \Gamma^c)$的一组基构造一组代表元，其中$\Gamma^c$是$\partial \Gamma$的一个指定子集。 为了实现这一目标，我们提出了一种新颖的基于图的算法，具有两个关键特性：它适用于不可定向曲面，从而推广了先前的方法，并且其最坏情况下的时间复杂度在网格的边数上是线性的，该网格对 $\mathcal{K}$ 进行三角剖分以得到 $\Gamma$。 重要的是，此算法作为关键的预处理步骤，解决了积分方程公式化中的边界元离散化遇到的低频失效问题。 显示英文摘要

摘要： Given a compact surface $\Gamma$ embedded in $\mathbb R^3$ with boundary $\partial \Gamma$, our goal is to construct a set of representatives for a basis of the relative cohomology group $H^1(\Gamma, \partial \Gamma^c)$, where $\Gamma^c$ is a specified subset of $\partial \Gamma$. To achieve this, we propose a novel graph-based algorithm with two key features: it is applicable to non-orientable surfaces, thereby generalizing previous approaches, and it has a worst-case time complexity that is linear in the number of edges of the mesh $\mathcal{K}$ triangulating $\Gamma$. Importantly, this algorithm serves as a critical pre-processing step to address the low-frequency breakdown encountered in boundary element discretizations of integral equation formulations.