EAGER (G&V): Exploring Morse Theoretic Tools for Automatic Mesh Generation and Simulation on Surfaces

摘要模拟真实的物理现象，如流体相互作用和变形体，已经成为计算物理和计算机动画中不可或缺的一部分。这样的模拟为娱乐业带来了令人惊叹的视觉效果，但也导致了在不同领域的新发现，如天体物理学、能源生产或理解全球气候。然而，在计算机上运行这些模拟之前，域的数学表示必须离散化，以便最小化计算误差(即，获得准确的物理结果)。现代模拟分辨率的提高使得这一问题变得越来越重要，特别是对于需要定期重新划分网格或需要全自动方法的模拟。目前在离散化和计算耦合方面的实践存在明显的弱点。计算工具通常需要特定的单元形状，例如六面体，这过度约束了离散化。另一方面，网格化质量通常由几何量来衡量，这些几何量仅提供与整体仿真性能的有限连接。这项研究正在展示一种新的方法。理论数学首次被用来开发一种明显依赖于模拟产生的标量场结构的离散化方案。关键的洞察是，一种拓扑结构，莫尔斯-斯梅尔(MS)复合体，充当基于给定标量场的域的自然四边形分解，称为背景函数。后台功能用作对来自模拟的密钥信息进行编码的机制。然后，MS复合体充当与输入域在几何上重合的粗网格，同时将其自身与模拟属性对齐。最后，通过优化和细分，生成局部适应分辨率的细粒度网格。这产生了使用较少元素更准确地跟踪目标模拟的离散化。

AbstractThe simulation of realistic physical phenomena, such as fluid interactions and deformable bodies, has become an indispensable part of both computational physics and computer animation. Such simulations produce stunning visual effects for the entertainment industry but also lead to new discoveries in diverse fields, such as astrophysics, energy production, or understanding the global climate. However, prior to running these simulations on a computer, the mathematical representation of the domain must be discretized in order to minimize computational errors (i.e., to obtain accurate physical results). The increased resolution of modern simulations is making this an increasingly important issue, especially for simulations requiring periodic remeshing or necessitating a fully automated approach.Current practice in the coupling of discretization and computation has significant weaknesses. The computational tools often demand specific element shapes, e.g. hexahedra, over-constraining the discretization. On the other hand, meshing quality is generally measured by geometric quantities that provide only a limited connection to overall simulation performance. This research is demonstrating a new approach. Theoretical mathematics is used to develop, for the first time, a discretization scheme that is explicitly dependent on the structure of scalar fields generated by the simulation. The key insight is that a topological structure, the Morse-Smale (MS) complex, acts as a natural quadrilateral decomposition of a domain based on a given scalar field, called a background function. The background function behaves as a mechanism for encoding key information from a simulation. The MS complex then acts as a coarse mesh that coincides geometrically with the input domain while aligning itself with simulation properties. Finally, through optimization and subdivision, fine-grained meshes are generated that adapt locally to the resolution needed. This produces a discretization that more accurately follows the target simulations using fewer elements.