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

流体相互作用、变形体等真实感物理现象的模拟已成为计算物理和计算机动画不可缺少的组成部分。这种模拟为娱乐业带来了令人惊叹的视觉效果，但也导致了不同领域的新发现，如天体物理学，能源生产或了解全球气候。然而，在计算机上运行这些模拟之前，域的数学表示必须被离散化以使计算误差最小化（即，以获得准确的物理结果）。现代模拟的分辨率不断提高，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完全自动化方法的模拟。计算工具通常需要特定的单元形状，例如六面体，过度约束离散化。另一方面，网格化质量通常通过几何量来测量，这些几何量仅提供与整体仿真性能的有限联系。这项研究展示了一种新的方法。理论数学是用来开发，第一次，一个离散化方案，是明确依赖于模拟产生的标量场的结构。关键的见解是，一个拓扑结构，莫尔斯-斯梅尔（MS）复杂，作为一个自然的四边形分解域的基础上给定的标量场，称为背景函数。背景函数的行为就像一种对模拟中的关键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一致的粗网格，同时将其自身与模拟属性对齐。最后，通过优化和细分，生成细粒度网格，局部适应所需的分辨率。这产生了一个离散化，使用更少的元素更准确地遵循目标模拟。

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.