Computational Methods for Simulating Flows and Structures with Complex Dynamic Geometry

水的复杂飞溅和面团的捏合都是液体和固体经历严重变形的常见例子。这种过程无处不在，包括材料形状的快速、戏剧性和频繁的变化，这导致了对能够无缝复制它们的计算机模拟技术的巨大需求。这样的算法将是创建真正身临其境的、物理上逼真的虚拟环境所必需的，这是计算机图形学研究的长期重大挑战。为了实用，这些算法必须同时对困难的输入具有健壮性、物理准确性和计算效率。计算网格，有时称为线框模型，是几十年来在计算机图形学中流行的数字对象和材料的简明表示。尽管它们有许多经过充分研究的优点，但现有算法的局限性阻碍了它们在对经历根本形状变化(即，改变几何图形，特别是拓扑)的材料进行动画制作方面的广泛使用。例如，基于网格的方法往往无法完全处理关键的误差来源，如数值不精确、交点纠缠、网格质量差和复杂的网格编辑；这种缺乏强大的数学保证的方法不可避免地导致脆弱的算法可能无法预测地崩溃。幸运的是，最近的进展使基于网格的严重、几何复杂变形的实用动画更接近实现。这项研究的三个主题利用了这一机会：(I)开发动态演变三角曲面的改进方法；(Ii)研究利用新工具生成任意多面体切割单元网格的流体和固体模拟技术；以及(Iii)推导基于网格的新颖技术，以制作由复杂分支结构或混合维度组件(实体、曲面、曲线)组成的材料的动画，即非流形几何。通过这项研究，一个不同的学生团队将接受跨学科的培训，包括流体和固体力学、计算数学和计算机图形学。这一独特的技能组合在数字媒体(即虚拟和增强现实、视觉效果、游戏开发、广告等)中都非常有价值。以及工程和应用数学的许多领域。拟议的议程将产生开源软件工具，以比目前可用的更高的准确性、健壮性和效率来模拟重要的流体和固体现象，立即为加拿大和全球的数字媒体部门提供更快、更逼真的动画能力。此外，由于将要开发的几何和物理技术的基本性质，它还将为未来在从生物物理到3D打印到地球物理等无数学科中出现高度变形材料的计算研究提供坚实的基础。

The intricate splashing of water and the kneading of bread dough are familiar examples of severe deformations undergone by liquids and solids. The ubiquity of such processes, involving rapid, dramatic, and frequent changes in a material's shape, has led to tremendous demand for computer simulation techniques that can seamlessly reproduce them. Such algorithms will be necessary for the creation of truly immersive, physically realistic virtual environments, a longstanding grand challenge of computer graphics research. To be practically useful, these algorithms must be at once robust to difficult inputs, physically accurate, and computationally efficient. Computational meshes, sometimes called wireframe models, are concise representations of digital objects and materials that have been prevalent in computer graphics for decades. Despite their many well-studied benefits, limitations of existing algorithms have prevented their widespread use in animating materials undergoing radical shape changes (i.e., changing geometry and especially topology). For example, mesh-based approaches have often failed to completely handle key sources of error, such as numerical imprecision, tangled intersections, poor quality meshes, and complex mesh edits; this dearth of strong mathematical guarantees leads inevitably to fragile algorithms that can crash unpredictably. Fortunately, recent advances have brought practical mesh-based animation of severe, geometrically complex deformations closer to being realized. The three themes of this research capitalize on this opportunity by: (I) developing improved methods for dynamically evolving triangulated surfaces; (II) investigating fluid and solid simulation techniques that exploit new tools for generating arbitrary polyhedral cut-cell meshes, and (III) deriving novel mesh-based techniques to animate materials composed of complex branching structures or mixed-dimensional components (solids, surfaces, curves), known as non-manifold geometry. Through this research a diverse student team will receive interdisciplinary training spanning fluid and solid mechanics, computational mathematics, and computer graphics. This unique skill set is highly valuable both in digital media (i.e., virtual and augmented reality, visual effects, game development, advertising, etc.) and many areas of engineering and applied mathematics. The proposed agenda will yield open-source software tools to simulate important fluid and solid phenomena with greater accuracy, robustness, and efficiency than is currently available, immediately providing faster and more realistic animation capabilities to the digital media sector in Canada and globally. Furthermore, because of the fundamental nature of the geometric and physical techniques to be developed, it will also provide a firm foundation for future computational investigations in the countless disciplines where highly deforming materials arise, from biophysics to 3D printing to geophysics and more.