Computational Methods for Simulating Flows and Structures with Complex Dynamic Geometry

水的复杂飞溅和面团的揉面是液体和固体经历严重变形的常见例子。这种过程的普遍存在，涉及材料形状的快速，戏剧性和频繁的变化，导致了对计算机模拟技术的巨大需求，可以无缝地再现它们。这样的算法对于创建真正沉浸式的、物理上真实的虚拟环境是必要的，这是计算机图形学研究的一个长期的巨大挑战。为了实用，这些算法必须对困难的输入具有鲁棒性，物理上准确，计算效率高。计算网格，有时称为线框模型，是数字对象和材料的简洁表示，在计算机图形学中已经流行了几十年。尽管它们有许多被充分研究的好处，但现有算法的局限性阻止了它们在动画材料中的广泛使用，这些材料经历了根本的形状变化（即，改变几何形状，尤其是拓扑结构）。例如，基于网格的方法通常无法完全处理关键的错误来源，例如数值不精确，纠缠的交叉，质量差的网格和复杂的网格编辑;这种缺乏强有力的数学保证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，最近的进展带来了实际的基于网格的动画严重，几何复杂的变形更接近实现。本研究的三个主题利用这个机会：（一）发展动态演化三角化曲面的改进方法;（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）衍生新的基于网格的技术，以动画化由复杂分支结构或混合维组件组成的材料（固体，曲面，曲线），称为非流形几何。通过这项研究，一个多元化的学生团队将接受跨学科的培训，涵盖流体和固体力学，计算数学和计算机图形学。这种独特的技能集在数字媒体（即，虚拟和增强现实、视觉效果、游戏开发、广告等）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，以比目前更高的准确性，鲁棒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，更逼真的动画功能。此外，由于要开发的几何和物理技术的基本性质，它还将为未来在高度变形材料出现的无数学科中的计算研究提供坚实的基础，从生物物理学到3D打印到生物物理学等等。

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.