Automated Geometric Modeling and Analysis

从弹跳球的运动到原子中电子的复杂行为，偏微分方程(PDE)在我们的世界的数学描述中被普遍使用。由于它们的多功能性，偏微分方程组还被用于无数学科中来计算分析；即使用计算机来求解偏微分方程组以获得虚拟的模拟行为。例如，偏微分方程的解在建筑学中用来评估结构的健壮性，在医学上用来预测和研究治疗的效果，在生物学上用来计算不可观测的量，在量子物理学中用来描述系统的波动状态。有限元方法是求解偏微分方程组最常用的方法，特别是在与结构、热分析或流体力学相关的学科中。一个PDE求解器不需要任何与数学或计算机科学相关的知识。应该要求用户只提供输入域边界、管理模型和边界条件。通过这种设置，PDE求解器系统应该计算域中每个点的解，并将其呈现给用户。令人惊讶的是，这不是现有的商业和开源软件的情况，在这些软件中，需要乏味、不直观和耗时的手动交互才能获得结果。例如，要从给定的输入模型获得解，需要数周的时间手动删除三角形并移动顶点才能用四面体填充输入模型。这种手动程序在处理大量模拟时会带来基本问题，这种做法近年来在机器学习中变得流行起来。在处理大型集合时，需要一个全自动的PDE解算器；手动调整参数并希望获得解决方案是不可想象的。我们的长期目标是开发一个直观和易于使用的分析管道，只需要输入边界、控制方程和边界条件。有了这个必要的输入，PDE求解器将产生一个有效、高效和健壮的解。开发这样的管道是一项雄心勃勃的工作，需要考虑不同的物理行为(例如，机械变形、流体、接触、波)和领域(例如，数学、物理、计算机科学)。因此，在接下来的五年里，我们计划专注于以下目标：1)从光滑的曲线域生成稳健的网格；2)开发新的标准来补偿低质量的曲面网格；3)将碰撞响应扩展到曲线几何图形；4)解决流体中的复杂物理现象或流固相互作用；以及5)将新的稳健的曲线管道集成到现有的封装中。我的HQP将致力于解决现实世界的问题，并开发实用的解决方案；这对行业和研究都是基础。我相信，我的研究与其他领域的交叉将使世界各地的其他研究人员受益，并提高加拿大研究的声望和知名度。

Partial differential equations (PDEs) are ubiquitously used in the mathematical description of our world, from the movement of bouncing balls to the complex behavior of an electron in an atom. Due to their versatility, PDEs are also used in countless disciplines to compute analyses; i.e., to use of a computer to solve the PDE in order to obtain a virtual simulated behavior. For instance, the solution of PDEs is used in architecture to evaluate structural soundness, in medicine to anticipate and study the effects of treatments, in biology to compute non-observable quantities, and in quantum physics to describe the wave state of a system. The finite element method is the most commonly used method to solve PDEs, particularly in disciplines related to structural and thermal analysis or fluid dynamics. A PDE solver should not require any knowledge related to mathematics or computer science. A user should be asked to provide only the input domain boundaries, the governing model, and the boundary conditions. With this setup, a PDE solver system should compute the solution for every point in the domain and present it to the user. Surprisingly, this is not the case for existing commercial and open-source software, where tedious, unintuitive, and time-consuming manual interactions are needed to obtain a result. For instance, to get a solution from a given input model, it takes weeks of manually removing triangles and moving vertices to fill the input model with tetrahedra. Such manual procedures pose fundamental problems when processing a large number of simulations, a practice that became popular in recent years with machine learning. When processing large collections, a fully automated PDE solver is necessary; it is inconceivable to manually tweak parameters and hope to obtain a solution. Our long-term goal is to develop an intuitive and easy-to-use analysis pipeline that requires only the input boundary, the governing equations, and the boundary conditions. With this essential input, the PDE solver will produce a valid, efficient, and robust solution. Developing such a pipeline is an ambitious effort that requires considering different physical behaviors (e.g., mechanical deformation, fluids, contacts, waves) and fields (e.g., mathematics, physics, computer science). Thus, in the next five years, we plan to focus on the following objectives: 1) generate robust meshes from smooth curved domains; 2) develop new criteria to compensate for low-quality curved meshes; 3) extend collision response to curved geometries; 4) address complex physical phenomena in fluids or fluid-structure interaction; and 5) integrate the new robust curved pipeline in existing packages. My HQPs will work on real-world problems and develop practical solutions; this is fundamental for both industry and for research. I believe that my research's intersection with other fields will benefit other researchers worldwide and increase the prestige and visibility of Canadian research.