Automated Geometric Modeling and Analysis

自动几何建模和分析

• 批准号: RGPIN-2021-03707
• 负责人: Schneider, Teseo
• 金额: $ 1.75万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741861
• 关键词: Automated; Geometric; Modeling; Analysis

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.

从弹跳球的运动到原子中电子的复杂行为，偏微分方程（PDEs）在我们的世界的数学描述中无处不在。由于它们的多功能性，偏微分方程也被用于无数学科的计算分析；即利用计算机求解PDE，以获得虚拟的模拟行为。例如，偏微分方程的解决方案用于建筑评估结构的稳健性，在医学上用于预测和研究治疗的效果，在生物学上用于计算不可观察的量，在量子物理学中用于描述系统的波态。有限元法是求解偏微分方程最常用的方法，特别是在结构和热分析或流体动力学相关的学科中。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、控制模型和边界条件。通过这种设置，PDE求解器系统应该计算域中每个点的解决方案，并将其呈现给用户。令人惊讶的是，对于现有的商业和开源软件来说，情况并非如此，在这些软件中，需要繁琐、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中获得解决方案，需要花费数周的时间手动删除三角形和移动顶点，以便用四面体填充输入模型。在处理大量模拟时，这种人工过程会带来根本性的问题，近年来，这种做法随着机器学习而变得流行。当处理大型集合时，全自动PDE求解器是必要的；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且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。有了这些基本输入，PDE求解器将生成一个有效、高效和健壮的解决方案。开发这样的管道是一项雄心勃勃的努力，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，在未来五年内，我们计划重点关注以下目标：1)从光滑曲面域生成鲁棒网格；2)制定补偿低质量曲面网格的新准则；3)将碰撞响应扩展到弯曲几何；4)研究流体或流固相互作用中的复杂物理现象；5)在现有封装中集成新的坚固弯曲管道。我的hqp将研究现实世界的问题，并制定切实可行的解决方案；这对工业和研究都是至关重要的。我相信，我的研究与其他领域的交叉将有利于世界各地的其他研究人员，并提高加拿大研究的声望和知名度。