Algorithms for continuum processes on complex, moving surfaces

复杂移动表面上连续过程的算法

• 批准号: RGPIN-2016-04361
• 负责人: Ruuth, Steven
• 金额: $ 1.89万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=681363
• 关键词: Algorithms; continuum; processes; complex; moving

***Partial differential equation (PDE) models for continuum processes arise throughout the applied and natural sciences. There has been a great effort made to develop numerical methods for many important classes of PDEs in one, two or three spatial dimensions. However, a remarkable variety of problems involve differential equations on curved surfaces. Numerical methods for PDEs on curved surfaces are much more complicated than in the standard Cartesian coordinate spaces R2 or R3. A popular class of methods for solving such PDEs are the embedding methods. In recent research, we introduced the closest point method, which is an embedding method that decouples surface geometry from the underlying differential operators. This research is part of the work that led to the 2011 Germund Dahquist Prize. *******The closest point method solves rather general PDEs on smooth, stationary surfaces in an accurate manner, using standard numerical PDE methods in the embedding space. However, many practical problems involve nonsmooth, moving surfaces. In this grant, a number of geometric enhancements to the closest point method will be introduced: (1) Research will be conducted on algorithms for general PDEs on surfaces with folds, junctions and other nonsmooth features, (2) robust methods for approximating PDEs on complex, evolving surfaces will be developed, (3) research on accurate methods for general PDEs on point cloud surfaces will also be carried out. Alongside these geometric improvements, efficiency enhancements will be developed using domain decomposition and optimized time-stepping methods. Analytical research will also be conducted to provide new insight into the convergence of the method, especially in situations where its performance is unexpectedly accurate. Throughout, highly qualified personnel will participate in the development of mathematical software, the analysis of numerical algorithms, and the design of numerical methods to solve PDEs on surfaces. This provides valuable experience for a broad range of careers in academic institutions and technology-related industries.*******The proposed research will seek algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while making strong use of existing techniques and software in 3D. It is anticipated that the methods obtained under this grant will enable researchers and end-users to numerically investigate realistic models of general continuum processes on complex moving surfaces with great accuracy.**** *****

* 连续过程的偏微分方程（PDE）模型出现在整个应用和自然科学中。人们已经做出了很大的努力，以发展的数值方法，许多重要的类偏微分方程在一个，两个或三个空间维。然而，一个显着的各种各样的问题涉及微分方程的曲面。 曲面上偏微分方程的数值方法比标准笛卡尔坐标空间R2或R3中的复杂得多。 求解此类偏微分方程的一类流行方法是嵌入方法。 在最近的研究中，我们引入了最近点方法，这是一种嵌入方法，它将曲面几何从底层微分算子中分离出来。 这项研究是2011年Germund Dahquist奖的一部分。 ******* 最近点方法在嵌入空间中使用标准数值PDE方法，以精确的方式在光滑，静止的表面上求解相当一般的PDE。 然而，许多实际问题涉及非光滑的移动表面。 在本基金中，将介绍一系列对最近点方法的几何增强：（1）将研究具有折叠，连接和其他非光滑特征的表面上的一般偏微分方程的算法，（2）将开发用于逼近复杂，不断变化的表面上的偏微分方程的鲁棒方法，（3）还将研究点云表面上的一般偏微分方程的精确方法。 除了这些几何改进之外，还将使用区域分解和优化的时间步进方法来提高效率。 还将进行分析研究，以提供对该方法收敛性的新见解，特别是在其性能出乎意料地准确的情况下。 在整个过程中，高素质的人员将参与数学软件的开发，数值算法的分析，以及数值方法的设计，以解决表面上的偏微分方程。这为学术机构和技术相关行业的广泛职业生涯提供了宝贵的经验。拟议的研究将寻求准确和高效的算法和软件，但在某种意义上说，它们是简单的，它们尽可能均匀地计算不同连续模型的解决方案，同时充分利用现有的3D技术和软件。 预计根据该补助金获得的方法将使研究人员和最终用户能够以极高的精度对复杂移动表面上的一般连续过程的现实模型进行数值研究。 *****