Efficient algorithms for evolving continuum processes on curved surfaces

曲面上演化连续过程的高效算法

• 批准号: RGPIN-2022-03302
• 负责人: Ruuth, Steven
• 金额: $ 3.5万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758573
• 关键词: Efficient; algorithms; evolving; continuum; processes

Continuum processes on surfaces are essential components to a remarkable variety of modern applications.  These include the physics-based modelling of computer-animated objects, the understanding and characterization of shape, the application and enhancement of texture on objects, and the mapping of cortical change in Alzheimer's disease.  Partial differential equations (PDEs) are the fundamental tools for formulating mathematical algorithms for continuum processes on flat spaces and curved surfaces.  However, working with such equations is much more complicated when the processes occur on curved surfaces rather than on standard Cartesian coordinate spaces.  As a consequence, the algorithms and software needed to solve the underlying equations are often poorly understood, inefficient, or simply unavailable. My long term vision is the development of efficient algorithms and software for general continuum models (involving standard as well as degenerate differential operators,  constraints, integrals, etc.) on general geometries (static or moving, open or closed, piecewise smooth or point cloud, and of arbitrary co-dimension within some general embedding space).  Consistent with this, we have introduced and developed closest point methods.  Such methods have the advantage of dramatically simplifying complex problems into the two standard problems of interpolation and continuum evolution.  To date, most work on closest point methods has focused on the numerical approximation of PDEs on certain smooth, moving surfaces and on the practical application of the method by end-users.  In the proposed research, we (i) conduct the first detailed analysis of the original explicit closest point method, (ii) analyze and develop parallel algorithms and software for the CPM, (iii) derive efficient time-evolution strategies, (iv) extend methods to the approximation of new flows of practical interest, and (v) construct maps between surfaces thereby enabling new, efficient methods for the processing of surfaces. The program of research develops 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 leveraging the use of existing standard algorithms and software in 3D. It improves the efficiency of methods in current use, conducts analysis for the improved understanding of existing and new methods, and enables the numerical approximation of surface processes that cannot presently be computed. It also develops the first domain decomposition software for the parallel computing of solutions to some of the most frequently occurring problems on moving surfaces.  As a consequence, the methods and software developed under this grant will enable researchers and end-users to numerically investigate new and realistic models of general continuum processes on complex moving surfaces with high accuracy.

表面连续过程是各种现代应用的重要组成部分。  这些包括基于物理的计算机动画对象建模、形状的理解和表征、对象纹理的应用和增强，以及阿尔茨海默病中皮质变化的映射。  偏微分方程 (PDE) 是制定平面空间和曲面连续介质过程数学算法的基本工具。  然而，当过程发生在曲面上而不是标准笛卡尔坐标空间上时，处理此类方程要复杂得多。  因此，求解基础方程所需的算法和软件通常难以理解、效率低下或根本不可用。我的长期愿景是为通用连续体模型（涉及标准和简并微分算子、约束、积分等）开发有效的算法和软件，适用于通用几何形状（静态或移动、开放或封闭、分段平滑或点云以及某些通用嵌入空间内的任意共同维度）。  与此相一致，我们引入并开发了最近点方法。  此类方法的优点是可以将复杂问题极大地简化为插值和连续演化两个标准问题。  迄今为止，大多数有关最近点方法的工作都集中在某些光滑、移动表面上的偏微分方程的数值近似以及最终用户对该方法的实际应用。  在拟议的研究中，我们（i）对原始显式最近点方法进行了首次详细分析，（ii）分析和开发CPM的并行算法和软件，（iii）导出有效的时间演化策略，（iv）将方法扩展到实际感兴趣的新流的近似，以及（v）在表面之间构建映射，从而为表面处理提供新的、有效的方法。该研究项目开发的算法和软件既准确又高效，同时又尽可能统一地计算不同连续体模型的解决方案，同时利用现有的 3D 标准算法和软件，因此很简单。它提高了当前使用的方法的效率，进行分析以加深对现有方法和新方法的理解，并能够对目前无法计算的表面过程进行数值近似。它还开发了第一个域分解软件，用于并行计算移动表面上一些最常见问题的解决方案。  因此，在这笔赠款下开发的方法和软件将使研究人员和最终用户能够以高精度数值研究复杂移动表面上的一般连续过程的新的和现实的模型。