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.

表面上的连续过程是多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的建模，形状的理解和表征，对物体上纹理的应用和增强以及阿尔茨海默氏病皮质变化的映射。部分微分方程（PDE）是制定数学算法的基本工具，用于在平坦空间和弯曲表面上继续过程。但是，当过程发生在弯曲表面而不是标准的笛卡尔坐标空间上时，使用此类方程式要复杂得多。结果，解决基础方程所需的算法和软件通常对理解，效率低下或根本不可用。我的长期视力是开发一般连续模型的有效算法和软件（涉及标准和差异差异），运算符，约束，积分等）在一般几何（静态或移动，开放或封闭，零件平滑或点平滑或点云）上，以及在某些一般嵌入空间中的任意偶发性）。与此相一致，我们引入并开发了闭合点方法。这种方法具有动态简化复杂问题的优势，使其插值的两个标准问题和持续进化。迄今为止，大多数封闭点方法的工作都集中在某些平滑，移动的表面以及最终用户对方法的实际应用上的PDE的数值近似。在拟议的研究中，我们（i）对原始明确的最接近点方法进行了首次详细分析，（ii）分析和开发了CPM的平行算法和软件，（iii）得出有效的时间进化策略，（iv）将方法扩展到实践兴趣的新流量的近似，以及（v）在此构建新方法，并构建了新的方法。精确有效的研究开发算法和软件的计划很简单，因为它们在利用3D中使用现有的标准算法和软件的同时，尽可能均匀地对不同的持续模型进行解决方案。它提高了方法在当前使用中的效率，对现有方法和新方法的理解进行了分析，并且还发展了无法计算的表面过程的数值近似。它还开发了第一个域分解软件，用于将解决方案并行计算到移动表面上一些最常出现的问题。结果，该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准确性地研究复杂移动表面上的一般连续性过程的新的和现实的模型。