Algorithms for approximating continuum processes on surfaces

表面连续过程的近似算法

• 批准号: 227823-2011
• 负责人: Ruuth, Steven
• 金额: $ 2.4万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570327
• 关键词: Algorithms; approximating; continuum; processes; surfaces

Continuous processes are ubiquitous throughout the applied and natural sciences. Because analytical solutions are rarely possible, the practical importance of accurate and efficient algorithms for the corresponding partial differential equation (PDE) models cannot be overemphasized. There has been a great effort made to develop numerical methods for many important classes of PDEs. These efforts focus on finding solutions in one, two or three spatial dimensions. But problems involving general differential equations also arise on two-dimensional curved surfaces or one-dimensional curved filaments. For example, PDEs on surfaces can be used to place a texture on a computer generated surface, or to enhance or restore a damaged pattern on a scanned surface. In machine recognition of objects, the solution of surface PDEs can be used to characterize the shapes of objects. In material science such equations have been used to examine phase change of a material on a curved surface, while in theoretical biology, PDEs on surfaces arise as part of the modeling bone pathologies such as osteoporosis. Despite the widespread occurrence of PDEs on curved surfaces there is still a need for a systematic approach for effectively computing the solution of general PDEs on general surfaces. Our research will continue the development and analysis of algorithms for solving PDEs on surfaces. The focus is on methods which (1) apply to general surfaces, (2) apply to general PDEs, (3) are accurate and efficient, and (4) are simple in the sense that they treat different PDEs in uniform fashion by making use of standard, existing techniques and software. It is anticipated that the methods obtained under this research grant will enable researchers and end-users to numerically investigate realistic models of general surface processes with great accuracy.

在整个应用和自然科学中，连续过程无处不在。由于很少可能进行分析解决方案，因此不能过分强调准确有效的算法对于相应的部分微分方程（PDE）模型的实际重要性。为许多重要类别的PDE类别开发数值方法，已经做出了巨大的努力。这些努力着重于在一个，两个或三个空间维度中找到解决方案。但是涉及一般微分方程的问题也出现在二维弯曲表面或一维弯曲细丝上。例如，表面上的PDE可用于将纹理放在计算机生成的表面上，或在扫描的表面上增强或恢复受损的图案。 在机器识别对象时，表面PDE的解决方案可用于表征对象的形状。 在材料科学中，这种方程已用于检查曲面上材料的相变，而在理论生物学中，表面上的PDE是建模骨骼病理（例如骨质疏松症）的一部分。 尽管PDE在弯曲表面上的广泛发生，但仍需要采用系统的方法来有效计算一般PDE在一般表面上的解决方案。 我们的研究将继续开发和分析以求解表面上PDE的算法。重点是（1）适用于一般表面，（2）适用于一般PDE的方法，（3）是准确有效的，并且（4）在某种意义上很简单，因为它们通过使用标准的现有技术和软件来以统一的方式处理不同的PDE。可以预料，根据本研究赠款获得的方法将使研究人员和最终用户能够以极高的准确性来研究一般表面过程的现实模型。