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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=524118
• 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.

连续过程在应用科学和自然科学中无处不在。由于解析解很少可能实现，因此相应的偏微分方程 (PDE) 模型的准确且高效的算法的实际重要性怎么强调也不为过。人们付出了巨大的努力来开发许多重要类型偏微分方程的数值方法。这些努力的重点是寻找一个、两个或三个空间维度的解决方案。但涉及一般微分方程的问题也出现在二维曲面或一维弯曲细丝上。例如，表面上的偏微分方程可用于在计算机生成的表面上放置纹理，或者增强或恢复扫描表面上损坏的图案。 在物体的机器识别中，表面偏微分方程的解可用于表征物体的形状。 在材料科学中，此类方程已用于检查弯曲表面上材料的相变，而在理论生物学中，表面上的偏微分方程作为骨质疏松症等骨病理学建模的一部分而出现。 尽管偏微分方程在曲面上广泛存在，但仍然需要一种系统方法来有效计算一般曲面上的一般偏微分方程的解。