SGER: Nonlinear Diffusion on a Sphere

SGER：球体上的非线性扩散

• 批准号: 0908470
• 负责人: Peter Palffy-Muhoray
• 金额: $ 13.5万
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908470&HistoricalAwards=false
• 关键词: SGER; Nonlinear; Diffusion; Sphere

Palffy-MuhorayDMS-0908470 A wide variety of problems in soft condensed matter physics require the solution of a nonlinear diffusion equation on a sphere. Examples are the phase separation of lipids in giant vesicles, the time evolution of liquid crystalline order, and the distribution of defects that determine the shapes of vesicles. The goal of this project is to develop a simple and effective scheme for the numerical description of nonlinear diffusion on a sphere. The first specific problem to be considered is the solution of the Cahn-Hilliard equation on the unit sphere. The investigators solve the Cahn-Hilliard and related equations numerically, using real space discretization. Numerical simulations are sensitive to details of the underlying lattice. Because it is not possible to construct a regular lattice on a sphere (if the number of points 20), they plan to use a random lattice, obtained by placing points randomly on the surface of the sphere. In devising a strategy towards this goal, three fascinating topics emerge: Voronoi tessellation of the sphere to identity the nearest neighbors of each lattice site; stable and efficient scheme to approximate differential operators; and regularizing the grid. This project provides a new algorithm to identify nearest neighbors among random points on a sphere. This is useful for solving a wide variety of problems, ranging from dominance regions of military bases around the world and the locating of fuel depots to modeling the behavior of vesicles and related biological objects. In addition, it gives insights into numerical stability associated with the approximants of differential operators on random lattices, and regularization strategies for improving random grids. It enables the effective numerical solution of nonlinear diffusion equations on a sphere, and thereby makes possible numerical descriptions of a wide variety of problems in soft condensed matter physics. An important aspect of the project is to get students and postdocs from underrepresented groups actively involved in this exciting research.

PALFFY-MUHORAYDMS-0908470软凝结物理学中的多种问题需要在球体上解决非线性扩散方程。 例如，巨型囊泡中脂质的相分离，液晶级的时间演变以及确定囊泡形状的缺陷分布。 该项目的目的是为无线性扩散在球体上的数值描述开发一个简单有效的方案。 要考虑的第一个特定问题是单位球体上Cahn-Hilliard方程的解决方案。 研究人员使用实际空间离散化以数值方式求解了Cahn-Hilliard和相关方程。 数值模拟对基础晶格的细节敏感。 由于无法在球体上构造常规晶格（如果点20的数量），他们计划使用随机的晶格，该晶格是通过将点随机放置在球体表面上而获得的。 在为这一目标制定策略时，出现了三个引人入胜的主题：沃罗诺伊（Voronoi）的voronoi tessellation tosheration to the Identity是每个晶格遗址最近的邻居的身份；稳定而有效的方案，以近似差异操作员；并规范网格。 该项目提供了一种新算法，以识别球体上随机点之间最近的邻居。 这对于解决各种问题很有用，从世界各地的军事基地的优势地区到燃料库的位置到建模囊泡的行为和相关的生物学对象。 此外，它可以洞悉与随机晶格上差分运算符的近似值相关的数值稳定性，以及改善随机网格的正则化策略。 它实现了球体上非线性扩散方程的有效数值解决方案，从而使软凝块物理学中各种各样的问题都可能描述了数值描述。 该项目的一个重要方面是让积极参与这项激动人心的研究的学生和博士后。