Solving two and three dimensional Interface Problems using Non-body-fitted Grids

使用非贴身网格解决二维和三维界面问题

• 批准号: 1317994
• 负责人: Songming Hou
• 金额: $ 12万
• 依托单位: Louisiana Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1317994&HistoricalAwards=false
• 关键词: Solving; two; three; dimensional; Interface

A typical elliptic interface problem is cast as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled with interface conditions, such as jumps in the solution and in the flux across the interface. In many situations, such as when the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture the complex geometry of the interface and the jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. The PI proposes a non-traditional finite element method to solve the elliptic and elasticity interface problems. The following are the proposed projects: 1. solving 3D elliptic interface problems; 2. solving 3D elasticity interface problems; 3. solving 3D multi-domain interface problems; 4. solving 2D multi-domain elasticity interface problems; 5. solving moving interface problems; 6. interdisciplinary applications. Interface problems have applications in electromagnetics, material science, fluid dynamics and other areas. For example, the interface could be the molecular surface of a protein, the water-oil interface, or similar. Therefore, by improving numerical techniques used to model interfaces, the proposed research will have substantial impact on interdisciplinary applications. Further, although the research requires advanced techniques, the concept of an interface exists in everyday life, which means that student interest can be stimulated in applied mathematics seminars.

典型的椭圆界面问题被施放为分段定义的椭圆形偏微分方程（PDE），在不同区域与接口条件相结合，例如在解决方案中的跳跃和跨界面的通量中的跳跃。在许多情况下，例如界面移动时，挑战是如何在不生成身体拟合网格的情况下准确，健壮，有效地解决此类问题。关键问题是如何在固定网格上有效地捕获接口的复杂几何形状和跨接口上的跳跃条件，而接口与网格没有对齐，并且PDE在接口上无效。 PI提出了一种非传统有限元方法来解决椭圆形和弹性界面问题。以下是拟议的项目：1。解决3D椭圆界面问题； 2。解决3D弹性接口问题； 3。解决3D多域界面问题； 4。解决2D多域弹性接口问题； 5。解决移动接口问题； 6。跨学科申请。界面问题在电磁，材料科学，流体动力学和其他领域中应用。例如，界面可以是蛋白质，水油界面的分子表面，也可以是类似的。因此，通过改进用于建模界面的数值技术，拟议的研究将对跨学科应用产生重大影响。此外，尽管研究需要先进的技术，但在日常生活中存在界面的概念，这意味着可以在应用数学研讨会中刺激学生的兴趣。