Immersd Finite Element Methods for Interface Problems

用于解决界面问题的 Immersd 有限元方法

• 批准号: 1016313
• 负责人: Tao Lin
• 金额: $ 21万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016313&HistoricalAwards=false
• 关键词: Immersd; Finite; Element; Methods; Interface

Most of the published research results about immersed finite element(IFE) methods deal with 2nd order elliptic interface problems. This project plans to extend the research on developing and analyzing IFE methods for interface problems to more sophisticated partial differential equations such as the Stokes system and the linear elasticity system which are of great importance in many applications of engineering and sciences. The proposed research consists of three key modules which complement each other. The first part is to develop new IFE functions and integrate them into modern finite element techniques for solving Stokes and linear elasticity interface problems. The intent is to find suitable IFE functions locally in interface elements that can handle interface jump conditions required by the interface problems and other conditions such as the inf-sup stability condition required by the finite element formulations to be used. The proposed research will result in efficient and robust IFE methods with an emphasis on DG formulations that simplify local h-, p-, and hp- refinements on Cartesian meshes. The second module is the theoretical analysis of IFEs, starting from their interpolation approximation capabilities and deriving error bounds for IFE solutions. The challenges are that the traditional analysis techniques have a limited use here. For example, using equivalent quotient norm in the scaling argument leads to an estimate for IFE interpolation useless for further deriving estimate of IFE solution unless it can be shown that the constants in the error bounds are independent of the interface. Also, 2D and 3D IFE methods are essentially non-confirming methods whose error estimations are often more complicated. The third module is about the applications of IFE methods. In addition to improving the IFE solver for the particle-in-cell simulator, this project will investigate the applications of IFE methods to be developed to multi-fluid Stokes flow problems and the multi-material shape/topology optimization problems involving the linear elasticity.The IFE methods to be developed in this project can provide new and efficient simulation tools that can use structured/Cartesian meshes to solve challenging interface problems involving multi-scale and multi-physics with nontrivial interfaces in many areas of engineering and science, including flow problems, electromagnetic problems, shape/topology optimization problems, to name just a few. This research project will proceed in harmony with the development of IFE software packages to verify and support the theory, and address complex and realistic problems from a variety of disciplines. This strategy will enable theoretical innovation to become practice much more quickly than traditionally possible. The proposed research projects will have a great potential to impact on numerical simulations in the design/research of ion-propulsion engines for interplanetary deep space travel, optimal packaging of electronic devices, efficient and better image reconstruction in computer tomography, non-destructive/non-invasive detection of suspicious materials in security check, design of optimal shapes for lighter and stronger structures, and many other application areas of great federal interests.

关于沉浸式元素（IFE）方法的大多数已发表的研究结果涉及第二阶椭圆界面问题。该项目计划将开发和分析界面问题的IFE方法的研究扩展到更复杂的部分微分方程，例如Stokes系统和线性弹性系统，这些方程在许多工程和科学应用中都非常重要。拟议的研究由三个关键模块组成，它们相互补充。第一部分是开发新的IFE功能，并将它们集成到解决Stokes和线性弹性接口问题的现代有限元技术中。目的是在接口元素中找到合适的IFE功能，这些功能可以处理接口问题和其他条件所需的接口跳转条件，例如要使用的有限元公式所需的INF-SUP稳定性条件。拟议的研究将导致有效且健壮的方法，重点是简化笛卡尔网格上局部H-，P-和HP-细化的DG制剂。第二个模块是对IFE的理论分析，从其插值近似功能开始，并为IFE解决方案得出误差界限。挑战是，这里的传统分析技术在这里使用有限。例如，在缩放参数中使用等效的标准会导致IFE插值无用的估计值，以进一步衍生IFE解决方案的估计值，除非可以证明误差范围中的常数与接口无关。同样，2D和3D IFE方法本质上是非引起不引人注目的方法，其误差估计通常更为复杂。第三个模块是关于IFE方法的应用。 In addition to improving the IFE solver for the particle-in-cell simulator, this project will investigate the applications of IFE methods to be developed to multi-fluid Stokes flow problems and the multi-material shape/topology optimization problems involving the linear elasticity.The IFE methods to be developed in this project can provide new and efficient simulation tools that can use structured/Cartesian meshes to solve challenging interface problems involving multi-scale and在许多工程和科学领域中具有非平凡界面的多物理学，包括流动问题，电磁问题，形状/拓扑优化问题，仅举几例。该研究项目将与IFE软件包的开发和谐相处，以验证和支持该理论，并解决各种学科的复杂和现实问题。该策略将使理论创新能够比传统上可能更快地成为实践。拟议的研究项目将具有对数值模拟的巨大潜力，可以在ION - 渗流引擎的设计/研究中进行数值模拟，用于行径深空间旅行，电子设备的最佳包装，计算机断层扫描中的有效和更好的图像重建，无孔隙/非侵入性/无创材料的可疑材料在安全性领域中的可疑材料对最佳构造的可疑材料以及最佳构造的功能，以及较强的较强的较强的较强的较强的较强的薄板和较强的型号。