Topics of Immersed Finite Element Methods

浸入式有限元方法主题

• 批准号: 2005272
• 负责人: Xu Zhang
• 金额: $ 4.98万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005272&HistoricalAwards=false
• 关键词: Topics; Immersed; Finite; Element; Methods

Interface problems are ubiquitous. When simulations involve multiple materials or multi-physics, interface problems arise. Many real-world problems in fluid mechanics, material science, mechanical engineering, and biomedical engineering are modeled by three-dimensional interface problems. The immersed finite element methods (IFEM) are a class of numerical methods for solving interface problems on interface-unfitted meshes. Two interrelated problems will be investigated in this research project. The first problem aims to design a self-adaptive IFEM based on the a posteriori error estimation. The second problem focuses on development, implementation, and analysis of three-dimensional IFEM.The first problem concerns the study of both residual-based and recovery-based error estimation for various immersed finite element discretizations. These include the immersed finite element approximation in conforming, nonconforming and discontinuous Galerkin frameworks. Rigorous mathematical analysis will be carried out for the reliability and efficiency error estimates of IFEM. The second problem focuses on interface problems of three spatial dimensions. It aims to develop an innovative approach to efficiently construct the three-dimensional immersed finite element functions. These immersed finite element functions will be implemented in various numerical schemes for three-dimensional interface problems. Theoretically, both a priori and a posteriori error estimates will be conducted for new IFEM schemes. Computationally, a three-dimensional IFEM software package will be developed with the feature of adaptive mesh refinement.

界面问题无处不在。当模拟涉及多种材料或多物理学时，会出现接口问题。流体力学，材料科学，机械工程和生物医学工程中的许多现实世界问题都是由三维界面问题建模的。沉浸式有限元方法（IFEM）是一类数值方法，用于在接口未拟合网格上求解接口问题。该研究项目将研究两个相互关联的问题。第一个问题旨在根据A后验误差估计设计自适应IFEM。第二个问题的重点是对三维IFEM的开发，实施和分析。第一个问题涉及对各种沉浸式有限元离散化的基于剩余和基于恢复的误差估计的研究。这些包括构象，不合格和不连续的盖尔金框架中的沉浸式有限元近似。将对IFEM的可靠性和效率误差估计进行严格的数学分析。第二个问题的重点是三个空间维度的界面问题。它旨在开发一种创新的方法来有效构建三维浸入式有限元函数。这些沉浸式有限元功能将在各种数值方案中实现，以解决三维界面问题。从理论上讲，将针对新的IFEM方案进行先验和后验错误估计。在计算上，将开发一个三维IFEM软件包，并具有自适应网状精炼的特征。

项目成果

A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

• DOI: 10.3934/era.2021032
• 发表时间: 2021
• 期刊: Electronic Research Archive
• 影响因子: 0.8
• 作者: Derrick Jones;Xu Zhang

A class of nonconforming immersed finite element methods for Stokes interface problems

• DOI: 10.1016/j.cam.2021.113493
• 发表时间: 2021
• 期刊: J. Comput. Appl. Math.
• 作者: Derrick Jones;Xu Zhang

A P2-P1 PARTIALLY PENALIZED IMMERSED FINITE ELEMENT METHOD FOR STOKES INTERFACE PROBLEMS

• 发表时间: 2021
• 作者: Yuan Chen