Novel Ideas and Analysis for Interface and Fluid-Structure Interaction Problems and Applications

研究界面和流固相互作用问题，例如涉及油和水混合物、气泡、冰和水界面、肿瘤生长或细胞变形的问题，具有许多实际应用。在这样的系统上进行实验通常是昂贵的，而计算模拟为研究这些具有挑战性的问题提供了另一种选择。本研究项目的目标是建立这样的系统模型，并设计有效和实用的计算算法来模拟、解决和控制这些问题。新开发的增强浸入界面方法将应用于多相流和流固耦合的几个重要应用。研究者将开发和传播实现这些方法的软件包。研究生将参与研究。本项目涉及基于结构网格的界面和流固相互作用问题的一些新思路的发展和分析。正在开发的方法将得到严格的数学分析和数值实验的支持。应用包括界面问题的最优控制、组织力学模型中与细胞生物学耦合的流固相互作用以及裂纹尖端的扩展。这些方法基于结构化网格，例如笛卡尔网格，这些网格不一定与二维和三维的接口对齐。项目包括：(1)用Stokes方程或Navier-Stokes方程模拟的流体流动与用Darcy定律模拟的多孔介质之间的流体结构相互作用的新增强方法；(2)一种具有全局精确解的边界或界面精确梯度计算新框架；(3)一种新的二阶对称、一致、无参数浸入有限元法；(4)非齐次跳跃条件下椭圆界面问题的一种新的无svd增广IFE方法；(5)不同层（多孔介质、Stokes流、Navier-Stokes流）和尺度下细胞变形新方法的应用；(6)基于Mumford-Shah最小值法的裂纹扩展数值模拟。该项目将对教育产生积极影响，吸引研究生和博士后研究人员在该领域进行研究。项目的一些组成部分将被设计为本科项目。该项目将产生对计算科学有用的软件，涉及不连续和奇点，自由边界/移动界面，多相和多物理，以及不规则域问题。

Studying interface and fluid-structure interaction problems, such as those involving oil and water mixtures, gas bubbles, ice and water interfaces, tumor growth, or cell deformation, has many practical applications. It is often costly to carry out experiments on such systems, and computational simulation provides an alternative for study of these challenging problems. The objectives of this research project are to model such systems and to design efficient and practical computational algorithms to simulate, solve, and control those problems. Newly developed augmented immersed interface methods will be applied to several important applications in multi-phase flows and fluid-structure interactions. The investigator will develop and disseminate software packages implementing these methods. Graduate students will be involved in the research. This project concerns the development and analysis of some new ideas for interface and fluid-structure interaction problems based on structured meshes. The methods under development will be supported by rigorous mathematical analysis and numerical experiments. Applications include optimal control of interface problems, fluid-structure interactions in modeling tissue mechanics coupled with cell biology, and tip propagation of a crack. The methods are based on structured meshes such as Cartesian meshes that are not necessarily aligned with the interface in two and three dimensions. The projects include: (1) new augmented methods for a fluid structure interaction between a fluid flow modeled by Stokes or Navier-Stokes equations and a porous medium modeled by the Darcy's law; (2) a new computational framework for accurate gradient computation on a boundary or interface with accurate solution globally; (3) a new second-order symmetric, consistent, and parameter-free immersed finite element method; (4) a new SVD-free augmented IFE method for elliptic interface problems with non-homogeneous jump conditions; (5) applications of the new methods for cell deformation in different layers (porous media, Stokes flow, Navier-Stokes flow) and scales; and (6)numerical simulations of crack propagation based on the Mumford-Shah minimizer. This project will have positive effect on education by attracting graduate students and postdoctoral researchers to conduct research in this area. Some components of the project will be designed as undergraduate projects. The project will produce software useful to computational science involving discontinuities and singularities, free-boundary/moving interfaces, multi-phase and multi-physics, and irregular domain problems.