Advanced Modeling, Numerical Studies and Analysis of Fluid-Structure Interaction Problems

该项目的目的是开发先进的建模和新的数值技术，以有效地执行稳定，精确，和先进的模拟一种类型的动态流固耦合（FSI）问题与可能的大旋转和可变形的弹性结构。流固耦合问题是指流体力与结构变形/旋转相互作用的问题，在空气动力学、生物工程、流体力学等科学和工程领域中有着重要的作用。然而，由于其强非线性耦合和多学科性质，对这些问题的全面研究仍然是一个挑战，因此正确的数学模型方程，以精确地展示FSI的基本特性变得更加重要。对于大多数流固耦合问题，模型方程的解析解是不可能获得的，而实验室实验的范围也受到限制;因此，为了研究流体与固体之间复杂相互作用所涉及的基础物理，数学建模和数值模拟变得更加必要和有前途。由于流固耦合问题数学模型的复杂性，目前的求解技术还远远不能令人满意，因此迫切需要更高效、更鲁棒的数值方法，而这类多物理场流固耦合问题要得到高效、精确的求解还有很长的路要走，该提案将致力于为以下三种方法开发先进的建模和新的数值技术：任意拉格朗日-欧拉（ALE）方法、虚拟区域方法和全欧拉相位场方法。这些方法仍然具有适定性，稳定性和/或收敛性分析的问题，以及一些关键的数值困难所造成的不匹配的网格上的接口，非线性耦合，和不连续/退化系数。本计画的目标就是针对这些困难，发展和分析适当的离散化方案、稳健的迭代方法和高性能的计算技术，以稳定、快速和精确地收敛于流固耦合模型的离散化系统。需要发展的先进的、新颖的模拟和数值计算技术包括：（1）。介绍了一种新的流体中旋转变形弹性结构的流固耦合模型，以及基于ALE方法和特征线法（MOC）的高效数值计算方法。（二）、不可压流体和可压结构流固耦合问题的一种新的虚拟区域方法及其适定性分析。（三）、一种新的稳定隐式格式，用于处理欧拉描述的退化结构动量方程。 新开发的数值技术将立即实施，以丰富我们的内部代码，并验证我们的FSI模型与获得的数值解。人们希望，在这个拟议的项目中开发的有效的数值技术将导致一个到两个数量级的改进，目前现有的流固耦合求解器。工业应用是这项研究的自然成果，因为它在工业中的广泛应用，以及我的合作者与流体力学和生物工程领域的工程师的密切联系。

The purpose of this project is to develop advanced modeling and novel numerical techniques in order to effectively perform stable, precise, and state of the art simulations for a type of dynamic fluid-structure interaction (FSI) problem with a possibly large rotational and deformable elastic structure. Coupled fluid-structure problems, which are characterized by the interaction of fluid forces and structural deformations/rotations, play prominent roles in many scientific and engineering fields such as aerodynamics, bio-engineering and hydrodynamics. Yet, a comprehensive study of such problems remains a challenge due to their strongly nonlinear coupling and multidisciplinary nature, so a correct mathematical model equation to precisely demonstrate the basic characteristics of FSI becomes more important. For most FSI problems, analytical solutions to the model equations are impossible to obtain, whereas laboratory experiments are also limited in scope; thus to investigate the fundamental physics involved in the complex interaction between fluids and solids, mathematical modeling and numerical simulations become more necessary and promising. Because of the complexity of the underlying mathematical model of FSI problems, current solution techniques are still far from being satisfactory, and therefore more efficient and robust numerical techniques are urgently needed.While there is still a long way before such multiphysics FSI problems can be completely solved in an efficient and precise manner, this proposal will be devoted to the development of advanced modeling and novel numerical techniques for the following three methods: Arbitrary Lagrangian-Eulerian (ALE) method, fictitious domain method, and full Eulerian-phase field method. These methods still possess the problems of well-posedness, stability and/or convergence analysis, as well as a number of critical numerical difficulties caused by the mismatched grid on the interface, nonlinear coupling, and discontinuous/degenerate coefficients. The goal of this project is to address these difficulties, develop and analyze proper discretization schemes, robust iterative methods, and high performance computing techniques to solve the discretized system of FSI model in the sense of stable, fast and accurate convergence. The advanced and novel modeling and numerical techniques to be developed include: (1). A newly developed FSI model for a rotational and deformable elastic structure that is immersed in fluid, and its efficient numerical method with an ALE approach and method of characteristics (MOC). (2). A new fictitious domain method for FSI problem with incompressible fluid and compressible structure, and its well-posedness analysis. (3). A new stable implicit scheme for a full Eulerian FSI model with phase field formulation to deal with the degenerate structural momentum equation in Eulerian description. Newly developed numerical techniques will be immediately implemented to enrich our in-house codes and to validate our FSI model with the obtained numerical solutions. It is hoped that the effective numerical techniques developed in this proposed project will result in one to two orders of magnitude of improvement over current existing FSI solvers. Industrial applications are natural outcomes of this research because of its extensive applications in industry, and the close ties of my collaborators with engineers in the fields of hydrodynamics and bio-engineering.