Divergence-Free Hybridizable Discontinuous Galerkin Methods for the Incompressible Navier-Stokes Equations on Moving Domains and Their Application to Fluid-Structure Interaction

流固耦合（FSI）问题可以出现在许多工程领域，并且是许多系统设计中的关键考虑因素，例如航空航天工业中飞机机翼的稳定性和响应，生物医学应用中动脉中的血液流动，以及土木工程中桥梁和高层建筑对风的响应。这些问题往往太复杂，无法用解析的方法解决，所以必须用数值模拟的方法来分析。高效、准确、鲁棒的数值方法解决FSI问题对这些应用至关重要。在这个项目中，PI将集中精力开发、分析和实现高阶强质量守恒有限元方法，用于运动域上的不可压缩流动问题，并将其应用于FSI问题。本项目为研究生提供研究训练机会。PI将(1)使用任意拉格朗日-欧拉（ALE）框架开发一种新的无散度杂交不连续Galerkin （div-free-HDG）格式，用于运动域上的不可压缩Navier-Stokes （INS）方程；(2)将ALE-div-free-HDG流体求解器扩展到模拟不可压缩流体与可压缩或不可压缩超弹性结构相互作用的运动域FSI问题。固定网格INS的无跳水hdg方案具有高阶精度、全局和局部守恒性、能量稳定性、压力鲁棒性、数值耗散最小和计算效率高等特点。将无俯冲hdg方案推广到ALE框架下的运动域问题，由于其中使用了散度一致的速度有限元空间，因此提出了进一步的理论和数值挑战。PI将使用一种新的无俯冲hdg方案的替代公式，该方案不需要速度有限元空间的散度一致性，可以有效地适应用于移动域INS和FSI问题的ALE框架。这项工作将是第一次系统地研究ALE强质量守恒有限元方法在移动域惯性系统和FSI问题上的应用。对这些方法进行了严格的稳定性分析，并对其收敛性进行了研究。将研究进一步提高这些方案效率的各种策略。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluid-structure interaction (FSI) problems can occur in many fields of engineering, and are a crucial consideration in the design of many systems, such as stability and response of aircraft wings in the aerospace industry, flow of blood through arteries in biomedical applications, and response of bridges and tall buildings to winds in civil engineering. These problems are often too complex to solve analytically and so they have to be analyzed by means of numerical simulations. Efficient, accurate, and robust numerical methods to solve FSI problems are of fundamental importance to these applications. In this project, the PI will concentrate his efforts on the development, analysis, and implementation of high-order strongly mass-conserving finite element methods for incompressible flow problems on moving domains and their application to FSI problems. This project provides research training opportunities for graduate students.The PI will (1) develop a novel divergence-free hybridizable discontinuous Galerkin (div-free-HDG) scheme for the incompressible Navier-Stokes (INS) equations on moving domains using the Arbitrary Lagrangian-Eulerian (ALE) framework, (2) extend the ALE-div-free-HDG fluid solver to moving-domain FSI problems that model the interaction of incompressible fluids with compressible or incompressible hyperelastic structures. The div-free-HDG scheme for INS on fixed meshes enjoys features such as higher-order accuracy, global and local conservation properties, energy-stability, pressure robustness, minimal amount of numerical dissipation and computational efficiency. The extension of the div-free-HDG scheme to moving domain problems within the ALE framework poses further theoretical and numerical challenges due to the divergence-conforming velocity finite element space used therein. The PI will use a novel alternative formulation of the div-free-HDG scheme, which does not require the divergence conformity of the velocity finite element space, that can be efficiently adapted to the ALE framework for moving domain INS and FSI problems. This work will be the first systematic study of ALE strongly mass-conserving finite element methods for moving domain INS and FSI problems. Rigorous stability analysis will be derived for the methods and their convergence properties will be investigated. Various strategies to further improve the efficiency of these schemes will be studied.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.