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)问题存在于许多工程领域，在许多系统的设计中都是一个重要的考虑因素，例如航空航天工业中飞机机翼的稳定性和响应，生物医学应用中血液在动脉中的流动，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，无法解析解决，因此必须通过数值模拟的方式进行分析。求解FSI问题的高效、准确和稳健的数值方法对这些应用是至关重要的。在这个项目中，PI将致力于运动区域上不可压缩流动问题的高阶强质量守恒有限元方法的开发、分析和实现，以及它们在FSI问题中的应用。该项目为研究生提供了研究培训的机会。PI将(1)利用任意拉格朗日-欧拉(ALE)框架发展一种新的运动区域上不可压缩Navier-Stokes(INS)方程的无散度杂交间断Galerkin(DIV-Free-HDG)格式，(2)将ALE-DIV-HDG流体求解器扩展到移动区域FSI问题，模拟不可压缩流体与可压缩或不可压缩超弹性结构的相互作用。该格式具有高精度、全局和局部守恒性、能量稳定性、压力稳健性、最小数值耗散量和计算效率等特点。将无div-HDG格式推广到ALE框架内的移动域问题，由于其中使用的是散度协调的速度有限元空间，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div-HDG格式，该格式不需要速度有限元空间的散度一致性，可以有效地适应移动区域INS和FSI问题的ALE框架。这将是首次系统地研究动域INS和FSI问题的ALE强质量守恒有限元方法。我们将对这些方法进行严格的稳定性分析，并研究它们的收敛性质。将研究进一步提高这些计划效率的各种策略。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.