标题： 一种稳定且收敛的流体-固体相互作用的Hodge分解方法 显示英文标题

标题： A stable and convergent method for Hodge decomposition of fluid-solid interaction

摘要： 流体-固体相互作用由于其强非线性和多学科性质而一直是一个具有挑战性的课题。 许多用于求解FSI问题的数值方法在非收敛和数值不稳定性方面遇到了困难。 尽管有广泛的研究，但开发一种既能保证收敛性又能保证稳定性的方法仍然是一项挑战。 我们在此工作中的讨论仅限于粘性不可压缩流体流动与刚体之间的相互作用。 我们采用Gibou和Min提出的单块方法，该方法导致了扩展的Hodge投影。 该投影不仅更新流体矢量场，还更新固体速度。 我们推导了扩展Hodge投影与具有非局部Robin边界条件的泊松方程之间的等价性。 我们通过证明泊松方程弱解的存在性、唯一性和正则性，展示了Hodge投影是唯一且正交的。 此外，我们还证明了投影的稳定性，即投影不会增加流体和固体的总动能。 此外，我们讨论了一种数值方法，作为Hodge投影的离散类比，然后我们证明了唯一分解和正交性在离散情况下也成立。 作为我们的主要结果之一，我们证明了数值解至少具有第一阶精度的收敛性。 我们在二维和三维情况下进行了数值实验，验证了我们的分析和论点。 显示英文摘要

摘要： Fluid-solid interaction has been a challenging subject due to their strong nonlinearity and multidisciplinary nature. Many of the numerical methods for solving FSI problems have struggled with non-convergence and numerical instability. In spite of comprehensive studies, it has been still a challenge to develop a method that guarantees both convergence and stability. Our discussion in this work is restricted to the interaction of viscous incompressible fluid flow and a rigid body. We take the monolithic approach by Gibou and Min that results in an extended Hodge projection. The projection updates not only the fluid vector field but also the solid velocities. We derive the equivalence of the extended Hodge projection to the Poisson equation with non-local Robin boundary condition. We prove the existence, uniqueness, and regularity for the weak solution of the Poisson equation, through which the Hodge projection is shown to be unique and orthogonal. Also, we show the stability of the projection in a sense that the projection does not increase the total kinetic energy of fluid and solid. Also, we discusse a numerical method as a discrete analogue to the Hodge projection, then we show that the unique decomposition and orthogonality also hold in the discrete setting. As one of our main results, we prove that the numerical solution is convergent with at least the first order accuracy. We carry out numerical experiments in two and three dimensions, which validate our analysis and arguments.