We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.

中文翻译：

---

一种用于移动具有质量的结构的增强型浸入式界面方法。

我们提出了一种增强浸入界面方法，用于模拟不可压缩流体中具有质量的可变形结构的动力学。流体由二维 Navier-Stokes 方程建模。由于质量引起的结构加速度与流速和压力有关。为简单起见，假定结构的表面张力为常数。在我们的方法中，我们将未知加速度视为唯一的增广变量，以便可以应用增广沉浸式界面方法。我们使用改进的投影方法，可以强制执行对应于未知加速度的压力跳跃条件。加速度必须与沿界面的流动加速度相匹配。所提出的增强方法针对具有固定界面的精确解进行了测试。它表明增强方法在空间上具有二阶收敛性。还研究了具有质量的可变形圆形结构的动力学。它表明流固系统具有双稳态：较小雷诺数的静止状态和较大雷诺数的振荡状态。观察结果与文献中的一致。