In this paper we propose a novel and general approach to design semi-implicit methods for the simulation of fluid-structure interaction problems in a fully Eulerian framework. In order to properly present the new method, we focus on the two-dimensional version of the general model developed to describe full membrane elasticity. The approach consists in treating the elastic source term by writing an evolution equation on the structure stress tensor, even if it is nonlinear. Then, it is possible to show that its semi-implicit discretization allows us to add to the linear system of the Navier-Stokes equations some consistent dissipation terms that depend on the local deformation and stiffness of the membrane. Due to the linearly implicit discretization, the approach does not need iterative solvers and can be easily applied to any Eulerian framework for fluid-structure interaction. Its stability properties are studied by performing a Von Neumann analysis on a simplified one-dimensional model and proving that, thanks to the additional dissipation, the discretized coupled system is unconditionally stable. Several numerical experiments are shown for two-dimensional problems by comparing the new method to the original explicit scheme and studying the effect of structure stiffness and mesh refinement on the membrane dynamics. The newly designed scheme is able to relax the time step restrictions that affect the explicit method and reduce crucially the computational costs, especially when very stiff membranes are under consideration.

翻译：本文提出一种新颖且通用的方法，用于在全欧拉框架下设计模拟流固耦合问题的半隐式格式。为恰当阐述新方法，我们聚焦于描述完整膜弹性的通用模型的二维版本。该方法的核心在于，即使弹性源项是非线性的，仍通过建立结构应力张量的演化方程对其进行处理。由此可证明，其半隐式离散化使得我们能够在纳维-斯托克斯方程的线性系统中添加与膜局部变形和刚度相关的一致性耗散项。由于采用线性隐式离散化，该方法无需迭代求解器，且能简便地应用于任何欧拉框架下的流固耦合问题。我们通过一维简化模型进行冯·诺依曼稳定性分析，并证明得益于额外耗散，离散化的耦合系统是无条件稳定的。针对二维问题，我们通过将新方法与原始显式格式对比，并研究结构刚度和网格细化对膜动力学的影响，展示了多项数值实验。新设计的格式能够缓解显式方法中的时间步长限制，显著降低计算成本，尤其在处理高刚度膜时效果更为突出。