This work focuses on the derivation and the analysis of a novel, strongly-coupled partitioned method for fluid–structure interaction problems. The flow is assumed to be viscous and incompressible, and the structure is modeled using linear elastodynamics equations. We assume that the structure is thick, i.e., modeled using the same number of spatial dimensions as fluid. Our newly developed numerical method is based on Robin boundary conditions, as well as on the refactorization of the Cauchy’s one-legged ‘\(\theta \)-like’ method, written as a sequence of Backward Euler–Forward Euler steps used to discretize the problem in time. This family of methods, parametrized by \(\theta \), is B-stable for any \(\theta \in [\frac{1}{2} , 1]\) and second-order accurate for \(\theta = \frac{1}{2} + {\mathcal {O}}(\tau )\), where \(\tau \) is the time step. In the proposed algorithm, the fluid and structure sub-problems, discretized using the Backward Euler scheme, are first solved iteratively until convergence. Then, the variables are linearly extrapolated, equivalent to solving Forward Euler problems. We prove that the iterative procedure is convergent, and that the proposed method is stable provided \(\theta \in [\frac{1}{2},1]\). Numerical examples, based on the finite element discretization in space, explore convergence rates using different values of parameters in the problem, and compare our method to other strongly-coupled partitioned schemes from the literature. We also compare our method to both a monolithic and a non-iterative partitioned solver on a benchmark problem with parameters within the physiological range of blood flow, obtaining an excellent agreement with the monolithic scheme.

这项工作着重于推导和分析一种新颖的，强耦合的分区方法来解决流固耦合问题。假定流动是粘性的和不可压缩的，并且使用线性弹性动力学方程对结构进行建模。我们假设结构是厚​​的，即使用与流体相同数量的空间尺寸进行建模。我们新开发的数值方法是基于Robin边界条件，以及基于柯西单腿“ \（\ theta \） -类”方法的重构，该方法写为用于离散化的Backward Euler–Forward Euler步骤的序列问题及时出现。由\（\ theta \）参数化的方法系列对于任何\（\ [[frac {1} {2}，1] \）中的\（\ theta \）都是B稳定的和\（\ theta = \ frac {1} {2} + {\ mathcal {O}}（\ tau）\）的二阶精度，其中\（\ tau \）是时间步长。在提出的算法中，首先迭代求解使用Backward Euler方案离散化的流体和结构子问题，直到收敛为止。然后，对变量进行线性外推，相当于求解正向欧拉问题。我们证明了迭代过程是收敛的，并且在提供\（\ theta \ in [\ frac {1} {2}，1] \）的情况下，所提出的方法是稳定的。。数值示例基于空间中的有限元离散化，使用问题中不同参数值探索收敛速度，并将我们的方法与文献中的其他强耦合分区方案进行比较。我们还将比较方法与基准问题的整体式和非迭代分区求解器，其基准参数在血流的生理范围内，从而与整体式方案取得了很好的一致性。