Refactorization of Cauchy’s Method: A Second-Order Partitioned Method for Fluid–Thick Structure Interaction Problems

Abstract

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.