Adaptive, second-order, unconditionally stable partitioned method for fluid–structure interaction

We propose a novel, time adaptive, strongly-coupled partitioned method for the interaction between a viscous, incompressible fluid and a thin elastic structure. The time integration is based on the refactorized Cauchy’s one-legged ‘θ−like’ method, which consists of a backward Euler method using a θτn-time step and a forward Euler method using a (1−θ)τn-time step. The bulk of the computation is done by the backward Euler method, as the forward Euler step is equivalent to (and implemented as) a linear extrapolation. The variable τn-time step integration scheme is combined with the partitioned, kinematically coupled β−scheme, used to decouple the fluid and structure sub-problems. In the backward Euler step, the two sub-problems are solved in a partitioned sequential manner, and iterated until convergence. Then, the fluid and structure sub-problems are post-processed/extrapolated in the forward Euler step, and finally the τn-time step is adapted. The refactorized Cauchy’s one-legged ‘θ−like’ method used in the development of the proposed method is equivalent to the midpoint rule when θ=12, in which case the method is non-dissipative and second-order accurate. We prove that the sub-iterative process of our algorithm is linearly convergent, and that the method is unconditionally stable when θ≥12. The numerical examples explore the properties of the method when both fixed and variable time steps are used, and in both cases shown an excellent agreement with the reference solution.