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 = 1 2 , 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 ≥ 1 2 . 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.