Abstract Summary
Fluid-structure interaction of thin, flexible structures is extremely common in nature and engineering, from leaf fluttering to parachute inflation. Numerical simulation of those systems is of interest in many fields of engineering but is challenging for a number of reasons; the low bending rigidity of the structure and the usually small thickness result in very large deformations. Additionally, the added mass effect can be significant and lead to inherent difficulties for partitioned fluid-structure interaction simulations. This paper presents a general and robust method for fluid-structure interaction of thin structures undergoing large displacement and large added mass effects by coupling an immersed boundary method with a finite element shell model. Special care is taken in the immersed boundary method to ensure the pressure boundary condition is enforced for deforming bodies via a variable coefficient Poisson's equation. The method can accurately simulate the fluid velocity and pressure induced by dynamic bodies undergoing large displacements using a computationally efficient pressure projection finite volume solver. The structural solver can be applied to bending and membrane-related problems, making our partitioned solver very general. To simulate problems under large added mass effects with our partitioned approach, we use a strongly coupled algorithm with an iterative Quasi-Newton method to determine the fluid-structure balance at the interface. We avoid the expensive computation of the inverse Jacobian within this root-finding iteration by constructing it from input-output pairs. These pairs are formed using the interface displacement and traction vector from the previous time steps. The solution to this under-determined system is achieved through a least square approach. The resulting Interface Quasi Newton from an Inverse Least Square problem (IQN ILS) scheme is efficient and can be applied to complex problems where the large computational cost of computing the inverse Jacobian is avoided. To demonstrate the efficiency of this coupled solver, we first simulated the quasi-steady flow around a membrane airfoil, showing near second-order convergence in space of our algorithm and very good agreement with the reference data. We then demonstrate the stability and accuracy of our solver for a flapping flag with vanishing bending stiffness. Flapping frequency and amplitude agree well with the literature. With an inverted flag, we then show that the solver is efficient and accurate under large added mass effects, with excellent agreement with the different flapping modes observed in the literature and their transition regions. Finally, we apply our coupled solver to the flapping of a flexible swimmer in a quiescent flow. We replicate experiments where a tapered flexible foil undergoes pure heave, applied to its leading edge with the trailing edge free to oscillate. The system's flexibility results in propulsion for a range of frequencies, and we compare it to the experimental findings. We perform both 2 and 3-dimensional simulations of the system. We show the correlation between the trailing edge flapping amplitude and propulsive efficiency. Finally, we use modal decomposition to distinguish between various structural responses at various frequencies.
