High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

Abstract

SummaryThis paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

The aeroelastic characteristics of high aspect ratio wing

“…Structure equation is shown as follows 18–20 where [M] is the generalized mass matrix of wing, [G] is the structural damping matrix, [K] is the generalized stiffness matrix of wing, r(t) is the generalized force vector, and x(t) is the generalized displacement vector. The damping matrix [G] is constructed using mass matrix.…”

“…Various literatures have focused their studies on this case. 16–20 Due to the large aspect ratio, wing usually undergoes severe bending deformation under aerodynamic force, leading to the displacement of tip several meters, hence, the mesh regeneration 21,22 method is adopted instead of mesh deformation. 23–26 In the part of message transferring from fluid to structure, the local-triangle searching method (LTS) was adopted.…”

Adjacency-based, non-intrusive model reduction for Vortex-Induced Vibrations

“…[33]. We thus assume that after substituting equation ( 11) in (10), the resulting operators can be approximated by sparse matrices, with non-zero elements only in the positions corresponding to adjacent nodes of any given internal grid node and thus, any matrix row.…”

Vortex-induced vibrations: a soft coral feeding strategy?

“…2015) and for classical FSI problems such as airfoil plunging and pitching, and a flexible strip at the rear of a square cylinder (Cori et al. 2015). The code was also validated and produced dynamical results consistent with the data in the literature for several flow-induced vibrations including VIVs of circular (Etienne & Pelletier 2012) and square (Hay et al.…”

“…First, we computed the FSI problem in an ALE framework using the in-house solver CADYF (Etienne et al 2009). This code was verified with the method of manufactured solutions (Hay et al 2014;Yu et al 2015) and for classical FSI problems such as airfoil plunging and pitching, and a flexible strip at the rear of a square cylinder (Cori et al 2015). The code was also validated and produced dynamical results consistent with the data in the literature for several flow-induced vibrations including VIVs of circular (Etienne & Pelletier 2012) and square (Hay et al 2015a) cylinders, and wake-induced vibrations of cylinders in tandem (Yu et al 2016).…”