High-order implicit Runge-Kutta time integrators for fluid-structure interactions

• Published in: International journal for numerical methods in fluids Vol. 78; no. 7; pp. 385 - 412
• Main Authors: Cori, Jean-François, Etienne, Stephane, Garon, Andre, Pelletier, Dominique
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 07.07.2015; Wiley Subscription Services, Inc
• Subjects: Accuracy; arbitrary Lagrangian-Eulerian; Finite element method; Fluid-structure interaction; fluid-structure interactions; implicit Runge-Kutta; Integrators; Mathematical analysis; Numerical analysis; Runge-Kutta method; Temporal logic
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4020

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 efficiency of implicit Runge–Kutta integrators of order up to five is tested for application in the field of fluid‐structure interactions. These schemes are used for both media in a fully coupled algorithm. Spatial discretization is obtained through the finite element method. It is shown that these schemes perform well at solving classical fluid‐structure problems.