Comput Mech (2008) 43:61–72 DOI 10.1007/s00466-008-0255-5 ORIGINAL PAPER Fixed-point fluid–structure interaction solvers with dynamic relaxation Ulrich Küttler · Wolfgang A. Wall Received: 31 October 2007 / Accepted: 20 January 2008 / Published online: 22 February 2008 © Springer-Verlag 2008 Abstract A fixed-point fluid–structure interaction (FSI) solver with dynamic relaxation is revisited. New develop- ments and insights gained in recent years motivated us to present an FSI solver with simplicity and robustness in a wide range of applications. Particular emphasis is placed on the calculation of the relaxation parameter by both Aitken’s ∆ 2 method and the method of steepest descent. These methods have shown to be crucial ingredients for efficient FSI simu- lations. Keywords Fluid–structure interaction · Fixed-point solver · Dirichlet–Neumman partitioning · Strong coupling 1 Introduction The development of numerical solvers for fluid–structure interaction (FSI) problems has been an active area of research in the last decade. Reliable FSI solvers are demanded in areas as diverse as aeroelasticity [8, 21], civil engineering [41] or hemodynamics [2, 16]. And as wide as the possible field of application are the requirements the solvers are confronted with: aerodynamics applications can couple a light compress- ible fluid to a stiff structure (e.g. aircraft wing) or a light incompressible fluid to a very light structure (e.g. parachute or sail), whereas hemodynamics simulations couple incom- pressible fluids and flexible structures with very comparable density. Thus there cannot be one FSI solver that fits all needs. Instead a variety of solution procedures is needed. A particular interesting class of FSI problems, that is the appropriate model in many areas, is the interaction of U. Küttler · W. A. Wall (B ) Chair of Computational Mechanics, TU Munich, Boltzmannstr. 15, 85747 Garching, Germany e-mail: wall@lnm.mw.tum.de incompressible fluids with structures that undergo large deformations. In that case both fields, the fluid and the struc- ture, present computational challenges on their own. And still the coupling is a nontrivial task, as the large structural deformations have a huge impact on the fluid field’s size and the coupled solution. Oftentimes there are sophisticated field solvers available that should be reused. This presents a fur- ther constraint to a possible coupling scheme. There are, of course, monolithic solvers that treat the nonlinear coupled problem in one go [2, 18, 19, 36], however these approaches require access to the field solver internals and cannot be pur- sued with black box solvers. The same goes for partitioned approaches that mimic the behavior of monolithic solvers [7]. One possible partitioning strategy that enables the reuse of existing field solvers is the Dirichlet–Neumann partition- ing, the predominant partition approach for FSI solvers. The most popular coupling methods are fixed-point methods [27, 30, 40] and interface Newton Krylov methods [10, 15, 16]. Other solvers suggested include a block-Newton solver with finite differenced off-diagonal blocks [25] and solvers based on vector extrapolation methods [26, 37]. See also [33, 34] for a FSI solver framework that includes partitioned block- iterative and quasi-direct coupling solvers as well as mono- lithic solvers. The most basic and yet highly efficient approach among this variety of methods is the fixed-point method with dynamic relaxation as suggested in [27, 40]. It is extremely easy to implement and surprisingly efficient and robust. Hence, in many cases this is the method of choice and it is an especially preferred method to use if a new attempt at FSI or other coupled problems is pursued. Unfortunately, however, the method has never been published in a journal but only in two hardly available conference proceedings [27, 40] so far. This paper finally provides the detailed treatment of both the Aitken relaxation method and the relaxation via the method 123