Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on an Eulerian Variational Formulation

Abstract

Abstract. We propose a general variational framework for the adaptive finite element approximation of fluid-structure interaction problems. The modeling is based on an Eulerian description of the (incompressible) fluid as well as the (elastic) structure dynamics. This is achieved by tracking the movement of the initial positions of all 'material' points. In this approach the deformation appears as a primary variable in an Eulerian framework. Our approach uses a technique which is similar to the Level Set method in so far that it also tracks initial data, in our case the set of Initial Positions, and from this determines to which 'phase' a point belongs. To avoid the need for reinitialization of the initial position set, we employ the harmonic continuation of the structure velocity field into the fluid domain. Based on this monolithic model of the fluid-structure interaction we apply the dual weighted residual method for goal-oriented a posteriori error estimation and mesh adaptation to fluid-structure interaction problems. Results from nonstationary examples are presented.

Goal-oriented error estimation for fluid–structure interaction problems

“…Fick, Brummelen and Zee [FBZ10] analyzed the coupling of a potential flow with a lower dimensional boundary structure with regard to sensitivity based error estimation and pointed out the role of the coupling conditions for the adjoint solutions. Dunne [Dun07] and [DRR10] have first implementations of goal-oriented adaptation fluid-structure interactions with complex nonlinear models including two-dimensional fluid and solid problems. To our understanding, there is still no proper analysis of the adjoint problems appearing in sensitivity based error estimation or optimization of fluid-structure interaction problems.…”

Finite Element Method for Fluid Structure Interaction with hp-Adaptivity

“…The effectiveness of the hp-adaptive implementation is investigated on a simple stationary test case problem, namely the modified version of the classical lid driven cavity problem as used in [4]. Fig.…”

“…Substantial early progress in this respect is arguably made by Dunne and Rannacher and Richter and Wick who proposed h-adaptive methods with goal-oriented error estimators [4]. Of particular interest is more recent effort by van der Zee et al The authors have implemented an hp-adaptive method for FSI problem which is also based on a goal-oriented error estimator.…”

Free Vibration Analysis of a Liquid in a Circular Cylindrical Rigid Tank Using the Hierarchical Finite Element Method

“…In the Eulerian approach (Haroun, 1983, Haroun, 1984, Veletsos and al, 1990, Balendra 1982, Hughes et al, 1981, Soria and Casadei, 1997, Legay and al, 2006, Dunne, 2007, Cottet and al, 2008 Rannacher and Richter, 2010, Sanches and coda, 2010, He and Qiao, 2011, Wick, 2013, the velocity potential, the pressure, or the velocity describes the behavior of the fluid, while the displacements represent the movements of the structure. The solution of the coupled system can then be obtained by solving the two systems separately, but by considering the effects of the interaction in an iterative way.…”