An Adaptive Finite Element Method for Fluid-Structure Interaction Problems Based on a Fully Eulerian Formulation

In this article, we continue the investigation of the general variational framework for the adaptive finite element approximation of fluid-structure interaction problems proposed in Dunne [12] and Dunne & Rannacher [13]. The modeling is based on an Eulerian description of the (incompressible) fluid as well as the (elastic) structure dynamics. This approach uses a technique which is similar to the Level Set method in the simulation of two-phase flows in so far that it also tracks initial data and from this determines to which “phase” a point belongs. The advantage is that, in contrast to the common ALE approach, the computation takes place on fixed, though dynamically adapted, reference meshes what avoids the critical degeneration in case of large deformation or boundary contact of the structure. Based on this monolithic model of the fluid-structure interaction, we apply the Dual Weighted Residual (DWR) Method for goal-oriented a posteriori error estimation and mesh adaptation to fluid-structure interaction problems. Several test examples are presented in order to illustrate the potential of this approach.

[1]  J. Tinsley Oden,et al.  Finite Elements, Computational Aspects, Vol. III. , 1985 .

[2]  Roland Becker,et al.  A Two-Level Stabilization Scheme for the Navier-Stokes Equations , 2004 .

[3]  Rolf Rannacher,et al.  Finite Element Methods for the Incompressible Navier-Stokes Equations , 2000 .

[4]  Roland Becker,et al.  Multigrid techniques for finite elements on locally refined meshes , 2000 .

[5]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[6]  Thomas Dunne,et al.  An Eulerian approach to fluid–structure interaction and goal‐oriented mesh adaptation , 2006 .

[7]  M. Schmich.  Adaptive Finite Element Methods for Computing Nonstationary Incompressible Flows , 2009 .

[8]  Noel J. Walkington,et al.  Digital Object Identifier (DOI) 10.1007/s002050100158 An Eulerian Description of Fluids Containing Visco-Elastic Particles , 2022 .

[9]  S. Turek,et al.  Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow , 2006 .

[10]  Harold A. Buetow,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[11]  Rolf Rannacher,et al.  NUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION BASED ON MONOLITHIC VARIATIONAL FORMULATIONS , 2010 .

[12]  Hans-Joachim Bungartz,et al.  Fluid-structure interaction : modelling, simulation, optimisation , 2006 .

[13]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[14]  Roland Becker,et al.  A finite element pressure gradient stabilization¶for the Stokes equations based on local projections , 2001 .

[15]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[16]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[17]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[18]  Stefan Turek,et al.  Efficient Solvers for Incompressible Flow Problems - An Algorithmic and Computational Approach , 1999, Lecture Notes in Computational Science and Engineering.

[19]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[20]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[21]  T. Wick,et al.  Finite elements for fluid–structure interaction in ALE and fully Eulerian coordinates , 2010 .

[22]  Boris Vexler,et al.  Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations , 2007, SIAM J. Sci. Comput..

[23]  Boris Vexler,et al.  Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems , 2007, SIAM J. Control. Optim..

[24]  Stefan Turek,et al.  A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics , 2006 .