An energy stable monolithic Eulerian fluid‐structure finite element method

Summary When written in an Eulerian frame, the conservation laws of continuum mechanics are similar for fluids and solids leading to a single set of variables for a monolithic formulation. Such formulations are well adapted to large displacement fluid-structure configurations, but stability is a challenging problem because of moving geometries. In this article the method is presented; time implicit discretizations are proposed with iterative algorithms well posed at each step, at least for small displacements; stability is discussed for an implicit in time finite element method in space by showing that energy decreases with time. The key numerical ingredient is the Characterics-Galerkin method coupled with a powerful mesh generator. A numerical section discusses implementation issues and presents a few simple tests. It is also shown that contacts are easily handled by extending the method to variational inequalities. This paper deals only with incompressible neo-Hookean Mooney-Rivlin hyperelastic material in two dimensions in a Newtonian fluid; but the method is not limited to these; compressible and 3D cases will be presented later. This article is protected by copyright. All rights reserved.

[1]  Thomas Richter,et al.  Fluid‐Structure Interactions in ALE and Fully Eulerian Coordinates , 2010 .

[2]  O. Pironneau.  Numerical Study of a Monolithic Fluid–Structure Formulation , 2016 .

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

[4]  Thomas Wick,et al.  Fully Eulerian fluid-structure interaction for time-dependent problems , 2013 .

[5]  Thomas Richter,et al.  A Fully Eulerian formulation for fluid-structure-interaction problems , 2013, J. Comput. Phys..

[6]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[7]  Yvon Maday,et al.  A high order characteristics method for the incompressible Navier—Stokes equations , 1994 .

[8]  Marina Vidrascu,et al.  Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures , 2013 .

[9]  Kazufumi Ito,et al.  Semi–Smooth Newton Methods for Variational Inequalities of the First Kind , 2003 .

[10]  M. Vanninathan,et al.  A fluid–structure model coupling the Navier–Stokes equations and the Lamé system , 2014 .

[11]  Stefan Turek,et al.  A MONOLITHIC FEM SOLVER FOR AN ALE FORMULATION OF FLUID-STRUCTURE INTERACTION WITH CONFIGURATION FOR NUMERICAL BENCHMARKING , 2006 .

[12]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[13]  Jie Liu,et al.  A second-order changing-connectivity ALE scheme and its application to FSI with large convection of fluids and near contact of structures , 2016, J. Comput. Phys..

[14]  I. Babuska,et al.  The plate paradox for hard and soft support , 1990 .

[15]  T. Belytschko,et al.  The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns , 2006 .

[16]  Daniele Boffi,et al.  The Finite Element Immersed Boundary Method with Distributed Lagrange Multiplier , 2014, SIAM J. Numer. Anal..

[17]  Muriel Boulakia.  Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid , 2003 .

[18]  P. Tallec,et al.  Fluid structure interaction with large structural displacements , 2001 .

[19]  Stefan Frei,et al.  Eulerian finite element methods for interface problems and fluid-structure interactions , 2016 .

[20]  Daniel Coutand,et al.  Motion of an Elastic Solid inside an Incompressible Viscous Fluid , 2005 .

[21]  Annalisa Quaini,et al.  Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement , 2012, J. Comput. Phys..

[22]  Olivier Pironneau,et al.  Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems , 2017 .

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