An Energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme

The conservation laws of continuum mechanics, written in an Eulerian frame, do not distinguish fluids and solids, except in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown monolithic methods for fluid structure interactions (FSI for short) are built. In this paper such a formulation is analysed when the solid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galerkin method render this approach feasible and reasonably robust. In this paper the method and its discretisation are presented, stability is discussed through an energy estimate. A numerical section discusses implementation issues and presents a few simple tests.

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

[2]  S. Antman.  Nonlinear problems of elasticity , 1994 .

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

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

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

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

[7]  K. Bathe,et al.  FINITE ELEMENT FORMULATIONS FOR LARGE DEFORMATION DYNAMIC ANALYSIS , 1975 .

[8]  H. Saunders.  Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

[9]  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..

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

[11]  K. Bathe.  Finite Element Procedures , 1995 .

[12]  Thierry Coupez,et al.  Implicit Boundary and Adaptive Anisotropic Meshing , 2015 .

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

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

[15]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[16]  O. Pironneau.  Finite Element Methods for Fluids , 1990 .

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

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

[19]  Rolf Rannacher,et al.  Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on an Eulerian Variational Formulation , 2006 .

[20]  Patrice Hauret,et al.  Méthodes numériques pour la dynamique des structures non linéaires incompressibles à deux échelles , 2004 .

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

[22]  I-Shih Liu,et al.  Successive linear approximation for finite elasticity , 2010 .

[23]  Rolf Rannacher,et al.  An Adaptive Finite Element Method for Fluid-Structure Interaction Problems Based on a Fully Eulerian Formulation , 2011 .

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

[25]  G. Cottet,et al.  EULERIAN FORMULATION AND LEVEL SET MODELS FOR INCOMPRESSIBLE FLUID-STRUCTURE INTERACTION , 2008 .

[26]  J. Boyle,et al.  Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches , 2008 .

[27]  F. NOBILE,et al.  An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions , 2008, SIAM J. Sci. Comput..

[28]  Frédéric Hecht,et al.  An energy stable monolithic Eulerian fluid‐structure finite element method , 2017 .

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

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

[31]  J. Marsden,et al.  Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems , 2000 .

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

[33]  Olivier Pironneau,et al.  Variational Analysis and Aerospace Engineering Mathematical Challenges for the Aerospace of the Future , 2016 .

[34]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[35]  S. Léger.  Méthode lagrangienne actualisée pour des problèmes hyperélastiques en très grandes déformations , 2014 .

[36]  A. Quarteroni,et al.  Multiscale models of the vascular system , 2009 .