A full Eulerian finite difference approach for solving fluid-structure coupling problems

A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation [Hirt, Nichols, J. Comput. Phys. 39 (1981) 201], which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Rivlin materials. In this paper, various verifications and validations of the present full Eulerian method, which solves the fluid and solid motions on a fixed grid, are demonstrated, and the numerical accuracy involved in the fluid-structure coupling problems is examined.

[1]  Tong Gao,et al.  Deformation of elastic particles in viscous shear flow , 2008, J. Comput. Phys..

[2]  L. Mahadevan,et al.  Soft lubrication: The elastohydrodynamics of nonconforming and conforming contacts , 2004, cond-mat/0412509.

[3]  Kazuo Kashiyama,et al.  Eulerian formulation using stabilized finite element method for large deformation solid dynamics , 2007 .

[4]  Y. Yuki,et al.  Efficient Immersed Boundary Method for Strong Interaction Problem of Arbitrary Shape Object with the Self-Induced Flow , 2007 .

[5]  Shu Takagi,et al.  An immersed boundary method for restricted diffusion with permeable interfaces , 2009, J. Comput. Phys..

[6]  Lucy T. Zhang,et al.  Immersed finite element method , 2004 .

[7]  Hong Zhao,et al.  A fixed-mesh method for incompressible flow-structure systems with finite solid deformations , 2008, J. Comput. Phys..

[8]  Toshiaki Hisada,et al.  Fluid–structure interaction analysis of the two-dimensional flag-in-wind problem by an interface-tracking ALE finite element method , 2007 .

[9]  C. Pozrikidis,et al.  Effect of membrane bending stiffness on the deformation of capsules in simple shear flow , 2001, Journal of Fluid Mechanics.

[10]  Jiun-Shyan Chen,et al.  A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory , 1996 .

[11]  R. D. Wood,et al.  Nonlinear Continuum Mechanics for Finite Element Analysis , 1997 .

[12]  R. Glowinski,et al.  A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow , 2001 .

[13]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[14]  S. Mittal,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II: Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , 1992 .

[15]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[16]  Ryutaro Himeno,et al.  Blood Flow Simulator using Medical Images without Mesh Generation , 2006 .

[17]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[18]  Toshio Kobayashi,et al.  Influence of wall elasticity in patient-specific hemodynamic simulations , 2007 .

[19]  Toshiaki Hisada,et al.  Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method. , 2004, Biophysical journal.

[20]  Takeo Kajishima.  Conservation Properties of Finite Difference Method for Convection. , 1994 .

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

[22]  Takeo Kajishima,et al.  A conservative momentum‐exchange algorithm for interaction problem between fluid and deformable particles , 2010 .

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

[24]  Antonio Huerta,et al.  Viscous flow with large free surface motion , 1988 .

[25]  T. Tezduyar,et al.  Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling , 2008 .

[26]  Morton E. Gurtin,et al.  The thermodynamics of constrained materials , 1973 .

[27]  P. Woodward,et al.  SLIC (Simple Line Interface Calculation) , 1976 .

[28]  Antonio Huerta,et al.  Viscous Flow Structure Interaction , 1988 .

[29]  R. Glowinski,et al.  A distributed Lagrange multiplier/fictitious domain method for particulate flows , 1999 .

[30]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[31]  Feng Xiao,et al.  A simple algebraic interface capturing scheme using hyperbolic tangent function , 2005 .

[32]  Yoichiro Matsumoto,et al.  The deformation behavior of multiple red blood cells in a capillary vessel. , 2009, Journal of biomechanical engineering.

[33]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[34]  J. C. Simo,et al.  Variational and projection methods for the volume constraint in finite deformation elasto-plasticity , 1985 .

[35]  Wheeler,et al.  Phase-field model for isothermal phase transitions in binary alloys. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[36]  S. Takeuchi,et al.  Full Eulerian simulations of biconcave neo-Hookean particles in a Poiseuille flow , 2010 .

[37]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .

[38]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[39]  Fpt Frank Baaijens,et al.  A Eulerian approach to the finite element modelling of neo-Hookean rubber material , 1994 .

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

[41]  Ted Belytschko,et al.  Fluid-structure interaction , 1980 .

[42]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[43]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[44]  N. Phan-Thien,et al.  Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions , 2005 .

[45]  L YoungsD,et al.  Time-dependent multi-material flow with large fluid distortion. , 1982 .

[46]  Koichi Suzumori,et al.  Feasibility study of an electrostatic actuator by finite element interaction analysis , 1999 .

[47]  Satya N. Atluri,et al.  Embedded Localized Strain Zone Constitutive Model in Finite Strain and Finite Rotation , 1995 .

[48]  C. Peskin,et al.  Implicit second-order immersed boundary methods with boundary mass , 2008 .

[49]  F. Xiao,et al.  Three-dimensional numerical simulation of flows with complex geometries in a regular Cartesian grid and its application to blood flow in cerebral artery with multiple aneurysms , 2005 .

[50]  J. Sethian,et al.  LEVEL SET METHODS FOR FLUID INTERFACES , 2003 .

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

[52]  S. Takeuchi,et al.  An implicit full Eulerian method for the fluid–structure interaction problem , 2011 .

[53]  Toshiaki Hisada,et al.  Analysis of Fluid-Structure Interaction Problems with Structural Buckling and Large Domain Changes by ALE Finite Element Method. , 2001 .

[54]  Howard H. Hu.  Direct simulation of flows of solid-liquid mixtures , 1996 .

[55]  S. Zaleski,et al.  Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows , 1999 .

[56]  T. Kajishima,et al.  Turbulence Structure of Particle-Laden Flow in a Vertical Plane Channel Due to Vortex Shedding , 2001 .

[57]  A. Popel,et al.  Large deformation of red blood cell ghosts in a simple shear flow. , 1998, Physics of fluids.

[58]  D. Durran.  Numerical Methods for Fluid Dynamics , 2010 .

[59]  T. E. TezduyarAerospace,et al.  3d Simulation of Fluid-particle Interactions with the Number of Particles Reaching 100 , 1996 .

[60]  Antonio J. Gil,et al.  Structural analysis of prestressed Saint Venant-Kirchhoff hyperelastic membranes subjected to moderate strains , 2006 .

[61]  G. Tryggvason,et al.  Computational Methods for Multiphase Flow: Immersed boundary methods for fluid interfaces , 2007 .

[62]  H. S. Udaykumar,et al.  An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces , 2003 .

[63]  Lucy T. Zhang,et al.  Imposing rigidity constraints on immersed objects in unsteady fluid flows , 2008 .

[64]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[65]  H. Sung,et al.  An immersed boundary method for fluid–flexible structure interaction , 2009 .

[66]  Thomas J. R. Hughes,et al.  Finite element modeling of blood flow in arteries , 1998 .

[67]  Gretar Tryggvason,et al.  Computational Methods for Multiphase Flow: Frontmatter , 2007 .

[68]  Andrea Prosperetti,et al.  PHYSALIS: a new method for particle simulation part II: two-dimensional Navier--Stokes flow around cylinders , 2003 .

[69]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[70]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[71]  C. Pozrikidis.  Axisymmetric motion of a file of red blood cells through capillaries , 2005 .

[72]  Boo Cheong Khoo,et al.  An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries , 2009, J. Comput. Phys..

[73]  Andrea Prosperetti,et al.  A second-order method for three-dimensional particle simulation , 2005 .

[74]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[75]  Zhaosheng Yu.  A DLM/FD method for fluid/flexible-body interactions , 2005 .

[76]  Lucy T. Zhang.  Immersed finite element method for fluid-structure interactions , 2007 .

[77]  Ryutaro Himeno,et al.  An image-based computational fluid dynamic method for haemodynamic simulation , 2002 .

[78]  J. A. Trapp,et al.  Reinforced materials with thermo-mechanical constraints , 1971 .

[79]  K. Bathe,et al.  An arbitrary lagrangian-eulerian velocity potential formulation for fluid-structure interaction , 1993 .

[80]  Zhilin Li,et al.  The immersed interface method for the Navier-Stokes equations with singular forces , 2001 .

[81]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[82]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[83]  Tayfun E. Tezduyar,et al.  Fluid-structure interactions of a cross parachute: Numerical simulation , 2001 .

[84]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[85]  C. Peskin.  Flow patterns around heart valves: A numerical method , 1972 .

[86]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[87]  T. Kajishima,et al.  Interaction between particle clusters and particle-induced turbulence , 2002 .

[88]  A. A. Amsden,et al.  A simplified MAC technique for incompressible fluid flow calculations , 1970 .

[89]  David Farrell,et al.  Immersed finite element method and its applications to biological systems. , 2006, Computer methods in applied mechanics and engineering.

[90]  T. Gültop,et al.  On the propagation of acceleration waves in incompressible hyperelastic solids , 2003 .

[91]  R. Rivlin.  Large Elastic Deformations of Isotropic Materials , 1997 .

[92]  Takeo Kajishima,et al.  Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations , 2007, J. Comput. Phys..

[93]  Toshio Kobayashi,et al.  Numerical Simulation System for Blood Flow in the Cerebral Artery Using CT Imaging Data , 2001 .

[94]  Charles A. Taylor,et al.  A coupled momentum method for modeling blood flow in three-dimensional deformable arteries , 2006 .

[95]  Thomas J. R. Hughes,et al.  A space-time formulation for multiscale phenomena , 1996 .

[96]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .