From Immersed Boundary Method to Immersed Continuum Methods

The objective of this paper is to present an overview of the newly proposed immersed continuum method in conjunction with the traditional treatment of fluidstructure interaction problems, the immersed boundary method, the extended immersed boundary method, the immersed finite element method, and the fictitious domain method. In particular, the key aspects of the immersed continuum method in comparison with the immersed boundary method are discussed. The immersed continuum method retains the same strategies employed in the extended immersed boundary method and the immersed finite element method, namely, the independent solid mesh moves on top of a fixed or prescribed background fluid mesh, and employs fully implicit time integration with a matrix-free combination of NewtonRaphson and GMRES iterative solution procedures. Therefore, the immersed continuum method is capable of handling compressible fluid interacting with compressible solid. Several numerical examples are also presented to demonstrate that the proposed immersed continuum method is a good candidate for multi-scale and multi-physics modeling platform.

[1]  P. Leyland,et al.  Analysis of Fluid-Structure Interactions , 1997 .

[2]  Xiaodong Wang.  Velocity/pressure mixed finite element and finite volume formulation with ALE descriptions for nonlinear fluid-structure interaction problems , 2000 .

[3]  R. Ohayon,et al.  Substructure variational analysis of the vibrations of coupled fluid–structure systems. Finite element results , 1979 .

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

[5]  J. Halleux,et al.  An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions , 1982 .

[6]  Weizhu Bao,et al.  ON THE INF SUP CONDITION OF MIXED FINITE ELEMENT FORMULATIONS FOR ACOUSTIC FLUIDS , 2001 .

[7]  Omar Ghattas,et al.  Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem , 1998 .

[8]  Shaofan Li,et al.  Reproducing kernel hierarchical partition of unity, Part I—formulation and theory , 1999 .

[9]  J. P. Beyer.  A computational model of the cochlea using the immersed boundary method , 1992 .

[10]  Chiang C. Mei,et al.  NUMERICAL METHODS IN WATER-WAVE DIFFRACTION AND RADIATION , 1978 .

[11]  B. Wetton,et al.  Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes , 1999 .

[12]  G. C. Everstine.  A symmetric potential formulation for fluid-structure interaction , 1981 .

[13]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[14]  Ted Belytschko,et al.  Two-dimensional fluid-structure impact computations with regularization , 1981 .

[15]  K. C. Park,et al.  Partitioned Transient Analysis Procedures for Coupled-Field Problems. , 1979 .

[16]  Wing Kam Liu,et al.  Finite element procedures for fluid-structure interactions and application to liquid storage tanks , 1981 .

[17]  C S Peskin,et al.  Computer-assisted design of butterfly bileaflet valves for the mitral position. , 1985, Scandinavian journal of thoracic and cardiovascular surgery.

[18]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[19]  Yves Ousset,et al.  A displacement method for the analysis of vibrations of coupled fluid-structure systems , 1978 .

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

[21]  Michael Shelley,et al.  Simulating the dynamics and interactions of flexible fibers in Stokes flows , 2004 .

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

[23]  K. Bathe,et al.  Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential , 1985 .

[24]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[25]  Robert Dillon,et al.  Modeling Biofilm Processes Using the Immersed Boundary Method , 1996 .

[26]  C. Farhat,et al.  Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and two-dimensional application , 1995 .

[27]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[28]  Omar Ghattas,et al.  A variational finite element method for stationary nonlinear fluid-solid interaction , 1995 .

[29]  Roger Ohayon,et al.  Fluid-Structure Interaction: Applied Numerical Methods , 1995 .

[30]  K. Bathe,et al.  DISPLACEMENT/PRESSURE BASED MIXED FINITE ELEMENT FORMULATIONS FOR ACOUSTIC FLUID–STRUCTURE INTERACTION PROBLEMS , 1997 .

[31]  G. C. Feng,et al.  Fluid-Structure Finite Element Vibrational Analysis , 1974 .

[32]  D. Yue,et al.  Some properties of a hybrid element method for water waves , 1979 .

[33]  J. Happel,et al.  Low Reynolds number hydrodynamics , 1965 .

[34]  Wing Kam Liu,et al.  Wavelet and multiple scale reproducing kernel methods , 1995 .

[35]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[36]  Ted Belytschko,et al.  Advances in multiple scale kernel particle methods , 1996 .

[37]  Tayfan E. Tezduyar,et al.  Stabilized Finite Element Formulations for Incompressible Flow Computations , 1991 .

[38]  Robert L. Taylor,et al.  Vibration analysis of fluid-solid systems using a finite element displacement formulation , 1990 .

[39]  Wing Kam Liu,et al.  Extended immersed boundary method using FEM and RKPM , 2004 .

[40]  C. Farhat,et al.  Mixed explicit/implicit time integration of coupled aeroelastic problems: Three‐field formulation, geometric conservation and distributed solution , 1995 .

[41]  Klaus-Jürgen Bathe,et al.  Finite element analysis of incompressible and compressible fluid flows with free surfaces and structural interactions , 1995 .

[42]  Jun Zhang,et al.  Drag reduction through self-similar bending of a flexible body , 2002, Nature.

[43]  C. Felippa,et al.  MIXED VARIATIONAL FORMULATION OF FINITE ELEMENT ANALYSIS OF ACOUSTOELASTIC/SLOSH FLUID-STRUCTURE INTERACTION , 1990 .

[44]  Ted Belytschko,et al.  A fluid-structure finite element method for the analysis of reactor safety problems , 1976 .

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

[46]  Thomas J. R. Hughes,et al.  An arbitrary Lagrangian-Eulerian finite rigid element method for interaction of fluid and a rigid body , 1992 .

[47]  L. Fauci,et al.  A computational model of aquatic animal locomotion , 1988 .

[48]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[49]  Dalin Tang,et al.  A multi-physics growth model with fluid–structure interactions for blood flow and re-stenosis in rat vein grafts: A growth model for blood flow and re-stenosis in grafts , 2003 .

[50]  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 .

[51]  K. Bathe,et al.  Finite element developments for general fluid flows with structural interactions , 2004 .

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

[53]  Lucy T. Zhang,et al.  Coupling of Navier–Stokes equations with protein molecular dynamics and its application to hemodynamics , 2004 .

[54]  Tayfun E. Tezduyar,et al.  Finite element methods for flow problems with moving boundaries and interfaces , 2001 .

[55]  Lorraine G. Olson,et al.  A study of displacement-based fluid finite elements for calculating frequencies of fluid and fluid-structure systems , 1983 .