PARALLEL EULERIAN METHODS FOR FLOWS WITH ELASTIC MEMBRANES

We propose and study parallel numerical algorithms for simulation of flows with elastic membranes, such as red blood cells. The challenges of modeling such flows include accurately resolving the interface motion and enforcing the appropriate interface mechanics. We propose a phase field Eulerian method for implicitly capturing the location of the dynamic interface between two fluids. In addition, for flows with elastic membranes separating the fluids, we present a phase field formulation for the membrane stresses. The method we propose avoids the computational geometry complexities associated with fully Lagrangian or front-tracking Eulerian methods. These complexities are particularly pronounced on parallel computers. The phase field formulation is discretized by a combined continuous/discontinuous Galerkin method, which lends itself to an inherently parallel implementation. We present a block Schur-complement preconditioner for the coupled membrane-flow system that neutralizes the ill-conditioning due to disparate material properties. Parallel performance results demonstrate the efficacy of this preconditioner.

[1]  Zhiliang Xu,et al.  Conservative Front Tracking with Improved Accuracy , 2003, SIAM J. Numer. Anal..

[2]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[3]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[4]  D. Zorin,et al.  A fast solver for the Stokes equations with distributed forces in complex geometries , 2004 .

[5]  C. Pozrikidis,et al.  Numerical Simulation of the Flow-Induced Deformation of Red Blood Cells , 2003, Annals of Biomedical Engineering.

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

[7]  Wei Shyy,et al.  Computational Modeling for Fluid Flow and Interfacial Transport (Dover Books on Engineering) , 1993 .

[8]  B. Griffith,et al.  A design improvement strategy for axial blood pumps using computational fluid dynamics. , 1996, ASAIO journal (1992).

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

[10]  C. Pozrikidis,et al.  Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow , 1995, Journal of Fluid Mechanics.

[11]  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 .

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

[13]  Daniel D. Joseph Interrogation of Direct Numerical Simulation of Solid-Liquid Flow , 1999 .

[14]  James F. Antaki,et al.  CFD-based design optimization of a three-dimensional rotary blood pump , 1996 .

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

[16]  R. Skalak,et al.  Flow of axisymmetric red blood cells in narrow capillaries , 1986, Journal of Fluid Mechanics.

[17]  P. Colella,et al.  A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains , 1998 .

[18]  Guy E. Blelloch,et al.  A PARALLEL DYNAMIC-MESH LAGRANGIAN METHOD FOR SIMULATION OF FLOWS WITH DYNAMIC INTERFACES , 2000, ACM/IEEE SC 2000 Conference (SC'00).

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

[20]  S. Zaleski,et al.  DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW , 1999 .

[21]  C. Pozrikidis Hydrodynamics of Liquid Capsules Enclosed by Elastic Membranes , 2001 .

[22]  J. Sethian Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[23]  J. Whiteman The Mathematics of Finite Elements and Applications. , 1983 .

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

[25]  A. Wathen,et al.  Schur complement preconditioners for the Navier–Stokes equations , 2002 .

[26]  Yiftah Navot,et al.  Elastic membranes in viscous shear flow , 1998 .

[27]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .