An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.

[1]  Ming-Chih Lai,et al.  A simple implementation of the immersed interface methods for Stokes flows with singular forces , 2008 .

[2]  Cornelis W. Oosterlee,et al.  Multigrid Methods for the Stokes System , 2006, Computing in Science & Engineering.

[3]  Ying Shan,et al.  Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing , 2008, J. Comput. Phys..

[4]  J. Thomas Beale,et al.  ON THE ACCURACY OF FINITE DIFFERENCE METHODS FOR ELLIPTIC PROBLEMS WITH INTERFACES , 2006 .

[5]  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 .

[6]  J. E. Gómez,et al.  A multipole direct and indirect BEM for 2D cavity flow at low Reynolds number , 1997 .

[7]  Randall J. LeVeque,et al.  An Immersed Interface Method for Incompressible Navier-Stokes Equations , 2003, SIAM J. Sci. Comput..

[8]  Eric Yu Tau.  Numerical solution of the steady Stokes equations , 1992 .

[9]  Andreas Wiegmann,et al.  The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions , 2000, SIAM J. Numer. Anal..

[10]  D. Calhoun.  A Cartesian Grid Method for Solving the Two-Dimensional Streamfunction-Vorticity Equations in Irregular Regions , 2002 .

[11]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[12]  J. Ferziger,et al.  A ghost-cell immersed boundary method for flow in complex geometry , 2002 .

[13]  H. Power,et al.  Numerical simulation of the motion and deformation of a non-Newtonian shear-thinning drop suspended in a Newtonian circular Couette flow using DR-BEM , 2009 .

[14]  H. Elman.  Preconditioners for saddle point problems arising in computational fluid dynamics , 2002 .

[15]  H. Elman.  Multigrid and Krylov subspace methods for the discrete Stokes equations , 1994 .

[16]  Christoph Börgers,et al.  Domain imbedding methods for the Stokes equations , 1990 .

[17]  Boo Cheong Khoo,et al.  An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model , 2007, J. Comput. Phys..

[18]  Vita Rutka,et al.  A staggered grid‐based explicit jump immersed interface method for two‐dimensional Stokes flows , 2008 .

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

[20]  Christoph BiJrgers.  Domain Imbedding Methods for the Stokes Equations , 1990 .

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

[22]  Zhilin Li,et al.  A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow , 2008, Adv. Comput. Math..

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

[24]  Cheng Wang,et al.  A Fast Finite Differenc Method For Solving Navier-Stokes Equations on Irregular Domains , 2003 .

[25]  F. Lien,et al.  A robust and efficient hybrid cut-cell/ghost-cell method with adaptive mesh refinement for moving boundaries on irregular domains , 2008 .

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

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

[28]  Ricardo Cortez,et al.  The Method of Regularized Stokeslets , 2001, SIAM J. Sci. Comput..

[29]  Boo Cheong Khoo,et al.  An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries , 2006, J. Comput. Phys..

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

[31]  Ahmed H. Sameh,et al.  An Efficient Iterative Method for the Generalized Stokes Problem , 1998, SIAM J. Sci. Comput..

[32]  K. Ito,et al.  An augmented approach for Stokes equations with a discontinuous viscosity and singular forces , 2007 .

[33]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

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

[35]  C. Pozrikidis.  Boundary Integral and Singularity Methods for Linearized Viscous Flow: Index , 1992 .

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

[37]  Ruo Li,et al.  Moving Mesh Finite Element Methods for the Incompressible Navier-Stokes Equations , 2005, SIAM J. Sci. Comput..

[38]  W. Shyy,et al.  Regular Article: An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries , 1999 .

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

[40]  Z. J. Wang,et al.  A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow , 2003 .

[41]  Jörg Peters,et al.  Fast Iterative Solvers for Discrete Stokes Equations , 2005, SIAM J. Sci. Comput..

[42]  Boo Cheong Khoo,et al.  An indirect boundary element method for three-dimensional explosion bubbles , 2004 .

[43]  Odd M. Faltinsen,et al.  A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries , 2008, J. Comput. Phys..

[44]  Maxim A. Olshanskii,et al.  Effective preconditioning of Uzawa type schemes for a generalized Stokes problem , 2000, Numerische Mathematik.

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

[46]  S. Green,et al.  Simulating the Motion of Flexible Pulp Fibres Using the Immersed Boundary Method , 1998 .

[47]  Tim Colonius,et al.  The immersed boundary method: A projection approach , 2007, J. Comput. Phys..

[48]  Anita T. Layton.  An efficient numerical method for the two-fluid Stokes equations with a moving immersed boundary , 2008 .

[49]  Charles S. Peskin,et al.  Stability and Instability in the Computation of Flows with Moving Immersed Boundaries: A Comparison of Three Methods , 1992, SIAM J. Sci. Comput..

[50]  Dongho Shin,et al.  Fast solvers for finite difference approximations for the stokes and navier-stokes equations , 1996, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[51]  H. S. Udaykumar,et al.  A Sharp Interface Cartesian Grid Methodfor Simulating Flows with ComplexMoving Boundaries , 2001 .

[52]  Aaron L. Fogelson,et al.  Continuum models of platelet aggregation: formulation and mechanical properties , 1992 .

[53]  Aaron L. Fogelson,et al.  Computational Methods for Continuum Models of Platelet Aggregation , 1999 .

[54]  H. Fasel,et al.  A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains , 2005 .

[55]  A. Fogelson.  A MATHEMATICAL MODEL AND NUMERICAL METHOD FOR STUDYING PLATELET ADHESION AND AGGREGATION DURING BLOOD CLOTTING , 1984 .

[56]  P. Swarztrauber,et al.  Efficient Fortran subprograms for the solution of separable elliptic partial differential equations , 1979 .

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

[58]  Z. Jane Wang,et al.  An immersed interface method for simulating the interaction of a fluid with moving boundaries , 2006, J. Comput. Phys..