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An adaptive version of the immersed boundary method. (English) Zbl 0953.76069

Summary: We present a computational setting for the immersed boundary method employing an adaptive mesh refinement. Enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested, progressively finer rectangular grid patches which dynamically follow the immersed boundary motion. The set of equations describing the interaction between a non-stationary, viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method, specially designed for locally refined meshes, to an implicit formulation of the immersed boundary method.
The main contributions of this work concern the formulation and implementation of a multilevel self-adaptive version of the immersed boundary method on locally refined meshes. This approach is tested on a particular two-dimensional model problem, for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary, and on the associated uniform mesh, built with the resolution of the finest level.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] Agresar, G., A Computational Environment for the Study of Circulating Cell Mechanics and Adhesion (1996)
[2] Agresar, G.; Linderman, J. J.; Tryggvason, G.; Powell, K. G., An adaptive, cartesian, front-tracking method for the motion, deformation and adhesion of circulating cells, J. Comput. Phys., 143, 346 (1998) · Zbl 0935.76047
[4] Bell, J.; Marcus, D. L., A second-order projection method for variable density flows, J. Comput. Phys., 101, 334 (1992) · Zbl 0759.76045
[5] Bell, J.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257 (1989) · Zbl 0681.76030
[6] Berger, M. J., Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations (1982)
[7] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64 (1989) · Zbl 0665.76070
[8] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partical differential equations, J. Comput. Phys., 53, 484 (1984) · Zbl 0536.65071
[9] Berger, M. J.; Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE Trans. Systems, Man, Cybernet., 21, 1278 (September/October 1991)
[10] Beyer, R. P., A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98, 145 (1992) · Zbl 0744.76128
[11] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 745 (1968) · Zbl 0198.50103
[12] Chorin, A. J., On the convergence of discrete approximations to the Navier-Stokes equations, Comm. Pure Appl. Math., 23, 341 (1969) · Zbl 0184.20103
[14] Fauci, L. J., Interaction of oscillating filaments—A computational study, J. Comput. Phys., 86, 294 (1990) · Zbl 0682.76105
[15] Fogelson, A. L., A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56, 111 (1984) · Zbl 0558.92009
[17] Haj-Hariri, H.; Shi, Q.; Borhan, A., Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers, Phys. Fluids, 9, 845 (1997)
[18] Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluids with free surfaces, Phys. Fluids, 8, 251 (1965)
[19] Howell, L. H.; Bell, J. B., An adaptive mesh projection method for viscous incompressible flow, SIAM J. Sci. Comput., 18, 996 (July 1997)
[20] LeVeque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019 (August 1994) · Zbl 0811.65083
[21] LeVeque, R. J.; Li, Z., Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension (February 1995)
[22] Li, Z., The Immersed Interface Method—A Numerical Approach for Partial Differential Equations with Interfaces (1994)
[23] Li, Z., Immersed Interface Method for Moving Interface Problems (May 1995)
[24] Mayo, A. A.; Peskin, C. S., An implicit numerical method for fluid dynamics problems with immersed elastic boundaries, Contemp. Math., 141, 261 (1993) · Zbl 0787.76055
[25] Minion, M. L., Two Methods for the Study of Vortex Patch Evolution on Locally Refined Grids (May 1994)
[26] Peskin, C. S., Flow Patterns around Heart Valves: A Digital Computer Method for Solving the Equations of Motion (July 1972)
[27] Peskin, C. S., Flow patterns around heart valves: A numerical method, J. Comput. Phys., 10, 252 (1972) · Zbl 0244.92002
[28] Peskin, C. S.; McQueen, D. M., A three-dimensional computational method for blood flow in the heart. I. Immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys., 81, 372 (1989) · Zbl 0668.76159
[29] Peskin, C. S.; McQueen, D. M., Computational biofluid dynamics, Contemp. Math., 141, 161 (1993) · Zbl 0786.76108
[31] Peyret, R.; Taylor, T. D., Computational Methods for Fluid Flow (1990) · Zbl 0717.76003
[32] Printz, B. F., Computer Modeling of Blood Flow through the Heart during the Complete Cardiac Cycle (1992)
[33] Quirk, J. J., An Adaptive Mesh Refinement Algorithm for Computational Shock Hydrodynamics (1991)
[34] Roma, A. M., A Multilevel Self-Adaptive Version of the Immersed Boundary Method (January 1996)
[35] Rosar, M. E., A Three-Dimensional Computer Model for Fluid Flow through a Collapsible Tube (June 1994)
[38] Tu, C.; Peskin, C. S., Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods, SIAM J. Sci. Statist. Comput., 13, 1361 (1992) · Zbl 0760.76067
[39] Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible flows, J. Comput. Phys., 100, 25 (1992) · Zbl 0758.76047
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