zbMATH — the first resource for mathematics

A fixed-mesh method for incompressible flow-structure systems with finite solid deformations. (English) Zbl 1329.74313
Summary: A fixed-mesh algorithm is proposed for simulating flow-structure interactions such as those occurring in biological systems, in which both the fluid and solid are incompressible and the solid deformations are large. Several of the well-known difficulties in simulating such flow-structure interactions are avoided by formulating a single set of equations of motion on a fixed Eulerian mesh. The solid’s deformation is tracked to compute elastic stresses by an overlapping Lagrangian mesh. In this way, the flow-structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system. These forces, which depend on the solid elastic stress distribution, are computed on the Lagrangian mesh by a standard finite-element method and then transferred to the fixed Eulerian mesh, where the joint momentum and continuity equations are solved by a finite-difference method. The constitutive model for the solid can be quite general. For the force transfer, standard immersed-boundary and immersed-interface methods can be used and are demonstrated. We have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum; the interface is still effectively sharp for this approach. The spatial convergence of the method is observed to be between first- and second-order, as in most immersed-boundary methods for membrane flows. The algorithm is demonstrated by the simulations of an advected elastic disk, a flexible leaflet in an oscillating flow, and a model of a swimming jellyfish.

74S30 Other numerical methods in solid mechanics (MSC2010)
76M25 Other numerical methods (fluid mechanics) (MSC2010)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI
[1] 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, 2, 346-380, (1998) · Zbl 0935.76047
[2] S. Bayyuk, K.G. Powell, B. van Leer, A simulation technique for 2-D unsteady inviscid flows around arbitrary moving and deforming bodies of arbitrary geometry, in: AIAA 11th Computational Fluid Dynamics Conference, Orlando, Florida, July 6-9 1993, pp. 1013-1024.
[3] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. comput. phys., 124, 2, 449-464, (1996) · Zbl 0847.76048
[4] Cummings, J.; Aivazis, M.; Samtaney, R.; Radovitzky, R.; Mauch, S.; Meiron, D., A virtual test facility for the simulation of dynamic response in materials, J. supercomput., 23, 39-59, (2002) · Zbl 0994.68536
[5] Dabiri, J.O.; Colon, S.P.; Costello, J.H.; Charib, M., Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analysis, J. exp. biol., 208, 1257-1265, (2005)
[6] Dasgupta, G., Integration within polygonal finite elements, J. aerospace eng., 9, (2003)
[7] Deiterding, R.; Radovitzky, R.; Mauch, S.P.; Noels, L.; Cummings, J.C.; Meiron, D.I., A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading, Eng. comput., 22, 325-347, (2006)
[8] Dunne, T., An Eulerian approach to fluid – structure interaction and goal-oriented mesh adaptation, Int. J. numer. methods fluids, 51, 1017-1039, (2006) · Zbl 1158.76400
[9] Fauci, L.J.; Peskin, C.S., A computational model of aquatic animal locomotion, J. comput. phys., 77, 1, 85-108, (1988) · Zbl 0641.76140
[10] Freund, J.B., Leukocyte margination in a model microvessel, Phys. fluids, 19, 023301, (2007) · Zbl 1146.76385
[11] Glowinski, R.; Pan, T.W.; Hesla, T.I.; Joseph, D.D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 25, 755-794, (1999) · Zbl 1137.76592
[12] Glowinski, R.; Pan, T.W.; Hesla, T.I.; Joseph, D.D.; Périaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys., 169, 363-426, (2001) · Zbl 1047.76097
[13] Goda, K., A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows, J. comput. phys., 30, 76-95, (1979) · Zbl 0405.76017
[14] J.A. Greenough, V. Beckner, R.B. Pember, W.Y. Crutchfield, J.B. Bell, P. Colella, An adaptive multifluid interface-capturing method for compressible flow in complex geometries, in: AIAA 26th Computational Fluid Dynamics Conference, San Diego, California, 1995.
[15] Hackbusch, W., Coupled problems in microsystem technology, ()
[16] Harlow, F.H.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. fluids, 8, 12, 2182-2189, (1965) · Zbl 1180.76043
[17] Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. comput. phys., 14, 3, 227-253, (1974) · Zbl 0292.76018
[18] D.J. Holdych, D. Noble, R.B. Secor, Quadrature rules for triangular and tetrahedral elements with generalized functions, Int. J. Numer. Methods Eng., in press, doi:10.1002/nme.2123. · Zbl 1167.74043
[19] Hu, H.H.; Joseph, D.D.; Crochet, M.J., Direct simulation of fluid particle motions, Theor. comp. fluid dyn., 3, 5, 285-306, (1992) · Zbl 0754.76054
[20] Hu, H.H.; Patankar, N.A.; Zhu, M.Y., Direct numerical simulations of fluid – solid systems using the arbitrary lagrangian – eulerian technique, J. comput. phys., 169, 2, 427-462, (2001) · Zbl 1047.76571
[21] Hughes, T.J., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs · Zbl 0634.73056
[22] Jaiman, R.K.; Jiao, X.; Geubelle, P.H.; Loth, E., Assessment of conservative load transfer for fluid – solid interface with non-matching meshes, Int. J. numer. methods eng., 64, 2014-2038, (2005) · Zbl 1122.74544
[23] Jaiman, R.K.; Jiao, X.; Geubellec, P.H.; Loth, E., Conservative load transfer along curved fluid – solid interface with non-matching meshes, J. comput. phys., 218, 1, 372-397, (2006) · Zbl 1158.76405
[24] Jiao, X.; Heath, M.T., Common-refinement based data transfer between nonmatching meshes in multiphysics simulations, Int. J. numer. methods eng., 14, 6, 379-402, (2004)
[25] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. comput. phys., 171, 1, 132-150, (2001) · Zbl 1057.76039
[26] Kim, J.; Moin, P., Application of a fractional-step method to incompressible navier – stokes equations, J. comput. phys., 59, 308-323, (1985) · Zbl 0582.76038
[27] Lai, M.; Peskin, C.S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. comput. phys., 160, 2, 705-719, (2000) · Zbl 0954.76066
[28] L. Lee, Immersed Interface Methods for Incompressible Flow with Moving Interfaces, Ph.D. Thesis, Department of Applied Mathematics, University of Washington, 2002.
[29] Lee, L.; LeVeque, R.J., An immersed interface method for incompressible navier – stokes equations, SIAM J. sci. comput., 25, 3, 832-856, (2003) · Zbl 1163.65322
[30] LeVeque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 4, 1019-1044, (1994) · Zbl 0811.65083
[31] LeVeque, R.J.; Li, Z., Immersed interface method for Stokes flow with elastic boundaries or surface tension, SIAM J. numer. anal., 18, 3, 709-735, (1997) · Zbl 0879.76061
[32] Li, Z., Immersed interface methods for moving interface problems, Numer. algorithms, 14, 269-293, (1997) · Zbl 0886.65096
[33] Liu, W.K.; Liu, Y.; Farrell, D.; Zhang, L.; Wang, X.S.; Fukui, Y.; Patankar, N.; Zhang, Y.; Bajaj, C., Immersed finite element method and its applications to biological systems, Comput. methods appl. mech. eng., 195, 1722-1749, (2006) · Zbl 1178.76232
[34] Losasso, F.; Gibou, F.; Fedkiw, R., Simulating water and smoke with an octree data structure, ACM tog, 23, 457-462, (2004)
[35] Mayao, 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
[36] McCorquodale, P.; Colella, P.; Johansen, H.A., Cartesian grid embedded boundary method for the heat equation on irregular domains, J. comput. phys., 173, 620-635, (2001) · Zbl 0991.65099
[37] McOwen, R.C., Partial differential equations: methods and equations, (1996), Prentice Hall · Zbl 0849.35001
[38] Min, C.H.; Gibou, F., Geometric integration over irregular domains with application to level set methods, J. comput. phys., 226, 2, 1432-1443, (2007) · Zbl 1125.65021
[39] Olśhanskii, M.A.; Staroverov, V.M., On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid, Int. J. numer. methods fluids, 33, 499-534, (2000) · Zbl 0959.76062
[40] Peskin, C.S., Flow patterns around heart valves: a numerical method, J. comput. phys., 10, 2, 252-271, (1972) · Zbl 0244.92002
[41] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 3, 220-252, (1977) · Zbl 0403.76100
[42] Pozrikidis, C., Boundary integral and singularity methods for linearized viscous flow, (1992), Cambridge University Press Cambridge · Zbl 0772.76005
[43] A.M. Roma, A Multilevel Self-adaptive Version of the Immersed Boundary Method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996.
[44] Roma, A.M.; Peskin, C.S.; Berger, M.J., An adaptive version of the immersed boundary method, J. comput. phys., 153, 2, 509-534, (1999) · Zbl 0953.76069
[45] Spalart, P.R.; Moser, R.D.; Rogers, M.M., Spectral methods for the navier – stokes equations with one infinite and two periodic directions, J. comput. phys., 96, 2, 297-324, (1991) · Zbl 0726.76074
[46] Sulsky, D.; Brackbill, J.U., A numerical method for suspension flow, J. comput. phys., 96, 2, 339-368, (1991) · Zbl 0727.76082
[47] Sulsky, D.; Chen, Z.; Schreyer, H.L., A particle method for history-dependent materials, Comput. methods appl. mech. eng., 118, 179-196, (1994) · Zbl 0851.73078
[48] Taira, K.; Colonius, T., The immersed boundary method: a projection approach, J. comput. phys., 225, 2, 2118-2137, (2007) · Zbl 1343.76027
[49] Udaykumar, H.S.; Mittal, R.; Rampunggoon, P.; Khanna, A., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. comput. phys., 174, 1, 380, (2001) · Zbl 1106.76428
[50] Udaykumar, H.S.; Tran, L.; Belk, D.M.; Vanden, K.J., An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces, J. comput. phys., 186, 1, 136-177, (2003) · Zbl 1047.76558
[51] Verstappen, R.W.C.P.; Veldman, A.E.P., Symmetry-preserving discretization of turbulent flow, J. comput. phys., 187, 1, 343-368, (2003) · Zbl 1062.76542
[52] Wang, X.; Liu, W.K., Extended immersed boundary method using FEM and RKPM, Comput. methods appl. mech. eng., 193, 1305-1321, (2004) · Zbl 1060.74676
[53] Xu, S.; Wang, Z.J., Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation, SIAM J. sci. comput., 27, 6, 1948-1980, (2006) · Zbl 1136.76346
[54] Yu, Z., A DLM/FD method for fluid/flexible-body interactions, J. comput. phys., 207, 1-27, (2005) · Zbl 1177.76304
[55] H. Zhao, A Fixed-mesh Flow-Structure Solver for Biological Systems with Large Solid Deformations, Ph.D. Thesis, Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 2006.
[56] X. Zhong, A new higher-order immersed interface method for multi-phase flow simulation, AIAA Paper, 2006-1294, 2006.
[57] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates part 1: the recovery technique, Int. J. numer. methods eng., 333, 7, 1331-1364, (1992) · Zbl 0769.73084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.