×

zbMATH — the first resource for mathematics

Fluid-structure interaction with applications in biomechanics. (English) Zbl 1145.74010
Summary: In bioengineering applications problems of flow interacting with elastic solid are very common. We formulate the problem of interaction for an incompressible fluid and an incompressible elastic material in a fully coupled arbitrary Lagrangian-Eulerian formulation. The mathematical description and numerical schemes are designed in such a way that more complicated constitutive relations (and more realistic for bioengineering applications) can be incorporated easily. The whole domain of interest is treated as one continuum, and the same discretization in space (FEM) and time (Crank-Nicholson) is used for both, solid and fluid, parts. The resulting nonlinear algebraic system is solved by an approximate Newton method. The combination of second-order discretization and fully coupled solution method gives a method with high accuracy and robustness. To demonstrate the flexibility of this numerical approach, we apply the same method to a mixture-based model of an elastic material with perfusion which also falls into the category of fluid-structure interactions. A few simple example calculations with simple material models and large deformations of the solid part are presented.

MSC:
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74L15 Biomechanical solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
92C10 Biomechanics
Software:
PETSc; SPLIB
PDF BibTeX Cite
Full Text: DOI
References:
[1] Almeida, E.S.; Spilker, R.L., Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues, Comput. meth. appl. mech. eng., 151, 513-538, (1998) · Zbl 0920.73350
[2] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, G.M. Knepley, L.C. McInnes, F.B. Smith, H. Zhang, PETSc: Portable Extensible toolkit for Scientific computation \(\langle\)http://www.mcs.anl.gov/pets⟩.
[3] Barrett, R.; Berry, M.; Chan, T.F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; Van der Vorst, H., Templates for the solution of linear systems: building blocks for iterative methods, (1994), SIAM Philadelphia, PA
[4] R. Bramley, X. Wang, SPLIB: A Library of Iterative Methods for Sparse Linear Systems, Department of Computer Science, Indiana University, Bloomington, IN, 1997 \(\langle\)http://www.cs.indiana.edu/ftp/bramley/splib.tar.gz⟩.
[5] Costa, K.D.; Hunter, P.J.; Rogers, J.M.; Guccione, J.M.; Waldman, L.K.; McCulloch, A.D., A three-dimensional finite element method for large elastic deformations of ventricular myocardum: I—cylindrical and spherical polar coordinates, Trans. ASME J. biomech. eng., 118, 4, 452-463, (1996)
[6] Costa, K.D.; Hunter, P.J.; Wayne, J.S.; Waldman, L.K.; Guccione, J.M.; McCulloch, A.D., A three-dimensional finite element method for large elastic deformations of ventricular myocardum: II—prolate spheroidal coordinates, Trans. ASME J. biomech. eng., 118, 4, 464-472, (1996)
[7] Dai, F.; Rajagopal, K.R., Diffusion of fluids through transversely isotropic solids, Acta mech., 82, 61-98, (1990) · Zbl 0717.76106
[8] Donzelli, P.S.; Spilker, R.L.; Baehmann, P.L.; Niu, Q.; Shephard, M.S., Automated adaptive analysis of the biphasic equations for soft tissue mechanics using a posteriori error indicators, Int. J. numer. methods eng., 34, 3, 1015-1033, (1992) · Zbl 0787.73063
[9] Farhat, C.; Lesoinne, M.; Maman, N., Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution, Int. J. numer. methods fluids, 21, 10, 807-835, (1995), Finite Element Methods in Large-scale Computational Fluid Dynamics, Tokyo, 1994 · Zbl 0865.76038
[10] Fung, Y.C., Biomechanics: mechanical properties of living tissues, (1993), Springer New York, NY
[11] Geller, S.; Krafczyk, M.; Tölke, J.; Turek, S.; Hron, J., Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows, Comput. & fluids, 35, 888-897, (2006) · Zbl 1177.76313
[12] Gurtin, M.E., Topics in finite elasticity, (1981), SIAM Philadelphia, PA
[13] Heil, M., Stokes flow in collapsible tubes: computation and experiment, J. fluid mech., 353, 285-312, (1997) · Zbl 0922.76113
[14] Heil, M., Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem, Int. J. numer. methods fluids, 28, 2, 243-265, (1998) · Zbl 0916.73054
[15] Hron, J.; Humphrey, J.D.; Rajagopal, K.R., Material identification of nonlinear solids infused with a fluid, Math. mech. solids, 7, 6, 629-646, (2002) · Zbl 1045.74023
[16] Humphrey, J.D.; Strumpf, R.K.; Yin, F.C.P., Determination of a constitutive relation for passive myocardium: I. A new functional form, J. biomech. eng., 112, 3, 333-339, (1990)
[17] Humphrey, J.D.; Strumpf, R.K.; Yin, F.C.P., Determination of a constitutive relation for passive myocardium: II. parameter estimation, J. biomech. eng., 112, 3, 340-346, (1990)
[18] Koobus, B.; Farhat, C., Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes, Comput. methods appl. mech. eng., 170, 1-2, 103-129, (1999) · Zbl 0943.76055
[19] Kwan, M.K.; Lai, M.W.; Mow, V.C., A finite deformation theory for cartilage and other soft hydrated connective tissues—I, equilibrium results, J. biomech., 23, 2, 145-155, (1990)
[20] Le Tallec, P.; Mani, S., Numerical analysis of a linearised fluid-structure interaction problem, Numer. math., 87, 2, 317-354, (2000) · Zbl 0998.76050
[21] Levenston, M.E.; Frank, E.H.; Grodzinsky, A.J., Variationally derived 3-field finite element formulations for quasistatic poroelastic analysis of hydrated biological tissues, Comput. meth. appl. mech. eng., 156, 231-246, (1998) · Zbl 0963.74064
[22] Maurel, W.; Wu, Y.; Thalmann, N.M.; Thalmann, D., Biomechanical models for soft tissue simulation, ESPRIT basic research series, (1998), Springer Berlin
[23] Miga, M.I.; Paulsen, K.D.; Kennedy, F.E.; Hoopes, P.J.; Hartov, D.W.; Roberts, A., Modeling surgical loads to account for subsurface tissue deformation during stereotactic neurosurgery, (), 501-511
[24] Oomens, C.W.J.; van Campen, D.H., A mixture approach to the mechanics of skin, J. biomech., 20, 9, 877-885, (1987)
[25] Paulsen, K.D.; Miga, M.I.; Kennedy, F.E.; Hoopes, P.J.; Hartov, A.; Roberts, D.W., A computational model for tracking subsurface tissue deformation during stereotactic neurosurgery, IEEE trans. biomed. eng., 46, 2, 213-225, (1999)
[26] Penrose, J.M.T.; Staples, C.J., Implicit fluid-structure coupling for simulation of cardiovascular problems, Int. J. numer. methods fluids, 40, 467-478, (2002) · Zbl 1042.76071
[27] Peskin, C.S., Numerical analysis of blood flow in the heart, J. computat. phys., 25, 3, 220-252, (1977) · Zbl 0403.76100
[28] C.S. Peskin, The fluid dynamics of heart valves: experimental, theoretical, and computational methods, in: Annual Review of Fluid Mechanics, vol. 14, Annual Reviews, Palo Alto, CA, 1982, pp. 235-259. · Zbl 0488.76129
[29] Peskin, C.S.; McQueen, D.M., Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. comput. phys., 37, 1, 113-132, (1980) · Zbl 0447.92009
[30] 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, 2, 372-405, (1989) · Zbl 0668.76159
[31] Quarteroni, A., Modeling the cardiovascular system: a mathematical challenge, (), 961-972 · Zbl 1036.92013
[32] Quarteroni, A.; Tuveri, M.; Veneziani, A., Computational vascular fluid dynamics: problems, models and methods, Comput. visual. sci., 2, 4, 163-197, (2000) · Zbl 1096.76042
[33] Rajagopal, K.R.; Tao, L., Mechanics of mixtures, (1995), World Scientific Publishing Co. Inc. River Edge, NJ · Zbl 0941.74500
[34] Reynolds, R.A.; Humphrey, J.D., Steady diffusion within a finitely extended mixture slab, Math. mech. solids, 3, 2, 127-147, (1998) · Zbl 1001.74549
[35] Rumpf, M., On equilibria in the interaction of fluids and elastic solids, (), 136-158 · Zbl 0930.35176
[36] Sackinger, P.A.; Schunk, P.R.; Rao, R.R., A newton – raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation, J. comput. phys., 125, 1, 83-103, (1996) · Zbl 0853.65138
[37] J.J. Shi, Application of theory of a Newtonian fluid and an isotropic non-linear elastic solid to diffusion problems, Ph.D. Thesis, University of Michigan, Ann Arbor, 1973.
[38] Spilker, R.L.; Suh, J.K., Formulation and evaluation of a finite element model for the biphasic model of hydrated soft tissues, Comput. struct., front. comput. mech., 35, 4, 425-439, (1990) · Zbl 0727.73061
[39] Spilker, R.L.; Suh, J.K.; Mow, V.C., Finite element formulation of the nonlinear biphasic model for articular cartilage and hydrated soft tissues including strain-dependent permeability, Comput. meth. bioeng., 9, 81-92, (1988)
[40] Suh, J.K.; Spilker, R.L.; Holmes, M.H., Penalty finite element analysis for non-linear mechanics of biphasic hydrated soft tissue under large deformation, Int. J. numer. methods eng., 32, 7, 1411-1439, (1991) · Zbl 0763.73057
[41] C. Truesdell, A First Course in Rational Continuum Mechanics, vol. 1, second ed., Academic Press, Boston, MA, 1991.
[42] Vankan, W.J.; Huyghe, J.M.; Janssen, J.D.; Huson, A., Poroelasticity of saturated solids with an application to blood perfusion, Int. J. eng. sci., 34, 9, 1019-1031, (1996) · Zbl 0899.73443
[43] Vankan, W.J.; Huyghe, J.M.; Janssen, J.D.; Huson, A., Finite element analysis of blood flow through biological tissue, Int. J. eng. sci., 35, 4, 375-385, (1997) · Zbl 0899.73534
[44] Vermilyea, M.E.; Spilker, R.L., Hybrid and mixed-penalty finite elements for 3-d analysis of soft hydrated tissue, Int. J. numer. meth. eng., 36, 24, 4223-4243, (1993) · Zbl 0799.76048
[45] Zoppou, C.; Barry, S.I.; Mercer, G.N., Dynamics of human milk extraction: a comparative study of breast-feeding and breast pumping, Bull. math. biol., 59, 5, 953-973, (1997) · Zbl 0897.92010
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.