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A NURBS-based immersed methodology for fluid-structure interaction. (English) Zbl 1423.74261

Summary: We introduce an isogeometric, immersed, and fully-implicit formulation for fluid-structure interaction (FSI). The method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids, which are a common case in various fields, such as, for example, biomechanics. In our FSI method, we utilize an Eulerian mesh on the whole domain and a Lagrangian mesh on the solid domain. The Lagrangian mesh is arbitrarily located on top of the Eulerian mesh in a non-conforming fashion. Due to the formulation of our problem, based on the Immersed Finite Element Method (IFEM), we do not need mesh update or remeshing algorithms. The fluid-structure interface is the boundary of the Lagrangian mesh, but cuts arbitrarily the Eulerian mesh. The generalized-\( \alpha\) method is used for time discretization and NURBS-based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains. The information transfer between the two meshes is carried out using the NURBS functions, which avoids the use of the so-called discretized delta functions. The higher order and especially the higher continuity of NURBS functions allow us to deal with severe mesh distortion in the Lagrangian mesh in comparison with classical \(\mathcal{C}^0\) linear piecewise functions as we prove numerically. Our numerical solutions attain good agreement with theoretical data for free-falling objects in two and three dimensions, which confirms the feasibility of our methodology.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
65D17 Computer-aided design (modeling of curves and surfaces)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74D10 Nonlinear constitutive equations for materials with memory
76D99 Incompressible viscous fluids

Software:

PETSc; ISOGAT; PetIGA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E., Computational fluid-structure interaction: methods and applications, (2012), John Wiley & Sons · Zbl 1286.74001
[2] Zhang, Y.; Bazilevs, Y.; Goswami, S.; Bajaj, C.; Hughes, T. J.R., Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow, Comput. Methods Appl. Mech. Engrg., 196, 2943-2959, (2007) · Zbl 1121.76076
[3] Bazilevs, Y.; Hsu, M.-C.; Zhang, Y.; Wang, W.; Kvamsdal, T.; Hentschel, S.; Isaksen, J., Computational vascular fluid-structure interaction: methodology and application to cerebral aneurysms, Biomech. Model. Mechanobiol., 9, 481-498, (2010)
[4] Takizawa, K.; Tezduyar, T. E., Computational methods for parachute fluid-structure interactions, Arch. Comput. Methods Eng., 19, 125-169, (2012) · Zbl 1354.76113
[5] Takizawa, K.; Henicke, B.; Tezduyar, T. E.; Hsu, M.-C.; Bazilevs, Y., Stabilized space-time computation of wind-turbine rotor aerodynamics, Comput. Mech., 48, 333-344, (2011) · Zbl 1398.76127
[6] Liu, W. K.; Liu, Y.; Farrell, D.; Zhang, L.; Wang, X.; Fukui, Y.; Patankar, N.; Zhang, Y.; Bajaj, C.; Lee, J.; Hong, J.; Chen, X.; Hsu, H., Immersed finite element method and its applications to biological systems, Comput. Methods Appl. Mech. Engrg., 195, 1722-1749, (2006) · Zbl 1178.76232
[7] Felippa, C. A.; Park, K.; Farhat, C., Partitioned analysis of coupled mechanical systems, Comput. Methods Appl. Mech. Engrg., 190, 3247-3270, (2001) · Zbl 0985.76075
[8] Küttler, U.; Wall, W. A., Fixed-point fluid-structure interaction solvers with dynamic relaxation, Comput. Mech., 43, 61-72, (2008) · Zbl 1236.74284
[9] Brummelen, E. H.; Geuzaine, P., Fundamentals of fluid-structure interaction, (2010), Wiley Online Library
[10] Badia, S.; Quaini, A.; Quarteroni, A., Splitting methods based on algebraic factorization for fluid-structure interaction, SIAM J. Sci. Comput., 30, 1778-1805, (2008) · Zbl 1368.74021
[11] Bueno, J.; Bona-Casas, C.; Bazilevs, Y.; Gomez, H., Interaction of complex fluids and solids: theory, algorithms and application to phase-change-driven implosion, Comput. Mech., (2014), in press · Zbl 1325.76114
[12] J. Donéa, P. Fasoli-Stella, S. Giuliani, Lagrangian and Eulerian finite element techniques for transient fluid-structure interaction problems, in: Transactions of the 4th International Conference on Structural Mechanics in Reactor Technology Volume B: Thermal and Fluid/Structure Dynamics Analysis B(1/2), 1977, pp. 1-12.
[13] Belytschko, T.; Kennedy, J. M., Computer models for subassembly simulation, Nucl. Eng. Des., 49, 17-38, (1978)
[14] Hughes, T. J.R.; Liu, W. K.; Zimmermann, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29, 329-349, (1981) · Zbl 0482.76039
[15] Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms, and computations, Comput. Mech., 43, 3-37, (2008) · Zbl 1169.74015
[16] Peskin, C., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 252-271, (1972) · Zbl 0244.92002
[17] McQueen, D.; Peskin, C., Computer-assisted design of butterfly bileaflet valves for the mitral position, Scand. J. Thorac. Cardiovasc. Surg., 19, 139-148, (1985)
[18] Beyer, R., A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98, 145-162, (1992) · Zbl 0744.76128
[19] Dillon, R.; Fauci, L.; Fogelson, A.; Gaver, D., Modeling biofilm processes using the immersed boundary method, J. Comput. Phys., 129, 57-73, (1996) · Zbl 0867.76100
[20] Fauci, L.; Peskin, C., A computational model of aquatic animal locomotion, J. Comput. Phys., 77, 85-108, (1988) · Zbl 0641.76140
[21] Zhang, L.; Gerstenberger, A.; Wang, X.; Liu, W. K., Immersed finite element method, Comput. Methods Appl. Mech. Engrg., 193, 2051-2067, (2004) · Zbl 1067.76576
[22] Liu, W. K.; Kim, D.; Tang, S., Mathematical foundations of the immersed finite element method, Comput. Mech., 39, 211-222, (2007) · Zbl 1178.74170
[23] Gay, M.; Zhang, L.; Liu, W. K., Stent modeling using immersed finite element method, Comput. Methods Appl. Mech. Engrg., 195, 4358-4370, (2006) · Zbl 1175.74081
[24] Liu, Y.; Zhang, L.; Wang, X.; Liu, W. K., Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics, Internat. J. Numer. Methods Fluids, 46, 1237-1252, (2004) · Zbl 1135.92302
[25] Liu, Y.; Liu, W. K., Rheology of red blood cell aggregation by computer simulation, J. Comput. Phys., 220, 139-154, (2006) · Zbl 1102.92010
[26] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[27] Gomez, H.; Calo, V. M.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197, 4333-4352, (2008) · Zbl 1194.74524
[28] Gomez, H.; Hughes, T. J.; Nogueira, X.; Calo, V. M., Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations, Comput. Methods Appl. Mech. Engrg., 199, 1828-1840, (2010) · Zbl 1231.76191
[29] Gomez, H.; París, J., Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinsky equation, Phys. Rev. E, 83, 046702, (2011)
[30] Gomez, H.; Nogueira, X., A new space-time discretization for the Swift-Hohenberg equation that strictly respects the Lyapunov functional, Commun. Nonlinear Sci. Numer. Simul., 17, 4930-4946, (2012) · Zbl 1352.76109
[31] Gomez, H.; Nogueira, X., An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249, 52-61, (2012) · Zbl 1348.74280
[32] Thiele, U.; Archer, A. J.; Robbins, M. J.; Gomez, H.; Knobloch, E., Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity, Phys. Rev. E, 87, 042915, (2013)
[33] Gomez, H.; Cueto-Felgueroso, L.; Juanes, R., Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium, J. Comput. Phys., 238, 217-239, (2013)
[34] Dedè, L.; Borden, M. J.; Hughes, T. J., Isogeometric analysis for topology optimization with a phase field model, Arch. Comput. Methods Eng., 19, 427-465, (2012) · Zbl 1354.74224
[35] Kiendl, J.; Bletzinger, K.-U.; Linhard, J.; Wüchner, R., Isogeometric shell analysis with Kirchhoff-love elements, Comput. Methods Appl. Mech. Engrg., 198, 3902-3914, (2009) · Zbl 1231.74422
[36] Lipton, S.; Evans, J.; Bazilevs, Y.; Elguedj, T.; Hughes, T. J.R., Robustness of isogeometric structural discretizations under severe mesh distortion, Comput. Methods Appl. Mech. Engrg., 199, 357-373, (2010) · Zbl 1227.74112
[37] Akkerman, I.; Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41, 371-378, (2008) · Zbl 1162.76355
[38] Evans, J. A.; Bazilevs, Y.; Babuška, I.; Hughes, T. J., \(n\)-widths, sup-infs, and optimality ratios for the \(k\)-version of the isogeometric finite element method, Comput. Methods Appl. Mech. Engrg., 198, 1726-1741, (2009) · Zbl 1227.65093
[39] Bazilevs, Y.; Calo, V. M.; Zhang, Y.; Hughes, T. J.R., Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput. Mech., 38, 310-322, (2006) · Zbl 1161.74020
[40] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Hughes, T. J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197, 173-201, (2007) · Zbl 1169.76352
[41] Evans, J. A.; Hughes, T. J., Isogeometric divergence-conforming \(b\)-splines for the unsteady Navier-Stokes equations, J. Comput. Phys., 241, 141-167, (2013) · Zbl 1349.76054
[42] Gomez, H.; Hughes, T. J.R., Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230, 5310-5327, (2011) · Zbl 1419.76439
[43] Dhote, R.; Gomez, H.; Melnik, R.; Zu, J., Isogeometric analysis of a dynamic thermo-mechanical phase-field model applied to shape memory alloys, Comput. Mech., 1-16, (2013) · Zbl 1398.74322
[44] Vilanova, G.; Colominas, I.; Gomez, H., Capillary networks in tumor angiogenesis: from discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis, Int. J. Numer. Methods Biomed. Eng., 29, 1015-1037, (2013)
[45] Vilanova, G.; Colominas, I.; Gomez, H., Coupling of discrete random walks and continuous modeling for three-dimensional tumor-induced angiogenesis, Comput. Mech., 1-16, (2013)
[46] Liu, J.; Gomez, H.; Evans, J. A.; Hughes, T. J.; Landis, C. M., Functional entropy variables: a new methodology for deriving thermodynamically consistent algorithms for complex fluids, with particular reference to the isothermal Navier-Stokes-Korteweg equations, J. Comput. Phys., 248, 47-86, (2013) · Zbl 1349.76237
[47] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Engrg., 195, 5257-5296, (2006) · Zbl 1119.74024
[48] Hughes, T. J.; Reali, A.; Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of \(p\)-method finite elements with \(k\)-method NURBS, Comput. Methods Appl. Mech. Engrg., 197, 49, 4104-4124, (2008) · Zbl 1194.74114
[49] Elguedj, T.; Bazilevs, Y.; Calo, V. M.; Hughes, T. J., \(\overline{B}\) and \(\overline{F}\) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements, Comput. Methods Appl. Mech. Engrg., 197, 2732-2762, (2008) · Zbl 1194.74518
[50] Benson, D.; Bazilevs, Y.; Hsu, M.; Hughes, T. J.R., Isogeometric shell analysis: the Reissner-Mindlin shell, Comput. Methods Appl. Mech. Engrg., 199, 276-289, (2010) · Zbl 1227.74107
[51] Kiendl, J.; Bazilevs, Y.; Hsu, M.-C.; Wüchner, R.; Bletzinger, K.-U., The bending strip method for isogeometric analysis of Kirchhoff-love shell structures comprised of multiple patches, Comput. Methods Appl. Mech. Engrg., 199, 2403-2416, (2010) · Zbl 1231.74482
[52] De Lorenzis, L.; Temizer, I.; Wriggers, P.; Zavarise, G., A large deformation frictional contact formulation using NURBS-based isogeometric analysis, Internat. J. Numer. Methods Engrg., 87, 1278-1300, (2011) · Zbl 1242.74104
[53] Dimitri, R.; De Lorenzis, L.; Scott, M.; Wriggers, P.; Taylor, R.; Zavarise, G., Isogeometric large deformation frictionless contact using \(t\)-splines, Comput. Methods Appl. Mech. Engrg., 269, 394-414, (2014) · Zbl 1296.74071
[54] Dimitri, R.; De Lorenzis, L.; Wriggers, P.; Zavarise, G., NURBS-and \(t\)-spline-based isogeometric cohesive zone modeling of interface debonding, Comput. Mech., 1-20, (2014) · Zbl 1398.74323
[55] Wall, W.; Frenzel, M.; Cyron, C., Isogeometric structural shape optimization, Comput. Methods Appl. Mech. Engrg., 197, 2976-2988, (2008) · Zbl 1194.74263
[56] Buffa, A.; Sangalli, G.; Vázquez, R., Isogeometric analysis in electromagnetics: \(B\)-splines approximation, Comput. Methods Appl. Mech. Engrg., 199, 1143-1152, (2010) · Zbl 1227.78026
[57] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis toward integration of CAD and FEA, (2009), Wiley · Zbl 1378.65009
[58] Hesch, C.; Gil, A.; Arranz Carreño, A.; Bonet, J.; Betsch, P., A mortar approach for fluid-structure interaction problems: immersed strategies for deformable and rigid bodies, Comput. Methods Appl. Mech. Engrg., 278, 853-882, (2014) · Zbl 1423.74889
[59] Hughes, T. J., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127, 387-401, (1995) · Zbl 0866.76044
[60] Hughes, T. J.R.; Feijóo, G.; Mazzei, L.; Quincy, J.-B., The variational multiscale method—a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166, 3-24, (1998) · Zbl 1017.65525
[61] Hughes, T. J.R.; Sangalli, G., Variational multiscale analysis: the fine-scale green’s function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal., 45, 539-557, (2007) · Zbl 1152.65111
[62] Chung, J.; Hulbert, G., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-method, J. Appl. Mech., 60, 371-375, (1993) · Zbl 0775.73337
[63] Jansen, K.; Whiting, C.; Hulbert, G., Generalized-\(\alpha\) method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Comput. Methods Appl. Mech. Engrg., 190, 305-319, (2000) · Zbl 0973.76048
[64] Brooks, A. N.; Hughes, T. J., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259, (1982) · Zbl 0497.76041
[65] Marsden, J. E.; Hughes, T. J., Mathematical foundations of elasticity, (1994), Courier Dover Publications · Zbl 0545.73031
[66] Lions, P.-L., Mathematical topics in fluid mechanics. vol. 1. incompressible models, (1996), Oxford Science · Zbl 0866.76002
[67] Hsu, M.-C.; Kamensky, D.; Bazilevs, Y.; Sacks, M.; Hughes, T., Fluid-structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation, Comput. Mech., 54, 1055-1071, (2014) · Zbl 1311.74039
[68] Kamensky, D.; Hsu, M.-C.; Schillinger, D.; Evans, J. A.; Aggarwal, A.; Bazilevs, Y.; Sacks, M. S.; Hughes, T. J., An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves, Comput. Methods Appl. Mech. Engrg., 284, 1005-1053, (2015), Also available as ICES Report 14-12 · Zbl 1423.74273
[69] Simo, J. C.; Hughes, T. J.R., Computational inelasticity, (2008), Springer · Zbl 0934.74003
[70] Auricchio, F.; Da Veiga, L.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 2075-2107, (2010) · Zbl 1226.65091
[71] Schillinger, D.; Evans, J.; Reali, A.; Scott, M.; Hughes, T. J.R., Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Engrg., 267, 170-232, (2013) · Zbl 1286.65174
[72] Gomez, H.; Reali, A.; Sangalli, G., Accurate, efficient, and (iso) geometrically flexible collocation methods for phase-field models, J. Comput. Phys., 262, 153-171, (2014) · Zbl 1349.82084
[73] Gomez, H.; Reali, A., An isogeometric collocation approach for Bernoulli-Euler beams and krichhoff plates, Comput. Methods Appl. Mech. Engrg., 284, 623-636, (2015) · Zbl 1423.74553
[74] Codina, R.; Principe, J.; Guasch, O.; Badia, S., Time dependent subscales in the stabilized finite element approximation of incompressible flow problems, Comput. Methods Appl. Mech. Engrg., 196, 2413-2430, (2007) · Zbl 1173.76335
[75] De Boor, C., A practical guide to splines, vol. 27, (1978), Springer-Verlag New York · Zbl 0406.41003
[76] Jansen, K. E.; Collis, S. S.; Whiting, C.; Shaki, F., A better consistency for low-order stabilized finite element methods, Comput. Methods Appl. Mech. Engrg., 174, 153-170, (1999) · Zbl 0956.76044
[77] Hesch, C.; Gil, A.; Arranz Carreño, A.; Bonet, J., On continuum immersed strategies for fluid-structure interaction, Comput. Methods Appl. Mech. Engrg., 247-248, 51-64, (2012) · Zbl 1352.76055
[78] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869, (1986) · Zbl 0599.65018
[79] Dupont, T.; Kendall, R. P.; Rachford, H., An approximate factorization procedure for solving self-adjoint elliptic difference equations, SIAM J. Numer. Anal., 5, 559-573, (1968) · Zbl 0174.47603
[80] Oliphant, T. A., An implicit, numerical method for solving two-dimensional time-dependent diffusion problems, Quart. Appl. Math., 19, 221-229, (1961) · Zbl 0105.10702
[81] Chan, T. F.; Van der Vorst, H. A., Approximate and incomplete factorizations, (1997), Springer · Zbl 0865.65015
[82] N. Collier, L. Dalcin, V.M. Calo, PetIGA: high-performance isogeometric analysis, preprint arXiv:1305.4452.
[83] Bernal, M.; Calo, V. M.; Collier, N.; Espinosa, G.; Fuentes, F.; Mahecha, J., Isogeometric analysis of hyperelastic materials using petiga, Procedia Comput. Sci., 18, 1604-1613, (2013)
[84] Vignal, P. A.; Collier, N.; Calo, V. M., Phase field modeling using petiga, Procedia Comput. Sci., 18, 1614-1623, (2013)
[85] S. Balay, M.F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, K. Rupp, B.F. Smith, H. Zhang, PETSc Web Page, 2014. http://www.mcs.anl.gov/petsc.
[86] Balay, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Rupp, K.; Smith, B. F.; Zhang, H., Petsc users manual, tech. rep. ANL-95/11—revision 3.4, (2013), Argonne National Laboratory
[87] Happel, J.; Brenner, H., Low Reynolds number hydrodynamics—with special applications to particulate media, (1983), Springer
[88] Stokes, G., On the effect of internal friction of fluids on the motion of pendulums, Trans. Cambridge Phil. Soc., 9, 6-106, (1851)
[89] Bohlin, X., On the drag on a rigid sphere moving in a viscous fluid inside a cylindrical tube, Trans. Roy. Inst. Teck., 155, 1-63, (1960)
[90] Francis, A., Wall effect in falling ball method for viscosity, Physics, 4, 403-406, (1933)
[91] Vuong, A.; Heinrich, C.; Simeon, B., ISOGAT: a 2D tutorial MATLAB code for isogeometric analysis, Comput. Aided Geom. Design, 27, 644-655, (2010) · Zbl 1205.65319
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