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A unified arbitrary Lagrangian-Eulerian model for fluid-structure interaction problems involving flows in flexible channels. (English) Zbl 07477392

Summary: In this work a finite element-based model for analyzing incompressible flows in flexible channels is presented. The model treats the fluid-solid interaction problem in a monolithic way, where the governing equations for both sub-domains are solved on a single moving grid taking advantage of an arbitrary Lagrangian/Eulerian framework (ALE). The unified implementation of the governing equations for both sub-domains is developed, where these are distinguished only in terms of the mesh-moving strategy and the constitutive equation coefficients. The unified formulation is derived considering a Newtonian incompressible fluid and a hypoelastic solid. Hypoelastic constitutive law is based on the strain rate and thus naturally facilitates employing velocity as a kinematic variable in the solid. Unifying the form of the governing equations and defining a semi-Lagrangian interface mesh-motion algorithm, one obtains the coupled problem formulated in terms of a unique kinematic variable. Resulting monolithic system is characterized by reduced variable heterogeneity resembling that of a single-media problem. The model used in conjunction with algebraic multigrid linear solver exhibits attractive convergence rates. The model is tested using a 2D and a 3D example.

MSC:

76Mxx Basic methods in fluid mechanics
74Fxx Coupling of solid mechanics with other effects
74Sxx Numerical and other methods in solid mechanics

Software:

AMGCL; GitHub; KRATOS
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Full Text: DOI

References:

[1] Hou, G.; Wang, J.; Layton, A., Numerical methods for fluid-structure interaction—a review, Commun. Comput. Phys., 12, 2, 337-377 (2012) · Zbl 1373.76001 · doi:10.4208/cicp.291210.290411s
[2] Hron,J., Turek,S.: A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics. In: Fluid-structure interaction. Springer, pp 146-170 (2006) · Zbl 1323.74086
[3] Gee, M.; Küttler, U.; Wall, W., Truly monolithic algebraic multigrid for fluid-structure interaction, Int. J. Numer. Methods Eng., 85, 8, 987-1016 (2011) · Zbl 1217.74121 · doi:10.1002/nme.3001
[4] Richter, T., A monolithic geometric multigrid solver for fluid-structure interactions in ALE formulation, Int. J. Numer. Methods Eng., 104, 5, 372-390 (2015) · Zbl 1352.76066 · doi:10.1002/nme.4943
[5] Aulisa, E.; Bna, S.; Bornia, G., A monolithic ALE Newton-Krylov solver with multigrid-Richardson-Schwarz preconditioning for incompressible fluid-structure interaction, Comput. Fluids, 174, 213-228 (2018) · Zbl 1410.76147 · doi:10.1016/j.compfluid.2018.08.003
[6] Langer, U.; Yang, H., Numerical simulation of fluid-structure interaction problems with hyperelastic models: a monolithic approach, Math. Comput. Simul., 145, 186-208 (2018) · Zbl 1485.74097 · doi:10.1016/j.matcom.2016.07.008
[7] Mayr, M.; Noll, M.; Gee, M., A hybrid interface preconditioner for monolithic fluid-structure interaction solvers, Adv. Model. Simul. Eng. Sci., 7, 1-33 (2020) · doi:10.1186/s40323-020-00150-9
[8] Idelsohn, SR; Marti, J.; Limache, A.; Oñate, E., Unified Lagrangian formulation for elastic solids and incompressible fluids. Application to fluid-structure interaction problems via the PFEM, Comput. Methods Appl. Mech. Eng., 197, 1762-1776 (2008) · Zbl 1194.74415 · doi:10.1016/j.cma.2007.06.004
[9] Idelsohn, SR; Marti, J.; Souto-Iglesias, A.; Oñate, E., Interaction between an elastic structure and free-surface flows: experimental versus numerical comparisons using the PFEM, Comput. Mech., 43, 1, 125-132 (2008) · Zbl 1177.74140 · doi:10.1007/s00466-008-0245-7
[10] Franci, A.; Oñate, E.; Carbonell, JM, Unified Lagrangian formulation for solid and fluid mechanics and FSI problems, Comput. Methods Appl. Mech. Eng., 298, 520-547 (2016) · Zbl 1423.76230 · doi:10.1016/j.cma.2015.09.023
[11] Zhu, M.; Scott, MH, Unified fractional step method for Lagrangian analysis of quasi-incompressible fluid and nonlinear structure interaction using the pfem, Int. J. Numer. Methods Eng., 109, 9, 1219-1236 (2017) · doi:10.1002/nme.5321
[12] Ryzhakov, PB; Oñate, E., A finite element model for fluid-structure interaction problems involving closed membranes, internal and external fluids, Comput. Methods Appl. Mech. Eng., 326, 422-445 (2017) · Zbl 1439.74115 · doi:10.1016/j.cma.2017.08.014
[13] Ryzhakov, PB; Marti, J.; Idelsohn, SR; Oñate, E., Fast fluid-structure interaction simulations using a displacement-based finite element model equipped with an explicit streamline integration prediction, Comput. Methods Appl. Mech. Eng., 315, 1080-1097 (2017) · Zbl 1439.76093 · doi:10.1016/j.cma.2016.12.003
[14] Ryzhakov, PB; Oñate, E.; Rossi, R.; Idelsohn, S., Improving mass conservation in simulation of incompressible flows, Int. J. Numer. Methods Eng., 90, 12, 1435-1451 (2012) · Zbl 1246.76059 · doi:10.1002/nme.3370
[15] Cremonesi, M.; Meduri, S.; Perego, US, Lagrangian-Eulerian enforcement of non-homogeneous boundary conditions in the particle finite element method, Comput. Part. Mech., 7, 41-56 (2019) · doi:10.1007/s40571-019-00245-0
[16] Turek, S., Hron, J., Madlik, M., Razzaq, M., Wobker, H., Acker, J.F.: Numerical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics. In: Fluid Structure Interaction II. Springer, pp. 193-220 (2011) · Zbl 1210.76118
[17] Chen, H.Y., Zhu, L., Huo, Y., Liu, Y., Kassab, G.S.: Fluid-structure interaction (fsi) modeling in the cardiovascular system. In: Computational Cardiovascular Mechanics. Springer, pp. 141-157 (2010)
[18] Takizawa, K.; Bazilevs, Y.; Tezduyar, TE; Long, CC; Marsden, AL; Schjodt, K., ST and ALE-VMS methods for patient-specific cardiovascular fluid mechanics modeling, Math. Models Methods Appl. Sci., 24, 12, 2437-2486 (2014) · Zbl 1296.76113 · doi:10.1142/S0218202514500250
[19] Mayr, M.; Klöppel, T.; Wall, WA; Gee, MW, A temporal consistent monolithic approach to fluid-structure interaction enabling single field predictors, SIAM J. Sci. Comput., 37, 1, B30-B59 (2015) · Zbl 1330.74159 · doi:10.1137/140953253
[20] Lozovskiy, A.; Olshanskii, MA; Vassilevski, YV, Analysis and assessment of a monolithic FSI finite element method, Comput. Fluids, 179, 277-288 (2019) · Zbl 1411.76068 · doi:10.1016/j.compfluid.2018.11.004
[21] Eken, A.; Sahin, M., A parallel monolithic algorithm for the numerical simulation of large-scale fluid structure interaction problems, Int. J. Numer. Methods Fluids, 80, 12, 687-714 (2016) · doi:10.1002/fld.4169
[22] Donea, J., Huerta, A., Ponthot, J.-Ph., Rodríguez-Ferran, A.: Arbitrary Lagrangian-Eulerian Methods (Encyclopedia of Computational Mechanics). Wiley online library edition (2004)
[23] Donea, J., Huerta, A.: Finite element method for flow problems. J. Wiley edition (2003)
[24] Truesdell, C.A., Noll, W.: The non-linear field theories of mechanics. Handbuch der Physik, Springer, Berlin. Bd III/3 (1992) · Zbl 0779.73004
[25] Pironneau,O.: Numerical study of a monolithic fluid-structure formulation. In Variational Analysis and Aerospace Engineering. Springer, pp. 401-420 (2016)
[26] Marti, J.; Idelsohn, SR; Limache, A.; Calvo, N.; D’Elia, J., A fully coupled particle method for quasi incompressible fluid-hypoelastic structure interactions, Mec. Comput., 25, 809-827 (2006)
[27] Wood, L.; Bossak, M.; Zienkiewicz, O., An alpha modification of Newmark’s method, Int. J. Numer. Methods Eng., 15, 10, 1562-1566 (1980) · Zbl 0441.73106 · doi:10.1002/nme.1620151011
[28] Codina, R., A stabilized finite element method for generalized stationary incompressible flows, Comput. Methods Appl. Mech. Eng., 190, 20-21, 2681-2706 (2001) · Zbl 0996.76045 · doi:10.1016/S0045-7825(00)00260-7
[29] Ryzhakov, P.; Cotela, J.; Rossi, R.; Oñate, E., A two-step monolithic method for the efficient simulation of incompressible flows, Int. J. Numer. Methods Fluids, 74, 12, 919-934 (2014) · Zbl 1455.76093 · doi:10.1002/fld.3881
[30] Zhu, M.; Scott, MH, Improved fractional step method for simulating fluid-structure interaction using the PFEM, Int. J. Numer. Methods Eng., 99, 12, 925-944 (2014) · Zbl 1352.74463 · doi:10.1002/nme.4727
[31] Batina, JT, Unsteady Euler airfoil solutions using unstructured dynamic meshes, AIAA J., 28, 1381-1388 (1990) · doi:10.2514/3.25229
[32] Behr, M.; Tezduyar, TE, Finite element solution strategies for large-scale flow simulations, Comput. Methods Appl. Mech. Eng., 112, 1, 3-24 (1994) · Zbl 0846.76041 · doi:10.1016/0045-7825(94)90016-7
[33] Selim, MM; Koomullil, RP, Mesh deformation approaches—a survey, J. Phys. Math., 7, 2, 1-9 (2016)
[34] Helenbrook, BT, Mesh deformation using the biharmonic operator, Int. J. Numer. Methods Eng., 56, 7, 1007-1021 (2003) · Zbl 1047.76044 · doi:10.1002/nme.595
[35] Field, DA, Laplacian smoothing and Delaunay triangulations, Commun. Appl. Numer. Methods, 4, 6, 709-712 (1988) · Zbl 0664.65107 · doi:10.1002/cnm.1630040603
[36] Löhner, R.; Yang, C., Improved ALE mesh velocities for moving bodies, Commun. Numer. Methods Eng., 12, 10, 599-608 (1996) · Zbl 0858.76042 · doi:10.1002/(SICI)1099-0887(199610)12:10<599::AID-CNM1>3.0.CO;2-Q
[37] Masud, A.; Hughes, TJR, A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems, Comput. Methods Appl. Mech. Eng., 146, 1, 91-126 (1997) · Zbl 0899.76259 · doi:10.1016/S0045-7825(96)01222-4
[38] Slyngstad, A.: Verification and validation of a monolithic fluid-structure interaction solver in fenics. A comparison of mesh lifting operators. Master’s thesis (2017)
[39] Wick, T., Fluid-structure interactions using different mesh motion techniques, Comput. Struct., 89, 13-14, 1456-1467 (2011) · doi:10.1016/j.compstruc.2011.02.019
[40] Bazilevs, Y.; Takizawa, K.; Tezduyar, TE, Challenges and directions in computational fluid-structure interaction, Math. Models Methods Appl. Sci., 23, 2, 215-221 (2013) · Zbl 1261.76025 · doi:10.1142/S0218202513400010
[41] Dadvand, P.; Rossi, R.; Oñate, E., An object-oriented environment for developing finite element codes for multi-disciplinary applications, Arch. Comput. Methods Eng., 17, 3, 253-297 (2010) · Zbl 1360.76130 · doi:10.1007/s11831-010-9045-2
[42] Kratos Multiphysics at GitHub. https://github.com/KratosMultiphysics/Kratos. Accessed 10 Oct 2021
[43] Demidov, D., Amgcl: an efficient, flexible, and extensible algebraic multigrid implementation, Lobachevskii J. Math., 40, 5, 535-546 (2019) · Zbl 1452.65426 · doi:10.1134/S1995080219050056
[44] Ryzhakov, P.; Soudah, E.; Dialami, N., Computational modeling of the fluid flow and the flexible intimal flap in type B aortic dissection via a monolithic arbitrary Lagrangian/Eulerian fluid-structure interaction model, Int. J. Numer. Methods Biomed. Eng., 35, 11, e3239 (2019) · doi:10.1002/cnm.3239
[45] Mok, D.P.: Partitionierte Lösungsansätze in der Strukturdynamik und der Fluid-Struktur-Interaktion. Ph.D. thesis, Universität Stuttgart (2001)
[46] Valdés Vázquez, J.G.: Nonlinear Analysis of Orthotropic Membrane and Shell Structures Including Fluid-Structure Interaction. Ph.D. thesis, Universitat Politècnica de Catalunya (2007)
[47] Apostolatos, A.; De Nayer, G.; Bletzinger, K.; Breuer, M.; Wüchner, R., Systematic evaluation of the interface description for fluid-structure interaction simulations using the isogeometric mortar-based mapping, J. Fluids Struct., 86, 368-399 (2019) · doi:10.1016/j.jfluidstructs.2019.02.012
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