Abstract
To address the computational challenges associated with contact between moving interfaces, such as those in cardiovascular fluid–structure interaction (FSI), parachute FSI, and flapping-wing aerodynamics, we introduce a space–time (ST) interface-tracking method that can deal with topology change (TC). In cardiovascular FSI, our primary target is heart valves. The method is a new version of the deforming-spatial-domain/stabilized space–time (DSD/SST) method, and we call it ST-TC. It includes a master–slave system that maintains the connectivity of the “parent” mesh when there is contact between the moving interfaces. It is an efficient, practical alternative to using unstructured ST meshes, but without giving up on the accurate representation of the interface or consistent representation of the interface motion. We explain the method with conceptual examples and present 2D test computations with models representative of the classes of problems we are targeting.
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References
Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29:329–349
Ohayon R (2001) Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems. Comput Methods Appl Mech Eng 190:3009–3019
van Brummelen EH, de Borst R (2005) On the nonnormality of subiteration for a fluid–structure interaction problem. SIAM J Sci Comput 27:599–621
Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38:310–322
Khurram RA, Masud A (2006) A multiscale/stabilized formulation of the incompressible Navier-Stokes equations for moving boundary flows and fluid–structure interaction. Comput Mech 38:403–416
Lohner R, Cebral JR, Yang C, Baum JD, Mestreau EL, Soto O (2006) Extending the range of applicability of the loose coupling approach for FSI simulations. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction, vol 53. Lecture notes in computational science and engineering, pp 82–100. Springer, Berlin
Bletzinger K-U, Wuchner R, Kupzok A (2006) Algorithmic treatment of shells and free form-membranes in FSI. In: Bungartz H-J, Schafer M (eds) Fluid–structure interaction, vol 53. Lecture notes in computational science and engineering, pp 336–355. Springer, Berlin
Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43:3–37
Dettmer WG, Peric D (2008) On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction. Comput Mech 43:81–90
Bazilevs Y, Gohean JR, Hughes TJR, Moser RD, Zhang Y (2009) Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik (2000) left ventricular assist device. Comput Methods Appl Mech Eng 198:3534–3550
Bazilevs Y, Hsu M-C, Benson D, Sankaran S, Marsden A (2009) Computational fluid–structure interaction: methods and application to a total cavopulmonary connection. Comput Mech 45:77–89
Calderer R, Masud A (2010) A multiscale stabilized ALE formulation for incompressible flows with moving boundaries. Comput Mech 46:185–197
Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Liang X, Kvamsdal T, Brekken R, Isaksen J (2010) A fully-coupled fluid–structure interaction simulation of cerebral aneurysms. Comput Mech 46:3–16
Bazilevs Y, Hsu M-C, Zhang Y, Wang W, Kvamsdal T, Hentschel S, Isaksen J (2010) Computational fluid–structure interaction: methods and application to cerebral aneurysms. Biomech Model Mechanobiol 9:481–498
Bazilevs Y, Hsu M-C, Akkerman I, Wright S, Takizawa K, Henicke B, Spielman T, Tezduyar TE (2011) 3D simulation of wind turbine rotors at full scale. Part I: Geometry modeling and aerodynamics. Int J Numer Methods Fluids 65:207–235. doi:10.1002/fld.2400
Bazilevs Y, Hsu M-C, Kiendl J, Wüchner R, Bletzinger K-U (2011) 3D simulation of wind turbine rotors at full scale. Part II: fluid–structure interaction modeling with composite blades. Int J Numer Methods Fluids 65:236–253
Akkerman I, Bazilevs Y, Kees CE, Farthing MW (2011) Isogeometric analysis of free-surface flow. J Comput Phys 230:4137–4152
Hsu M-C, Bazilevs Y (2011) Blood vessel tissue prestress modeling for vascular fluid–structure interaction simulations. Finite Elem Anal Des 47:593–599
Nagaoka S, Nakabayashi Y, Yagawa G, Kim YJ (2011) Accurate fluid–structure interaction computations using elements without mid-side nodes. Comput Mech 48:269–276. doi:10.1007/s00466-011-0620-7
Bazilevs Y, Hsu M-C, Takizawa K, Tezduyar TE (2012) ALE-VMS and ST-VMS methods for computer modeling of wind-turbine rotor aerodynamics and fluid–structure interaction. Math Models Methods Appl Sci 22:1230002. doi:10.1142/S0218202512300025
Akkerman I, Bazilevs Y, Benson DJ, Farthing MW, Kees CE (2012) Free-surface flow and fluid–object interaction modeling with emphasis on ship hydrodynamics. J Appl Mech 79:010905
Hsu M-C, Akkerman I, Bazilevs Y (2012) Wind turbine aerodynamics using ALE-VMS: validation and role of weakly enforced boundary conditions. Comput Mech 50:499–511
Hsu M-C, Bazilevs Y (2012) Fluid–structure interaction modeling of wind turbines: simulating the full machine. Comput Mech 50:821–833
Akkerman I, Dunaway J, Kvandal J, Spinks J, Bazilevs Y (2012) Toward free-surface modeling of planing vessels: simulation of the Fridsma hull using ALE-VMS. Comput Mech 50:719–727
Minami S, Kawai H, Yoshimura S (2012) Parallel BDD-based monolithic approach for acoustic fluid–structure interaction. Comput Mech 50:707–718
Miras T, Schotte J-S, Ohayon R (2012) Energy approach for static and linearized dynamic studies of elastic structures containing incompressible liquids with capillarity: a theoretical formulation. Comput Mech 50:729–741
van Opstal TM, van Brummelen EH, de Borst R, Lewis MR (2012) A finite-element/boundary-element method for large-displacement fluid–structure interaction. Comput Mech 50:779–788
Yao JY, Liu GR, Narmoneva DA, Hinton RB, Zhang Z-Q (2012) Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves. Comput Mech 50:789–804
Larese A, Rossi R, Onate E, Idelsohn SR (2012) A coupled PFEM–Eulerian approach for the solution of porous FSI problems. Comput Mech 50:805–819
Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid–structure interaction: methods and applications. Wiley, London
Bazilevs Y, Takizawa K, Tezduyar TE (2013) Challenges and directions in computational fluid–structure interaction. Math Models Methods Appl Sci 23:215–221. doi:10.1142/S0218202513400010
Korobenko A, Hsu M-C, Akkerman I, Tippmann J, Bazilevs Y (2013) Structural mechanics modeling and FSI simulation of wind turbines. Math Models Methods Appl Sci 23:249–272
Yao JY, Liu GR, Qian D, Chen CL, Xu GX (2013) A moving-mesh gradient smoothing method for compressible CFD problems. Math Models Methods Appl Sci 23:273–305
Kamran K, Rossi R, Onate E, Idelsohn SR (2013) A compressible Lagrangian framework for modeling the fluid–structure interaction in the underwater implosion of an aluminum cylinder. Math Models Methods Appl Sci 23:339–367
Hsu M-C, Akkerman I, Bazilevs Y (2013) Finite element simulation of wind turbine aerodynamics: validation study using NREL phase VI experiment. Wind Energy. doi:10.1002/we.1599
Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44. doi:10.1016/S0065-2156(08)70153-4
Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces: the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94:339–351. doi:10.1016/0045-7825(92)90059-S
Tezduyar TE, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces: the deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94:353–371. doi:10.1016/0045-7825(92)90060-W
Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575. doi:10.1002/fld.505
Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900. doi:10.1002/fld.1430
Takizawa K, Tezduyar TE (2011) Multiscale space–time fluid–structure interaction techniques. Comput Mech 48:247–267. doi:10.1007/s00466-011-0571-z
Takizawa K, Tezduyar TE (2012) Space–time fluid–structure interaction methods. Math Models Methods Appl Sci 22:1230001. doi:10.1142/S0218202512300013
Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26:27–36. doi:10.1109/2.237441
Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119:73–94. doi:10.1016/0045-7825(94)00077-8
Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8:83–130. doi:10.1007/BF02897870
Tezduyar T, Aliabadi S, Behr M, Johnson A, Kalro V, Litke M (1996) Flow simulation and high performance computing. Comput Mech 18:397–412. doi:10.1007/BF00350249
Mittal S, Tezduyar TE (1995) Parallel finite element simulation of 3D incompressible flows: fluid–structure interactions. Int J Numer Methods Fluids 21:933–953. doi:10.1002/fld.1650211011
Takizawa K, Henicke B, Puntel A, Kostov N, Tezduyar TE (2012) Space–time techniques for computational aerodynamics modeling of flapping wings of an actual locust. Comput Mech 50:743–760. doi:10.1007/s00466-012-0759-x
Takizawa K, Kostov N, Puntel A, Henicke B, Tezduyar TE (2012) Space–time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle. Comput Mech 50:761–778. doi:10.1007/s00466-012-0758-y
Takizawa K, Henicke B, Tezduyar TE, Hsu M-C, Bazilevs Y (2011) Stabilized space–time computation of wind-turbine rotor aerodynamics. Comput Mech 48:333–344. doi:10.1007/s00466-011-0589-2
Takizawa K, Henicke B, Montes D, Tezduyar TE, Hsu M-C, Bazilevs Y (2011) Numerical-performance studies for the stabilized space–time computation of wind-turbine rotor aerodynamics. Comput Mech 48:647–657. doi:10.1007/s00466-011-0614-5
Takizawa K, Tezduyar TE, McIntyre S, Kostov N, Kolesar R, Habluetzel C (2013) Space–time VMS computation of wind-turbine rotor and tower aerodynamics. Comput Mech. doi:10.1007/s00466-013-0888-x
Takase S, Kashiyama K, Tanaka S, Tezduyar TE (2011) Space-time SUPG finite element computation of shallow-water flows with moving shorelines. Comput Mech 48:293–306. doi:10.1007/s00466-011-0618-1
Kalro V, Tezduyar TE (2000) A parallel 3D computational method for fluid–structure interactions in parachute systems. Comput Methods Appl Mech Eng 190:321–332. doi:10.1016/S0045-7825(00)00204-8
Tezduyar T, Osawa Y (2001) Fluid–structure interactions of a parachute crossing the far wake of an aircraft. Comput Methods Appl Mech Eng 191:717–726. doi:10.1016/S0045-7825(01)00311-5
Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space–time finite element techniques for computation of fluid–structure interactions. Comput Methods Appl Mech Eng 195:2002–2027. doi:10.1016/j.cma.2004.09.014
Manguoglu M, Sameh AH, Tezduyar TE, Sathe S (2008) A nested iterative scheme for computation of incompressible flows in long domains. Comput Mech 43:73–80. doi:10.1007/s00466-008-0276-0
Tezduyar TE, Sathe S, Pausewang J, Schwaab M, Christopher J, Crabtree J (2008) Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods. Comput Mech 43:39–49. doi:10.1007/s00466-008-0261-7
Tezduyar TE, Sathe S, Schwaab M, Pausewang J, Christopher J, Crabtree J (2008) Fluid–structure interaction modeling of ringsail parachutes. Comput Mech 43:133–142. doi:10.1007/s00466-008-0260-8
Sathe S, Tezduyar TE (2008) Modeling of fluid–structure interactions with the space–time finite elements: contact problems. Comput Mech 43:51–60. doi:10.1007/s00466-008-0299-6
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2008) Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling. Comput Mech 43:151–159. doi:10.1007/s00466-008-0325-8
Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Multiscale sequentially-coupled arterial FSI technique. Comput Mech 46:17–29. doi:10.1007/s00466-009-0423-2
Takizawa K, Moorman C, Wright S, Christopher J, Tezduyar TE (2010) Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions. Comput Mech 46:31–41. doi:10.1007/s00466-009-0425-0
Manguoglu M, Takizawa K, Sameh AH, Tezduyar TE (2010) Solution of linear systems in arterial fluid mechanics computations with boundary layer mesh refinement. Comput Mech 46:83–89. doi:10.1007/s00466-009-0426-z
Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE (2010) Role of 0D peripheral vasculature model in fluid–structure interaction modeling of aneurysms. Comput Mech 46:43–52. doi:10.1007/s00466-009-0439-7
Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space–time finite element computation of complex fluid–structure interactions. Int J Numer Methods Fluids 64:1201–1218. doi:10.1002/fld.2221
Takizawa K, Spielman T, Tezduyar TE (2011) Space–time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters. Comput Mech 48:345–364. doi:10.1007/s00466-011-0590-9
Manguoglu M, Takizawa K, Sameh AH, Tezduyar TE (2011) A parallel sparse algorithm targeting arterial fluid mechanics computations. Comput Mech 48:377–384. doi:10.1007/s00466-011-0619-0
Takizawa K, Tezduyar TE (2012) Computational methods for parachute fluid–structure interactions. Arch Comput Methods Eng 19:125–169. doi:10.1007/s11831-012-9070-4
Takizawa K, Bazilevs Y, Tezduyar TE (2012) Space–time and ALE-VMS techniques for patient-specific cardiovascular fluid–structure interaction modeling. Arch Comput Methods Eng 19:171–225. doi:10.1007/s11831-012-9071-3
Takizawa K, Fritze M, Montes D, Spielman T, Tezduyar TE (2012) Fluid–structure interaction modeling of ringsail parachutes with disreefing and modified geometric porosity. Comput Mech 50:835–854. doi:10.1007/s00466-012-0761-3
Takizawa K, Montes D, Fritze M, McIntyre S, Boben J, Tezduyar TE (2013) Methods for FSI modeling of spacecraft parachute dynamics and cover separation. Math Models Methods Appl Sci 23:307–338. doi:10.1142/S0218202513400058
Takizawa K, Tezduyar TE, Boben J, Kostov N, Boswell C, Buscher A (2013) Fluid–structure interaction modeling of clusters of spacecraft parachutes with modified geometric porosity. Comput Mech. doi:10.1007/s00466-013-0880-5
Akin JE, Tezduyar TE, Ungor M (2007) Computation of flow problems with the mixed interface-tracking/interface-capturing technique (MITICT). Comput Fluids 36:2–11. doi:10.1016/j.compfluid.2005.07.008
Cruchaga MA, Celentano DJ, Tezduyar TE (2007) A numerical model based on the mixed interface-tracking/interface-capturing technique (MITICT) for flows with fluid–solid and fluid–fluid interfaces. Int J Numer Methods Fluids 54:1021–1030. doi:10.1002/fld.1498
Wick T (2013) Coupling of fully Eulerian and arbitrary Lagrangian–Eulerian methods for fluid–structure interaction computations. Comput Mech. doi:10.1007/s00466-013-0866-3
Saad Y, Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195
Bazilevs Y, Hughes TJR (2008) NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput Mech 43:143–150
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Chichester
Acknowledgments
This work was supported in part by JST-CREST and Rice–Waseda research agreement (first and fourth authors). It was also supported in part by ARO Grant W911NF-12-1-0162 (second and third authors) and NASA Johnson Space Center Grant NNX13AD87G.
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Appendix A: Element degeneration
Appendix A: Element degeneration
1.1 A. 1 Degeneration identification
We identify the degeneration by utilizing an index set \(\mathbf {a} = (a_0, a_1,\ldots ,a_{n_{\mathrm {en}}-1})\), obtained by mapping from the nodal indices of the parent element domain, \(k = 0,1,\ldots ,n_{\mathrm {en}}-1\). In this mapping, when there is a degeneration, the lowest nodal index maps to itself as the master index, and the nodal indices of the slaves map to that master index. For example, for a quadrilateral element \((n_{\mathrm {en}}=4)\) with degeneration involving nodal indices 0 and 3, the mapping becomes \(\mathbf {a} = (0, 1, 2, 0)\). As another example, we note that the representation \(\mathbf {a} = (0, 0, 1, 3)\) would not be considered a valid one in our implementation, because the third mapped index 1 belongs to a slave that has already been mapped to 0. The valid representation in that case would be \(\mathbf {a} = (0, 0, 0, 3)\).
We serialize the index set \(\mathbf {a}\) with the serial ID number “\(s\)”, as given by the following expression:
We note that in our implementation strategy, initially the kind of invalid index sets mentioned above are included in the serialization, but eventually they are excluded from the list of acceptable index sets. We also note that, if needed, the index set can be recovered from the ID number with the following modulo operation:
1.2 A.2 Hexahedral element
Figures 35, 36, 37, 38 and 39 show the degeneration modes of the hexahedral element, excluding those that lead to zero volume. Table 2 shows, for the degeneration modes that do not lead to zero volume, the sets of serial IDs that can be obtained by rotating the parent element to different sets of nodal indices.
We note that an element with its ID number listed in Table 2 could still be invalid, not because of degeneration, but because it could possibly have a negative Jacobian.
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Takizawa, K., Tezduyar, T.E., Buscher, A. et al. Space–time interface-tracking with topology change (ST-TC). Comput Mech 54, 955–971 (2014). https://doi.org/10.1007/s00466-013-0935-7
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DOI: https://doi.org/10.1007/s00466-013-0935-7