A fully partitioned Lagrangian framework for FSI problems characterized by free surfaces, large solid deformations and displacements, and strong added-mass effects. (English) Zbl 1440.74380
Summary: In this work, a fully partitioned Lagrangian framework for the solution of fluid-structure interaction (FSI) problems involving free surfaces, large solid displacements and deformations, and strong added mass effects is presented. The fluid is simulated using the Particle Finite Element Method (PFEM), while Metafor, a large deformations nonlinear Finite Element code, is employed to simulate the solid part. The fully partitioned coupling is ensured through an Interface Quasi-Newton Inverse Least Squares (IQN-ILS) [J. Degroote, K.-J. Bathe and J. Vierendeels, “Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction”, Comput. Struct. 87, No. 11–12, 793–801 (2009; doi:10.1016/j.compstruc.2008.11.013)] strategy to avoid added mass effects. The Lagrangian particle nature of the PFEM allows the simulation of problems involving free surfaces and very large solid displacements, usually difficult to achieve with traditional body-fitted CFD techniques. We show that owing to the generality of its formulation the PFEM can be used as is in the framework of fully partitioned FSI coupling schemes, where minimal information (i.e. loads and displacements at the FSI interface) is exchanged between the fluid and the solid solvers. More importantly, we demonstrate that a fully partitioned PFEM-FEM coupling based on the IQN-ILS strategy allows the simulation of a very large spectrum of FSI problems without incurring added-mass instabilities. The performance of the IQN-ILS coupling strategy in a fully Lagrangian framework is also assessed and compared to more traditional approaches such as Block-Gauss-Seidel (BGS) iterations with Aitken relaxation. An extensive work of verification and benchmarking is proposed, aiming to encompass all the combinations of physical and numerical parameters possibly leading to added-mass instabilities, and testing the IQN-ILS strategy on different benchmarks beyond those already proposed in the literature. The coupling is performed through CUPyDO [the authors, “CUPyDO – an integrated Python environment for coupled fluid-structure simulations”, Adv. Eng. Softw. 128, 69–85 (2019; doi:10.1016/j.advengsoft.2018.05.007)], a general Python framework for partitioned FSI coupling.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76Dxx | Incompressible viscous fluids |
Keywords:
fluid-structure interaction; partitioned approaches; CUPyDO; added mass; free-surface flows; particle finite element methodReferences:
| [1] | Li, Z.; Leduc, J.; Nunez-Ramirez, J.; Combescure, A.; Marongiu, J. C., A non-intrusive partitioned approach to couple smoothed particle hydrodynamics and finite element methods for transient fluid – structure interaction problems with large interface motion, Comput. Mech., 55, 4, 697-718 (2015) · Zbl 1334.76081 |
| [2] | Habchi, C.; Russeil, S.; Bougeard, D.; Harion, J. L.; Lemenand, T.; Ghanem, A.; Valle, D. D.; Peerhossaini, H., Partitioned solver for strongly coupled fluid – structure interaction, Comput. & Fluids, 71, 306-319 (2013) · Zbl 1365.76155 |
| [3] | Dettmer, W.; Perić, D., A new staggered scheme for fluid – structure interaction, Internat. J. Numer. Methods Engrg., 93, 1, 1-22 (2013) · Zbl 1352.74471 |
| [4] | Nobile, F.; Vergara, C., Partitioned algorithms for fluid – structure interaction problems in haemodynamics, Milan J. Math., 80, 2, 443-467 (2012) · Zbl 1344.76099 |
| [5] | Wall, W. A.; Genkinger, S.; Ramm, E., A strong coupling partitioned approach for fluid – structure interaction with free surfaces, Comput. & Fluids, 36, 1, 169-183 (2007), Challenges and advances in flow simulation and modeling · Zbl 1181.76147 |
| [6] | Küttler, U.; Wall, W. A., Fixed-point fluid – structure interaction solvers with dynamic relaxation, Comput. Mech., 43, 1, 61-72 (2008) · Zbl 1236.74284 |
| [7] | Farhat, C.; van der Zee, K. G.; Geuzaine, P., Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity, Comput. Methods Appl. Mech. Engrg., 195, 17-18, 1973-2001 (2006) · Zbl 1178.76259 |
| [8] | Matthies, H. G.; Steindorf, J., Partitioned strong coupling algorithms for fluid – structure interaction, Comput. Struct., 81, 8-11, 805-812 (2003) |
| [9] | Hirt, C. W.; Nichols, B. D., Volume of Fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201-225 (1981) · Zbl 0462.76020 |
| [10] | Osher, S.; Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces (2006), Springer |
| [11] | Idelsohn, S. R.; Oñate, E.; Del Pin, F., The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Internat. J. Numer. Methods Engrg., 61, 964-989 (2004) · Zbl 1075.76576 |
| [12] | Oñate, E.; Idelsohn, S. R.; Del Pin, F.; Aubry, R., The particle finite element method. An overview, Int. J. Comput. Methods, 1, 2, 267-307 (2004) · Zbl 1182.76901 |
| [13] | Monaghan, J. J.; Gingold, R. A., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389 (1977) · Zbl 0421.76032 |
| [14] | Monaghan, J. J., Smoothed particle hydrodynamics, Rep. Progr. Phys., 68, 1703-1759 (2005) |
| [15] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 34, 2, 229-256 (1994) · Zbl 0796.73077 |
| [16] | Idelsohn, S. R.; 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. Engrg., 197, 1762-1776 (2008) · Zbl 1194.74415 |
| [17] | Franci, A.; Oñate, E.; Carbonell, J. M., Unified Lagrangian formulation for solid and fluid mechanics and FSI problems, Comput. Methods Appl. Mech. Engrg., 298, 520-547 (2016) · Zbl 1423.76230 |
| [18] | Zhu, M.; Scott, M. H., Improved fractional step method for simulating fluid – structure interaction using the PFEM, Internat. J. Numer. Methods Engrg., 99, 12, 925-944 (2014) · Zbl 1352.74463 |
| [19] | Meduri, S.; Cremonesi, M.; Perego, U.; Bettinotti, O.; Kurkchubasche, A.; Oancea, V., A partitioned fully explicit Lagrangian finite element method for highly nonlinear fluid – structure-interaction problems, Internat. J. Numer. Methods Engrg., 113, 43-64 (2017) · Zbl 07867255 |
| [20] | Thomas, D.; Cerquaglia, M.; Boman, R.; Economon, T.; Alonso, J.; Dimitriadis, G.; Terrapon, V., CUPyDO an integrated Python environment for coupled multi-physics simulations, Adv. Eng. Softw., 128, 69-85 (2019) |
| [21] | Edelsbrunner, H.; Mücke, E. P., Three-dimensional alpha shapes, ACM Trans. Graph., 13, 1, 43-72 (1994) · Zbl 0806.68107 |
| [22] | Franci, A.; Cremonesi, M., On the effect of standard PFEM remeshing on volume conservation in free-surface fluid flow problems, Comput. Part. Mech., 4, 3, 331-343 (2016) |
| [23] | Idelsohn, S. R.; Oñate, E., To mesh or not to mesh. That is the question..., Comput. Methods Appl. Mech. Engrg., 195, 4681-4696 (2006) · Zbl 1118.74051 |
| [24] | Cremonesi, M.; Meduri, S.; Perego, U.; Frangi, A., An explicit Lagrangian finite element method for free-surface weakly compressible flows, Comput. Part. Mech., 4, 3, 357-369 (2016) |
| [25] | Idelsohn, S. R.; Mier-Torrecilla, M.; Oñate, E., Multi-fluid flows with the particle finite element method, Comput. Methods Appl. Mech. Engrg., 198, 2750-2767 (2009) · Zbl 1228.76086 |
| [26] | Idelsohn, S. R.; Mier-Torrecilla, M.; Nigro, N. M.; Oñate, E., On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field, Comput. Mech., 46, 115-124 (2010) · Zbl 1301.76051 |
| [27] | Zhang, X.; Krabbenhoft, K.; Pedroso, D. M.; Lyamin, A. V.; Sheng, D.; Vicente da Silva, M.; Wang, D., Particle finite element analysis of large deformation and granular flow problems, Comput. Geotech., 54, 133-142 (2013) |
| [28] | Cremonesi, M.; Perego, U., A Lagrangian finite element approach for the simulation of water-waves induced by landslides, Comput. Struct., 89, 1086-1093 (2011) |
| [29] | Idelsohn, S. R.; 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 |
| [30] | Idelsohn, S. R.; Del Pin, F.; Rossi, R.; Oñate, E., Fluid – structure interaction problems with strong added-mass effect, Internat. J. Numer. Methods Engrg., 80, 1261-1294 (2009) · Zbl 1183.74059 |
| [31] | Oñate, E.; Franci, A.; Carbonell, J. M., A particle finite element method for analysis of industrial forming processes, Comput. Mech., 54, 85-107 (2014) · Zbl 1398.76121 |
| [32] | Oñate, E.; Rossi, R.; Idelsohn, S. R.; Butler, K. M., Melting and spread of polymers in fire with the particle finite element method, Internat. J. Numer. Methods Engrg., 81, 1046-1072 (2010) · Zbl 1183.76812 |
| [33] | Idelsohn, S. R.; Nigro, N. M.; Gimenez, J. M.; Rossi, R.; Marti, J. M., A fast and accurate method to solve the incompressible Navier-Stokes equations, Eng. Comput., 30, 2, 197-222 (2013) |
| [34] | Idelsohn, S. R.; Marti, J.; Becker, P.; Oñate, E., Analysis of multifluid flows with large time steps using the particle finite element method, Internat. J. Numer. Methods Fluids, 75, 9, 621-644 (2014) · Zbl 1455.76147 |
| [35] | Ryzhakov, P. B.; Marti, J.; Idelsohn, S. R.; 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. Engrg., 315, 1080-1097 (2017) · Zbl 1439.76093 |
| [36] | Cerquaglia, M.; Deliége, G.; Boman, R.; Terrapon, V.; Ponthot, J. P., Free-slip boundary conditions for simulating free-surface incompressible flows through the particle finite element method, Internat. J. Numer. Methods Engrg., 110, 10, 921-946 (2017) · Zbl 07866614 |
| [37] | . Metafor, A nonlinear finite element code, University of Liège. http://metafor.ltas.ulg.ac.be/; . Metafor, A nonlinear finite element code, University of Liège. http://metafor.ltas.ulg.ac.be/ |
| [38] | Belytschko, T.; Liu, W. K.; Moran, B., Nonlinear Finite Elements for Continua and Structures (2001), Wiley |
| [39] | Babuška, I.; Narasimhan, R., The Babuška-Brezzi condition and the patch test: an example, Comput. Methods Appl. Mech. Engrg., 140, 183-199 (1997) · Zbl 0883.65069 |
| [40] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Method (1991), Springer: Springer Berlin · Zbl 0788.73002 |
| [41] | Tezduyar, T. E.; Mittal, S.; Ray, S. E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. Methods Appl. Mech. Engrg., 95, 221-242 (1992) · Zbl 0756.76048 |
| [42] | Cremonesi, M.; Frangi, A.; Perego, U., A Lagrangian finite element approach for the analysis of fluid – structure interaction problems, Internat. J. Numer. Methods Engrg., 84, 5, 610-630 (2010) · Zbl 1202.74164 |
| [43] | Idelsohn, S. R.; Oñate, E., The challenge of mass conservation in the solution of free-surface flows with the fractional-step method: Problems and solutions, Int. J. Numer. Methods Biomed. Eng., 26, 1313-1330 (2010) · Zbl 1274.76191 |
| [44] | Ryzhakov, P.; Oñate, E.; Rossi, R.; Idelsohn, S. R., Improving mass conservation in simulation of incompressible flows, Internat. J. Numer. Methods Engrg., 90, 1435-1451 (2012) · Zbl 1246.76059 |
| [45] | Ponthot, J. P., Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes, Int. J. Plast., 18, 1, 91-126 (2002) · Zbl 1035.74012 |
| [46] | Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-\( \alpha\) method, J. Appl. Mech., 60, 2, 371-375 (1993) · Zbl 0775.73337 |
| [47] | Malkus, D. S.; Hughes, T. J., Mixed finite element methods — reduced and selective integration techniques: A unification of concepts, Comput. Methods Appl. Mech. Engrg., 15, 1, 63-81 (1978) · Zbl 0381.73075 |
| [48] | Simo, J. C.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 8, 1595-1638 (1990) · Zbl 0724.73222 |
| [49] | Bui, Q.; Papeleux, L.; Ponthot, J., Numerical simulation of springback using enhanced assumed strain elements, J. Mater Process. Technol., 153-154, 314-318 (2004) |
| [50] | Adam, L.; Ponthot, J. P., Thermomechanical modeling of metals at finite strains: First and mixed order finite elements, Int. J. Solids Struct., 42, 21, 5615-5655 (2005) · Zbl 1113.74418 |
| [51] | Donea, J.; Huerta, A.; Ponthot, J. P.; Rodríguez-Ferran, A., Arbitrary Lagrangian-Eulerian methods, (Encyclopedia of Computational Mechanics (2004), John Wiley & Sons), Ch. 14 |
| [52] | Boman, R.; Ponthot, J. P., Efficient ALE mesh management for 3D quasi-Eulerian problems, Internat. J. Numer. Methods Engrg., 92, 10, 857-890 (2012) · Zbl 1352.74328 |
| [53] | Koeune, R.; Ponthot, J. P., A one phase thermomechanical model for the numerical simulation of semi-solid material behavior. Application to thixoforming, Int. J. Plast., 58, 120-153 (2014) |
| [54] | Jeunechamps, P. P.; Ponthot, J. P., An efficient 3D implicit approach for the thermomechanical simulation of elastic-viscoplastic materials submitted to high strain rate and damage, Internat. J. Numer. Methods Engrg., 94, 10, 920-960 (2013) · Zbl 1352.74369 |
| [55] | Mengoni, M.; Ponthot, J. P., Isotropic continuum damage/repair model for alveolar bone remodeling, J. Comput. Appl. Math., 234, 7, 2036-2045 (2010) · Zbl 1191.92002 |
| [56] | Boman, R.; Ponthot, J. P., Finite element simulation of lubricated contact in rolling using the arbitrary Lagrangian-Eulerian formulation, Comput. Methods Appl. Mech. Engrg., 193, 39, 4323-4353 (2004) · Zbl 1198.74094 |
| [57] | Farhat, C.; Lesoinne, M.; Tallec, P. L., Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity, Comput. Methods Appl. Mech. Engrg., 157, 1, 95-114 (1998) · Zbl 0951.74015 |
| [58] | Causin, P.; Gerbeau, J.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid – structure problems, Comput. Methods Appl. Mech. Engrg., 194, 42-44, 4506-4527 (2005) · Zbl 1101.74027 |
| [59] | Degroote, J.; Bathe, K. J.; Vierendeels, J., Performance of a new partitioned procedure versus a monolithic procedure in fluid – structure interaction, Comput. Struct., 87, 11-12, 793-801 (2009) |
| [60] | Vierendeels, J.; Lanoye, L.; Degroote, J.; Verdonck, P., Implicit coupling of partitioned fluid – structure interaction problems with reduced order models, Comput. Struct., 85, 11-14, 970-976 (2007) |
| [61] | Bungartz, H. J.; Lindner, F.; Gatzhammer, B.; Mehl, M.; Scheufele, K.; Shukaev, A.; Uekermann, B., preCICE — a fully parallel library for multi-physics surface coupling, Comput. & Fluids, 141, 250-258 (2016), Advances in fluid – structure interaction · Zbl 1390.76004 |
| [62] | Beckert, A.; Wendland, H., Multivariate interpolation for fluid – structure-interaction problems using radial basis functions, Aerosp. Sci. Technol., 5, 2, 125-134 (2001) · Zbl 1034.74018 |
| [63] | de Boer, A.; van Zuijlen, A.; Bijl, H., Review of coupling methods for non-matching meshes, Comput. Methods Appl. Mech. Engrg., 196, 8, 1515-1525 (2007), Domain decomposition methods: recent advances and new challenges in engineering · Zbl 1173.74485 |
| [64] | Degroote, J.; Souto-Iglesias, A.; Paepegem, W. V.; Annerel, S.; Bruggeman, P.; Vierendeels, J., Partitioned simulation of the interaction between an elastic structure and free surface flow, Comput. Methods Appl. Mech. Engrg., 199, 33-36, 2085-2098 (2010) · Zbl 1231.74107 |
| [65] | Joosten, M. M.; Dettmer, W. G.; Perić, D., Analysis of the block Gauss-Seidel solution procedure for a strongly coupled model problem with reference to fluid – structure interaction, Internat. J. Numer. Methods Engrg., 78, 7, 757-778 (2009) · Zbl 1183.74347 |
| [66] | Wood, C.; Gil, A.; Hassan, O.; Bonet, J., Partitioned block-Gauss-Seidel coupling for dynamic fluid – structure interaction, Comput. Struct., 88, 23-24, 1367-1382 (2010), Special issue: Association of Computational Mechanics — United Kingdom |
| [67] | Rossi, R.; Oate, E., Analysis of some partitioned algorithms for fluid – structure interaction, Eng. Comput., 27, 1, 20-56 (2010) · Zbl 1284.74143 |
| [68] | Irons, B. M.; Tuck, R. C., A version of the Aitken accelerator for computer iteration, Internat. J. Numer. Methods Engrg., 1, 3, 275-277 (1969) · Zbl 0256.65021 |
| [69] | Degroote, J.; Bruggeman, P.; Haelterman, R.; Vierendeels, J., Stability of a coupling technique for partitioned solvers in FSI applications, Comput. Struct., 86, 23-24, 2224-2234 (2008) |
| [70] | Haelterman, R.; Bogaers, A.; Scheufele, K.; Uekermann, B.; Mehl, M., Improving the performance of the partitioned QN-ILS procedure for fluid – structure interaction problems: Filtering, Comput. Struct., 171, Supplement C, 9-17 (2016) |
| [71] | Förster, C.; Wall, W. A.; Ramm, E., Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 196, 7, 1278-1293 (2007) · Zbl 1173.74418 |
| [72] | Degroote, J.; Annerel, S.; Vierendeels, J., Stability analysis of Gauss-Seidel iterations in a partitioned simulation of fluid – structure interaction, Comput. Struct., 88, 5-6, 263-271 (2010) |
| [73] | Olivier, M.; Morissette, J. F.; Dumas, G., A fluid – structure interaction solver for nano-air-vehicle flapping wings, (Fluid Dynamics and Co-located Conferences (2009), American Institute of Aeronautics and Astronautics) |
| [74] | Dettmer, W.; Perić, D., A computational framework for fluid – structure interaction: Finite element formulation and applications, Comput. Methods Appl. Mech. Engrg., 195, 41-43, 5754-5779 (2006), John H. Argyris memorial issue. Part II · Zbl 1155.76354 |
| [75] | F. Palacios, M. Colonno, A. Aranake, A. Campos, S. Copeland, T. Economon, A. Lonkar, T. Lukaczyk, T. Taylor, J. Alonso, Stanford University Unstructured (SU2): An open-source integrated computational environment for multi-physics simulation and design, in: AIAA 51st Aerospace Sciences Meeting, Grapevine, TX, 7-10 January, 2013. http://dx.doi.org/10.2514/6.2013-287; F. Palacios, M. Colonno, A. Aranake, A. Campos, S. Copeland, T. Economon, A. Lonkar, T. Lukaczyk, T. Taylor, J. Alonso, Stanford University Unstructured (SU2): An open-source integrated computational environment for multi-physics simulation and design, in: AIAA 51st Aerospace Sciences Meeting, Grapevine, TX, 7-10 January, 2013. http://dx.doi.org/10.2514/6.2013-287 |
| [76] | Papanastasiou, T. C.; Malamataris, N.; Ellwood, K., A new outflow boundary condition, Internat. J. Numer. Methods Fluids, 14, 5, 587-608 (1992) · Zbl 0747.76039 |
| [77] | Griffiths, D. F., The ‘no boundary condition’ outflow boundary condition, Internat. J. Numer. Methods Fluids, 24, 4, 393-411 (1997) · Zbl 1113.76402 |
| [78] | Renardy, M., Imposing ‘no’ boundary condition at outflow: Why does it work?, Internat. J. Numer. Methods Fluids, 24, 4, 413-417 (1997) · Zbl 1113.76405 |
| [79] | Walhorn, E.; Kölke, A.; Hübner, B.; Dinkler, D., Fluid – structure coupling within a monolithic model involving free surface flows, Comput. Struct., 83, 2100-2111 (2005) |
| [80] | Liu, M.; Shao, J.; Li, H., Numerical simulation of hydro-elastic problems with smoothed particle hydrodynamics method, J. Hydrodynamics, 25, 5, 673-682 (2013) |
| [81] | Ryzhakov, P. B.; Rossi, R.; Idelsohn, S. R.; Oate, E., A monolithic Lagrangian approach for fluid – structure interaction problems, Comput. Mech., 46, 883-899 (2010) · Zbl 1344.74016 |
| [82] | Rafiee, A.; Thiagarajan, K. P., An SPH projection method for simulating fluid-hypoelastic structure interaction, Comput. Methods Appl. Mech. Engrg., 198, 33-36, 2785-2795 (2009) · Zbl 1228.76117 |
| [83] | S. Meduri, M. Cremonesi, U. Perego, A fully explicit fluid-structure interaction approach based on the PFEM, in: VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2017, pp. 299-306.; S. Meduri, M. Cremonesi, U. Perego, A fully explicit fluid-structure interaction approach based on the PFEM, in: VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2017, pp. 299-306. |
| [84] | Antoci, C.; Gallati, M.; Sibilla, S., Numerical simulation of fluid – structure interaction by SPH, Comput. Struct., 85, 11-14, 879-890 (2007) |
| [85] | Yang, Q.; Jones, V.; McCue, L., Free-surface flow interactions with deformable structures using an SPH-FEM model, Ocean Eng., 55, Supplement C, 136-147 (2012) |
| [86] | Antoci, C., Simulazione numerica dell’interazione fluido-struttura con la tecnica SPH (2006), Università degli studi di Pavia, (Ph.D. thesis) |
| [87] | Hesch, C.; Gil, A.; Carreo, A. A.; Bonet, J., On continuum immersed strategies for fluid – structure interaction, Comput. Methods Appl. Mech. Engrg., 247-248, 51-64 (2012) · Zbl 1352.76055 |
| [88] | Gil, A.; Carreo, A. A.; Bonet, J.; Hassan, O., The immersed structural potential method for haemodynamic applications, J. Comput. Phys., 229, 22, 8613-8641 (2010) · Zbl 1198.92010 |
| [89] | Wang, X.; Liu, W. K., Extended immersed boundary method using FEM and RKPM, Comput. Methods Appl. Mech. Engrg., 193, 12, 1305-1321 (2004), Meshfree methods: Recent advances and new applications · Zbl 1060.74676 |
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