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A Nitsche stabilized finite element method for embedded interfaces: application to fluid-structure interaction and rigid-body contact. (English) Zbl 1436.76029

Summary: In this article, an accurate and robust numerical formulation is presented for the simulation of the fluid-structure interaction in incompressible fluid flow. The incompressible Navier-Stokes equation is discretized with a stabilized finite element framework on the fixed Eulerian grid. Both symmetric and non-symmetric Nitsche’s methods are accessed and employed to weakly impose Dirichlet boundary condition along the interface embedded in the element together with the ghost penalty method stabilizing the solution jump across the element edges. An easy-to-implement and robust numerical integration scheme based on a projection approach is proposed. To the author’s knowledge, so far, there is no application of a projection-based approach in the field of numerical integration to deal with discontinuities. Therefore, the results presented in this article is considered as a pioneered and novel projection-based approach in the field of numerical integration to deal with embedded discontinuous function. A second-order staggered-partitioned scheme is employed to weakly couple the fluid and structure solvers. A second-order accurate and unconditionally stable time integration scheme is implemented for simulations. Accurate numerical results are obtained in the numerical examples and validation cases, including vortex-induced vibration (VIV), rotation, freely fall and rigid-body contact.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

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[1] Hirt, C.; Amsden, A. A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 3, 227-253 (1974) · Zbl 0292.76018
[2] Belytschko, T.; Kennedy, J. M., Computer models for subassembly simulation, Nucl. Eng. Des., 49, 1-2, 17-38 (1978)
[3] Belytschko, T.; Kennedy, J. M.; Schoeberle, D., Quasi-Eulerian finite element formulation for fluid-structure interaction, J. Press. Vessel Technol., 102, 1, 62-69 (1980)
[4] Hughes, T. J.; Liu, W. K.; Zimmermann, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29, 3, 329-349 (1981) · Zbl 0482.76039
[5] Donea, J.; Giuliani, S.; Halleux, J. P., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl. Mech. Eng., 33, 1, 689-723 (1982) · Zbl 0508.73063
[6] Peskin, C. S., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 2, 252-271 (1972) · Zbl 0244.92002
[7] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 3, 220-252 (1977) · Zbl 0403.76100
[8] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2002) · Zbl 1123.74309
[9] Lee, L.; LeVeque, R. J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 25, 3, 832-856 (2003) · Zbl 1163.65322
[10] Wang, X.; Liu, W. K., Extended immersed boundary method using FEM and RKPM, Comput. Methods Appl. Mech. Eng., 193, 12-14, 1305-1321 (2004) · Zbl 1060.74676
[11] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 239-261 (2005) · Zbl 1117.76049
[12] Shu, C.; Liu, N.; Chew, Y. T., A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate flow past a circular cylinder, J. Comput. Phys., 226, 2, 1607-1622 (2007) · Zbl 1173.76395
[13] Glowinski, R.; Pan, T. W.; Periaux, J., A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 112, 1-4, 133-148 (1994) · Zbl 0845.76069
[14] Glowinski, R.; Pan, T. W.; Hesla, T. I.; Joseph, D. D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiph. Flow, 25, 5, 755-794 (1999) · Zbl 1137.76592
[15] Van Loon, R.; Anderson, P. D.; Baaijens, F. P.; Van de Vosse, F. N., A three-dimensional fluid-structure interaction method for heart valve modelling, C. R. Mecanique, 333, 12, 856-866 (2005) · Zbl 1173.76319
[16] Gerstenberger, A.; Wall, W. A., An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction, Comput. Methods Appl. Mech. Eng., 197, 19, 1699-1714 (2008) · Zbl 1194.76117
[17] Mayer, U. M.; Popp, A.; Gerstenberger, A.; Wall, W. A., 3d fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach, Comput. Mech., 46, 1, 53-67 (2010) · Zbl 1301.74018
[18] Alauzet, F.; Fabrèges, B.; Fernández, M. A.; Landajuela, M., Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures, Comput. Methods Appl. Mech. Eng., 301, 300-335 (2016) · Zbl 1423.76201
[19] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Anal. Numér., 8, R2, 129-151 (1974) · Zbl 0338.90047
[20] Brezzi, F.; Bathe, K. J., A discourse on the stability conditions for mixed finite element formulations, Comput. Methods Appl. Mech. Eng., 82, 1-3, 27-57 (1990) · Zbl 0736.73062
[21] Nitsche, J., Über ein variationsprinzip zur lösung von dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, (Abhandlungen aus dem. Abhandlungen aus dem, Mathematischen Seminar der Universität Hamburg, vol. 36 (1971), Springer), 9-15 · Zbl 0229.65079
[22] Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math., 62, 4, 328-341 (2012) · Zbl 1316.65099
[23] Massing, A.; Larson, M. G.; Logg, A.; Rognes, M. E., A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61, 3, 604-628 (2014) · Zbl 1417.76028
[24] Dettmer, W. G.; Kadapa, C.; Perić, D., A stabilised immersed boundary method on hierarchical b-spline grids, Comput. Methods Appl. Mech. Eng., 311, 415-437 (2016) · Zbl 1439.76061
[25] Schillinger, D.; Harari, I.; Hsu, M. C.; Kamensky, D.; Stoter, S. K.; Yu, Y.; Zhao, Y., The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements, Comput. Methods Appl. Mech. Eng., 309, 625-652 (2016) · Zbl 1439.65192
[26] Kadapa, C.; Dettmer, W. G.; Perić, D., A stabilised immersed boundary method on hierarchical b-spline grids for fluid-rigid body interaction with solid-solid contact, Comput. Methods Appl. Mech. Eng., 318, 242-269 (2017) · Zbl 1439.74429
[27] Zou, Z.; Aquino, W.; Harari, I., Nitsche’s method for Helmholtz problems with embedded interfaces, Int. J. Numer. Methods Eng., 110, 7, 618-636 (2017) · Zbl 1365.35018
[28] Kadapa, C.; Dettmer, W. G.; Perić, D., A stabilised immersed framework on hierarchical b-spline grids for fluid-flexible structure interaction with solid-solid contact, Comput. Methods Appl. Mech. Eng., 335, 472-489 (2018) · Zbl 1440.74406
[29] Burman, E., Ghost penalty, C. R. Math., 348, 21-22, 1217-1220 (2010) · Zbl 1204.65142
[30] Liu, G. R., Mesh Free Methods: Moving Beyond the Finite Element Method (2002), CRC Press
[31] Belytschko, T.; Gracie, R.; Ventura, G., A review of extended/generalized finite element methods for material modeling, Model. Simul. Mater. Sci. Eng., 17, 4, Article 043001 pp. (2009)
[32] Samet, H., Applications of Spatial Data Structures: Computer Graphics, Image Processing, and Gis (1990), Addison-Wesley: Addison-Wesley Reading, Mass
[33] Berg, M. D.; Cheong, O.; Kreveld, M. V.; Overmars, M., Computational Geometry: Algorithms and Applications (2008), Springer-Verlag TELOS · Zbl 1140.68069
[34] Blom, F. J., A monolithical fluid-structure interaction algorithm applied to the piston problem, Comput. Methods Appl. Mech. Eng., 167, 3-4, 369-391 (1998) · Zbl 0948.76046
[35] Dettmer, W.; Perić, D., A computational framework for fluid-rigid body interaction: finite element formulation and applications, Comput. Methods Appl. Mech. Eng., 195, 13-16, 1633-1666 (2006) · Zbl 1123.76029
[36] Jaiman, R. K.; Guan, M. Z.; Miyanawala, T. P., Partitioned iterative and dynamic subgrid-scale methods for freely vibrating square-section structures at subcritical Reynolds number, Comput. Fluids, 133, 68-89 (2016) · Zbl 1390.76056
[37] Kadapa, C.; Dettmer, W. G.; Perić, D., A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical B-spline grids, Comput. Methods Appl. Mech. Eng., 301, 1-27 (2016) · Zbl 1423.76243
[38] Dettmer, W. G.; Perić, D., A new staggered scheme for fluid-structure interaction, Int. J. Numer. Methods Eng., 93, 1, 1-22 (2013) · Zbl 1352.74471
[39] Placzek, A.; Sigrist, J. F.; Hamdouni, A., Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: forced and free oscillations, Comput. Fluids, 38, 1, 80-100 (2009) · Zbl 1237.76029
[40] Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. Appl. Mech., 60, 2, 371-375 (1993) · Zbl 0775.73337
[41] Jansen, K. E.; Whiting, C. H.; Hulbert, G. M., A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Comput. Methods Appl. Mech. Eng., 190, 3, 305-319 (2000) · Zbl 0973.76048
[42] Heywood, J. G.; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 22, 5, 325-352 (1996) · Zbl 0863.76016
[43] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 1-3, 199-259 (1982) · Zbl 0497.76041
[44] Shakib, F.; Hughes, T. J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 89, 1-3, 141-219 (1991)
[45] 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. Eng., 95, 2, 221-242 (1992) · Zbl 0756.76048
[46] Franca, L. P.; Frey, S. L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 99, 2, 209-233 (1992) · Zbl 0765.76048
[47] Harari, I.; Hughes, T. J., What are c and h?: inequalities for the analysis and design of finite element methods, Comput. Methods Appl. Mech. Eng., 97, 2, 157-192 (1992) · Zbl 0764.73083
[48] Benk, J., Immersed boundary methods within a PDE toolbox on distributed memory systems (2012), Universitätsbibliothek der TU München, Ph.D. thesis
[49] Burman, E., A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J. Numer. Anal., 50, 4, 1959-1981 (2012) · Zbl 1262.65165
[50] Gregory, J., Quadratic Form Theory and Differential Equations, vol. 152 (1981), Elsevier
[51] Cook, R. D., Concepts and Applications of Finite Element Analysis (1981) · Zbl 0534.73056
[52] Vapnik, V. N., An overview of statistical learning theory, IEEE Trans. Neural Netw., 10, 5, 988-999 (1999)
[53] Suykens, J. A.K., Least Squares Support Vector Machines (2002), World Scientific: World Scientific River Edge, N.J, Singapore · Zbl 1017.93004
[54] Murty, M. N.; Raghava, R., Support Vector Machines and Perceptrons: Learning, Optimization, Classification, and Application to Social Networks (2016), Springer · Zbl 1365.68003
[55] Liu, B.; Jin, Y.; Magee, A.; Yiew, L.; Zhang, S., System identification of Abkowitz Model for ship maneuvering motion based on ε-support vector regression, (International Conference on Offshore Mechanics and Arctic Engineering, vol. 58844 (2019), American Society of Mechanical Engineers), V07AT06A067
[56] Lumley, J. L., The structure of inhomogeneous turbulence, (Yaglom, A. M.; Tatarski, V. I., Atmospheric Turbulence and Wave Propagation (1967)), 166-178
[57] Lumley, J. L., Stochastic Tools in Turbulence (2007), Courier Corporation · Zbl 1230.76004
[58] Schmid, P. J., Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28 (2010) · Zbl 1197.76091
[59] Jovanović, M. R.; Schmid, P. J.; Nichols, J. W., Sparsity-promoting dynamic mode decomposition, Phys. Fluids, 26, 2, Article 024103 pp. (2014)
[60] Liu, B.; Jaiman, R. K., Dynamics and stability of gap-flow interference in a vibrating side-by-side arrangement of two circular cylinders, J. Fluid Mech., 855, 804-838 (2018) · Zbl 1415.76193
[61] Wilkinson, J. H., The Algebraic Eigenvalue Problem, vol. 662 (1965), Oxford Clarendon · Zbl 0258.65037
[62] Mikhlin, S. G., Variational Methods in Mathematical Physics, vol. 50 (1964), Pergamon Press, [distributed by Macmillan, New York] · Zbl 0119.19002
[63] Ern, A.; Guermond, J. L., Theory and Practice of Finite Elements, vol. 159 (2013), Springer Science & Business Media
[64] Rätz, A.; Voigt, A., PDE’s on surfaces—a diffuse interface approach, Commun. Math. Sci., 4, 3, 575-590 (2006) · Zbl 1113.35092
[65] Li, X.; Lowengrub, J.; Rätz, A.; Voigt, A., Solving PDEs in complex geometries: a diffuse domain approach, Commun. Math. Sci., 7, 1, 81 (2009) · Zbl 1178.35027
[66] Nguyen, L. H.; Stoter, S. K.F.; Ruess, M.; Uribe, M. A.S.; Schillinger, D., The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries, Int. J. Numer. Methods Eng., 113, 4, 601-633 (2018)
[67] Tritton, D. J., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech., 6, 4, 547-567 (1959) · Zbl 0092.19502
[68] Coutanceau, M.; Bouard, R., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow, J. Fluid Mech., 79, 2, 231-256 (1977)
[69] Calhoun, D., A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176, 2, 231-275 (2002) · Zbl 1130.76371
[70] Russell, D.; Wang, Z. J., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191, 1, 177-205 (2003) · Zbl 1160.76389
[71] Li, Z.; Jaiman, R. K.; Khoo, B. C., An immersed interface method for flow past circular cylinder in the vicinity of a plane moving wall, Int. J. Numer. Methods Fluids, 81, 10, 611-639 (2016)
[72] Braza, M.; Chassaing, P. H.H. M.; Minh, H. H., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. Fluid Mech., 165, 79-130 (1986) · Zbl 0596.76047
[73] Liu, C.; Zheng, X.; Sung, C. H., Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys., 139, 1, 35-57 (1998) · Zbl 0908.76064
[74] Kadapa, C.; Dettmer, W. G.; Perić, D., A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical b-spline grids, Comput. Methods Appl. Mech. Eng., 301, 1-27 (2016) · Zbl 1423.76243
[75] Chen, Y. M.; Ou, Y. R.; Pearlstein, A. J., Development of the wake behind a circular cylinder impulsively started into rotatory and rectilinear motion, J. Fluid Mech., 253, 449-484 (1993) · Zbl 0809.76024
[76] Mittal, S.; Kumar, B., Flow past a rotating cylinder, J. Fluid Mech., 476, 303-334 (2003) · Zbl 1163.76442
[77] Liu, B.; Jaiman, R. K., Interaction dynamics of gap flow with vortex-induced vibration in side-by-side cylinder arrangement, Phys. Fluids, 28, 12, Article 127103 pp. (2016)
[78] Bao, Y.; Huang, C.; Zhou, D.; Tu, J.; Han, Z., Two-degree-of-freedom flow-induced vibrations on isolated and tandem cylinders with varying natural frequency ratios, J. Fluids Struct., 35, 50-75 (2012)
[79] Wan, D.; Turek, S., Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. Numer. Methods Fluids, 51, 5, 531-566 (2006) · Zbl 1145.76406
[80] Wang, Y.; Shu, C.; Teo, C. J.; Wu, J., An immersed boundary-lattice Boltzmann flux solver and its applications to fluid-structure interaction problems, J. Fluids Struct., 54, 440-465 (2015)
[81] Liska, S.; Colonius, T., A fast immersed boundary method for external incompressible viscous flows using lattice Green’s functions, J. Comput. Phys., 331, 257-279 (2017) · Zbl 1378.76083
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