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A monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction model. (English) Zbl 1489.65141

The authors have introduced a novel monolithic divergence-conforming HDG scheme fo a linear fluid-structure interaction (FSI) problem with a thick structure. They have presented divergence-conforming HDG spatial discretization of the linear FSI system. Further, they have perforemed a priori error analysis of the semidiscrete scheme. They have applied the Crank-Nicolson scheme for the time discretization. The resulting algebraic system is solved adopting a preconditioned MinRes method. Finally, they have simulated some numerical results to show the robust of their method with resepect to the mesh size, time step size and material parameters.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76D07 Stokes and related (Oseen, etc.) flows
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74B10 Linear elasticity with initial stresses
76M10 Finite element methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics

Software:

NGSolve; hypre; BoomerAMG
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Full Text: DOI arXiv

References:

[1] M. Ainsworth and G. Fu, Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations, J. Sci. Comput., 77 (2018), pp. 443-466. · Zbl 1407.65275
[2] S. Badia, A. Quaini, and A. Quarteroni, Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 4216-4232. · Zbl 1194.74058
[3] J. W. Banks, W. D. Henshaw, A. K. Kapila, and D. W. Schwendeman, An added-mass partition algorithm for fluid-structure interactions of compressible fluids and nonlinear solids, J. Comput. Phys., 305 (2016), pp. 1037-1064. · Zbl 1349.76428
[4] D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and Applications, Vol. 44, Springer-Verlag, Berlin, 2013. · Zbl 1277.65092
[5] J. H. Bramble and J. E. Pasciak, Iterative techniques for time dependent Stokes problems, Approxi. Theory Appl., 33 (1997), pp. 13-30. · Zbl 1030.76506
[6] M. Bukac, A. Seboldt, and C. Trenchea, Refactorization of Cauchy’s Method: A Second-Order Partitioned Method for Fluid-Thick Structure Interaction Problems, preprint, arXiv:2008.12979, 2020. · Zbl 1477.65155
[7] M. Bukač, S. Čanić, R. Glowinski, B. Muha, and A. Quaini, A modular, operator-splitting scheme for fluid-structure interaction problems with thick structures, Internat. J. Numer. Methods Fluids, 74 (2014), pp. 577-604. · Zbl 1455.76034
[8] H.-J. Bungartz and M. Schäfer, Fluid-structure Interaction: Modelling, Simulation, Optimisation, Vol. 53, Springer-Verlag, Berlin, 2006. · Zbl 1097.76002
[9] E. Burman and M. A. Fernández, Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 766-784. · Zbl 1229.76045
[10] J. Cahouet and J.-P. Chabard, Some fast 3D finite element solvers for the generalized Stokes problem, Internat. J. Numer. Methods Fluids, 8 (1988), pp. 869-895. · Zbl 0665.76038
[11] P. Causin, J. Gerbeau, and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 4506-4527. · Zbl 1101.74027
[12] B. Chabaud and B. Cockburn, Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp., 81 (2012), pp. 107-129. · Zbl 1251.65138
[13] S. K. Chakrabarti, Numerical Models in Fluid-Structure Interaction, WIT Press, Billerica, MA, 2005. · Zbl 1070.76002
[14] B. Cockburn, Discontinuous Galerkin methods for computational fluid dynamics; in Encyclopedia of Computational Mechanics, 2nd ed., Wiley, New York, 2018, pp. 1-63.
[15] B. Cockburn, N. C. Nguyen, and J. Peraire, HDG Methods for hyperbolic problems, in Handbook of Numerical Analysis, Vol. 17, Elsevier, Amsterdam, 2016, pp. 173-197.
[16] S. Deparis, M. Discacciati, G. Fourestey, and A. Quarteroni, Fluid-structure algorithms based on Steklov-Poincaré operators, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 5797-5812. · Zbl 1124.76026
[17] E. H. Dowell and K. C. Hall, Modeling of fluid-structure interaction, Ann. Rev. Fluid Mech., 33 (2001), pp. 445-490. · Zbl 1052.76059
[18] H. C. Elman, D. J. Silvester, and A. J. Wathen, Block preconditioners for the discrete incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 40 (2002), pp. 333-344. · Zbl 1019.76023
[19] R. D. Falgout and U. M. Yang, hYPRE: A library of high performance preconditioners, in Computational Science ICCS 2002, Springer-Verlag, Berlin, 2002, pp. 632-641. · Zbl 1056.65046
[20] C. Farhat, K. G. Van der Zee, and P. Geuzaine, Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 1973-2001. · Zbl 1178.76259
[21] M. A. Fernández and M. Landajuela, A fully decoupled scheme for the interaction of a thin-walled structure with an incompressible fluid, C. Re. Math., 351 (2013), pp. 161-164. · Zbl 1307.74031
[22] A. Figueroa, I. Vignon-Clementel, K. Jansen, T. Hughes, and C. Taylor, A coupled momentum method for modeling blood flow in three-dimensional deformable arteries, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 5685-5706. · Zbl 1126.76029
[23] G. Fu, Uniform Auxiliary Space Preconditioning for HDG Methods for Elliptic Operators with a Parameter Dependent Low Order Term, arXiv:2011.11828 [math.NA]. · Zbl 1478.65123
[24] G. Fu, Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for incompressible flow with moving boundaries and interfaces, Comput. Methods Appl. Mech. Engrg., 367 (2020), p. 113158. · Zbl 1442.76066
[25] G. Fu, C. Lehrenfeld, A. Linke, and T. Streckenbach, Locking Free and Gradient Robust \(H(div)\)-conforming HDG Methods for Linear Elasticity, preprint, arXiv:2001.08610 [math.NA], 2011. · Zbl 1460.65146
[26] M. W. Gee, U. Küttler, and W. A. Wall, Truly monolithic algebraic multigrid for fluid-structure interaction, Int. J. Numer. Methods Engrg., 85 (2011), pp. 987-1016. · Zbl 1217.74121
[27] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin, 2010. · Zbl 1192.65097
[28] V. E. Henson and U. M. Yang, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Meth., 41 (2002), pp. 155-177. · Zbl 0995.65128
[29] G. Hou, J. Wang, and A. Layton, Numerical methods for fluid-structure interaction-a review, Comm. Comput. Phys., 12 (2012), pp. 337-377. · Zbl 1373.76001
[30] B. Hübner, E. Walhorn, and D. Dinkler, A monolithic approach to fluid-structure interaction using space-time finite elements, Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 2087-2104. · Zbl 1067.74575
[31] I. C. F. Ipsen, A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23 (2001), pp. 1050-1051. · Zbl 0998.65049
[32] T. Klöppel, A. Popp, U. Küttler, and W. A. Wall, Fluid-structure interaction for non-conforming interfaces based on a dual mortar formulation, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 3111-3126. · Zbl 1230.74185
[33] A. La Spina, M. Kronbichler, M. Giacomini, W. A. Wall, and A. Huerta, A weakly compressible hybridizable discontinuous Galerkin formulation for fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg., 372 (2020), 113392. · Zbl 1506.74415
[34] C. Lehrenfeld, Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems, Diploma thesis, MathCCES/IGPM, RWTH Aachen, 2010.
[35] C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307 (2016), pp. 339-361. · Zbl 1439.76081
[36] M. Lukáčová-Medvid’ová, G. Rusnáková, and A. Hundertmark-Zaušková, Kinematic splitting algorithm for fluid-structure interaction in hemodynamics, Comput. Methods Appl. Mech. Engrg., 265 (2013), pp. 83-106. · Zbl 1286.76176
[37] K.-A. Mardal and R. Winther, Uniform preconditioners for the time dependent Stokes problem, Numer. Math., 98 (2004), pp. 305-327. · Zbl 1058.65101
[38] M. F. Murphy, G. H. Golub, and A. J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969-1972. · Zbl 0959.65063
[39] M. Neunteufel and J. Schöberl, Fluid-structure interaction with \(h(div)\)-conforming finite elements, Comp. Struct., 243 (2021), 106402. · Zbl 1506.74192
[40] N. C. Nguyen and J. Peraire, Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics, J. Comput. Phys., 231 (2012), pp. 5955-5988. · Zbl 1277.65082
[41] F. Nobile, Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics, Ph.D. thesis, École polytechnique fédérale de Lausanne, 2001.
[42] F. Nobile and C. Vergara, An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions, SIAM J. Sci. Comput., 30 (2008), pp. 731-763. · Zbl 1168.74038
[43] M. A. Olshanskii, J. Peters, and A. Reusken, Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations, Numer. Math., 105 (2006), pp. 159-191. · Zbl 1120.65059
[44] T. Richter, Fluid-structure Interactions: Models, Analysis and Finite Elements, Vol. 118, Springer-Verlag, Berlin, 2017. · Zbl 1374.76001
[45] S. Rugonyi and K.-J. Bathe, On finite element analysis of fluid flows fully coupled with structural interactions, Comput. Model. Engrg. Sci., 2 (2001), pp. 195-212.
[46] T. Rusten, P. S. Vassilevski, and R. Winther, Interior penalty preconditioners for mixed finite element approximations of elliptic problems, Math. Comp., 65 (1996), pp. 447-466. · Zbl 0857.65117
[47] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003. · Zbl 1031.65046
[48] J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, ASC report 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology, 2014.
[49] J. Sheldon, S. Miller, and J. Pitt, A hybridizable discontinuous Galerkin method for modeling fluid-structure interaction, J. Comput. Phys., 326 (2016), pp. 91-114. · Zbl 1373.74042
[50] J. Sheldon, S. Miller, and J. Pitt, An improved formulation for hybridizable discontinuous Galerkin fluid-structure interaction modeling with reduced computational expense, Commun. Comput. Phys., 24 (2018), pp. 1279-1299. · Zbl 1475.74122
[51] T. E. Tezduyar and S. Sathe, Modelling of fluid-structure interactions with the space-time finite elements: Solution techniques, Internat. J. Numer. Methods Fluids, 54 (2007), pp. 855-900. · Zbl 1144.74044
[52] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235. · Zbl 0857.65129
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