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A parallel Newton multigrid framework for monolithic fluid-structure interactions. (English) Zbl 07161485
Summary: We present a monolithic parallel Newton-multigrid solver for nonlinear nonstationary three dimensional fluid-structure interactions in arbitrary Lagrangian Eulerian (ALE) formulation. We start with a finite element discretization of the coupled problem, based on a remapping of the Navier-Stokes equations onto a fixed reference framework. The strongly coupled fluid-structure interaction problem is discretized with finite elements in space and finite differences in time. The resulting nonlinear and linear systems of equations are large and show a very high condition number. We present a novel Newton approach that is based on two essential ideas: First, a condensation of the solid deformation by exploiting the discretized velocity-deformation relation \(d_t \mathbf{u}=\mathbf{v} \), second, the Jacobian of the fluid-structure interaction system is simplified by neglecting all derivatives with respect to the ALE deformation, an approximation that has shown to have little impact. The resulting system of equations decouples into a joint momentum equation and into two separate equations for the deformation fields in solid and fluid. Besides a reduction of the problem sizes, the approximation has a positive effect on the conditioning of the systems such that multigrid solvers with simple smoothers like a parallel Vanka-iteration can be applied. We demonstrate the efficiency of the resulting solver infrastructure on a well-studied 2d test-case and we also introduce a challenging 3d problem.

76M Basic methods in fluid mechanics
74F Coupling of solid mechanics with other effects
76D Incompressible viscous fluids
65M Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[1] Amestoy, Pr; Guermouche, A.; L’Excellent, Jy; Pralet, S., Hybrid scheduling for the parallel solution of linear systems, Parallel Comput., 32, 2, 136-156 (2006)
[2] 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
[3] Becker, R.; Braack, M., Multigrid techniques for finite elements on locally refined meshes, Numer. Linear Algebra Appl., 7, 363-379 (2000) · Zbl 1051.65117
[4] Becker, R.; Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38, 4, 173-199 (2001) · Zbl 1008.76036
[5] Becker, R., Braack, M., Meidner, D., Richter, T., Vexler, B.: The finite element toolkit Gascoigne. http://www.gascoigne.uni-hd.de
[6] Braack, M.; Richter, T., Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements, Comput. Fluids, 35, 4, 372-392 (2006) · Zbl 1160.76364
[7] Brummelen, E.; Zee, K.; Borst, R., Space/time multigrid for a fluid-structure-interaction problem, Appl. Numer. Math., 58, 12, 1951-1971 (2008) · Zbl 1148.74046
[8] Bungartz, H.J., Schäfer, M. (eds.): Fluid-Structure Interaction. Modelling, Simulation, Optimisation. Lecture Notes in Computational Science and Engineering, vol. 53. Springer (2006). ISBN-10: 3-540-34595-7
[9] Bungartz, H.J., Schäfer, M. (eds.): Fluid-Structure Interaction II. Modelling, Simulation, Optimisation. Lecture Notes in Computational Science and Engineering. Springer (2010)
[10] Causin, P.; Gereau, J.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Eng., 194, 4506-4527 (2005) · Zbl 1101.74027
[11] Crosetto, P.; Deparis, S.; Fourestey, G.; Quarteroni, A., Parallel algorithms for fluid-structure interaction problems in haemodynamics, SIAM J. Sci. Comput., 33, 4, 1598-1622 (2011) · Zbl 1417.92008
[12] Davis, T., Umfpack, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Soft., 30, 2, 196-199 (2014) · Zbl 1072.65037
[13] Deparis, S.; Forti, D.; Grandperrin, G.; Quarteroni, A., Facsi: a block parallel preconditioner for fluid-structure interaction in hemodynamics, J. Comput. Phys., 327, 700-718 (2016) · Zbl 1373.74036
[14] Failer, L.: Optimal control for time dependent nonlinear fluid-structure interaction. Ph.D. thesis, Technische Universität München (2017)
[15] Failer, L.; Wick, T., Adaptive time-step control for nonlinear fluid-structure interaction, J. Comput. Phys., 366, 448-477 (2018) · Zbl 1406.76048
[16] Fernández, M.; Gerbeau, Jf; Formaggia, L.; Quarteroni, A.; Veneziani, A., Algorithms for fluid-structure interaction problems, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, MS & A, 307-346 (2009), Berlin: Springer, Berlin
[17] Fernández, M.; Moubachir, M., A newton method using exact Jacobians for solving fluid-structure coupling, Comput. Struct>, 83, 127-142 (2005)
[18] Frei, S.: Eulerian finite element methods for interface problems and fluid-structure interactions. Ph.D. thesis, Universität Heidelberg (2016). 10.11588/heidok.00021590
[19] Gee, M.; Küttler, U.; Wall, W., Truly monolithic algebraic multigrid for fluid-structure interaction, Int. J. Numer. Method Eng., 85, 987-1016 (2010) · Zbl 1217.74121
[20] Guennebaud, G., Jacob, B., et al.: Eigen v3. http://eigen.tuxfamily.org (2010)
[21] Heil, M.; Hazel, A.; Boyle, J., Solvers for large-displacement fluid-structure interaction problems: segregated vs monolithic approaches, Comput. Mech., 43, 91-101 (2008) · Zbl 1309.76126
[22] Heywood, J.; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Math. Fluids., 22, 325-352 (1992) · Zbl 0863.76016
[23] Hron, J.; Turek, S.; Bungartz, Hj; Schäfer, M., A monolithic FEM/Multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics, Fluid-Structure Interaction: Modeling, Simulation, Optimization, 146-170 (2006), Berlin: Springer, Berlin · Zbl 1323.74086
[24] Hron, J.; Turek, S.; Bungartz, Hj; Schäfer, M., Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, Fluid-Structure Interaction: Modeling, Simulation, Optimization, 371-385 (2006), Berlin: Springer, Berlin · Zbl 1323.76049
[25] Jodlbauer, D.; Langer, U.; Wick, T., Parallel block-preconditioned monolithic solvers for fluid-structure interaction problems, Int. J. Numer. Methods Eng., 117, 6, 623-643 (2019)
[26] Kimmritz, M.; Richter, T., Parallel multigrid method for finite element simulations of complex flow problems on locally refined meshes, Numer. Linear Algebra Appl., 18, 4, 615-636 (2010) · Zbl 1265.76043
[27] Klemm, M., Supinski, B., (eds.): OpenMP Application Programming Interface Specification Version 5.0. Independently published (2019)
[28] Langer, U., Yang, H.: Recent development of robust monolithic fluid-structure interaction solvers. In: Fluid-Structure Interaction. Modeling, Adaptive Discretization and Solvers. Radon Series on Computational and Applied Mathematics, vol. 20, pp. 169-192. de Gruyter (2017)
[29] Molnar, M.: Stabilisierte Finite Elemente für Strömungsprobleme auf bewegten Gebieten. Master’s thesis, Universität Heidelberg (2015)
[30] Pironneau, O.: An energy preserving monolithic eulerian fluid-structure numerical scheme. Chinese Annals of Mathematics 39, (2016). 10.1007/s11401-018-1061-9 · Zbl 1386.76104
[31] Pironneau, O.: An Energy stable Monolithic Eulerian Fluid-Structure Numerical Scheme with compressible materials (2019). arXiv:1607.08083
[32] Richter, T., A monolithic geometric multigrid solver for fluid-structure interactions in ALE formulation, Int. J. Numer. Meth. Eng., 104, 5, 372-390 (2015) · Zbl 1352.76066
[33] Richter, T., Fluid-structure Interactions. Models, Analysis and Finite Elements (2017), Berlin: Springer, Berlin · Zbl 1374.76001
[34] Richter, T.; Wick, T., Finite elements for fluid-structure interaction in ALE and Fully Eulerian coordinates, Comput. Methods Appl. Mech. Eng., 199, 41-44, 2633-2642 (2010) · Zbl 1231.74436
[35] Richter, T.; Wick, T.; Carraro, T.; Geiger, M.; Körkel, S.; Rannacher, R., On time discretizations of fluid-structure interactions, Multiple Shooting and Time Domain Decomposition Methods, 377-400 (2015), Berlin: Springer, Berlin · Zbl 1382.76195
[36] Turek, S., Hron, J., Madlik, M., Razzaq, M., Wobker, H., Acker, J.: Numerical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics. Technical report, Fakultät für Mathematik, TU Dortmund (2010). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 403 · Zbl 1210.76118
[37] Turek, S.; Rivkind, L.; Hron, J.; Glowinski, R., Numerical study of a modified time-stepping theta-scheme for incompressible flow simulations, J. Sci. Comput., 28, 2-3, 533-547 (2006) · Zbl 1158.76385
[38] Wall, W.: Fluid-structure interaction with stabilized finite elements. Ph.D. thesis, University of Stuttgart (1999). Urn:nbn:de:bsz:93-opus-6234
[39] Wick, T.: Personal communication. University of Hannover (September 2019)
[40] Yirgit, S.; Schäfer, M.; Heck, M., Grid movement techniques and their influence on laminar fluid-structure interaction rpoblems, J. Fluids Struct., 24, 6, 819-832 (2008)
[41] Zee, K.; Brummelen, E.; Borst, R., Goal-oriented error estimation and adaptivity for free-boundary problems: the domain-map linearization approach, SIAM J. Sci. Comput., 32, 2, 1074-1092 (2010) · Zbl 1210.35300
[42] Zee, K.; Brummelen, E.; Borst, R., Goal-oriented error estimation and adaptivity for free-boundary problems: the shape-linearization approach, SIAM J. Sci. Comput., 32, 2, 1093-1118 (2010) · Zbl 1209.35159
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