Parallel time-stepping for fluid–structure interactions

Nils Margenberg¹ ; Thomas Richter²

¹ Helmut Schmidt University, Hamburg, Germany.
² Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 20.

Résumé

We present a parallel time-stepping method for fluid–structure interactions. The interaction between the incompressible Navier-Stokes equations and a hyperelastic solid is formulated in a fully monolithic framework. Discretization in space is based on equal order finite element for all variables and a variant of the Crank-Nicolson scheme is used as second order time integrator. To accelerate the solution of the systems, we analyze a parallel-in time method. For different numerical test cases in 2d and in 3d we present the efficiency of the resulting solution approach. We also discuss some challenges and limitations that are connected to the special structure of fluid–structure interaction problem. In particular, we will investigate stability and dissipation effects of the time integration and their influence on the convergence of the parareal method. It turns out that especially processes based on an internal dynamics (e.g. driven by the vortex street around an elastic obstacle) cause great difficulties. Configurations however, which are driven by oscillatory problem data, are well-suited for parallel time stepping and allow for substantial speedups.

Détail

DOI : 10.1051/mmnp/2021005

Bibliographie

[1] G. Ariel, H. Nguyen and R. Tsai, θ-parareal schemes. Preprint https://arxiv.org/abs/1704.06882 (2017).

[2] E. Aubanel Scheduling of tasks in the parareal algorithm Parallel Comput 2011 172 182

[3] E. Aulisa, S. Bna, G. Bornia A monolithic ale Newton-Krylov solver with multigrid-Richardson-Schwarz preconditioning for incompressible fluid–structure interaction Comput. Fluids 2018 213 228

[4] G. Avalos, I. Lasiecka, R. Triggiani Higher regularity of a coupled parabolic hyperbolic fluid–structure interactive system Georgian Math. J 2008 402 437

[5] G. Avalos, R. Triggiani Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid–structure interaction Discrete Contin. Dyn. Syst. Ser. S 2009 417 447

[6] G. Avalos, R. Triggiani Fluid–structure interaction with and without internal dissipation of the structure: a contrast study in stability Evol. Equ. Control Theory 2013 563 598

[7] A.-M. Baudron, J.-J. Lautard, Y. Maday, M. Riahi, J. Salomon Parareal in time 3d numerical solver for the LWR benchmark neutron diffusion transient model J. Comput. Phys 2014 67 79

[8] R. Becker, M. Braack A finite element pressure gradient stabilization for the Stokes equations based on local projections Calcolo 2001 173 199

[9] R. Becker, M. Braack, D. Meidner, T. Richter, B. Vexler The finite element toolkit Gascoigne. 2020

[10] M. Besier, W. Wollner On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes Int. J. Numer. Math. Fluids 2012 1054 1064

[11] A. Blouza, L. Boudin, S.M. Kaber Parallel in time algorithms with reduction methods for solving chemical kinetics Commun. Appl. Math. Comput. Sci 2011 241 263

[12] P. Causin, J.F. Gereau, F. Nobile Added-mass effect in the design of partitioned algorithms for fluid–structure problems Comput. Methods Appl. Mech. Eng 2005 4506 4527

[13] R. Croce, D. Ruprecht and R. Krause, Parallel-in-space-and-time simulation of the three-dimensional, unsteady Navier-Stokes equations for incompressible flow. In Modeling, Simulation and Optimization of Complex Processes-HPSC 2012. Springer (2014) 13–23.

[14] J. Donea An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid–structure interactions Comput. Methods Appl. Mech. Eng 1982 689 723

[15] L. Failer, T. Richter A Newton multigrid framework for optimal control of fluid–structure interactions To appear in: Optim. Eng. 2020

[16] L. Failer, T. Richter A parallel Newton multigrid framework for monolithic fluid–structure interactions J. Sci. Comput 2020 28

[17] M.A. Fernández and J.-F. Gerbeau, Algorithms for fluid–structure interaction problems. In Cardiovascular Mathematics: Modelingand simulation of the circulatory system, volume 1, edited by L. Formaggia, A. Quarteroni, and A. Veneziani. MS A. Springer (2009) 307–346.

[18] P.F. Fischer, F. Hecht and Y. Maday, A parareal in time semi-implicit approximation of the Navier-stokes equations. In Domain decomposition methods in science and engineering. Springer (2005) 433–440.

[19] S. Frei, T. Richter A locally modified parametric finite element method for interface problems SIAM J. Numer. Anal 2014 2315 2334

[20] M.J. Gander, S. Vandewalle Analysis of the parareal time-parallel time-integration method SIAM J. Sci. Comput 2007 556 578

[21] M.W. Gee, U. Küttler, W.A. Wall Truly monolithic algebraic multigrid for fluid–structure interaction Int. J. Numer. Meth. Eng 2010 987 1016

[22] D. Hartmann, C. Lessig, N. Margenberg, T. Richter A neural network multigrid solver for the Navier-stokes equations Preprint 2020

[23] T. Haut, B. Wingate An asymptotic parallel-in-time method for highly oscillatory PDEs SIAM J. Sci. Comput 2014 A693 A713

[24] M. Heil, A.L. Hazel, J. Boyle Solvers for large-displacement fluid–structure interaction problems: Segregated vs. monolithic approaches Comput. Mech. 2008 91 101

[25] J.G. Heywood, R. Rannacher, S. Turek Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations Int. J. Numer. Math. Fluids 1992 325 352

[26] C.W. Hirt, A.A. Amsden, J.L. Cook An arbitrary Lagrangian-Eulerian computing method for all flow speeds J. Comp. Phys 1974 227 469

[27] J. Hron and S. Turek, A monolithic FEM/Multigrid solver for an ALE formulation of fluid–structure interaction with applications in biomechanics. In Fluid-structure Interaction: Modeling, Simulation, Optimization, edited by H.-J. Bungartz and M. Schäfer. Lecture Notes in Computational Science and Engineering. Springer (2006) 146–170.

[28] J. Hron and S. Turek, Proposal for numerical benchmarking of fluid–structure interaction between an elastic object and laminar incompressible flow. In Fluid-structure Interaction: Modeling, Simulation, Optimization, edited by H.-J. Bungartz and M. Schäfer. Lecture Notes in Computational Science and Engineering. Springer (2006) 371–385.

[29] T.J.R. Hughes, W.K. Liu, T.K. Zimmermann Lagrangian-Eulerian finite element formulations for incompressible viscous flows Comput. Methods Appl. Mech. Eng 1981 329 349

[30] A. Kreienbuehl, A. Naegel, D. Ruprecht, R. Speck, G. Wittum, R. Krause Numerical simulation of skin transport using parareal Comput. Visual. Sci 2015 99 108

[31] U. Langer and H. Yang, Recent development of robust monolithic fluid–structure interaction solvers. In fluid–structure Interactions. Modeling, Adaptive Discretization and Solvers. Vol. 20 of Radon Series on Computational and Applied Mathematics. de Gruyter (2017).

[32] M. Lenoir Optimal isoparametric finite elements and error estimates for domains involving curved boundaries SIAM J. Numer. Anal 1986 562 580

[33] J.-L. Lions, Y. Maday, G. Turinici Résolution d’edp par un schéma en temps C. R. Acad. Sci. Math 2001 661 668

[34] M. Luskin, R. Rannacher On the smoothing propoerty of the Crank-Nicholson scheme Appl. Anal 1982 117 135

[35] M. Molnar, Stabilisierte Finite Elemente für Strömungsprobleme auf bewegten Gebieten. Master’s thesis, Universität Heidelberg (2015).

[36] R. Rannacher Finite element solution of diffusion problems with irregular data Numer. Math 1984 309 327

[37] R. Rannacher On the stabilization of the Crank-Nicolson scheme for long time calculations Inpreparation 1986

[38] T. Richter A monolithic geometric multigrid solver for fluid–structure interactions in ALE formulation Int. J. Numer. Meth. Eng 2015 372 390

[39] T. Richter, Fluid–structure Interactions. Models, Analysis and Finite Elements. Vol. 118 of Lecture notes in computational science and engineering. Springer (2017).

[40] T. Richter, T. Wick Finite elements for fluid–structure interaction in ALE and Fully Eulerian coordinates Comput. Methods Appl. Mech. Eng 2010 2633 2642

[41] T. Richter and T. Wick, On time discretizations of fluid–structure interactions. In Multiple Shooting and Time Domain Decomposition Methods, edited by T. Carraro, M. Geiger, S. Körkel, and R. Rannacher. Vol. 9 of Contributions in Mathematical and Computational Science. Springer (2015) 377–400.

[42] T. Richter, S. Frei Second order time-stepping for parabolic interface problems with moving interfaces Model. Math. Anal. Numer 2017 1539 1560

[43] D. Ruprecht Wave propagation characteristics of parareal Comput. Visual. Sci 2018 1 17

[44] D. Samaddar, D.E. Newman, R. Sánchez Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm J. Comput. Phys 2010 6558 6573

[45] M. Soszynska and T. Richter, Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation. Preprint https://arxiv.org/abs/2007.05372 (2020).

[46] J. Steiner, D. Ruprecht, R. Speck and R. Krause, Convergence of parareal for the Navier-Stokes equations depending on the Reynolds number. In Numerical Mathematics and Advanced Applications - ENUMATH 2013. Edited by Assyr Abdulle, Simone Deparis, Daniel Kressner, Fabio Nobile, and Marco Picasso. Springer International Publishing, Cham (2015) 195–202.

[47] J.M.F. Trindade, J.C.F. Pereira Parallel-in-time simulation of the unsteady Navier–stokes equations for incompressible flow Int. J. Numer. Methods Fluids 2004 1123 1136

[48] S. Turek, J. Hron, M. Madlik, M. Razzaq, H. Wobker, J. Acker 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. Ergebnisberichte des Instituts für Angewandte Mathematik, Num. 403 2010

[49] S. Turek, L. Rivkind, J. Hron, R. Glowinski Numerical study of a modified time–stepping theta–scheme for incompressible flow simulations J. Sci. Comput 2006 533 547