×

Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation. (English) Zbl 1478.65088

Summary: We study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime. Therefore we present two decoupling methods, namely a partitioned relaxation scheme and a shooting method. Furthermore, we develop an a posteriori error estimator serving as a mean for an adaptive time-stepping procedure. The goal is to optimally balance the time-step sizes of the two subproblems. The error estimator is based on the dual weighted residual method and relies on the space-time Galerkin formulation of the coupled problem. As an example, we take a linear set-up with the heat equation coupled to the wave equation. We formulate the problem in a monolithic manner using the space-time framework. In numerical test cases, we demonstrate the efficiency of the solution process and we also validate the accuracy of the a posteriori error estimator and its use for controlling the time-step sizes.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35K05 Heat equation
35L05 Wave equation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Avalos, G.; Lasiecka, I.; Triggiani, R., Higher regularity of a coupled parabolic hyperbolic fluid-structure interactive system, Georgian Math. J., 15, 402-437 (2008) · Zbl 1157.35085 · doi:10.1515/GMJ.2008.403
[2] Becker, R., Rannacher, R.: An Optimal Control Approach to A Posteriori Error Estimation in Finite Element Methods, vol. 10, pp. 1-102. Cambridge University Press (2001) · Zbl 1105.65349
[3] Berger, M., Stability of interfaces with mesh refinement, Math. Comput., 45, 301-318 (1985) · Zbl 0647.65065 · doi:10.2307/2008126
[4] Blum, H.; Lisky, S.; Rannacher, R., A domain splitting algorithm for parabolic problems, Comput. Arch. Sci. Comput., 49, 1, 11-23 (1992) · Zbl 0767.65073 · doi:10.1007/BF02238647
[5] Burman, E.; Fernández, M., An unfitted nitsche method for incompressible fluid-structureinteraction using overlapping meshes, Comp. Meth. Appl. Mech. Eng., 279, 497-514 (2014) · Zbl 1423.74867 · doi:10.1016/j.cma.2014.07.007
[6] Causin, P.; Gereau, J.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comp. Meth. Appl. Mech. Eng., 194, 4506-4527 (2005) · Zbl 1101.74027 · doi:10.1016/j.cma.2004.12.005
[7] Collino, F.; Fouquet, M.; Joly, P., A conservative space-time mesh refinement method for the 1-D wave equation. Part I: construction, Numer. Math., 95, 197-221 (2003) · Zbl 1048.65089 · doi:10.1007/s00211-002-0446-5
[8] Collino, F.; Fouquet, M.; Joly, P., A conservative space-time mesh refinement method for the 1-D wave equation. Part II: analysis, Numer. Math., 95, 223-251 (2003) · Zbl 1036.65073 · doi:10.1007/s00211-002-0447-4
[9] Dawson, C.; Du, Q.; Dupont, T., A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. Comput., 57, 63-71 (1991) · Zbl 0732.65091 · doi:10.1090/S0025-5718-1991-1079011-4
[10] De Moerloose, L.; Taelman, L.; Segers, P.; Vierendeels, J.; Degroote, J., Analysis of several subcycling schemes in partitioned simulations of a strongly coupled fluid-structure interaction, Int. J. Num. Meth. Fluids, 89, 6, 181-195 (2018) · doi:10.1002/fld.4688
[11] Degroote, J.; Bathe, KJ; Vierendeels, J., Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction, Comput. Struct., 87, 793-801 (2009) · doi:10.1016/j.compstruc.2008.11.013
[12] Donea, J., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comp. Meth. Appl. Mech. Eng., 33, 689-723 (1982) · Zbl 0508.73063 · doi:10.1016/0045-7825(82)90128-1
[13] Du, Q.; Gunzburger, M.; Hou, L.; Lee, J., Analysis of a linear fluid-structure interaction problem, Discrete Continuous Dyn. Syst., 9, 3, 633-650 (2003) · Zbl 1039.35076 · doi:10.3934/dcds.2003.9.633
[14] Du, Q.; Gunzburger, M.; Hou, L.; Lee, J., Semidiscrete finite element approximation of a linearfluid-structure interaction problem, SIAM J. Numer. Anal., 42, 1, 1-29 (2004) · Zbl 1159.74343 · doi:10.1137/S0036142903408654
[15] Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 105-158 (1995) · Zbl 0829.65122
[16] Failer, L.; Meidner, D.; Vexler, B., Optimal control of a linear unsteady fluid-structure interaction problem, J. Optim. Theory Appl., 170, 1-27 (2016) · Zbl 1346.35033 · doi:10.1007/s10957-016-0930-1
[17] Faille, I., Nataf, F., Willien, F., Wolf, S.: Two local time stepping schemes for parabolic problems. In: Multiresolution and Adaptive Methods for Convection-Dominated Problems, ESAIM Proc., vol. 29, pp. 58-72. EDP Sci., Les Ulis (2009). doi:10.1051/proc/2009055 · Zbl 1181.65119
[18] Hron, J., Turek, S.: Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow. In: H.J. Bungartz, M. Schäfer (eds.) Fluid-Structure Interaction: Modeling, Simulation, Optimization, Lecture Notes in Computational Science and Engineering, pp. 371-385. Springer (2006) · Zbl 1323.76049
[19] Knoll, D.; Keyes, D., Jacobian-free Newton-Krylow methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-396 (2004) · Zbl 1036.65045 · doi:10.1016/j.jcp.2003.08.010
[20] Meidner, D.; Richter, T., Goal-oriented error estimation for the fractional step theta scheme, Comput. Methods Appl. Math., 14, 203-230 (2014) · Zbl 1284.65135 · doi:10.1515/cmam-2014-0002
[21] Nitsche, J., Über ein variationsprinzip zur lösung von dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36, 1, 9-15 (1971) · Zbl 0229.65079 · doi:10.1007/BF02995904
[22] Piperno, S., Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations, Int. J. Num. Meth. Fluids, 25, 1207-1226 (1997) · Zbl 0910.76065 · doi:10.1002/(SICI)1097-0363(19971130)25:10<1207::AID-FLD616>3.0.CO;2-R
[23] Piperno, S., Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems, ESAIM Math. Model. Numer. Anal., 40, 815-841 (2006) · Zbl 1121.78014 · doi:10.1051/m2an:2006035
[24] Richter, T.: Fluid-structure Interactions. Models, Analysis and Finite Elements, Lecture Notes in Computational Science and Engineering, vol. 118. Springer (2017) · Zbl 1374.76001
[25] Richter, T., Wick, T.: On time discretizations of fluid-structure interactions. In: T. Carraro, M. Geiger, S. Körkel, R. Rannacher (eds.) Multiple Shooting and Time Domain Decomposition Methods, Contributions in Mathematical and Computational Science, vol. 9, pp. 377-400. Springer (2015) · Zbl 1382.76195
[26] Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder. (with support by F. Durst, E. Krause and R. Rannacher). In: E. Hirschel (ed.) Flow Simulation with High-Performance Computers II. DFG Priority Research Program Results 1993-1995, no. 52 in Notes Numer. Fluid Mech., pp. 547-566. Vieweg, Wiesbaden (1996) · Zbl 0874.76070
[27] Schmich, M.; Vexler, B., Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Comput., 30, 369-393 (2008) · Zbl 1169.65098 · doi:10.1137/060670468
[28] Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, Computational Mathematics, vol. 25. Springer (1997) · Zbl 0884.65097
[29] Zhang, X.; Zuazua, E., Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184, 49-120 (2007) · Zbl 1178.74075 · doi:10.1007/s00205-006-0020-x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.