Applications of Mathematics, Vol. 63, No. 6, pp. 739-764, 2018

Space-time discontinuous Galerkin method for the solution of fluid-structure interaction

Monika Balázsová, Miloslav Feistauer, Jaromír Horáček, Martin Hadrava, Adam Kosík

Received May 2, 2018.   Published online November 19, 2018.

Abstract:  The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.
Keywords:  nonstationary compressible Navier-Stokes equations; time-dependent domain; arbitrary Lagrangian-Eulerian method; linear and nonlinear dynamic elasticity; space-time discontinuous Galerkin method; vocal folds vibrations
Classification MSC:  65M60, 65M99, 74B05, 74B20, 74F10
DOI:  10.21136/AM.2018.0139-18

PDF available at:  Springer   Institute of Mathematics CAS

[1] G. Akrivis, C. Makridakis: Galerkin time-stepping methods for nonlinear parabolic equations. M2AN, Math. Model. Numer. Anal. 38 (2004), 261-289. DOI 10.1051/m2an:2004013 | MR 2069147 | Zbl 1085.65094
[2] S. Badia, R. Codina: On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Int. J. Numer. Methods Eng. 72 (2007), 46-71. DOI 10.1002/nme.1998 | MR 2353132 | Zbl 1194.74361
[3] M. Balázsová, M. Feistauer: On the stability of the ALE space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. Appl. Math., Praha 60 (2015), 501-526. DOI 10.1007/s10492-015-0109-3 | MR 3396478 | Zbl 1363.65157
[4] M. Balázsová, M. Feistauer, M. Hadrava, A. Kosík: On the stability of the space-time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems. J. Numer. Math. 23 (2015), 211-233. DOI 10.1515/jnma-2015-0014 | MR 3420382 | Zbl 1327.65168
[5] D. Boffi, L. Gastaldi, L. Heltai: Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17 (2007), 1479-1505. DOI 10.1142/S0218202507002352 | MR 2359913 | Zbl 1186.76661
[6] J. Bonet, R. D. Wood: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (2008). DOI 10.1017/CBO9780511755446 | MR 2398580 | Zbl 1142.74002
[7] A. Bonito, I. Kyza, R. H. Nochetto: Time-discrete higher-order ALE formulations: stability. SIAM J. Numer. Anal. 51 (2013), 577-604. DOI 10.1137/120862715 | MR 3033024 | Zbl 1267.65114
[8] J. Česenek, M. Feistauer: Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 50 (2012), 1181-1206. DOI 10.1137/110828903 | MR 2970739 | Zbl 1312.65157
[9] J. Česenek, M. Feistauer, J. Horáček, V. Kučera, J. Prokopová: Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219 (2013), 7139-7150. DOI 10.1016/j.amc.2011.08.077 | MR 3030556 | Zbl 06299746
[10] J. Česenek, M. Feistauer, A. Kosík: DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM, Z. Angew. Math. Mech. 93 (2013), 387-402. DOI 10.1002/zamm.201100184 | MR 3069914 | Zbl 1277.74026
[11] K. Chrysafinos, N. J. Walkington: Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006), 349-366. DOI 10.1137/030602289 | MR 2217386 | Zbl 1112.65086
[12] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam (1978). DOI 10.1016/S0168-2024(08)70174-7 | MR 0520174 | Zbl 0383.65058
[13] P. G. Ciarlet: Mathematical Elasticity. Volume I: Three-Dimensional Elasticity. Studies in Mathematics and Its Applications 20, North-Holland, Amsterdam (1988). DOI 10.1016/S0168-2024(08)70050-X | MR 0936420 | Zbl 0648.73014
[14] T. A. Davis, I. S. Duff: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25 (1999), 1-20. DOI 10.1145/305658.287640 | MR 1697461 | Zbl 0962.65027
[15] P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics 35, Springer, Berlin (2004). DOI 10.1007/978-3-642-23899-4 | MR 2893875 | Zbl 1056.65051
[16] V. Dolejší, M. Feistauer: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham (2015). DOI 10.1007/978-3-319-19267-3 | MR 3363720 | Zbl 06467550
[17] V. Dolejší, M. Feistauer, C. Schwab: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul. 61 (2003), 333-346. DOI 10.1016/S0378-4754(02)00087-3 | MR 1984135 | Zbl 1013.65108
[18] J. Donea, S. Giuliani, J. P. Halleux: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33 (1982), 689-723. DOI 10.1016/0045-7825(82)90128-1 | Zbl 0508.73063 |
[19] K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations. Cambridge University Press, Cambridge (1996). MR 1414897 | Zbl 0946.65049
[20] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991), 43-77. DOI 10.1137/0728003 | MR 1083324 | Zbl 0732.65093
[21] D. Estep, S. Larsson: The discontinuous Galerkin method for semilinear parabolic problems. RAIRO, Modélisation Math. Anal. Numér. 27 (1993), 35-54. DOI 10.1051/m2an/1993270100351 | MR 1204627 | Zbl 0768.65065
[22] M. Feistauer, J. Česenek, J. Horáček, V. Kučera, J. Prokopová: DGFEM for the numerical solution of compressible flow in time dependent domains and applications to fluid-structure interaction. Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010 (J. C. F. Pereira, A. Sequeira, eds.). Lisbon, Portugal (published ellectronically) (2010).
[23] M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2003). MR 2261900 | Zbl 1028.76001
[24] M. Feistauer, J. Hájek, K. Švadlenka: Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. Appl. Math., Praha 52 (2007), 197-233. DOI 10.1007/s10492-007-0011-8 | MR 2316153 | Zbl 1164.65469
[25] M. Feistauer, J. Hasnedlová-Prokopová, J. Horáček, A. Kosík, V. Kučera: DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254 (2013), 17-30. DOI 10.1016/ | MR 3061063 | Zbl 1290.65089
[26] M. Feistauer, J. Horáček, V. Kučera, J. Prokopová: On numerical solution of compressible flow in time-dependent domains. Math. Bohem. 137 (2012), 1-16. MR 2978442 | Zbl 1249.65196
[27] M. Feistauer, V. Kučera: On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007), 208-221. DOI 10.1016/ | MR 2322268 | Zbl 1114.76042
[28] M. Feistauer, V. Kučera, K. Najzar, J. Prokopová: Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011), 251-288. DOI 10.1007/s00211-010-0348-x | MR 2754851 | Zbl 1211.65125
[29] M. Feistauer, V. Kučera, J. Prokopová: Discontinuous Galerkin solution of compressible flow in time-dependent domains. Math. Comput. Simul. 80 (2010), 1612-1623. DOI 10.1016/j.matcom.2009.01.020 | MR 2647255 | Zbl 05780120
[30] M. Á. Fernández, M. Moubachir: A Newton method using exact jacobians for solving fluid-structure coupling. Comput. Struct. 83 (2005), 127-142. DOI 10.1016/j.compstruc.2004.04.021
[31] L. Formaggia, F. Nobile: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999), 105-131. MR 1699243 | Zbl 0942.65113
[32] L. Gastaldi: A priori error estimates for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001), 123-156. DOI 10.1515/JNMA.2001.123 | MR 1836870 | Zbl 0988.65082
[33] J. Hasnedlová, M. Feistauer, J. Horáček, A. Kosík, V. Kučera: Numerical simulation of fluid-structure interaction of compressible flow and elastic structure. Computing 95 (2013), S343-S361. DOI 10.1007/s00607-012-0240-x | MR 3054377
[34] K. Khadra, P. Angot, S. Parneix, J.-P. Caltagirone: Fictiuous domain approach for numerical modelling of Navier-Stokes equations. Int. J. Numer. Methods Fluids 34 (2000), 651-684. DOI 10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-D | Zbl 1032.76041 |
[35] T. M. Richter: Goal-oriented error estimation for fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 223/224 (2012), 28-42. DOI 10.1016/j.cma.2012.02.014 | MR 2917479 | Zbl 1253.74037
[36] D. M. Schötzau: hp-DGFEM for parabolic evolution problems: Applications to diffusion and viscous incompressible fluid flow. Thesis (Dr.Sc.Math)-Eidgenoessische Technische Hochschule Zürich, ProQuest Dissertations Publishing (1999). MR 2715264
[37] D. Schötzau, C. Schwab: An $hp$ a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37 (2000), 207-232. DOI 10.1007/s100920070002 | MR 1812787 | Zbl 1012.65084
[38] V. Thomée: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics 25 Springer, Berlin (2006). DOI 10.1007/3-540-33122-0 | MR 2249024 | Zbl 1105.65102
[39] M. Vlasák, V. Dolejší, J. Hájek: A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equations 27 (2011), 1456-1482. DOI 10.1002/num.20591 | MR 2838303 | Zbl 1237.65105
[40] Z. Yang, D. J. Mavriplis: Unstructured dynamic meshes with higher-order time integration schemes for the unsteady Navier-Stokes equations. 43rd AIAA Aerospace Sciences Meeting and Exhibit. Reno, 2005, AIAA Paper, 1222. DOI 10.2514/6.2005-1222

Affiliations:   Monika Balázsová, Miloslav Feistauer, Martin Hadrava, Adam Kosík: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail:,,,; Jaromír Horáček: Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejškova 5, 182 00 Praha 8, Czech Republic, e-mail:

PDF available at: