Abstract
The subject of the paper is the numerical simulation of viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Navier-Stokes equations. The flow problem is coupled with the dynamical linear elasticity problem. Both problems are discretized in space by the discontinuous Galerkin (DG) finite element method using piecewise polynomial discontinuous approximations. The time discretization is carried out by the BDF scheme or the DG in time. The developed methods are tested by numerical experiments and applied to the solution of a fluid-structure interaction problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285 (1997)
Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31, 79–95 (1999)
Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Dover, New York (1996)
Česenek, J., Feistauer, M., Kosík, A.: DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM – Z. Angew. Math. Mech. 93(6–7), 387–402 (2013)
Davis, T.A., Duff, I.S.: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25, 1–19 (1999)
Dolejší, V.: Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows. Commun. Comput. Phys. 4, 231–274 (2008)
Dolejší, V., Feistauer, M., Schwab, C.: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul. 61, 333–346 (2003)
Dowell, E.H.: A Modern Course in Aeroelasticity. Kluwer, Dodrecht (1995)
Dubcová, L., Feistauer, M., Horáček, J., Sváček, P.: Numerical simulation of interaction between turbulent flow and a vibrating airfoil. Comput. Vis. Sci. 12, 207–225 (2009)
Feistauer, M.: Mathematical Methods in Fluid Dynamics. Longman, Harlow (1993)
Feistauer, M., Kučera, V.: On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224, 208–221 (2007)
Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford (2003)
Feistauer, M., Česenek, J., Horáček, J., Kučera, V., Prokopová, J.: DGFEM for the numerical solution of compressible flow in time dependent domains and applications to fluid-structure interaction. In: Pereira, J.C.F., Sequeira, A. (eds.) Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010, Lisbon (2010). ISBN 978-989-96778-1-4 (published ellectronically)
Feistauer, M., Horáček, J., Kučera, V., Prokopová, J.: On numerical solution of compressible flow in time-dependent domains. Math. Bohem. 137, 1–16 (2012)
Feistauer, M., Hasnedlová-Prokopová, J., Horáček, J., Kosík, A., Kučera, V.: DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254, 17–30 (2013)
Naudasher, E., Rockwell, D.: Flow-Induced Vibrations. A.A. Balkema, Rotterdam (1994)
Nomura, T., Hughes, T.J.R.: An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body. Comput. Methods Appl. Mech. Eng. 95, 115–138 (1992)
Sváček, P., Feistauer, M., Horáček, J.: Numerical simulation of flow induced airfoil vibrations with large amplitudes. J. Fluids Struct. 23, 391–411 (2007)
van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow. J. Comput. Phys. 182, 546–585 (2002)
Vijayasundaram, G.: Transonic flow simulation using upstream centered scheme of Godunov type in finite elements. J. Comput. Phys. 63, 416–433 (1986)
Vlasák, M., Dolejší, V., Hájek, J.: A priori estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Potential Differ. Eq. 27, 1456–1482 (2011)
Acknowledgements
This work was supported by the grants P101/11/0207 (J. Horáček) and 13-00522S (M. Feistauer) of the Czech Science Foundation. The work of M. Hadrava and A. Kosík was supported by the grant SVV-2014-260106, financed by the Charles University in Prague.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Feistauer, M., Česenek, J., Hadrava, M., Horáček, J., Kosík, A. (2014). Discontinuous Galerkin Method and Applications to Fluid-Structure Interaction Problems. In: Abgrall, R., Beaugendre, H., Congedo, P., Dobrzynski, C., Perrier, V., Ricchiuto, M. (eds) High Order Nonlinear Numerical Schemes for Evolutionary PDEs. Lecture Notes in Computational Science and Engineering, vol 99. Springer, Cham. https://doi.org/10.1007/978-3-319-05455-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-05455-1_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05454-4
Online ISBN: 978-3-319-05455-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)