Applications of Mathematics

, Volume 63, Issue 6, pp 739–764

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

• Monika Balázsová
• Miloslav Feistauer
• Jaromír Horáček
Article

## 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

## MSC 2010

65M60 65M99 74B05 74B20 74F10

## References

1. [1]
G. Akrivis, C. Makridakis: Galerkin time–stepping methods for nonlinear parabolic equations. M2AN, Math. Model. Numer. Anal. 38 (2004), 261–289.
2. [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.
3. [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.
4. [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.
5. [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.
6. [6]
J. Bonet, R. D. Wood: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 2008.
7. [7]
A. Bonito, I. Kyza, R. H. Nochetto: Time–discrete higher–order ALE formulations: stability. SIAM J. Numer. Anal. 51 (2013), 577–604.
8. [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.
9. [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.
10. [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.
11. [11]
K. Chrysafinos, N. J. Walkington: Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006), 349–366.
12. [12]
P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North–Holland Publishing Company, Amsterdam, 1978.Google Scholar
13. [13]
P. G. Ciarlet: Mathematical Elasticity. Volume I: Three–Dimensional Elasticity. Studies in Mathematics and Its Applications 20, North–Holland, Amsterdam, 1988.
14. [14]
T. A. Davis, I. S. Duff: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25 (1999), 1–20.
15. [15]
P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics 35, Springer, Berlin, 2004.
16. [16]
V. Dolejší, M. Feistauer: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham, 2015.
17. [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.
18. [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.
19. [19]
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations. Cambridge University Press, Cambridge, 1996.
20. [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.
21. [21]
D. Estep, S. Larsson: The discontinuous Galerkin method for semilinear parabolic problems. RAIRO, Modélisation Math. Anal. Numér. 27 (1993), 35–54.
22. [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.Google Scholar
23. [23]
M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2003.
24. [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.
25. [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.
26. [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.
27. [27]
M. Feistauer, V. Kučera: On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007), 208–221.
28. [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.
29. [29]
M. Feistauer, V. Kučera, J. Prokopová: Discontinuous Galerkin solution of compressible flow in time–dependent domains. Math. Comput. Simul. 80 (2010), 1612–1623.
30. [30]
M. Á. Fernández, M. Moubachir: A Newton method using exact jacobians for solving fluid–structure coupling. Comput. Struct. 83 (2005), 127–142.
31. [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.
32. [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.
33. [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.
34. [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.
35. [35]
T. M. Richter: Goal–oriented error estimation for fluid–structure interaction problems. Comput. Methods Appl. Mech. Eng. 223/224 (2012), 28–42.
36. [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.Google Scholar
37. [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.
38. [38]
V. Thomée: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics 25, Springer, Berlin, 2006.Google Scholar
39. [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.
40. [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.Google Scholar

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

## Authors and Affiliations

• Monika Balázsová
• 1
• Miloslav Feistauer
• 1
• Jaromír Horáček
• 2