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

Martin Balazovjech; Miloslav Feistauer; Jaromír Horáček; Martin Hadrava; Adam Kosík

Applications of Mathematics (2018)

  • Volume: 63, Issue: 6, page 739-764
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Balazovjech, Martin, et al. "Space-time discontinuous Galerkin method for the solution of fluid-structure interaction." Applications of Mathematics 63.6 (2018): 739-764. <http://eudml.org/doc/294825>.

@article{Balazovjech2018,
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.},
author = {Balazovjech, Martin, Feistauer, Miloslav, Horáček, Jaromír, Hadrava, Martin, Kosík, Adam},
journal = {Applications of Mathematics},
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},
language = {eng},
number = {6},
pages = {739-764},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Space-time discontinuous Galerkin method for the solution of fluid-structure interaction},
url = {http://eudml.org/doc/294825},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Balazovjech, Martin
AU - Feistauer, Miloslav
AU - Horáček, Jaromír
AU - Hadrava, Martin
AU - Kosík, Adam
TI - Space-time discontinuous Galerkin method for the solution of fluid-structure interaction
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 6
SP - 739
EP - 764
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/294825
ER -

References

top
  1. Akrivis, G., Makridakis, C., 10.1051/m2an:2004013, M2AN, Math. Model. Numer. Anal. 38 (2004), 261-289. (2004) Zbl1085.65094MR2069147DOI10.1051/m2an:2004013
  2. Badia, S., Codina, R., 10.1002/nme.1998, Int. J. Numer. Methods Eng. 72 (2007), 46-71. (2007) Zbl1194.74361MR2353132DOI10.1002/nme.1998
  3. Balázsová, M., Feistauer, M., 10.1007/s10492-015-0109-3, Appl. Math., Praha 60 (2015), 501-526. (2015) Zbl1363.65157MR3396478DOI10.1007/s10492-015-0109-3
  4. Balázsová, M., Feistauer, M., Hadrava, M., Kosík, A., 10.1515/jnma-2015-0014, J. Numer. Math. 23 (2015), 211-233. (2015) Zbl1327.65168MR3420382DOI10.1515/jnma-2015-0014
  5. Boffi, D., Gastaldi, L., Heltai, L., 10.1142/S0218202507002352, Math. Models Methods Appl. Sci. 17 (2007), 1479-1505. (2007) Zbl1186.76661MR2359913DOI10.1142/S0218202507002352
  6. Bonet, J., Wood, R. D., 10.1017/CBO9780511755446, Cambridge University Press, Cambridge (2008). (2008) Zbl1142.74002MR2398580DOI10.1017/CBO9780511755446
  7. Bonito, A., Kyza, I., Nochetto, R. H., 10.1137/120862715, SIAM J. Numer. Anal. 51 (2013), 577-604. (2013) Zbl1267.65114MR3033024DOI10.1137/120862715
  8. Česenek, J., Feistauer, M., 10.1137/110828903, SIAM J. Numer. Anal. 50 (2012), 1181-1206. (2012) Zbl1312.65157MR2970739DOI10.1137/110828903
  9. Česenek, J., Feistauer, M., Horáček, J., Kučera, V., Prokopová, J., 10.1016/j.amc.2011.08.077, Appl. Math. Comput. 219 (2013), 7139-7150. (2013) Zbl06299746MR3030556DOI10.1016/j.amc.2011.08.077
  10. Česenek, J., Feistauer, M., Kosík, A., 10.1002/zamm.201100184, ZAMM, Z. Angew. Math. Mech. 93 (2013), 387-402. (2013) Zbl1277.74026MR3069914DOI10.1002/zamm.201100184
  11. Chrysafinos, K., Walkington, N. J., 10.1137/030602289, SIAM J. Numer. Anal. 44 (2006), 349-366. (2006) Zbl1112.65086MR2217386DOI10.1137/030602289
  12. Ciarlet, P. G., 10.1016/S0168-2024(08)70174-7, Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam (1978). (1978) Zbl0383.65058MR0520174DOI10.1016/S0168-2024(08)70174-7
  13. Ciarlet, P. G., 10.1016/S0168-2024(08)70050-X, Studies in Mathematics and Its Applications 20, North-Holland, Amsterdam (1988). (1988) Zbl0648.73014MR0936420DOI10.1016/S0168-2024(08)70050-X
  14. Davis, T. A., Duff, I. S., 10.1145/305658.287640, ACM Trans. Math. Softw. 25 (1999), 1-20. (1999) Zbl0962.65027MR1697461DOI10.1145/305658.287640
  15. Deuflhard, P., 10.1007/978-3-642-23899-4, Springer Series in Computational Mathematics 35, Springer, Berlin (2004). (2004) Zbl1056.65051MR2893875DOI10.1007/978-3-642-23899-4
  16. Dolejší, V., Feistauer, M., 10.1007/978-3-319-19267-3, Springer Series in Computational Mathematics 48, Springer, Cham (2015). (2015) Zbl06467550MR3363720DOI10.1007/978-3-319-19267-3
  17. Dolejší, V., Feistauer, M., Schwab, C., 10.1016/S0378-4754(02)00087-3, Math. Comput. Simul. 61 (2003), 333-346. (2003) Zbl1013.65108MR1984135DOI10.1016/S0378-4754(02)00087-3
  18. Donea, J., Giuliani, S., Halleux, J. P., 10.1016/0045-7825(82)90128-1, Comput. Methods Appl. Mech. Eng. 33 (1982), 689-723. (1982) Zbl0508.73063DOI10.1016/0045-7825(82)90128-1
  19. Eriksson, K., Estep, D., Hansbo, P., Johnson, C., Computational Differential Equations, Cambridge University Press, Cambridge (1996). (1996) Zbl0946.65049MR1414897
  20. Eriksson, K., Johnson, C., 10.1137/0728003, SIAM J. Numer. Anal. 28 (1991), 43-77. (1991) Zbl0732.65093MR1083324DOI10.1137/0728003
  21. Estep, D., Larsson, S., 10.1051/m2an/1993270100351, RAIRO, Modélisation Math. Anal. Numér. 27 (1993), 35-54. (1993) Zbl0768.65065MR1204627DOI10.1051/m2an/1993270100351
  22. 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, Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010 J. C. F. Pereira, A. Sequeira Lisbon, Portugal (published ellectronically) (2010). (2010) MR2647255
  23. Feistauer, M., Felcman, J., Straškraba, I., Mathematical and Computational Methods for Compressible Flow, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2003). (2003) Zbl1028.76001MR2261900
  24. Feistauer, M., Hájek, J., Švadlenka, K., 10.1007/s10492-007-0011-8, Appl. Math., Praha 52 (2007), 197-233. (2007) Zbl1164.65469MR2316153DOI10.1007/s10492-007-0011-8
  25. Feistauer, M., Hasnedlová-Prokopová, J., Horáček, J., Kosík, A., Kučera, V., 10.1016/j.cam.2013.03.028, J. Comput. Appl. Math. 254 (2013), 17-30. (2013) Zbl1290.65089MR3061063DOI10.1016/j.cam.2013.03.028
  26. Feistauer, M., Horáček, J., Kučera, V., Prokopová, J., On numerical solution of compressible flow in time-dependent domains, Math. Bohem. 137 (2012), 1-16. (2012) Zbl1249.65196MR2978442
  27. Feistauer, M., Kučera, V., 10.1016/j.jcp.2007.01.035, J. Comput. Phys. 224 (2007), 208-221. (2007) Zbl1114.76042MR2322268DOI10.1016/j.jcp.2007.01.035
  28. Feistauer, M., Kučera, V., Najzar, K., Prokopová, J., 10.1007/s00211-010-0348-x, Numer. Math. 117 (2011), 251-288. (2011) Zbl1211.65125MR2754851DOI10.1007/s00211-010-0348-x
  29. Feistauer, M., Kučera, V., Prokopová, J., 10.1016/j.matcom.2009.01.020, Math. Comput. Simul. 80 (2010), 1612-1623. (2010) Zbl05780120MR2647255DOI10.1016/j.matcom.2009.01.020
  30. Fernández, M. Á., Moubachir, M., 10.1016/j.compstruc.2004.04.021, Comput. Struct. 83 (2005), 127-142. (2005) DOI10.1016/j.compstruc.2004.04.021
  31. Formaggia, L., Nobile, F., A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math. 7 (1999), 105-131. (1999) Zbl0942.65113MR1699243
  32. Gastaldi, L., 10.1515/JNMA.2001.123, East-West J. Numer. Math. 9 (2001), 123-156. (2001) Zbl0988.65082MR1836870DOI10.1515/JNMA.2001.123
  33. Hasnedlová, J., Feistauer, M., Horáček, J., Kosík, A., Kučera, V., 10.1007/s00607-012-0240-x, Computing 95 (2013), S343--S361. (2013) MR3054377DOI10.1007/s00607-012-0240-x
  34. Khadra, K., Angot, P., Parneix, S., Caltagirone, J.-P., 10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-D, Int. J. Numer. Methods Fluids 34 (2000), 651-684. (2000) Zbl1032.76041DOI10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-D
  35. Richter, T. M., 10.1016/j.cma.2012.02.014, Comput. Methods Appl. Mech. Eng. 223/224 (2012), 28-42. (2012) Zbl1253.74037MR2917479DOI10.1016/j.cma.2012.02.014
  36. Schötzau, D. M., 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). (1999) MR2715264
  37. Schötzau, D., Schwab, C., 10.1007/s100920070002, Calcolo 37 (2000), 207-232. (2000) Zbl1012.65084MR1812787DOI10.1007/s100920070002
  38. Thomée, V., 10.1007/3-540-33122-0, Springer Series in Computational Mathematics 25 Springer, Berlin (2006). (2006) Zbl1105.65102MR2249024DOI10.1007/3-540-33122-0
  39. Vlasák, M., Dolejší, V., Hájek, J., 10.1002/num.20591, Numer. Methods Partial Differ. Equations 27 (2011), 1456-1482. (2011) Zbl1237.65105MR2838303DOI10.1002/num.20591
  40. Yang, Z., Mavriplis, D. J., 10.2514/6.2005-1222, 43rd AIAA Aerospace Sciences Meeting and Exhibit Reno (2005), AIAA Paper, 1222. DOI10.2514/6.2005-1222

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.