Skip to main content
SpringerLink
Log in

Recent Developments in Spectral Element Simulations of Moving-Domain Problems

  • Chapter
  • First Online:
Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Part of the book series: Fields Institute Communications ((FIC,volume 79))

Abstract

Presented here are recent developments in spectral element methods for simulations of incompressible and low-Mach-number flows in domains with moving boundaries. Features include PDE-based mesh motion, implicit treatment of fluid–structure interaction based on a Green’s function decomposition, and an arbitrary Lagrangian-Eulerian formulation for low-Mach-number flows that includes an evolution equation for the background thermodynamic pressure. Several examples illustrate the basic principles introduced in the text.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We remark that \(\bar{A}\) governs the pressure in certain Navier-Stokes formulations when the system is closed. A pressure with zero mean is readily computed iteratively by projecting the constant mode out of the right-hand side and out of the pressure with each iteration.

  2. 2.

    In this section, we occasionally use “t” to represent the third coordinate in the reference domain \(\hat{\varOmega }\). It should not be confused with time because there is no temporal variation in the current context.

  3. 3.

    We remark that Patera’s original SEM paper [5] used a similar Green’s function approach to enforce the divergence-free constraint at domain boundaries.

References

  1. S. Hosseini, R. Vinuesa1, P. Schlatter, A. Hanifi, D. Henningson, Int. J. of Heat and Fluid Flow (submitted)

    Google Scholar 

  2. M. Schmitt, Direct numerical simulations in engine-like geometries. Ph.D. thesis, ETH Zurich (2014). Zurich, CH

    Google Scholar 

  3. M. Schmitt, K. Boulouchos, Int. J. of Engine Res. p. 1468087415619289 (2015)

    Google Scholar 

  4. M. Schmitt, C. Frouzakis, Y. Wright, A. Tomboulides, K. Boulouchos, Int. J. of Engine Res. 17(1), 63 (2016)

    Article  Google Scholar 

  5. A. Patera, J. Comput. Phys. 54, 468 (1984)

    Article  Google Scholar 

  6. S. Orszag, M. Israeli, M. Deville, J. Sci. Comp. 1, 75 (1986)

    Article  Google Scholar 

  7. Y. Maday, A. Patera, E. Rønquist, J. Sci. Comput. 5, 263 (1990)

    Article  MathSciNet  Google Scholar 

  8. A. Tomboulides, M. Israeli, G. Karniadakis, J. Sci. Comput. 4, 291 (1989)

    Article  MathSciNet  Google Scholar 

  9. J. Perot, J. Comp. Phys. 108, 51 (1993)

    Article  MathSciNet  Google Scholar 

  10. W. Couzy, Spectral element discretization of the unsteady Navier-Stokes equations and its iterative solution on parallel computers. Ph.D. thesis, Swiss Federal Institute of Technology-Lausanne (1995). Thesis nr. 1380

    Google Scholar 

  11. P. Fischer, J. Comput. Phys. 133, 84 (1997)

    Article  MathSciNet  Google Scholar 

  12. P. Fischer, J. Lottes, in Domain Decomposition Methods in Science and Engineering Series, ed. by R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund, J. Xu (Springer, Berlin, 2004)

    Google Scholar 

  13. J.W. Lottes, P.F. Fischer, J. Sci. Comput. 24, 45 (2005)

    Article  MathSciNet  Google Scholar 

  14. H. Tufo, P. Fischer, J. Parallel Distrib. Comput. 61, 151 (2001)

    Article  Google Scholar 

  15. P. Fischer, J. Lottes, W. Pointer, A. Siegel, J. Phys. Conf. Series 125, 012076 (2008)

    Article  Google Scholar 

  16. J. Lottes, Independent quality measures for symmetric AMG components. Tech. Rep. ANL/MCS-P1820-0111, Argonne National Laboratory, Argonne, IL, USA (2011)

    Google Scholar 

  17. J. Boyd, J. Comput. Phys. 143, 283 (1998)

    Article  MathSciNet  Google Scholar 

  18. P. Fischer, J. Mullen, Comptes rendus de l’Académie des sciences, Série I- Analyse numérique 332, 265 (2001)

    Google Scholar 

  19. J. Malm, P. Schlatter, P. Fischer, D. Henningson, J. Sci. Comp. 57, 254 (2013)

    Article  Google Scholar 

  20. H. Tufo, P. Fischer, in Proc. of the ACM/IEEE SC99 Conf. on High Performance Networking and Computing, Gordon Bell Prize (IEEE Computer Soc., CDROM, 1999)

    Google Scholar 

  21. L. Ho, A Legendre spectral element method for simulation of incompressible unsteady viscous free-surface flows. Ph.D. thesis, Massachusetts Institute of Technology (1989). Cambridge, MA.

    Google Scholar 

  22. L. Ho, Y. Maday, A. Patera, E. Rønquist, Comput. Methods Appl. Mech. Engrg. 80, 65 (1990)

    Article  MathSciNet  Google Scholar 

  23. L. Ho, A. Patera, Comput. Methods Appl. Mech. Engng. 80, 355 (1990)

    Article  Google Scholar 

  24. Y. Maday, A. Patera, in State-of-the-Art Surveys in Computational Mechanics, ed. by A. Noor, J. Oden (ASME, New York, 1989), pp. 71–143

    Google Scholar 

  25. M. Deville, P. Fischer, E. Mund, High-Order Methods for Incompressible Fluid Flow (Cambridge University Press, Cambridge, 2002)

    Book  MATH  Google Scholar 

  26. P. Fischer, A. Patera, J. Comput. Phys. 92, 380 (1991)

    Article  Google Scholar 

  27. J. Guermond, P. Minev, J. Shen, Comput. Methods Appl. Mech. Engrg. 195, 6011 (2006)

    Article  MathSciNet  Google Scholar 

  28. A.G. Tomboulides, J.C.Y. Lee, S.A. Orszag, J. Sci. Comp. 12, 139 (1997)

    Article  Google Scholar 

  29. A. Tomboulides, S. Orszag, J. Comput. Phys. 146(691–706) (1998)

    Google Scholar 

  30. B.T. Chu, X. Kovasznay, J. Fluid Mech. 3(5), 494 (1958)

    Article  MathSciNet  Google Scholar 

  31. R.G. Rehm, H.R. Baum, J. Res. Nat. Bur. Stand. 83(3), 97 (1978)

    Article  Google Scholar 

  32. A. Majda, J. Sethian, Combust. Sci. Tech. 42(3–4), 185 (1985)

    Article  Google Scholar 

  33. G. Byrne, A. Hindmarsh, Int. J. High Perform. Comput. Appl. 13, 354 (1999)

    Article  Google Scholar 

  34. S. Orszag, M. Israeli, M. Deville, J. Sci. Comp. 1, 75 (1986)

    Article  Google Scholar 

  35. J. Donea, A. Huerta, J.P. Ponthot, A. Rodriguez-Ferran, Encyclopedia of computational mechanics DOI: 10.1002/0470091355.ecm009, 1:14 (2004)

    Google Scholar 

  36. P. Fischer, N. Miller, H. Tufo, in Parallel Solution of Partial Differential Equations, ed. by P. Bjørstad, M. Luskin (Springer, Berlin, 2000), pp. 158–180

    Google Scholar 

  37. S. Orszag, J. Comput. Phys. 37, 70 (1980)

    Article  MathSciNet  Google Scholar 

  38. P. Fischer, Spectral element solution of the navier-stokes equations on high performance distributed-memory parallel processors. Ph.D. thesis, Massachusetts Institute of Technology (1989). Cambridge, MA.

    Google Scholar 

  39. D. Giannakis, P. Fischer, R. Rosner, J. Comput. Phys. 228, 1188 (2009)

    Article  MathSciNet  Google Scholar 

  40. A. Masud, T.J.R. Hughes, Comput. Methods Appl. Mech. Engrg. 146, 91 (1997)

    Article  MathSciNet  Google Scholar 

  41. H. Kanchi, A. Masud, Int. J. Numer. Methods Fluids 54, 923 (2007)

    Article  Google Scholar 

  42. P. Fischer, Comput. Methods Appl. Mech. Engrg. 163, 193 (1998)

    Article  MathSciNet  Google Scholar 

  43. J. Hron, S. Turek, A Monolithic FEM/Multigrid Solver for an ALE Formulation of Fluid-Structure Interaction with Applications in Biomechanics, Lecture Notes in Computational Science and Engineering, vol. 53 (Springer, 2010)

    Google Scholar 

  44. J.F. Gerbeau, F. Nobile, P. Causin, Comput. Methods Appl. Mech. Engrg. 194, 4506 (2005)

    Article  MathSciNet  Google Scholar 

  45. M. Fernandez, J. Gerbeau, C. Grandmont, Comptes rendus de l’Académie des sciences, Série I- Analyse numérique 342, 279 (2006)

    Google Scholar 

  46. C. Farhat, A. Rallu, K. Wang, T. Belytschko, Int. J. Numer. Methods Eng. 84, 73 (2010)

    Article  Google Scholar 

  47. J. Banks, W.D. Henshaw, B. Sjögreen, J. Comput. Phys. 231(17), 5854 (2013)

    Article  Google Scholar 

  48. J. Banks, W.D. Henshaw, D.W. Schwendeman, J. Comput. Phys. 269, 108 (2014)

    Article  MathSciNet  Google Scholar 

  49. J. Banks, W.D. Henshaw, D.W. Schwendeman, J. Comput. Phys. 268, 399 (2014)

    Article  MathSciNet  Google Scholar 

  50. Fernández, M. Landajuela, M. Vidrascu, J. Comput. Phys. 297, 156 (2015)

    Article  MathSciNet  Google Scholar 

  51. P. Bearman, J. Fluids and Structures 27, 648 (2010)

    Article  Google Scholar 

  52. R. Tumkur, R. Calderer, A. Masud, A. Pearlstein, L. Bergman, A. Vakakis, J. Fluids and Structures 40, 214 (2013)

    Article  Google Scholar 

  53. P. Fischer, in 22nd AIAA Computational Fluid Dynamics Conference, AIAA Aviation (AIAA 2015-3049, 2015)

    Google Scholar 

  54. O. Walsh, in The NSE II-Theory and Numerical Methods, ed. by J. Heywood, K. Masuda, R. Rautmann, V. Solonikkov (Springer, 1992), pp. 306–309

    Google Scholar 

  55. T. Bjontegaard, E.M. Rønquist, Comput. Methods Appl. Mech Engng. 197(51), 4763–4773 (2008)

    Article  Google Scholar 

  56. Kee, R.J., F.M. Rupley, J.A. Miller, M.E. Coltrin, J.F. Grcar, E. Meeks, H.K. Moffat, A.E. Lutz, G. DixonLewis, M.D. Smooke, J. Warnatz, G.H. Evans, R.S. Larson, R.E. Mitchell, L.R. Petzold, W.C. Reynolds, M. Caracotsios, W.E. Stewart, P. Glarborg, C. Wang,, O. Adigun, CHEMKIN collection, Release 3.6. Tech. rep., Reaction Design, Inc., San Diego, CA (2000)

    Google Scholar 

  57. T.K. Prasanth, S. Mittal, J. Comput. Phys. 594, 463 (2008)

    Google Scholar 

  58. R. Tumkur, P. Fischer, L. Bergman, A. Vakakis, A. Pearlstein, submitted (2015)

    Google Scholar 

Download references

Acknowledgements

This material was based upon work supported by U.S. Department of Energy, Office of Science, the Office of Advanced Scientific Computing Research, under Contract DE-AC02-06CH11357. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility. The work of the second author was carried out at the Aerothermochemistry and Combustion Systems Laboratory, LAV-ETH Zurich.

Author information

Authors and Affiliations

  1. Argonne National Laboratory and University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Paul Fischer

  2. ETH Zurich, Zurich, Switzerland

    Martin Schmitt

  3. Bosch GmbH, Gasoline Systems, Schwieberdingen, Germany

    Martin Schmitt

  4. Argonne National Laboratory and Aristotle University of Thessaloniki, Thessaloniki, Greece

    Ananias Tomboulides

Authors
  1. Paul Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Schmitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ananias Tomboulides
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Fischer .

Editor information

Editors and Affiliations

  1. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  2. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  3. Departement de Mathematiques, Universite de Montreal, Montreal, Québec, Canada

    Jacques Belair

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Science+Business Media LLC

About this chapter

Cite this chapter

Fischer, P., Schmitt, M., Tomboulides, A. (2017). Recent Developments in Spectral Element Simulations of Moving-Domain Problems. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_7

Download citation

Publish with us

Policies and ethics

Search

Navigation