Skip to main content
SpringerLink
Log in

Algebraic Flux Correction and Geometric Conservation in ALE Computations

  • Chapter
Flux-Corrected Transport

Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

In this chapter, we describe the important role played by the so-called Geometric Conservation Law (GCL) in the design of Flux-Corrected Transport (FCT) methods for Arbitrary Lagrangian-Eulerian (ALE) applications. We propose a conservative synchronized remap algorithm applicable to arbitrary Lagrangian-Eulerian computations with nodal finite elements. Unique to the proposed method is the direct incorporation of the geometric conservation law (GCL) in the resulting numerical scheme. We show how the geometric conservation law allows the proposed method to inherit the positivity preserving and local extrema diminishing (LED) properties typical of FCT schemes for pure transport problems. The extension to systems of equations which typically arise in meteorological and compressible flow computations is performed by means of a synchronized strategy. The proposed approach also complements and extends the work of the first author on nodal-based methods for shock hydrodynamics, delivering a fully integrated suite of Lagrangian/remap algorithms for computations of compressible materials under extreme load conditions. Numerical tests in multiple dimensions show that the method is robust and accurate in typical computational scenarios.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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

References

  1. Aziz, A.K., Monk, P.: Continuous finite elements in space and time for the heat equation. Math. Comput. 144, 70–97 (1998)

    Google Scholar 

  2. Benson, D.J.: An efficient, accurate, simple ALE method for nonlinear finite element programs. Comput. Methods Appl. Mech. Eng. 72, 305–350 (1989)

    Article  ADS  MATH  Google Scholar 

  3. Benson, D.J.: Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Eng. 99, 235–394 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Benson, D.J.: Momentum advection on a staggered grid. J. Comput. Phys. 100, 143–162 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Boiarkine, O., Kuzmin, D., Čanić, S., Guidoboni, G., Mikelić, A.: A positivity-preserving ALE finite element scheme for convection-diffusion equations in moving domains. J. Comput. Phys. 230, 2896–2914 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Bonito, A., Kyza, I., Nochetto, R.H.: A priori error analysis of time discrete higher-order ALE formulations (2011, in preparation)

    Google Scholar 

  7. Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11, 38–69 (1973)

    Article  ADS  MATH  Google Scholar 

  8. Boris, J.P., Book, D.L.: Flux-corrected transport. II. Generalization of the method. J. Comput. Phys. 18, 248–283 (1975)

    Article  ADS  MATH  Google Scholar 

  9. Boris, J.P., Book, D.L.: Flux-corrected transport. III. Minimal error FCT algorithms. J. Comput. Phys. 20, 397–431 (1976)

    Article  ADS  MATH  Google Scholar 

  10. Donea, J., Giuliani, S., Halleux, J.P.: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33, 689–723 (1982)

    Article  ADS  MATH  Google Scholar 

  11. Dukowicz, J.K., Baumgardner, J.: Incremental remapping as a transportation/advection algorithm. J. Comput. Phys. 160, 318–335 (2000)

    Article  ADS  MATH  Google Scholar 

  12. Dukowicz, J.K., Kodis, J.W.: Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations. SIAM J. Sci. Stat. Comput. 8, 305–321 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dukowicz, J.K., Padial, N.: REMAP3D: A conservative three-dimensional remapping code. LA-12136-MS Report, Los Alamos National Laboratory, Los Alamos, NM, USA (1991)

    Google Scholar 

  14. Fletcher, C.A.J.: The group finite element formulation. Comput. Methods Appl. Mech. Eng. 37, 225–244 (1983). doi:10.1016/0045-7825(83)90122-6

    Article  MathSciNet  ADS  Google Scholar 

  15. Formaggia, L., Nobile, F.: A stability analysis for the arbitrary Lagrangian-Eulerian formulation with finite elements. East-West J. Numer. Math. 7(2), 105–131 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Formaggia, L., Nobile, F.: Stability analysis of second-order time accurate schemes for ALE-FEM. Comput. Methods Appl. Mech. Eng. 193, 4097–4116 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Forster, C., Wall, W.A., Ramm, E.: Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 196, 1278–1293 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  18. Fressmann, D., Wriggers, P.: Advection approaches for single- and multi-material arbitrary Lagrangian-Eulerian finite element procedures. Comput. Mech. 225, 153–190 (2007). doi:10.1007/s00466-005-0016-7

    Google Scholar 

  19. Galera, S., Maire, P.-H., Breil, J.: A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction. J. Comput. Phys. 229, 5755–5787 (2010). doi:10.1016/j.jcp.2010.04.019

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Hirt, C.W., Amsden, A.A., Cook, J.L.: An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253 (1974)

    Article  ADS  MATH  Google Scholar 

  21. Hulme, B.L.: Discrete Galerkin and related one-step methods for ordinary differential equations. Math. Comput. 26(120), 881–891 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jameson, A.: Computational algorithms for aerodynamic analysis and design. Appl. Numer. Math. 13, 743–776 (1993)

    Article  MathSciNet  Google Scholar 

  23. Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1. Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Int. J. Comput. Fluid Dyn. 4, 171–218 (1995)

    Article  Google Scholar 

  24. Jameson, A.: Computational algorithms for aerodynamic analysis and design. Int. J. Numer. Methods Fluids 20, 743–776 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Jamet, P.: Stability and convergence of a generalized Crank-Nicolson scheme on a variable mesh for the heat equation. SIAM J. Numer. Anal. 17, 530–539 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Knupp, P., Margolin, L.G., Shashkov, M.J.: Reference Jacobian optimization-based rezone strategies for arbitrary Lagrangian-Eulerian methods. J. Comput. Phys. 176, 93–128 (2002)

    Article  ADS  MATH  Google Scholar 

  27. Kucharik, M., Shaskov, M., Wendroff, B.: An efficient linearity-and-bound-preserving remapping method. J. Comput. Phys. 188, 462–471 (2003)

    Article  ADS  MATH  Google Scholar 

  28. Kucharik, M., Breil, J., Maire, P.-H., Berndt, M., Shashkov, M.J.: Hybrid remap for multi-material ALE. Comput. Fluids (2010). doi:10.1016/j.compfluid.2010.08.004

    Google Scholar 

  29. Kuzmin, D.: Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 228, 2517–2534 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Kuzmin, D., Turek, S.: Flux correction tools for finite elements. J. Comput. Phys. 175, 525–538 (2002). doi:10.1006/jcph.2001.6955

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Kuzmin, D., Möller, M., Turek, S.: High-resolution FEM-FCT schemes for multidimensional conservation laws. Comput. Methods Appl. Mech. Eng. 193, 4915–4946 (2003)

    Article  Google Scholar 

  32. Kuzmin, D., Möller, M., Turek, S.: Multidimensional FEM-FCT schemes for arbitrary time-stepping. Int. J. Numer. Methods Fluids 42, 265–295 (2003)

    Article  ADS  MATH  Google Scholar 

  33. Kuzmin, D., Löhner, R., Turek, S. (eds.): Flux-Corrected Transport: Principles, Algorithms and Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  34. Kuzmin, D., Möller, M., Shadid, J.N., Shashkov, M.J.: Failsafe flux limiting and constrained data projections for systems of conservation laws. J. Comput. Phys. 229, 8766–8779 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Lesoinne, M., Farhat, C.: Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Methods Appl. Mech. Eng. 134, 71–90 (1996)

    Article  ADS  MATH  Google Scholar 

  36. Liska, R., Shashkov, M.J., Vàchal, P., Wendroff, B.: Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian methods. J. Comput. Phys. 225, 1467–1497 (2010)

    Article  ADS  Google Scholar 

  37. Löhner, R., Morgan, K., Zienkiwicz, O.C.: The solution of non-linear hyperbolic equation systems by the finite element method. Int. J. Numer. Methods Fluids 4, 1043–1063 (1984). doi:10.1002/fld.1650041105

    Article  MATH  Google Scholar 

  38. Löhner, R., Morgan, K., Peraire, J., Vahdati, M.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 7, 1093–1109 (1987). doi:10.1002/fld.1650071007

    Article  MATH  Google Scholar 

  39. Löhner, R., Morgan, K., Vahdati, M., Boris, J.P., Book, D.L.: FEM-FCT: Combining unstructured grids with high resolution. Commun. Appl. Numer. Methods 4, 717–729 (1988). doi:10.1002/cnm.1530040605

    Article  MATH  Google Scholar 

  40. López Ortega, A., Scovazzi, G.: A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements. J. Comput. Phys. (2011). doi:10.1016/j.jcp.2011.05.005

    Google Scholar 

  41. Loubère, R., Shashkov, M.J.: A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian Eulerian methods. J. Comput. Phys. 209, 105–138 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Loubère, R., Maire, P.-H., Shashkov, M.J.: ReALE: A reconnection arbitrary Lagrangian-Eulerian method in cylindrical coordinates. Computers and Fluids (2010). doi:10.1016/j.compfluid.2010.08.024

  43. Loubère, R., Maire, P.-H., Shashkov, M.J., Breil, J., Galera, S.: ReALE: A reconnection-based arbitrary Lagrangian-Eulerian method. J. Comput. Phys. 229, 4724–4761 (2010). doi:10.1016/j.jcp.2010.03.011

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Love, E., Scovazzi, G.: On the angular momentum conservation and incremental objectivity properties of a predictor/multi-corrector method for Lagrangian shock hydrodynamics. Comput. Methods Appl. Mech. Eng. 198, 3207–3213 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Maire, P.-H., Breil, J., Galera, S.: A cell-centered arbitrary Lagrangian-Eulerian (ALE) method. Int. J. Numer. Methods Fluids 56, 1161–1166 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Margolin, L.G., Shashkov, M.: Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys. 184(1), 266–298 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Margolin, L.G., Shashkov, M.: Remapping, recovery and repair on a staggered grid. Comput. Methods Appl. Mech. Eng. 193(39–41), 4139–4155 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Masud, A., Hughes, T.J.R.: A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput. Methods Appl. Mech. Eng. 146, 91–126 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. Scovazzi, G.: A discourse on Galilean invariance and SUPG-type stabilization. Comput. Methods Appl. Mech. Eng. 196, 1108–1132 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Scovazzi, G.: Galilean invariance and stabilized methods for compressible flows. Int. J. Numer. Methods Fluids 54, 757–778 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Scovazzi, G.: Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations. Comput. Methods Appl. Mech. Eng. 196, 966–978 (2007)

    ADS  Google Scholar 

  52. Scovazzi, G., Hughes, T.J.R.: Lecture notes on continuum mechanics on arbitrary moving domains. Technical Report SAND-2007-6312P, Sandia National Laboratories (2007)

    Google Scholar 

  53. Scovazzi, G., Love, E.: A generalized view on Galilean invariance in stabilized compressible flow computations. Int. J. Numer. Methods Fluids 64, 1065–1083 (2010). doi:10.1002/fld.2417

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Scovazzi, G., Christon, M.A., Hughes, T.J.R., Shadid, J.N.: Stabilized shock hydrodynamics: I. A Lagrangian method. Comput. Methods Appl. Mech. Eng. 196, 923–966 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Scovazzi, G., Shadid, J.N., Love, E., Rider, W.J.: A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics. Comput. Methods Appl. Mech. Eng. 199, 3059–3100 (2010). doi:10.1016/j.cma.2010.03.027

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)

    MATH  Google Scholar 

  57. Shashkov, M., Wendroff, B.: The repair paradigm and application to conservation laws. J. Comput. Phys. 198, 265–277 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Shashkov, M.J., Lipnikov, K.: The error-minimization-based rezone strategy for arbitrary Lagrangian-Eulerian methods. Commun. Comput. Phys. 1, 53–81 (2005)

    Google Scholar 

  59. Smolarkiewicz, P., Margolin, L.G.: MPDATA: A finite-difference solver for geophysical ows. J. Comput. Phys. 140, 459–480 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Vàchal, P., Liska, R.: Sequential flux-corrected remapping for ale methods. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P. (eds.) Numerical Mathematics and Advanced Applications, ENUMATH 2005, pp. 671–679. Springer, Berlin (2006)

    Chapter  Google Scholar 

  61. Vàchal, P., Garimella, R.V., Shashkov, M.J.: Untangling of 2D meshes in ALE simulations. J. Comput. Phys. 196, 627–644 (2004)

    Article  ADS  MATH  Google Scholar 

  62. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335–362 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge, for very valuable discussions, Dr. M. Möller at University of Dortmund, Professor D. Kuzmin at the University of Erlangen-Nuremberg, Professor A. Bonito at Texas A&M University, Professor R.H. Nochetto at the University of Maryland, Dr. E. Love and Dr. J. Shadid at Sandia National Laboratories, and Dr. M. Shashkov at Los Alamos National Laboratory. A. López Ortega would also like to acknowledge his advisors D.I. Pullin and D.I. Meiron at the Graduate Aerospace Laboratories at California Institute of Technology.

A. López Ortega is supported by the PSAAP program, funded by the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA29613.

G. Scovazzi would like to acknowledge the continuing support of Dr. J. Stewart and Dr. S. Domino through Computer Science Research Foundation Grants at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Author information

Authors and Affiliations

  1. Numerical Analysis and Applications, Sandia National Laboratories, MS-1319, Albuquerque, NM, 87185-1319, USA

    Guglielmo Scovazzi

  2. Graduate Aerospace Laboratories, California Institute of Technology, MC 105-50, Pasadena, CA, 91125, USA

    Alejandro López Ortega

Authors
  1. Guglielmo Scovazzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alejandro López Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guglielmo Scovazzi .

Editor information

Editors and Affiliations

  1. , Chair of Applied Mathematics III, University of Erlangen-Nuremberg, Haberstr. 2, Erlangen, 91058, Germany

    Dmitri Kuzmin

  2. , School of Computational Sciences, George Mason University, 4400 University Drive MS 6A2, Fairfax, 22030, Virginia, USA

    Rainald Löhner

  3. Institute of Applied Mathematics, LS III, Dortmund University of Technology, Vogelpothsweg 87, Dortmund, 44227, Germany

    Stefan Turek

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Scovazzi, G., López Ortega, A. (2012). Algebraic Flux Correction and Geometric Conservation in ALE Computations. In: Kuzmin, D., Löhner, R., Turek, S. (eds) Flux-Corrected Transport. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4038-9_9

Download citation

Publish with us

Policies and ethics

Search

Navigation