Skip to main content
SpringerLink
Log in
Book cover

Multiscale Methods in Computational Mechanics pp 53–73Cite as

Variational Germano Approach for Multiscale Formulations

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 55))

Abstract

In this chapter the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convectionU? diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the leastsquares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluid, 3:1760–1765, 1991.

    Article  MATH  Google Scholar 

  2. A.A. Oberai and J. Wanderer. A dynamic approach for evaluating parameters in a numerical method. Int. J. Numer. Meth. Fluids, 62:50–71, 2005.

    MATH  MathSciNet  Google Scholar 

  3. A.A. Oberai and J. Wanderer. A dynamic multiscale viscosity method for the spectral approximation of conservation laws. Comput. Methods Appl. Mech. Engrg., 195:1778–1792, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  4. A.A. Oberai and J. Wanderer. Variational formulation of the Germano identity for the Navier-Stokes equations. Journal of Turbulence, 6(7): 2005.

    MATH  MathSciNet  Google Scholar 

  5. A.A. Oberai and J. Wanderer. Optimal numerical solution of PDEs using the variational Germano identity. Comput. Methods Appl. Mech. Engrg., 197:2948–2962, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Hoffman. Dynamic subgrid modeling for time dependent convection-diffusionreaction equations with fractal solutions. Int. J. Numer. Meth. Fluids, 40:583–592, 2002.

    Article  MATH  Google Scholar 

  7. E. Onate, J. Garcia, and S. Idelsohn. Computation of the stabilization parameter for the finite element solution of advection-diffusion problems. Int. J. Numer. Meth. Fluids, 25:1385–1407, 1997.

    Article  MATH  Google Scholar 

  8. T.J.R. Hughes. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg., 127:387–401, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  9. T.J.R. Hughes, G.R. Feijoo, L. Mazzei, and J.B. Quincy. The variational multiscale method - A paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg., 166:3–24, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  10. T.J.R. Hughes and G. Sangalli. Variational multiscale analysis: the fine-scale green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal, 45:539–557, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  11. A.N. Brooks and T.J.R. Hughes. Streamline Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier- Stokes equations. Comput. Methods Appl. Mech. Engrg., 32:199–259, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  12. L.P. Franca, S.L. Frey, and T.J.R. Hughes. Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg., 95:253–276, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  13. I. Harari and T.J.R. Hughes. What are C and h?: Inequalities for the analysis and design of finite element method. Comput. Methods Appl. Mech. Engrg., 97:157–192, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Akkerman. Adaptive variational multiscale formulations using the discrete Germano approach. Dissertation, University of Technology Delft, 2009.

    Google Scholar 

  15. I. Akkerman, K.G. van der Zee, and S.J. Hulshoff. A Variational Germano approach for Stabilized Finite element methods. Comput. Methods Appl. Mech. Engrg. Accepted.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural Engineering, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA, 92093, USA

    Ido Akkerman

  2. Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands

    Steven J. Hulshoff

  3. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, Austin, TX, 78712, USA

    Kris G. van der Zee

  4. Mechanical Engineering, Eindhoven University of Technology, 513, WH 1.125, 5600 MB, Eindhoven, The Netherlands

    René de Borst

Authors
  1. Ido Akkerman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven J. Hulshoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kris G. van der Zee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. René de Borst
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ido Akkerman .

Editor information

Editors and Affiliations

  1. Dept. Mechanical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

    René de Borst

  2. Inst. Mechanik, Lehrstuhl II, Univ. Stuttgart, Pfaffenwaldring 7, Stuttgart, 70569, Germany

    Ekkehard Ramm

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Akkerman, I., Hulshoff, S.J., van der Zee, K.G., de Borst, R. (2010). Variational Germano Approach for Multiscale Formulations. In: de Borst, R., Ramm, E. (eds) Multiscale Methods in Computational Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 55. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9809-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-90-481-9809-2_4

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-9808-5

  • Online ISBN: 978-90-481-9809-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation