Skip to main content
SpringerLink
Log in
Book cover

Model Reduction of Parametrized Systems pp 91–106Cite as

Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in Vibroacoustics

  • Chapter
  • First Online:

Part of the book series: MS&A ((MS&A,volume 17))

Abstract

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues.

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   119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   159.99
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

References

  1. Beirão Da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)

    Google Scholar 

  2. Ben Belgacem, F., Maday, Y.: The mortar finite element method for three dimensional finite elements. Math. Model. Numer. Anal. 31(2), 289–302 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Brezis, H. et al. (eds.) Nonlinear Partial Differential Equations and Their Applications, vol. XI, pp. 13–51. Collège de France, Paris (1994)

    Google Scholar 

  4. Brivadis, E., Buffa, A., Wohlmuth, B., Wunderlich, L.: Isogeometric mortar methods. Comput. Methods Appl. Mech. Eng. 284, 292–319 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric analysis. Towards Integration of CAD and FEA. Wiley, Chichester (2009)

    Book  MATH  Google Scholar 

  6. de Falco, C., Reali, A., Vázquez, R.: GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42(12), 1020–1034 (2011)

    Article  MATH  Google Scholar 

  7. Dittmann, M., Franke, M., Temizer, I., Hesch, C.: Isogeometric analysis and thermomechanical mortar contact problems. Comput. Methods Appl. Mech. Eng. 274, 192–212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dornisch, W., Vitucci, G., Klinkel, S.: The weak substitution method – an application of the mortar method for patch coupling in NURBS-based isogeometric analysis. Int. J. Numer. Methods Eng. 103(3), 205–234 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Drohmann, M., Haasdonk, B., Kaulmann, S., Ohlberger, M.: A software framework for reduced basis methods using DUNE -RB and RBmatlab. Advances in DUNE, pp. 77–88. Springer, Berlin (2012)

    Google Scholar 

  10. Fletcher, N.H., Rossing, T.: The Physics of Musical Instruments, 2nd edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

  11. Fumagalli, I., Manzoni, A., Parolini, N., Verani, M.: Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems. ESAIM: Math. Model. Numer. Anal. 50, 1857–1885 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gerner, A.L., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34(5), A2812–A2836 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glas, S., Urban, K.: On non-coercive variational inequalities. SIAM J. Numer. Anal. 52, 2250–2271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Glas, S., Urban, K.: Numerical investigations of an error bound for reduced basis approximations of noncoercice variational inequalities. IFAC-PapersOnLine 48(1), 721–726 (2015)

    Article  Google Scholar 

  15. Haasdonk, B., Salomon, J., Wohlmuth, B.: A reduced basis method for parametrized variational inequalities. SIAM J. Numer. Anal. 50, 2656–2676 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hesch, C., Betsch, P.: Isogeometric analysis and domain decomposition methods. Comput. Methods Appl. Mech. Eng. 213–216, 104–112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Horger, T., Wohlmuth, B., Dickopf, T.: Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems. ESAIM: Math. Model. Numer. Anal. 51, 443–465 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods. Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iapichino, L., Quarteroni, A., Rozza, G., Volkwein, S.: Reduced basis method for the Stokes equations in decomposable domains using greedy optimization. In: European Conference Mathematics in Industry, ECMI 2014, pp. 1–7 (2014)

    Google Scholar 

  21. Jansson, E.V.: Violin frequency response – bridge mobility and bridge feet distance. Appl. Acoust. 65(12), 1197–1205 (2004)

    Article  Google Scholar 

  22. Lovgren, A., Maday, Y., Ronquist, E.: A reduced basis element method for the steady Stokes problem. Math. Model. Numer. Anal. 40, 529–552 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C.R. Acad. Sci., Paris, Sér. I 331(2), 153–158 (2000)

    Google Scholar 

  24. Milani, R., Quarteroni, A., Rozza, G.: Reduced basis method for linear elasticity problems with many parameters. Comput. Methods Appl. Mech. Eng. 197(51–52), 4812–4829 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Negri, F., Manzoni, A., Rozza, G.: Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations. Comput. Math. Appl. 69(4), 319–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pau, G.: Reduced-basis method for band structure calculations. Phys. Rev. E 76, 046704 (2007)

    Article  Google Scholar 

  27. Pau, G.: Reduced basis method for simulation of nanodevices. Phys. Rev. B 78, 155425 (2008)

    Article  Google Scholar 

  28. Quarteroni, A.: Numerical Models for Differential Problems, MS&A, vol. 8, 2nd edn. Springer, Milan (2014)

    Google Scholar 

  29. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer, New York (2015)

    MATH  Google Scholar 

  30. Rand, O., Rovenski, V.: Analytical Methods in Anisotropic Elasticity: With Symbolic Computational Tools. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

  31. Ranz, T.: Ein feuchte- und temperaturabhängiger anisotroper Werkstoff: Holz. Beiträge zur Materialtheorie. Universität der Bundeswehr München (2007)

    Google Scholar 

  32. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods. Appl. Mech. Eng. 196, 1244–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    MATH  Google Scholar 

  34. Rozza, G., Huynh, D.B.P., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125(1), 115–152 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Seitz, A., Farah, P., Kremheller, J., Wohlmuth, B., Wall, W., Popp, A.: Isogeometric dual mortar methods for computational contact mechanics. Comput. Methods Appl. Mech. Eng. 301, 259–280 (2016)

    Article  MathSciNet  Google Scholar 

  36. Vallaghe, S., Huynh, D.P., Knezevic, D.J., Nguyen, T.L., Patera, A.T.: Component-based reduced basis for parametrized symmetric eigenproblems. Adv. Model. Simul. Eng. Sci. 2 (2015)

    Google Scholar 

  37. Woodhouse, J.: On the “bridge hill” of the violin. Acta Acust. United Acust. 91(1), 155–165 (2005)

    Google Scholar 

Download references

Acknowledgements

We would like to gratefully acknowledge the funds provided by the “Deutsche Forschungsgemeinschaft” under the contract/grant numbers: WO 671/11-1, WO 671/13-2 and WO 671/15-1 (within the Priority Programme SPP 1748, “Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis”).

Author information

Authors and Affiliations

  1. Institute for Numerical Mathematics, Technische Universität München, Boltzmannstraße 3, 85748, Garching b. München, Germany

    Thomas Horger, Barbara Wohlmuth & Linus Wunderlich

Authors
  1. Thomas Horger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Barbara Wohlmuth
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Linus Wunderlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Horger .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Horger, T., Wohlmuth, B., Wunderlich, L. (2017). Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in Vibroacoustics. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation