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ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

  • Andreas Buhr
  • Christian Engwer
  • Mario Ohlberger
  • Stephan Rave
Chapter
Part of the MS&A book series (MS&A, volume 17)

Abstract

The simulation method ArbiLoMod (Buhr et al., SIAM J. Sci. Comput. 2017, accepted) has the goal of providing users of Finite Element based simulation software with quick re-simulation after localized changes to the model under consideration. It generates a Reduced Order Model (ROM) for the full model without ever solving the full model. To this end, a localized variant of the Reduced Basis method is employed, solving only small localized problems in the generation of the reduced basis. The key to quick re-simulation lies in recycling most of the localized basis vectors after a localized model change. In this publication, ArbiLoMod’s local training algorithm is analyzed numerically for the non-coercive problem of time harmonic Maxwell’s equations in 2D, formulated in H(curl).

Notes

Acknowledgements

Andreas Buhr was supported by CST—Computer Simulation Technology AG. Stephan Rave was supported by the German Federal Ministry of Education and Research (BMBF) under contract number 05M13PMA.

References

  1. 1.
    Buhr, A., Engwer, C., Ohlberger, M., Rave, S.: ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput. (2017). AcceptedGoogle Scholar
  2. 2.
    Burgard, S., Sommer, A., Farle, O., Dyczij-Edlinger, R.: Reduced-order models of finite-element systems featuring shape and material parameters. Electromagnetics 34(3–4), 143–160 (2014)CrossRefGoogle Scholar
  3. 3.
    Carlberg, K., Bou-Mosleh, C., Farhat, C.: Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86(2), 155–181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y., Hesthaven, J.S., Maday, Y.: A seamless reduced basis element method for 2D Maxwell’s problem: an introduction. Spectral and High Order Methods for Partial Differential Equations: Selected papers from the ICOSAHOM ’09 Conference, June 22–26, Trondheim, Norway, pp. 141–152. Springer, Berlin (2011)Google Scholar
  5. 5.
    Chen, Y., Hesthaven, J.S., Maday, Y., Rodríguez, J.: Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(2), 970–996 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dahmen, W., Plesken, C., Welper,G.: Double greedy algorithms: reduced basis methods for transport dominated problems. ESAIM: Math. Model. Numer. Anal. 48(03), 623–663 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eftang, J.L., Patera, A.T.: Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng. 96(5), 269–302 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fares, M., Hesthaven, J.S., Maday, Y., Stamm, B.: The reduced basis method for the electric field integral equation. J. Comput. Phys. 230(14), 5532–5555 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hess, M.W., Benner, P.: Fast evaluation of time harmonic Maxwell’s equations using the reduced basis method. IEEE Trans. Microw. Theory Tech. 61(6), 2265–2274 (2013)CrossRefGoogle Scholar
  11. 11.
    Iapichino, L., Quarteroni, A., Rozza,G.: A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Eng. 221–222, 63–82 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Maday, Y., Rønquist, E.M.: A reduced-basis element method. J. Sci. Comput. 17(1/4), 447–459 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maier, I., Haasdonk, B.: A Dirichlet-Neumann reduced basis method for homogeneous domain decomposition problems. Appl. Numer. Math. 78, 31–48 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Maxwell, J.C.: On physical lines of force. Lond. Edinb. Dublin Philos. Mag. J. Sci. 21(139), 161–175 (1861)Google Scholar
  15. 15.
    Milk, R., Rave, S., Schindler, F.: pyMOR-generic algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5), S194–S216 (2016). doi:10.1137/15M1026614, https://doi.org/10.1137/15M1026614
  16. 16.
    Monk, P.: Finite Element Methods for Maxwell’s Equations Oxford University Press, Oxford (2003)Google Scholar
  17. 17.
    Nedelec, J.-C.: Mixed finite elements in r 3. Numer. Math. 35(3), 315–341 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ohlberger, M., Schindler, F.: Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6), A2865–A2895 (2015)CrossRefzbMATHGoogle Scholar
  19. 19.
    Phuong Huynh, D.B., Knezevic, D.J., Patera, A.T.: A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM: Math. Model. Numer. Anal. 47(1), 213–251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pomplun, J., Schmidt, F.: Accelerated a posteriori error estimation for the reduced basis method with application to 3d electromagnetic scattering problems. SIAM J. Sci. Comput. 32(2), 498–520 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181(1), 43–69 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Andreas Buhr
    • 1
  • Christian Engwer
    • 1
  • Mario Ohlberger
    • 1
  • Stephan Rave
    • 1
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany

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