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A Reduced Basis Method with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Application to Linear Elasticity

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Model Reduction of Parametrized Systems

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

Abstract

We present a reduced basis method for parametrized linear elasticity equations with two objectives: providing an error bound with respect to the exact weak solution of the PDE, as opposed to the typical finite-element “truth”, in the online stage; providing automatic adaptivity in both physical and parameter spaces in the offline stage. Our error bound builds on two ingredients: a minimum-residual mixed formulation with a built-in bound for the dual norm of the residual with respect to an infinite-dimensional function space; a combination of a minimum eigenvalue bound technique and the successive constraint method which provides a lower bound of the stability constant with respect to the infinite-dimensional function space. The automatic adaptivity combines spatial mesh adaptation and greedy parameter sampling for reduced bases and successive constraint method to yield a reliable online system in an efficient manner. We demonstrate the effectiveness of the approach for a parametrized linear elasticity problem with geometry transformations and parameter-dependent singularities induced by cracks.

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Acknowledgements

I would like to thank Prof. Anthony Patera of MIT for the many fruitful discussions. This work was supported by OSD/AFOSR/MURI Grant FA9550-09-1-0613, ONR Grant N00014-11-1-0713, and the University of Toronto Institute for Aerospace Studies (UTIAS).

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Authors and Affiliations

  1. University of Toronto Institute for Aerospace Studies, 4925 Dufferin St, Toronto, ON, Canada, M3H 5T6

    Masayuki Yano

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  1. Masayuki Yano
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Correspondence to Masayuki Yano .

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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

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Yano, M. (2017). A Reduced Basis Method with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Application to Linear Elasticity. 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_4

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