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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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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DOI: https://doi.org/10.1007/978-3-319-58786-8_4
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