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Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations

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Abstract

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity—due to the fourth order derivative terms, the non-linearity and the parameter dependence—provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

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Acknowledgements

The authors thank Dr. F. Ballarin (SISSA) for his great help with the RBniCS software and precious discussion. The authors thank Prof. A. T. Patera for the inspiring conversations and valuable time. This work was supported by European Union Funding for Research and Innovation through the European Research Council (project H2020 ERC CoG 2015 AROMA-CFD grant 681447, P.I. Prof. Gianluigi Rozza) by the INDAM-GNCS 2017-18 project “Advanced numerical methods combined with computational reduction techniques for parameterised PDEs and applications”, and by the MIT-FVG project ROM2S “Reduced Order Methods at MIT and SISSA”.

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  1. mathLab, Mathematics Area, SISSA, International School for Advanced Studies, via Bonomea 265, 34136, Trieste, Italy

    Federico Pichi & Gianluigi Rozza

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  1. Federico Pichi
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Correspondence to Gianluigi Rozza.

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Pichi, F., Rozza, G. Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations. J Sci Comput 81, 112–135 (2019). https://doi.org/10.1007/s10915-019-01003-3

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