Abstract
We present in detail three different quasi-Newton isogeometric algorithms for the treatment of free boundary problems. Two algorithms are based on standard Galerkin formulations, while the third is a fully-collocated scheme. With respect to standard approaches, isogeometric analysis enables the accurate description of curved geometries, and is thus particularly suitable for free boundary numerical simulation. We apply the algorithms and compare their performances to several benchmark tests, considering both Dirichlet and periodic boundary conditions. Our results constitute a starting point of an in-depth analysis of the Euler equations for incompressible fluids.
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Notes
- 1.
The strict positivity is not strictly necessary: If g < 0 one could, for instance, keep track of the sign of g in the numerical method directly. However, g has to have a definite sign everywhere on \(\Gamma _{\mathcal {V}}\).
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Acknowledgements
The authors would like to thank Rafael Vázquez for the suggestions regarding the implementation of the code. MM and GS were partially supported by the European Research Council through the FP7 Ideas Consolidator Grant HIGEOM n.616563 and by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Mathematics, University of Pavia”. FR was supported by grants no. 231668 and 250070 from the Norwegian Research Council. This support is gratefully acknowledged. MM and GS are members of the INdAM Research group GNCS.
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Montardini, M., Remonato, F., Sangalli, G. (2021). Isogeometric Methods for Free Boundary Problems. In: van Brummelen, H., Vuik, C., Möller, M., Verhoosel, C., Simeon, B., Jüttler, B. (eds) Isogeometric Analysis and Applications 2018. IGAA 2018. Lecture Notes in Computational Science and Engineering, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-030-49836-8_7
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