High-Order Variational Time Integrators for Particle Dynamics

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Abstract

The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to the simulation of brownout phenomena occurring in helicopter take-off and landing.

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  • 1. J. E. Marsden and M. West Discrete mechanics and variational integrators Acta Numer. vol. 10 pp. 357{514 2001.

  • 2. C. Kane J. E. Marsden M. Ortiz and M. West Variational integrators and the Newmark algorithmfor conservative and dissipative mechanical systems Internat. J. Numer. Methods Engrg. vol. 49 no. 10 pp. 1295{1325 2000.

  • 3. E. Hairer C. Lubich and G. Wanner Geometric numerical integration vol. 31 of Springer Series in Computational Mathematics. Springer Heidelberg 2010.

  • 4. J. Hall and M. Leok Spectral variational integrators Numer. Math. vol. 130 no. 4 pp. 681{740 2015.

  • 5. S. Ober-Blobaum and N. Saake Construction and analysis of higher order Galerkin variational integrators Adv. Comput. Math. vol. 41 no. 6 pp. 955{986 2015.

  • 6. S. Ober-Blobaum Galerkin variational integrators and modi_ed symplectic Runge-Kutta methods IMA J. Numer. Anal. vol. 37 no. 1 pp. 375{406 2017.

  • 7. R. Porc_u Metodi numerici e tecniche di programmazione per l'accelerazione di un modello di dinamicadi particelle non interagenti Master's thesis Politecnico di Milano 2013.

  • 8. L. D. Landau and E. M. Lifshitz Mechanics. Course of Theoretical Physics Vol. 1. Translatedfrom the Russian by J. B. Bell Pergamon Press Oxford-London-New York-Paris; Addison-WesleyPublishing Co. Inc. Reading Mass. 1960.

  • 9. V. I. Arnol'd Mathematical methods of classical mechanics vol. 60 of Graduate Texts in Mathematics. Springer-Verlag New York second ed. 1989. Translated from the Russian by K. Vogtmann and A. Weinstein.

  • 10. G. Auchmuty Optimal coercivity inequalities in W1;p() Proc. Roy. Soc. Edinburgh Sect. A vol. 135 no. 5 pp. 915{933 2005.

  • 11. C. Bernardi C. Canuto and Y. Maday Generalized inf-sup conditions for Chebyshev spectral approximationof the Stokes problem SIAM J. Numer. Anal. vol. 25 no. 6 pp. 1237{1271 1988.

  • 12. R. A. Nicolaides Existence uniqueness and approximation for generalized saddle point problems SIAM J. Numer. Anal. vol. 19 no. 2 pp. 349{357 1982.

  • 13. C. Canuto and A. Quarteroni Approximation results for orthogonal polynomials in Sobolev spaces Math. Comp. vol. 38 no. 157 pp. 67{86 1982.

  • 14. C. Canuto M. Y. Hussaini A. Quarteroni and T. A. A. Zang Spectral methods. Scientific Computation Berlin: Springer-Verlag 2006. Fundamentals in single domains.

  • 15. C. Phillips and R. E. Brown The effect of helicopter configuration on the fluid dynamics of brouwnout in 34th European Rotorcraft Forum 16-19 September 2008.

  • 16. E. Miglio N. Parolini M. Penati and R. Porcu GPU parallelization of brownout simulations with a non-interacting particles dynamic model MOX Report 29/2016 Politecnico di Milano 2016.

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CiteScore 2018: 0.95

SCImago Journal Rank (SJR) 2018: 0.324
Source Normalized Impact per Paper (SNIP) 2018: 0.73

Mathematical Citation Quotient (MCQ) 2018: 0.27

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