High-Order Variational Time Integrators for Particle Dynamics

E. Miglio; N. Parolini; M. Penati; R. Porcù

doi:10.2478/caim-2018-0015

.blurhash-client-img { display: none !important; }

High-Order Variational Time Integrators for Particle Dynamics

Communications in Applied and Industrial Mathematics

Volume 9 (2018): Issue 2 (December 2018)

By: E. Miglio, N. Parolini, M. Penati and R. Porcù

Open Access

|Dec 2018

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.

References

1. J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., vol. 10, pp. 357{514, 2001.10.1017/S096249290100006X
Search in Google Scholar Back to article
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, 2000.10.1002/1097-0207(20001210)49:10<;1295::AID-NME993>3.0.CO;2-W
Search in Google Scholar Back to article
3. E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, vol. 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010.
Search in Google Scholar Back to article
4. J. Hall and M. Leok, Spectral variational integrators, Numer. Math., vol. 130, no. 4, pp. 681{740, 2015.10.1007/s00211-014-0679-0
Search in Google Scholar Back to article
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, 2015.10.1007/s10444-014-9394-8
Search in Google Scholar Back to article
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, 2017.10.1093/imanum/drv062
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.10.1007/978-1-4757-2063-1
Search in Google Scholar Back to article
10. G. Auchmuty, Optimal coercivity inequalities in W1;p(), Proc. Roy. Soc. Edinburgh Sect. A, vol. 135, no. 5, pp. 915{933, 2005.10.1017/S0308210500004182
Search in Google Scholar Back to article
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, 1988.10.1137/0725070
Search in Google Scholar Back to article
12. R. A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal., vol. 19, no. 2, pp. 349{357, 1982.10.1137/0719021
Search in Google Scholar Back to article
13. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., vol. 38, no. 157, pp. 67{86, 1982.10.1090/S0025-5718-1982-0637287-3
Search in Google Scholar Back to article
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.10.1007/978-3-540-30726-6
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/caim-2018-0015 | Journal eISSN: 2038-0909

Journal RSS Feed

Language: English

Page range: 34 - 49

Submitted on: Oct 14, 2016

Accepted on: Oct 4, 2018

Published on: Dec 19, 2018

Published by: Italian Society for Applied and Industrial Mathemathics

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Variational integrators,

Galerkin method,

particle dynamics

Related subjects:

Mathematics,

Numerical and computational mathematics,

Applied mathematics

© 2018 E. Miglio, N. Parolini, M. Penati, R. Porcù, published by Italian Society for Applied and Industrial Mathemathics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 9 (2018): Issue 2 (December 2018)