Time integration errors and some new functionals for the dynamics of a free mass |
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| Affiliation: | 1. Department of Mathematics, Imperial College, London, United Kingdom;2. CNRS-LMD-IPSL, École Normale Supérieure de Paris/CNRS, Paris, France;3. Computing +Mathematical Sciences, Caltech, 1200 E. California Blvd, Pasadena, CA 91125, USA;4. School of Mathematics, Shanghai Jiao Tong University, Minhang District, Shanghai, 200240, China;5. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH–1015 Lausanne, Switzerland;6. Department of Precision and Microsystems Engineering, Faculty of 3ME, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, Netherlands;1. Université de Bretagne Sud, FRE CNRS 3744 IRDL, Centre de Recherche, Rue de Saint Maudé, BP 92116, 56321 Lorient, France;2. Department of Mechanical Engineering, Trakya University, 22180 Edirne, Turkey;3. Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA;1. Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600 036, India;2. Department of Mechanical Engineering, Texas A & M University, College Station, TX 77840, USA;3. Dipartimento di Ingegneria Industriale, Università di Perugia, 06125 Perugia, Italy;1. Mechanical Engineering Research Institute of the Russian Academy of Sciences, 85, Belinsky str., 603024, Nizhny Novgorod, Russia;2. Nizhny Novgorod Lobachevsky State University, 23, Gagarin av., 603950, Nizhny Novgorod, Russia;3. Department of Mathematical Modelling, Tver State University, Sadoviy per. 35, 170002 Tver, Russia;4. Institute of Problems in Mechanical Engineering, Bolshoy 61, V.O., Saint-Petersburg 199178, Russia;5. St. Petersburg State University, 7-9, Universitetskaya nab., V.O., Saint-Petersburg 199034, Russia;6. St. Petersburg State Polytechnical University, Polytechnicheskaya st., 29, SaintPetersburg 195251, Russia;1. School of Civil Engineering, The University of Queensland, St Lucia, Queensland 4072, Australia;2. Université de Bretagne Sud, IRDL, UBS, Lorient, Institut de Recherche Dupuy de Lôme, Centre de Recherche, Rue de Saint Maudé – BP, 92116, 56321 Lorient Cedex, France |
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| Abstract: | ![]()
We study the numerical integration of the Poisson second-order ordinary differential equation which describes, for instance, the dynamics of a free mass. Classical integration algorithms, when applied to such an equation, furnish solutions affected by a significant “drift” error, apparently not studied so far. In the first part of this work we define measures of such a drift. We then proceed to illustrate how to construct both classical and extended functionals for the equation of motion of a free mass with given initial conditions. These tools allow both the derivation of new variationally-based time integration algorithms for this problem, and, in some cases, the theoretical isolation of the source of the drift. While we prove that this particular error is unavoidable in any algorithmic solution of this problem, we also provide some new time integration algorithms, extensions at little added cost of classical methods, which permit to substantially improve numerical predictions. |
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