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Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints

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Abstract

In this work, variational integrators of higher order for dynamical systems with holonomic constraints are constructed and analyzed. The construction is based on approximating the configuration and the Lagrange multiplier via different polynomials. The splitting of the augmented Lagrangian in two parts enables the use of different quadrature formulas to approximate the integral of each part. Conditions are derived that ensure the linear independence of the higher order constrained discrete Euler-Lagrange equations and stiff accuracy. Time reversibility is investigated for the discrete flow on configuration level only as for the flow on configuration and momentum level. The fulfillment of the hidden constraints plays an important role for the time reversibility of the presented integrators. The order of convergence is investigated numerically. Order reduction of the momentum and the Lagrange multiplier compared to the order of the configuration occurs in general, but can be avoided by fulfilling the hidden constraints in a simple post processing step. Regarding efficiency versus accuracy a numerical analysis yields that higher orders increase the accuracy of the discrete solution substantially while the computational costs decrease. A comparison to the constrained Galerkin methods in Marsden and West (Acta Numerica 10, 357–514 2001) and the symplectic SPARK integrators of Jay (SIAM Journal on Numerical Analysis 45(5), 1814–1842 2007) reveals that the approach presented here is more general and thus allows for more flexibility in the design of the integrator.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Erlangen-Nuremberg, Immerwahrstrasse 1, 91058, Erlangen, Germany

    Theresa Wenger & Sigrid Leyendecker

  2. Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK

    Sina Ober-Blöbaum

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  1. Theresa Wenger
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  2. Sina Ober-Blöbaum
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  3. Sigrid Leyendecker
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Correspondence to Sigrid Leyendecker.

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Communicated by: Arieh Iserles

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Wenger, T., Ober-Blöbaum, S. & Leyendecker, S. Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints. Adv Comput Math 43, 1163–1195 (2017). https://doi.org/10.1007/s10444-017-9520-5

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  • DOI: https://doi.org/10.1007/s10444-017-9520-5

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