Variational time integrators

The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in de...

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Published in:	International journal for numerical methods in engineering Vol. 60; no. 1; pp. 153 - 212
Main Authors:	Lew, A., Marsden, J. E., Ortiz, M., West, M.
Format:	Journal Article
Language:	English
Published:	Chichester, UK John Wiley & Sons, Ltd 07.05.2004
Subjects:	Accuracy Algorithms Conservation Convergence discrete mechanics elastodynamics Finite element method geometric integration Integrators Mathematical analysis Mathematical models multi-time-step subcycling variational integrators
ISSN:	0029-5981, 1097-0207
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Abstract	The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speed‐ups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact path‐independent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the J‐integral at the tip of a crack in a finite element mesh. Copyright © 2004 John Wiley & Sons, Ltd.
AbstractList	The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speed-ups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact path-independent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the J-integral at the tip of a crack in a finite element mesh. The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speed‐ups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact path‐independent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the J‐integral at the tip of a crack in a finite element mesh. Copyright © 2004 John Wiley & Sons, Ltd.
Author	Marsden, J. E. Ortiz, M. Lew, A. West, M.
Author_xml	– sequence: 1 givenname: A. surname: Lew fullname: Lew, A. organization: Graduate Aeronautical Laboratories 105-50, California Institute of Technology, Pasadena, CA 91125, U.S.A – sequence: 2 givenname: J. E. surname: Marsden fullname: Marsden, J. E. organization: Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, U.S.A – sequence: 3 givenname: M. surname: Ortiz fullname: Ortiz, M. email: ortiz@aero.caltech.edu organization: Graduate Aeronautical Laboratories 105-50, California Institute of Technology, Pasadena, CA 91125, U.S.A – sequence: 4 givenname: M. surname: West fullname: West, M. organization: Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, U.S.A
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SubjectTerms	Accuracy Algorithms Conservation Convergence discrete mechanics elastodynamics Finite element method geometric integration Integrators Mathematical analysis Mathematical models multi-time-step subcycling variational integrators
Title	Variational time integrators
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