Momentum conserving symplectic integrators

Reich, Sebastian

doi:10.1016/0167-2789(94)90046-9

Cited by 63 publications

(41 citation statements)

References 10 publications

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“…In doing so, we use a scheme similar to that of McLachlan and Scovel 10 ͑see also Reich 11 ͒, in which one defines a momentum variable canonically conjugate to Q -the resulting Hamilton's equations of motion subject to the orthogonality constraint Q T Q ϭ 1 are then of the proper form for the implementation of efficient symplectic integrators such as SHAKE/RATTLE.…”

Section: The Rshake Methodsmentioning

confidence: 99%

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A symplectic method for rigid-body molecular simulation

Kol

Laird

Leimkuhler

1997

The Journal of Chemical Physics

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Rigid-body molecular dynamics simulations typically are performed in a quaternion representation. The nonseparable form of the Hamiltonian in quaternions prevents the use of a standard leapfrog ͑Verlet͒ integrator, so nonsymplectic Runge-Kutta, multistep, or extrapolation methods are generally used. This is unfortunate since symplectic methods like Verlet exhibit superior energy conservation in long-time integrations. In this article, we describe an alternative method, which we call RSHAKE ͑for rotation-SHAKE͒, in which the entire rotation matrix is evolved ͑using the scheme of McLachlan and Scovel ͓J. Nonlin. Sci. 16 233 ͑1995͔͒͒ in tandem with the particle positions. We employ a fast approximate Newton solver to preserve the orthogonality of the rotation matrix. We test our method on a system of soft-sphere dipoles and compare with quaternion evolution using a 4th-order predictor-corrector integrator. Although the short-time error of the quaternion algorithm is smaller for fixed time step than that for RSHAKE, the quaternion scheme exhibits an energy drift which is not observed in simulations with RSHAKE, hence a fixed energy tolerance can be achieved by using a larger time step. The superiority of RSHAKE increases with system size.

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Section: The Rshake Methodsmentioning

confidence: 99%

“…constraining the bond lengths ͑or angles͒, we constrain the rotation matrix to be orthogonal. 10,11 We also describe efficient iteration procedures for the nonlinear equations that must be solved at each time step. We will refer to this approach as RSHAKE ͑for rotation SHAKE͒.…”

Section: Introductionmentioning

confidence: 99%

A symplectic method for rigid-body molecular simulation

Kol

Laird

Leimkuhler

1997

The Journal of Chemical Physics

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“…Algorithmic conservation of the constraint invariants. Note that in the mG(k) method the original constraint equations in (34) have been di erentiated once with respect to time (cf. Remark 2.8).…”

Section: The Mixed Galerkin (Mg) Methodsmentioning

confidence: 99%

Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints

Betsch

Steinmann

2002

Numerical Meth Engineering

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SUMMARYA Galerkin-based discretization method for index 3 di erential algebraic equations pertaining to ÿnite-dimensional mechanical systems with holonomic constraints is proposed. In particular, the mixed Galerkin (mG) method is introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained Hamiltonian system, namely total energy and angular momentum. In addition to that, the constraints on the conÿguration level and on the velocity=momentum level are fulÿlled exactly. The application of the mG method to speciÿc mechanical systems such as the pendulum, rigid body dynamics and the coupled motion of rigid and exible bodies is presented. Related numerical examples are investigated to evaluate the numerical performance of the mG(1) and mG(2) method.

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“…In the field of MBS geometric integration, special attention is devoted to structure preserving methods that exploit rich geometric structure of rigid body rotational dynamics (see [7,16,25,58,67,73,84,85,87,89,90,97,104,135,146,147,152] and references cited therein). To this end, rigid body rotational dynamics is studied most conveniently as Lie-Poisson system that is defined on so * (3) (the dual space of so (3)).…”

Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

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Momentum conserving symplectic integrators

Cited by 63 publications

References 10 publications

A symplectic method for rigid-body molecular simulation

A symplectic method for rigid-body molecular simulation

Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints

Geometric methods and formulations in computational multibody system dynamics

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