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39
A TimeReversible VariableStepsize Integrator for Constrained Dynamics
, 1997
"... This article considers the design and implementation of variabletimestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, we then consider timereparameterization techniques employing a timereversible (symmetric) integration ..."
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This article considers the design and implementation of variabletimestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, we then consider timereparameterization techniques employing a timereversible (symmetric) integration method to solve the equations of motion. We give several numerical examples, including a simulation of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation.
Mechanical Systems Subject to Holonomic Constraints: DifferentialAlgebraic Formulations and Conservative Integration
"... The numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered. Schemes are introduced for the direct treatment of a differentialalgebraic form of the equations that preserve the constraints, the total energy, and other integrals su ..."
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The numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered. Schemes are introduced for the direct treatment of a differentialalgebraic form of the equations that preserve the constraints, the total energy, and other integrals such as linear and angular momentum arising from affine symmetries. Moreover, the schemes can be shown to preserve the property of timereversibility in an appropriate sense. An example is given to illustrate various aspects of the proposed methods. 1 Introduction The equations of motion for a conservative mechanical system subject to holonomic constraints can often be written in the differentialalgebraic [5] form q = D 2 H(q; p) p = \GammaD 1 H(q; p) \Gamma Dg(q) T 0 = g(q) 9 = ; (1) where q 2 IR n are the configuration variables, p 2 IR n are variables conjugate to the velocities q in an appropriate sense, g : IR n ! IR m (m ! n) is a smooth constraint function, 2 IR m ...
The midpoint scheme and variants for Hamiltonian systems: advantages and pitfalls
, 1997
"... The (implicit) midpoint scheme, like higher order Gausscollocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in mole ..."
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Cited by 11 (4 self)
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The (implicit) midpoint scheme, like higher order Gausscollocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or structural mechanics, because there is no stability restriction when it is applied to a simple harmonic oscillator. Although it is wellknown that the midpoint scheme may also exhibit instabilities in various stiff situations, one might still hope for good results when resonancetype instabilities are avoided. In this paper we investigate the suitability of the midpoint scheme for highly oscillatory, frictionless mechanical systems, where the stepsize k is much larger than the system's small parameter ", in case that the solution remains bounded as " ! 0. We show that in general one must require that k
On Some Difficulties in Integrating Highly Oscillatory Hamiltonian Systems
 LECTURE NOTES IN COMPUTATIONAL SCIENCE AND ENGINEERING
, 1997
"... The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the stepsize restrictions of explicit methods such as the Verlet met ..."
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Cited by 9 (2 self)
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The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the stepsize restrictions of explicit methods such as the Verlet method. Among these are multipletimestepping, constrained dynamics, and implicit methods. In this paper, we investigate the suitability of timereversible, semiimplicit methods. Here semiimplicit means that only the highly oscillatory part is integrated by an implicit method such as the midpoint method or an energyconserving variant of it. The hope is that such methods will allow one to use a stepsize k which is much larger than the period " of the fast oscillations. However, our results are not encouraging. Even in the absence of resonancetype instabilities, we show that in general one must require that k
Constraint preserving integrators for general nonlinear higher index DAEs
, 1995
"... This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. ..."
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Cited by 7 (3 self)
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This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs.
High Order Numerical Integrators for Differential Equations using Composition and Processing of low Order Methods
 APPL. NUMER. MATH
, 2000
"... In this paper we show how to build high order integrators for solving ordinary differential equations by composition of low order methods and using the processing technique. From a basic pth order method, \Psi p , one can obtain high order integrators in the processed form \Psi n = P KP \Gamma1 ( ..."
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Cited by 6 (2 self)
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In this paper we show how to build high order integrators for solving ordinary differential equations by composition of low order methods and using the processing technique. From a basic pth order method, \Psi p , one can obtain high order integrators in the processed form \Psi n = P KP \Gamma1 (n ? p) being both the processor P and the kernel K compositions of the basic method. The number of conditions for the kernel is drastically reduced if we compare with a standard composition. The particular case in which \Psi p is a symmetric scheme of order 2 and 4, respectively, is analysed, and new optimised 6th and 8th order integrators are built.
Symplectic Integrators for Systems of Rigid Bodies
 Integration Algorithms and Classical Mechanics, volume 10 of Fields Institute Communications. AMS
, 1996
"... Recent work reported in the literature suggest that for the longterm integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. This has especially been shown for the numerical treatment of conservative manyparticle systems as they arise ..."
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Recent work reported in the literature suggest that for the longterm integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. This has especially been shown for the numerical treatment of conservative manyparticle systems as they arise, e.g., in molecular dynamics and astronomy. In this paper we use recent results by McLachlan [10] and Reich [13] on the symplectic integration of LiePoisson systems to derive an explicit symplectic integrator for rigid bodies moving under the influence of external forces. In case that rigid bodies are interconnected by joints, the resulting holonomic constraints can be treated in the same way as described by Reich [12] for general Hamiltonian systems with holonomic constraints. Keywords: Hamiltonian systems, symplectic discretization, multibody systems AMS(MOS) Subject Classifications: 65L05, 7008, 70F20 1 Introduction Much recent research has gone into developing numerical discretizat...
Integration Methods for Molecular Dynamics
 IN MATHEMATICAL APPROACHES TO BIOMOLECULAR STRUCTURE AND DYNAMICS, IMA VOLUMES IN MATHEMATICS AND ITS APPLICATIONS
, 1996
"... Classical molecular dynamics simulation of a macromolecule requires the use of an efficient timestepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on m ..."
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Cited by 6 (2 self)
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Classical molecular dynamics simulation of a macromolecule requires the use of an efficient timestepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches.
Torsion Dynamics of Molecular Systems
 Phys. Rev. E
, 1996
"... Based on the concept of free energy, we derive a Hamiltonian formulation for molecular dynamics in torsion space. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced by B.R. Brooks, J. Zhou, and S. Reich in the context of molec ..."
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Cited by 6 (4 self)
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Based on the concept of free energy, we derive a Hamiltonian formulation for molecular dynamics in torsion space. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced by B.R. Brooks, J. Zhou, and S. Reich in the context of molecular dynamics. We consider a few simplifications that allow one to calculate the free energy analytically and to write the corresponding equations of motion as a constraint Hamiltonian system that can conveniently be discretized by the wellknown SHAKE algorithm. The additional computational costs, compared to using the orginal force field and constraining bondlengths and bondangles to their equilibrium value (hard constraints), amount, in general, to less than a complete force evaluation. We show for a single butane molecule that our Hamiltonian formulation yields the correct Boltzmann distribution in the torsion angle while the original Hamiltonian together with hard constraints on the b...
MultiMultiplier AmbientSpace Formulations of Constrained Dynamical Systems: The Case of Linearized Incompressible Elastodynamics
, 1999
"... Various formulations of the equations of motion for both finiteand infinitedimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambie ..."
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Cited by 6 (3 self)
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Various formulations of the equations of motion for both finiteand infinitedimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are not restricted to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space differ in each of the cases. For the model problem of linearized incompressible elastodynamics, we study three formulations and establish the wellposedness of one formulation corresponding to a homogeneous, isotropic material body with a specified traction on its boundary. 1 Introduction In this article we study formulations of the equations of motion for Lagrangian dynamical systems whose configuration space can b...