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39
Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems
 Internat. J. Numer. Methods Engrg
, 1999
"... The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum ..."
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Cited by 64 (36 self)
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The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior. This analytical result is verified through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d’Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which
Mechanical Integrators Derived from a Discrete Variational Principle
"... Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by ..."
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Cited by 61 (13 self)
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Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We girst present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing mechanical integrators for continuous Lagrangian systems. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.
Symplectic Numerical Integrators in Constrained Hamiltonian Systems
, 1994
"... : Recent work reported in the literature suggests that for the longtime integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic (or canonical) structure of the flow. Here we investigate the symplecticness of numerical integrators for constrained dynamics, su ..."
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Cited by 45 (7 self)
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: Recent work reported in the literature suggests that for the longtime integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic (or canonical) structure of the flow. Here we investigate the symplecticness of numerical integrators for constrained dynamics, such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies. This leads to a constrained Hamiltonian system of smaller dimension. Previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space. Here it is shown that the very popular Verlet method with SHAKEtype constraints is equivalent to the same method with RATTLEtype constraints and that the latter is symplectic and time reversible. (This assumes that the iteration is carried to convergence.) We also demonstrate the global convergence of the Verlet scheme in the presence of SHAKEtype and RATTLE...
Geometric numerical integration illustrated by the StörmerVerlet method
, 2003
"... The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric nume ..."
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Cited by 27 (4 self)
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The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a crosssection of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent longtime behaviour of the method: longtime energy conservation, linear error growth and preservation of invariant tori in nearintegrable systems, a discrete virial theorem, and preservation of adiabatic invariants.
Equivariant constrained symplectic integration
 J. Nonlinear Sci
, 1995
"... We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting ..."
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Cited by 23 (3 self)
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We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for LiePoisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum. 1.
Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation
 ANNU. REV. BIOPHYS. BIOMOL. STRUCT
, 1997
"... Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stab ..."
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Cited by 23 (9 self)
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Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stability. This timestep problem is challenging if strictly faster methods with the same allatom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multipletimescale contexts—where the fast motions are rapidly decaying or largely decoupled from others—have not been as successful for biomolecules, where vibrational coupling is strong. This review examines general issues that limit the timestep and describes available methods (constrained, reducedvariable, implicit, symplectic, multipletimestep, and normalmodebased schemes). A section compares results of selected integrators for a model dipeptide, assessing physical and numerical performance. Included is our dual timestep method LN, which relies on an approximate linearization of the equations of motion every �t interval (5 fs or less), the solution of which is obtained by explicit integration at the inner timestep �τ (e.g., 0.5 fs). LN is computationally competitive, providing 4–5 speedup factors, and results are in good agreement, in comparison to 0.5 fs trajectories. These collective algorithmic efforts help fill the gap between the time range that can be simulated and the timespans of major biological interest (milliseconds and longer). Still, only a hierarchy of models and methods, along with
Exploiting Invariants In The Numerical Solution Of Multipoint Boundary Value Problems For DAE
 SIAM J. Sci. Comp
, 1998
"... This paper presents a new approach to the numerical solution of boundary value problems for higher index differential algebraic equations. Invariants known for the original DAE as well as invariants of the reduced index 1 formulation are exploited to stabilize initial value problem integration, deri ..."
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Cited by 19 (11 self)
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This paper presents a new approach to the numerical solution of boundary value problems for higher index differential algebraic equations. Invariants known for the original DAE as well as invariants of the reduced index 1 formulation are exploited to stabilize initial value problem integration, derivative generation and nonlinear and linear systems solution of an enhanced multiple shooting method. Extensions to collocation are given. Applications are presented for two important problem classes: parameter estimation in multibody systems given in descriptor form, and singular and state constrained optimal control problems. In particular, generalizations of the "internal numerical differentiation" technique to DAE with invariants and a new multistage least squares decomposition technique for DAE boundary value problems are developed, which are implemented in the multiple shooting code PARFIT and in the collocation code COLFIT. Further, a method is described which minimizes the number of...
Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems
 Physica D
, 1995
"... We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential e ..."
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Cited by 17 (8 self)
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We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small stepsize, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bondangle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion...
A Symplectic Integrator for Riemannian Manifolds
, 1996
"... this article we show how such an ambient Riemannian metric can be used to frame the popular "leapfrog" method in the category of Riemannian manifolds, thus creating new intrinsic methods applicable to mechanical systems with configuration spaces as general as homogeneous spaces of semisimple Lie gro ..."
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Cited by 14 (2 self)
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this article we show how such an ambient Riemannian metric can be used to frame the popular "leapfrog" method in the category of Riemannian manifolds, thus creating new intrinsic methods applicable to mechanical systems with configuration spaces as general as homogeneous spaces of semisimple Lie groups. This new method, shown in Figure (1), is implicit, second order, timereversing, symplectic and respects those symmetries of the system which are also isometries of the ambient metric. For arbitrary potential energy, but where the kinetic energy metric of the mechanical system is proportional to the ambient metric, our method is explicit, and is equivalent to the splitting method corresponding to the splitting of the Hamiltonian into kinetic and potential energy. 2. Context and Notation
Structure Preservation For Constrained Dynamics With Super Partitioned Additive RungeKutta Methods
 SIAM J. Sci. Comput
, 1998
"... A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is ..."
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Cited by 13 (9 self)
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A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is presented. A new class of integrators is defined: the super partitioned additive RungeKutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structurepreservation, the class of sstage Lobatto IIIABCC* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.