Time Integration and Discrete Hamiltonian Systems

González, Óscar

doi:10.1007/978-1-4612-1246-1_10

Cited by 146 publications

(268 citation statements)

References 19 publications

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“…r The second perspective is the extension towards to the nonlinear case as it has been done in some special case of the mid-point rule in [5,29,47]. Using the special form of the gyroscopic forces if the mass matrix depends on q and the fact that some forces derived from a smooth potential, we may use the discrete derivative introduced in [25] to extend the result of the paper in a nonlinear setting.…”

Section: Resultsmentioning

confidence: 99%

“…The numerical scheme (25) has been designed such that it deals, as the Moreau-Jean scheme with the contact forces and impact through their associated impulses P k+1 in a fully implicit way. In this manner, we ensure that the scheme will be consistent when the time-step vanishes if an impact occurs.…”

Section: Nonsmooth Newmark and Generalized-α Schemementioning

confidence: 99%

“…Proposition 6.5. The variation of the "algorithmic" energy H over a time-step performed by the nonsmooth HHT scheme (scheme (25) with α m = 0 and α f = α) is…”

Section: The Nonsmooth Hht Casementioning

confidence: 99%

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Energy conservation and dissipation properties of time‐integration methods for nonsmooth elastodynamics with contact

Acary

2015

Z Angew Math Mech

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This research report is devoted to the study of the conservation and the dissipation properties of the mechanical energy of several time-integration methods dedicated to the elastodynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized-α schemes leads to energy blow-up, we study two schemes dedicated to the time-integration of nonsmooth systems with contact: the Moreau-Jean scheme and the nonsmooth generalized-α scheme. The energy conservation and dissipation properties of the Moreau-Jean is firstly shown. In a second step, the nonsmooth generalized-α scheme is studied by adapting the previous works of Krenk and Høgsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber-Hughes-Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact. Mots-clés :Dynamique du contact, impact, mécanique numérique du contact, intégration numérique en temps, conservation d'énergie, propriétés de dissipation.Energy conservation and dissipation properties for the nonsmooth elastodynamics with contact 3

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Section: Resultsmentioning

confidence: 99%

Section: Nonsmooth Newmark and Generalized-α Schemementioning

confidence: 99%

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Energy conservation and dissipation properties of time‐integration methods for nonsmooth elastodynamics with contact

Acary

2015

Z Angew Math Mech

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“…cT ik 1T D 0 and ik 1T D ik 2T D 0 because there is no penalty spring in the tangential direction, see the slip condition in Equation (25). Therefore, the equation that provides the infinite (therefore undetermined) relations between ik 1N , ik 2N is:…”

Section: Discrete Total Bodies' Energy Balancementioning

confidence: 99%

An energy consistent frictional dissipating algorithm for particle contact problems

Bravo

Pérez-Aparicio

Laursen

2012

Numerical Meth Engineering

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SUMMARY An energy frictional dissipating algorithm (EFDA) for time integration of Coulomb frictional impact‐contact problems is presented. With the use of the penalty method, and in the context of a conserving framework, linear and angular momenta are conserved, and energy is consistently dissipated. Previously published formulations were stable, forcing the energy dissipation to be monotonic to prevent unstable energy growth. The shortcoming of many was that they were not able to reproduce the real kinematics and dissipation of physical processes, provided by analytical formulations and experiments. EFDA formulates a conserving framework on the basis of a physical energy dissipation estimator. This framework uses an enhanced penalty contact model based on a spring and a dashpot, enforcing physical frictional energy dissipation, controlling gap vibrations, and modifying the velocities and contact forces during each time step. The result is that the dissipated energy, kinematics, and contact forces are consistent with the expected physical behavior. Energy frictional dissipating algorithm has been applied to four rigid‐body frictional problems using the discrete element method. The first problem is the analysis of a disk moving on a flat rough surface; the second problem analyzes the kinematics and energy dissipation of elliptical particle impacts. Numerical solutions are compared with analytical ones in both problems. The motion and impact of two disks moving on a semicircular surface are studied in the third problem. Finally, the fourth problem simulates the collapse of a two‐dimensional granular column, which is compared with the experimental results. Copyright © 2012 John Wiley & Sons, Ltd.

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“…However, this property is linked specifically to linear elasticity with quadratic strains. An important extension of the energy conservation algorithms to more general nonlinear problems was made by Gonzalez [15,16], who generalized the representation of the internal force to include an additive discrete energy gradient term, much in the vein of the gradient representation in optimization algorithms, [17]. An early scalar version of the secant style conservative integration was presented by Greenspan [18], and discrete gradient integrations methods have been developed for classic mechanics [19], nonlinear elasticity [20], multibody dynamics [21], and mathematical physics [22].In the energy-momentum approach, the algorithms are typically formulated for an undamped system.…”

mentioning

confidence: 99%

Global format for energy–momentum based time integration in nonlinear dynamics

Krenk

2014

Numerical Meth Engineering

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SUMMARYA global format is developed for momentum and energy consistent time integration of second order dynamic systems with general non-linear stiffness. The algorithm is formulated by integrating the state space equations of motion over the time increment. The internal force is first represented in fourth order form consisting of the end-point mean value plus a term containing the stiffness matrix increment. This form gives energy conservation for systems with internal energy as a quartic function of the displacement components. This representation is then extended to general energy conservation via a discrete gradient representation. The present procedure works directly with the internal force and the stiffness matrix at the time integration interval end-points, and in contrast to previous energy conserving algorithms does not require any special form of the energy function nor use of mean-value products at the element level or explicit use of a geometric stiffness matrix. An optional monotonic algorithmic damping, increasing with response frequency, is developed in terms of a single damping parameter. In the solution procedure the velocity is eliminated and the non-linear iterations are based on the displacement components alone. The procedure represents an energy consistent alternative to available collocation methods, with an equally simple implementation.

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Time Integration and Discrete Hamiltonian Systems

Cited by 146 publications

References 19 publications

Energy conservation and dissipation properties of time‐integration methods for nonsmooth elastodynamics with contact

Energy conservation and dissipation properties of time‐integration methods for nonsmooth elastodynamics with contact

An energy consistent frictional dissipating algorithm for particle contact problems

Global format for energy–momentum based time integration in nonlinear dynamics

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