On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry

González, Óscar; Simo, J. C.

doi:10.1016/0045-7825(96)01009-2

Cited by 155 publications

(139 citation statements)

References 12 publications

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“…Selecting () ( ) ( ) ( ) ( ) (22). Note that this selection is only possible for fully-actuated systems since it includes the inverse of the input port matrix.…”

Section: ( )mentioning

confidence: 99%

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Some results on disturbance attenuation for Hamiltonian systems via direct discrete‐time design

Yalçın

Goren-Sumer

Astolfi

2014

Intl J Robust & Nonlinear

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SUMMARYThe disturbance attenuation and robust disturbance attenuation problems for Hamiltonian systems in the discrete-time setting are considered and some new results are presented. The new results are derived utilizing the recently presented dissipativity equality obtained by adding the dissipation rate function to the classical dissipativity inequality. A selection of the dissipation rate function yields the new results. These results include a condition on the dissipation structure of the system to achieve the desired disturbance attenuation level and gives direct construction of optimal control laws for any desired disturbance attenuation level. The results remove the need to solve Hamilton-Jacobi-Isaacs inequalities.

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“…Selecting () ( ) ( ) ( ) ( ) (22). Note that this selection is only possible for fully-actuated systems since it includes the inverse of the input port matrix.…”

Section: ( )mentioning

confidence: 99%

“…We finally recall the discrete gradient conditions [22], since a gradient-based discrete time model of the considered class of Hamiltonian systems is used.…”

Section: Remarkmentioning

confidence: 99%

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Some results on disturbance attenuation for Hamiltonian systems via direct discrete‐time design

Yalçın

Goren-Sumer

Astolfi

2014

Intl J Robust & Nonlinear

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“…Based on the concept of discrete derivative introduced in (González and Simó, 1996) in the context of Hamiltonian systems, it is possible to obtain a conserving algorithm for (6), which exactly conserves the total energy and the linear and angular momentum.…”

Section: Conserving Penalty Formulationmentioning

confidence: 99%

A Conservative Augmented Lagrangian Algorithm for the Dynamics of Constrained Mechanical Systems

Orden

Ortega

2006

Mechanics Based Design of Structures and Machines

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The motion of many practical mechanical systems is often constrained. An important example is the dynamics of multibody systems, where the numerical solution of this type of systems faces several difficulties. A strategy for to solve this type of problems is the augmented Lagrange formulation, which allows the use of numerical integrators for ODEs, combined with an update scheme for the algebraic variables, accomplishing exact fulfillment of the constraints. This work focuses on the design of a conservative version of this augmented Lagrangian formulation for holonomic constraints, proposing a numerical procedure that exhibits excellent stability.

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“…Amongst the latter, contributions were made by Simo, Tarnow & Wong [1992], Gonzalez & Simo [1996] and Gonzalez [1996]. The approach in this paper paves the way towards constructing symplecticenergy-momentum integrators.…”

Section: Introductionmentioning

confidence: 99%

Asynchronous Variational Integrators

Lew¹,

Ortiz²

Geometry, Mechanics, and Dynamics

145

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We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton's variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly.To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated.Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.

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On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry

Cited by 155 publications

References 12 publications

Some results on disturbance attenuation for Hamiltonian systems via direct discrete‐time design

Some results on disturbance attenuation for Hamiltonian systems via direct discrete‐time design

A Conservative Augmented Lagrangian Algorithm for the Dynamics of Constrained Mechanical Systems

Asynchronous Variational Integrators

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