Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

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“…Eqs. (15) and (16)) is straightforward. The expression of the G-equivariant force in the current context is given by…”

Section: Generalization and Preservation Of Momentamentioning

confidence: 92%

“…To accommodate the preservation of linear and angular momenta as discussed in Eqs. (15) and (16), Eq. (55) must be modified as indicated next: Let G be a Lie group with algebra g and coalgebra g * , which acts on the configuration space Q ⊆ R 3n by means of the action χ : G × Q → Q.…”

Section: Generalization and Preservation Of Momentamentioning

confidence: 99%

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

2019

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“…A comparison between the numerical performance of VI's and energy-momentum schemes when applied to the numerical simulation of flexible multibody dynamics is presented in Betsch et al (2010). The solvability of some geometric integrators for multibody systems is analyzed in (Kobilarov 2014).…”

Section: Variational Integrationmentioning

confidence: 99%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

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“…However, the approach is not limited to energy-momentum conserving schemes. Hence, any class of implicit time stepping schemes such as variational symplectic integrators or dissipating schemes are easily adapted, see [4].…”

Section: Introductionmentioning

confidence: 99%

A flexible approach to the simulation of hybrid multibody systems

Sänger

Betsch

2010

Proc Appl Math and Mech

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The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics.In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1].The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied.

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Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics

Cited by 25 publications

References 38 publications

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

A Brief Introduction to Variational Integrators

A flexible approach to the simulation of hybrid multibody systems

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