Nonsmooth Lagrangian Mechanics and Variational Collision Integrators

Fetecau, Razvan C.; Marsden, Jerrold E.; Ortiz, Michael; West, Matthew

doi:10.1137/s1111111102406038

Cited by 133 publications

(148 citation statements)

References 54 publications

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“…In the particular case of classical mechanics, that is, when spacetime reduces to time, our integrator recovers the collision variational integrators of Fetecau et al [42].…”

Section: Introductionsupporting

confidence: 63%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

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“…In the particular case of classical mechanics, that is, when spacetime reduces to time, our integrator recovers the collision variational integrators of Fetecau et al [42].…”

Section: Introductionsupporting

confidence: 63%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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“…Thus we obtain this time the extension of (3.7) where virtual velocities are considered, and the extension of (5.45) for the case of impacting motions. We note in passing that the arguments that yield [13,Equation (1a)] (which is the same as…”

Section: + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t)mentioning

confidence: 99%

“…) with missing mass matrix and minus sign in the right-hand side) are spurious: there is a shortcoming in the reasoning in [13] because of the use of the condition δ q (t k ) +q(t k )δt k ∈ T q(t k ) bd( ), whileq(·) jumps at t k , and T q(t k ) bd( ) denotes the tangent plane at q(t k ) (and not the tangent cone) to the boundary of the admissible domain (the correct way to derive the material page 385 in [13], is in Section 6 of reference 724 by Leine et al).…”

Section: + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t) L + Y(t)mentioning

confidence: 99%

Erratum to: Nonsmooth Mechanics

Brogliato¹

2016

Communications and Control Engineering

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“…Hamilton's principle in nonsmooth mechanics has been studied by many authors [2,5]. For our conservative system, the classical form still holds:…”

Section: Nonsmooth Formulation and Impact Equations 31 Equations Of mentioning

confidence: 99%

“…The impact event is assumed to happen in an infinitesimal time. The displacement is continuous but the velocity is discontinuous [2,3,4,5], which read…”

Section: Nonsmooth Formulation and Impact Equations 31 Equations Of mentioning

confidence: 99%

Smooth and Non-Smooth Solutions for Geometrically Exact Beams Subjected to Impact

Nguyen¹,

Hjiaj²,

Sansour³

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. Shock and Impact problems have been the focus of research for decades in many engineering areas such as structural engineering, automobile industry, aerospace, multibody dynamics. In general, unlike classical dynamics, impact requires specials techniques to correctly model the underlying physics. The solutions for the problems of impact have been developed in different ways by many authors with each of them using different hypothesis. However, formulations of impact of beams have received limited attention. The solution methods for impact can be divided into two main groups: smooth and non-smooth approaches. In this paper we will develop both smooth and nonsmooth formulations for impact between projectiles and beams. The adopted beam kinematics is geometrically exact. In smooth formulations, the velocities and forces are continuous functions. The impact forces is determined by the corresponding deformations of the projectile, the deformation of beam is calculated with the applied impact force. This approach is appropriate when the projectile is deformable and it reflects the reality of many impact scenarios. When it comes to impact between rigid projectile and deformable beams, the problem becomes more complex and should be treated using non-smooth formulations where the velocities and the impact forces may have jumps, thus, they are discontinuous. Set-valued force laws are used to account for unilateral conditions on both displacement and velocities. The appropriate framework is differntial measure combined with convex analysis The coefficient of restitution plays also an important role. The integration is also split into smooth and non-smooth part. Though, having stable time integration schemes for geometrically exact beams is not a straightforward task. For both formulations, the use of a newly developed energy-momentum time integration scheme conserve perfectly the total energy and the momentum of the system during and after the impact. A range of numerical applications will be delivered to show excellent performances of the new formulations.1169

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Nonsmooth Lagrangian Mechanics and Variational Collision Integrators

Cited by 133 publications

References 54 publications

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Erratum to: Nonsmooth Mechanics

Smooth and Non-Smooth Solutions for Geometrically Exact Beams Subjected to Impact

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