Multisymplectic Lie group variational integrator for a geometrically exact beam in

Demoures, François; Gay–Balmaz, François; Kobilarov, Marin; Raţiu, Tudor S.

doi:10.1016/j.cnsns.2014.02.032

Cited by 39 publications

(29 citation statements)

References 20 publications

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“…Such a variational discretization has been carried out in [16] following the multisymplectic discretization of [47]. This approach allows for symplecticity in both the space and time evolutions, as well as a discrete version of covariant Noether theorem (that implies the classical one).…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

Discrete variational Lie group formulation of geometrically exact beam dynamics

et al. 2014

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The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically.

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Section: Comparison With Other Methodsmentioning

confidence: 99%

Discrete variational Lie group formulation of geometrically exact beam dynamics

et al. 2014

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“…An explicit, second-order accurate VI that can be identified with a Lie-group, symplectic, partitioned RungeKutta method for finite element discretizations of geometrically exact rods is presented in (Mata 2015). The formulation of VI's for spatial beams and plates is carried out in (Demoures 2012;Demoures et al 2014). In (Nichols and Murphey 2008) a VI for simulating the dynamics of cable structures is formulated.…”

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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“…We are in the process of developing a variational integrator for this problem, however, we expect that such a development will be quite intricate. Symplectic and multisymplectic variational integrators for a geometrically exact rod without fluid motion have been only derived recently, see [71] and [72], and the fluid motion presents the substantial difficulty of introducing right-invariant terms in the problem. Multisymplectic discretization for tube conveying incompressible fluid has been considered in [65].…”

Section: Remark 24 (On Further Equation Simplification)mentioning

confidence: 99%

Geometric Theory of Flexible and Expandable Tubes Conveying Fluid: Equations, Solutions and Shock Waves

Gay–Balmaz¹,

Putkaradze²

2018

J Nonlinear Sci

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We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes, and compute several examples of particular solutions. The theory is derived from a variational treatment of Cosserat rod theory extended to incorporate expandable walls and moving flow inside the tube. The results presented here are useful for biological flows and industrial applications involving high speed motion of gas in flexible tubes.

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Multisymplectic Lie group variational integrator for a geometrically exact beam in

Cited by 39 publications

References 20 publications

Discrete variational Lie group formulation of geometrically exact beam dynamics

Discrete variational Lie group formulation of geometrically exact beam dynamics

A Brief Introduction to Variational Integrators

Geometric Theory of Flexible and Expandable Tubes Conveying Fluid: Equations, Solutions and Shock Waves

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