General techniques for constructing variational integrators

Leok, Melvin; Shingel, Tatiana

doi:10.1007/s11464-012-0190-9

Cited by 49 publications

(74 citation statements)

References 25 publications

(26 reference statements)

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“…The choice of the Taylor method as the underlying one-step method has the advantage that it only requires one to precompute the prolongation of the Euler-Lagrange vector field once at the initial time, and the computational cost is not increased appreciably by having to compute the numerical solution at multiple quadrature nodes, since that only requires a polynomial evaluation. This efficiency in evaluation improves upon the methods outlined in [10] and [11], which utilized collocation and the shooting-method, respectively. Example 1.…”

Section: Lemmamentioning

confidence: 83%

“…The simulations compare the discrete Lagrangian form of the Taylor variational integrator (TVI), the discrete right Hamiltonian form of the Taylor variational integrator (HTVI), the symmetric Taylor variational integrator of 4th order (SV4), Taylor's method, and the Runge-Kutta shooting variational integrators (ShVI) (see [11]). Overall, high-order Taylor methods perform quite well in terms of computational time versus global error.…”

Section: Comparison Of Methodsmentioning

confidence: 99%

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Lagrangian and Hamiltonian Taylor variational integrators

2017

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Abstract. In this paper, we present a variational integrator that is based on an approximation of the Euler-Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator introduced in [11]. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.

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Section: Lemmamentioning

confidence: 83%

Section: Comparison Of Methodsmentioning

confidence: 99%

Lagrangian and Hamiltonian Taylor variational integrators

2017

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“…Dealing with these issues, in their presentation of discrete mechanics, Marsden and West [39] defined an exact discrete Lagrangian L E d associated to any smooth Lagrangian L. They observed that if another discrete Lagrangian L d approximates the exact discrete Lagrangian to order p, then the corresponding discrete Hamiltonian flow that solves the discrete Euler-Lagrange equations, is order p accurate when used as a one-step numerical scheme to approximate smooth solutions of Euler-Lagrange equations. Leok and Shingel [35] develop a systematic method to construct discrete lagrangians (and variational principles) from smooth Lagrangians in mechanics, using a choice of a numerical quadrature formula, together with a choice of a finite-dimensional function space or a one-step numerical scheme. The order of the quadrature formula and numerical scheme determines the order of accuracy and momentum-conservation properties of the associated variational integrators.…”

Section: Proofmentioning

confidence: 99%

“…A systematic study is done for the case of covariant Lagrangian densities on a Lie group, using the exponential mapping, or the Cayley transform on SO(3) as retraction mapping (the latter improves the computational cost, keeping the order of accuracy), and fixing an arbitrary quadrature rule with several nodes. In [46] it is suggested that this variational error analysis may be extended to discrete field theories, and that one may develop general techniques for constructing variational integrators for field theories, extending in this way the theory of variational error analysis introduced by [39] and explored in [35], among others. Moreover [46] also remarks that a corresponding theory of variational error analysis for the discretization of field theories may also rely on a deeper understanding of the associated boundary Lagrangians and how they serve as generating functionals for multisymplectic relations.…”

Section: Proofmentioning

confidence: 99%

Discrete formulation for the dynamics of rods deforming in space

Casimiro

Rodrigo

2019

Journal of Mathematical Physics

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The movement of rods in an euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth lagrangian density is invariant by some group of transformations, there exist corresponding conserved Noether currents. Generally numerical schemes dealing with PDEs fail to reflect these conservation properties.We describe the main ingredients needed to create, from the smooth lagrangian density, a variational principle for discrete motions of a discrete rod, with corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle, using a reduced forward difference operator. We show how this introduces a discrete lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.

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“…The essential aspects of VI's can be reviewed in Marsden and Wendlandt 1997; and in (Leok and Shingel 2012a;Leok 2005) general techniques for constructing variational integrators are provided. Spectral variational integrators are described in (Hall and Leok 2014a) and prolongation-collocation methods in (Leok and Shingel 2012b).…”

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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General techniques for constructing variational integrators

Cited by 49 publications

References 25 publications

Lagrangian and Hamiltonian Taylor variational integrators

Lagrangian and Hamiltonian Taylor variational integrators

Discrete formulation for the dynamics of rods deforming in space

A Brief Introduction to Variational Integrators

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