Variational time integrators

Lew, Adrián J.; Marsden, Jerrold E.; Ortiz, Michael; West, Matthew

doi:10.1002/nme.958

Cited by 212 publications

(219 citation statements)

References 45 publications

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“…We can construct a variational integrator for a mechanical system with Lagrangian L(q, v)-here assumed to be time-independent for simplicity-by considering the action for the system over a small interval of time (Farr and Bertschinger 2007;Lew et al 2004):…”

Section: Variational Integratorsmentioning

confidence: 99%

Variational integrators for almost-integrable systems

Farr

2008

Celest Mech Dyn Astr

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We construct several variational integrators-integrators based on a discrete variational principle-for systems with Lagrangians of the form L = L A + εL B , with ε ≪ 1, where L A describes an integrable system. These integrators exploit that ε ≪ 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system. Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems, but their construction and error analysis is significantly simpler in the variational framework. One novel method we present, involving a weighted time-averaging of the perturbing terms, removes all errors from the integration at O (ε). This last method is implicit, and involves evaluating a potentially expensive time-integral, but for some systems and some error tolerances it can significantly outperform traditional simulation methods.

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Section: Variational Integratorsmentioning

confidence: 99%

Variational integrators for almost-integrable systems

Farr

2008

Celest Mech Dyn Astr

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“…First, we quickly review, from Marsden et al [82] and Lew et al [80,81], some facts about multisymplectic variational integrators for smooth unconstrained problems. Then, we develop our approach for two different classes of constrained problems, using the generalized Lagrange multiplier approach.…”

Section: Multisymplectic Variational Integrator For Nonsmooth Mechanimentioning

confidence: 99%

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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“…However, the spatial derivatives are taken on the geometry at time t i . Although we have no proof that the newly proposed scheme (10) possesses the properties of symplectic integrators (as defined by Lew et al 2004), it will be referred to as symplectic-like scheme in the following. Similar schemes are known to have good energy-conserving properties on the long term, and have been widely used, e.g., in celestial mechanics and mechanical engineering (Lew et al 2004).…”

Section: Free Surface Time Steppingmentioning

confidence: 99%

Simulation of floating structure dynamics in waves by implicit coupling of a fully non-linear potential flow model and a rigid body motion approach

Dombre

Benoît

Violeau

et al. 2014

J. Ocean Eng. Mar. Energy

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We demonstrate the accuracy and convergence of a new numerical model solving wave-structure interactions based on the fully non-linear potential flow (FNPF) theory coupled to a rigid body motion approach. This work extends an earlier model proposed by Guerber et al. (Eng Anal Bound Elements 36(7):1151-1163, 2012), restricted to fully submerged structures, by allowing to solve for freely floating bodies on the free surface. Although we are currently extending the model to three dimensions (3D), the work reported here only considers two-dimensional (2D) problems. We first introduce the FNPF model, We then detail the numerical scheme used for coupling the FNPF model to the motion of a floating rigid body. Moreover, we propose a new numerical strategy for advancing the free surface front inspired by symplectic integrators, which achieves a much better performance for energy conservation. The developed algorithm is first applied to forced motion cases, for which analytical and experimental results can be found in the literature and used as benchmarks. The accuracy of the numerical solution for the fluid and applied forces is then discussed for cases with small or large amplitude motion. In the latter case, a preliminary investigation of non-linear effects is performed for the classical application of a semi-circular heaving cylinder, by comparing the computed hydrodynamic force to the experimental measurements of Yamashita (J Soc Nav Arch 141: [61][62][63][64][65][66][67][68][69][70] 1977). In particular, the comparison of the magnitude of the force harmonics, up to the third order, shows the importance of simulating non-linear interactions, which become important as the ratio of the radius of the cylinder over the wavelength increases. In a second set of applications, we assess the model accuracy in dealing with freely floating bodies. As a first test case, we consider the decaying motion of a freely heaving horizontal circular cylinder released from a non-equilibrium position above the free surface. In this more demanding computations, we verify that total energy fluid-plus-body motion is well conserved, which confirms the accuracy of the fluid-structure interaction algorithm. As a second test case, we consider the free motion of a rectangular barge in waves and compute the first-order response amplitude operators.

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Variational time integrators

Cited by 212 publications

References 45 publications

Variational integrators for almost-integrable systems

Variational integrators for almost-integrable systems

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Simulation of floating structure dynamics in waves by implicit coupling of a fully non-linear potential flow model and a rigid body motion approach

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