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126
Discrete Euler-Poincaré and Lie-Poisson equations
- Nonlinearity
, 1999
"... Abstract. In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are G-invariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the ..."
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Cited by 55 (7 self)
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Abstract. In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are G-invariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG,and a discrete Langragian L: G × G → R is constructed in such a way that the Ginvariance property is preserved. Reduction by G results in new “variational” principle for the reduced Lagrangian ℓ: G → R, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in [MPS 98, WM 97] which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G =SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser-
Multi-Symplectic Runge-Kutta Collocation Methods for Hamiltonian Wave Equations
- J. Comput. Phys
, 1999
"... A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss-Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to nume ..."
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Cited by 42 (7 self)
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A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss-Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for Hamiltonian ODEs. We also discuss the issue of conservation of energy and momentum. Since time discretization by a Gauss-Legendre method is computational rather expensive, we suggest several semi-explicit multi-symplectic methods based on Gauss-Legendre collocation in space and explicit or linearly implicit symplectic discretizations in time. 1 Introduction The scalar wave equation @ tt u = @ xx u \Gamma V 0 (u); (x; t) 2 U ae R 2 ; (1) V : R ! R some smooth function, is an example of a multi-symplectic Hamiltonian PDE [3] of type M@ t z +K@ x z = r z S...
Nonsmooth Lagrangian mechanics and variational collision integrators
- SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum ..."
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Cited by 36 (9 self)
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Abstract. Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.
Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top
, 1998
"... We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous int ..."
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Cited by 30 (1 self)
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We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics, – the Lagrange top. We recall the derivation of the Euler–Poinsot equations of motion both in the frame moving with the body and in the rest frame (the latter ones being less widely known). We find a discrete time Lagrange function turning into the known continuous time Lagrangian in the continuous limit, and elaborate both descriptions of the resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuous time system, the integrals of motion being deformed. The discrete time Lax representations are also found. Kirchhoff’s kinetic analogy between elastic curves and motions of the Lagrange top is also generalised to the discrete context.
Asynchronous contact mechanics
"... We develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of well-posed problems (a) have no interpenetrations, (b) obey causality, momentum- and energy-conservation laws, and (c ..."
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Cited by 28 (8 self)
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We develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of well-posed problems (a) have no interpenetrations, (b) obey causality, momentum- and energy-conservation laws, and (c) complete in finite time. We achieve these guarantees through a novel synthesis of asynchronous variational integrators, kinetic data structures, and a discretization of the contact barrier potential by an infinite sum of nested quadratic potentials. In a series of two- and threedimensional examples, we illustrate that this method more easily handles challenging problems involving complex contact geometries, sharp features, and sliding during extremely tight contact.
Structure-preserving model reduction for mechanical systems
, 2003
"... This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loève expansion, balancing, and wavel ..."
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Cited by 26 (4 self)
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This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.
2000], Reduction in principal fiber bundles: Covariant Euler-Poincaré equations
- Proc. Amer. Math. Soc
"... Abstract. Let π: P → M n be a principal G-bundle, and let L: J 1 P → Λ n (M) be a G-invariant Lagrangian density. We obtain the Euler-Poincaré equations for the reduced Lagrangian l defined on C(P), the bundle of connections on P. 1. ..."
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Cited by 24 (6 self)
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Abstract. Let π: P → M n be a principal G-bundle, and let L: J 1 P → Λ n (M) be a G-invariant Lagrangian density. We obtain the Euler-Poincaré equations for the reduced Lagrangian l defined on C(P), the bundle of connections on P. 1.
Multisymplectic box schemes and the Korteweg–de Vries equation
, 2003
"... We develop and compare some geometric integrators for the Korteweg-de Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, ..."
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Cited by 22 (4 self)
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We develop and compare some geometric integrators for the Korteweg-de Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are susceptible to developing small wiggles. At the same instances the fully implicit and multisymplectic Preissmann scheme, written as a 12-point box scheme, stays smooth. This is accounted for by the ability of the box scheme to preserve the shape of the dispersion relation of any hyperbolic system for all ∆x and ∆t. We also develop a simplified 8-point version of this box scheme which maintains its advantageous features.
