Prolongation-collocation variational integrators

Leok, Melvin; Shingel, Tatiana

doi:10.1093/imanum/drr042

Cited by 24 publications

(35 citation statements)

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“…www.zamm-journal.org Recently, Leok and Shingel [50] have proposed a variational integrator based on Hermite finite elements in time. Their formulation is derived from a prolongation-collocation approach: In addition to the discrete Euler-Lagrange equations this method accounts for the system's equation of motion in strong form,…”

Section: Relation To Other Methodsmentioning

confidence: 99%

“…Recently, Leok and Shingel have proposed a variational integrator based on Hermite finite elements in time. Their formulation is derived from a prolongation‐collocation approach: In addition to the discrete Euler‐Lagrange equations this method accounts for the system's equation of motion in strong form,

m {\overset{x}{true}}^{t} false(t_{•} false) + f [x^{t} (t_{•})] = 0, • \in {n, n + 1} .

For cubic Hermite shape functions — as they are used in our schemes — the velocities,

{\hat{boldv}}_{n}

and

{\hat{boldv}}_{n + 1}

, are computed from Eq.…”

Section: Temporal Discretizationmentioning

confidence: 99%

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C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

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Section: Relation To Other Methodsmentioning

confidence: 99%

m {\overset{x}{true}}^{t} false(t_{•} false) + f [x^{t} (t_{•})] = 0, • \in {n, n + 1} .

For cubic Hermite shape functions — as they are used in our schemes — the velocities,

{\hat{boldv}}_{n}

and

{\hat{boldv}}_{n + 1}

, are computed from Eq.…”

Section: Temporal Discretizationmentioning

confidence: 99%

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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“…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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“…It can be shown that the midpoint approximation is a second order method with an error of the order O(h 2+1 ), see Marsden and West (2001, p. 402). The use of more sophisticated approximation schemes than (2.4) can however provide approximations of the exact flow of arbitrarily high order (see, e.g., Hairer, Lubich, & Wanner, 2005;Leokand & Shingel, 2011).…”

Section: The Variational Integrators Approach To Discretizationmentioning

confidence: 99%

A variational integrators approach to second order modeling and identification of linear mechanical systems

2014

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Prolongation-collocation variational integrators

Cited by 24 publications

References 24 publications

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

A Brief Introduction to Variational Integrators

A variational integrators approach to second order modeling and identification of linear mechanical systems

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Prolongation-collocation variational integrators

Cited by 24 publications

References 24 publications

C1‐continuous space‐time discretization based on Hamilton's law of varying action

C1‐continuous space‐time discretization based on Hamilton's law of varying action

A Brief Introduction to Variational Integrators

A variational integrators approach to second order modeling and identification of linear mechanical systems

Contact Info

Product

Resources

About

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action