Variational integrators for electric circuits

Ober‐Blöbaum, Sina; Tao, Molei; Owhadi, Houman

doi:10.1002/pamm.201110380

Cited by 2 publications

(2 citation statements)

References 28 publications

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“…Other applications of variational integration can be reviewed, e.g., in (Kharevych et al 2006;Kraus 2013;Ober-Blöbauma et al 2013;Stern and Grinspun 2009). …”

Section: Variational Integrationmentioning

confidence: 98%

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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“…Other applications of variational integration can be reviewed, e.g., in (Kharevych et al 2006;Kraus 2013;Ober-Blöbauma et al 2013;Stern and Grinspun 2009). …”

Section: Variational Integrationmentioning

confidence: 98%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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“…Variational integrators are automatically symplectic and momentum preserving. They have been developed since the 1960's in optimal control theory and the 1970's in mechanics ( [15][16][17][18][19][20][21][22], see [13] for a more complete historical overview). Note that variational integrators can be extended to solve efficiently non-variational problems by embeeding the latter into a larger Lagrangian system [23,24].…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Variational integrators for electric circuits

Cited by 2 publications

References 28 publications

A Brief Introduction to Variational Integrators

A Brief Introduction to Variational Integrators

A review of some geometric integrators

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