Lee-Variational Integrator

2010 ◽  
pp. 581-616
Author(s):  
Kang Feng ◽  
Mengzhao Qin
Keyword(s):  
Variational Integrator

Discrete-time modeling and control of a boost converter by means of a variational integrator and sliding modes

2014 ◽  
Vol 351 (1) ◽  
pp. 315-339 ◽  
Author(s):  
Jorge Rivera ◽  
Florentino Chavira ◽  
Alexander Loukianov ◽  
Susana Ortega ◽  
Juan J. Raygoza
Keyword(s):  
Discrete Time ◽  
Boost Converter ◽  
Sliding Modes ◽  
Modeling And Control ◽  
Variational Integrator ◽  
Time Modeling ◽  
And Control ◽  
Discrete Time Modeling

scholarly journals Variational integrator for the rotating shallow‐water equations on the sphere

2019 ◽  
Vol 145 (720) ◽  
pp. 1070-1088 ◽  
Author(s):  
Rüdiger Brecht ◽  
Werner Bauer ◽  
Alexander Bihlo ◽  
François Gay‐Balmaz ◽  
Scott MacLachlan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Variational Integrator ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

scholarly journals Multisymplectic Lie group variational integrator for a geometrically exact beam in

2014 ◽  
Vol 19 (10) ◽  
pp. 3492-3512 ◽  
Author(s):  
François Demoures ◽  
François Gay-Balmaz ◽  
Marin Kobilarov ◽  
Tudor S. Ratiu
Keyword(s):  
Lie Group ◽  
Variational Integrator ◽  
Geometrically Exact Beam

scholarly journals On Converting Any One-Step Method to a Variational Integrator of the Same Order

2009 ◽  
Author(s):  
George W. Patrick ◽  
Charles Cuell ◽  
Raymond J. Spiteri ◽  
William Zhang
Keyword(s):  
Molecular Dynamics ◽  
Integration Method ◽  
Variational Integrators ◽  
Step Method ◽  
Flexible Approach ◽  
Variational Integrator ◽  
One Step ◽  
The One

In the formalism of constrained mechanics, such as that which underlies the SHAKE and RATTLE methods of molecular dynamics, we present an algorithm to convert any one-step integration method to a variational integrator of the same order. The one-step method is arbitrary, and the conversion can be automated, resulting in a powerful and flexible approach to the generation of novel variational integrators.

scholarly journals Fully Kinetic Simulation of Ion-Temperature-Gradient Instability in Tokamaks

2018 ◽  
Author(s):  
Youjun Hu ◽  
Matthew T. Miecnikowski ◽  
Yang Chen ◽  
Scott E. Parker
Keyword(s):  
Temperature Gradient ◽  
Time Scale ◽  
Low Frequency ◽  
Magnetic Configuration ◽  
Short Time Scale ◽  
Ion Temperature ◽  
Ion Temperature Gradient ◽  
Cyclotron Motion ◽  
Drift Motion ◽  
Variational Integrator

The feasibility of using full ion kinetics, instead of gyrokinetics, in simulating low-frequency Ion-Temperature-Gradient (ITG) instabilities in tokamaks has recently been demonstrated by Sturdevant et al. [Physics of Plasmas 24, 081207 (2017)]. In that work, a variational integrator was developed to integrate the full orbits of ions in toroidal geometry, which proved to be accurate in capturing both the short-time scale cyclotron motion and long time scale drift motion. The present work extends that work in three aspects. First, we implement a relatively simple full orbit integrator, the Boris integrator, and demonstrate that the accuracy of this integrator is also sufficient for simulation of ITG instabilities. Second, the equilibrium magnetic configuration is extended to general toroidal configuration specified numerically, enabling simulation of realistic equilibria reconstructed from tokamak experiments. Third, we extend that work to the nonlinear regime and investigate the nonlinear saturation of ITG instabilities. To verify the new numerical implementation of the orbit integrator and magnetic configuration, the linear electrostatic ITG frequency and growth rate are compared with those given in Sturdevant's work and good agreement is found.

scholarly journals Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Xiaojia Xiang ◽  
Lizhen Wu ◽  
Lincheng Shen ◽  
Jie Li
Keyword(s):  
Rigid Body ◽  
Numerical Method ◽  
Lie Algebra ◽  
Collocation Method ◽  
Attitude Control ◽  
Variational Integrator ◽  
Body Attitude ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral ◽  
Configuration Equation

The collocation method is extended to the special orthogonal group SO(3) with application to optimal attitude control (OAC) of a rigid body. A left-invariant rigid-body attitude dynamical model on SO(3) is established. For the left invariance of the attitude configuration equation in body-fixed frame, a geometrically exact numerical method on SO(3), referred to as the geometric collocation method, is proposed by deriving the equivalent Lie algebra equation inso(3)of the left-invariant configuration equation. When compared with the general Gauss pseudo-spectral method, the explicit RKMK, and Lie group variational integrator having the same order and stepsize in numerical tests for evolving a free-floating rigid-body attitude dynamics, the proposed method is higher in accuracy, time performance, and structural conservativeness. In addition, the numerical method is applied to solve a constrained OAC problem on SO(3). The optimal control problem is transcribed into a nonlinear programming problem, in which the equivalent Lie algebra equation is being considered as the defect constraints instead of the configuration equation. The transcription method is coordinate-free and does not need chart switching or special handling of singularities. More importantly, with the numerical advantage of the geometric collocation method, the proposed OAC method may generate satisfying convergence rate.

scholarly journals Variational Integrators for Structure-Preserving Filtering

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Jarvis Schultz ◽  
Kathrin Flaßkamp ◽  
Todd D. Murphey
Keyword(s):  
Kalman Filter ◽  
Control Systems ◽  
Discrete Time ◽  
Variational Integrators ◽  
Digital Control System ◽  
Variational Integrator ◽  
Energy Behavior ◽  
Term Energy ◽  
Estimation And Filtering

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discrete-time approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, and constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefits from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

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