Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 300 papers for “Variational integrator”

Simpson’s Variational Integrator for Systems with Quadratic Lagrangians

Juan Antonio Rojas-Quintero, François Dubois, José Guadalupe Cabrera-Díaz · Axioms · 2024

This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.

Cited by 2Open access

A variational integrator for the Discrete Element Method

David N. de Klerk, Thomas Shire, Zhiwei Gao et al. · Journal of Computational Physics · 2022

A novel implicit integration scheme for the Discrete Element Method (DEM) based on the variational integrator approach is presented. The numerical solver provides a fully dynamical description that, notably, reduces to an energy minimisation scheme in the quasi-static limit. A detailed derivation of the numerical method is presented for the Hookean contact model and tested against an established open source DEM package that uses the velocity-Verlet integration scheme. These tests compare results for a single collision, long-term stability and statistical quantities of ensembles of particles. Numerically, the proposed integration method demonstrates equivalent accuracy to the velocity-Verlet method.

Cited by 8Open access

Satellite Dynamics Simulator Development Using Lie Group Variational Integrator

Dae‐Young Lee, John Springmann, Sara Spangelo et al. · AIAA Modeling and Simulation Technologies Conference · 2011

Simulation technology is becoming increasingly crucial in the design and optimization of satellites due to the difficulties in testing and verifying system parameters on the ground. Computationally tractable and accurate methods are required in order to test satellite parameters in the complex and dynamic space environment. Although various satellite teams have developed simulation tools, many suffer from inaccurate numerical integrators, resulting in their simulations being of low fidelity for long duration simulations. This paper presents a MATLAB/Simulink-based simulator which includes high fidelity integration and modeling for accurate and relatively quick results. The simulator includes an energy-preserving variational integrator for both translational and rotational dynamics. A Lie Group Variational Integrator is used for the rotational dynamics, which enforces an orthogonality constraint for improved accuracy. This approach requires less computational time relative to other integration methods such as Runge-Kutta method for the same level of integration accuracy. The simulator includes perturbations to the orbital motion and attitude, including Earth oblateness, aerodynamic drag, solar pressure, gravity gradient, and residual dipole. The simulator also includes an advanced hysteresis model for improved modeling of magnetic attitude control systems. Simulation results are provided for a representative small satellite mission in low earth orbit with a passive magnetic stabilization control system. We compare the novel integration and hysteresis techniques to conventional simulators for long duration simulations for a realistic mission scenario.

Cited by 16

The Construction and Applications of Bayesian Variational Integrator Networks

Momoko OTAKE, Linyu Peng · Keisan Rikigaku Koenkai koen ronbunshu/Keisan Rikigaku Kouenkai kouen rombunshuu · 2022

In numerical analysis, the conservation of structure, i.e., whether the essential properties of physical phenomena can be reproduced numerically, is of great importance. On the other hand, neural networks that can learn conservation laws has recently received great attention. These models take important structure of physical problems into consideration, which are often ignored in traditional models that directly learn using data, consequently exhibiting long-term accurate prediction and interpretability. In the current study, we construct neural network models based on variational integrator, a typical kind of structure-preserving method. In addition, we introduce Bayesian framework to measure uncertainty of the model and evaluate the models quantitively. Illustrative examples are studied.

Cited by 0Open access

A Simple Variational Integrator for General Holonomic Mechanical Systems

Ulrich Mutze · 2010

For an arbitrary holonomic mechanical system an integrator is derived by applying an extended principle of stationary action to the manifold of those system paths which are parabolic with respect to the system of generalized coordinates under consideration. A modified variational derivative of the action integral is shown to agree with the force that the environment of the system exerts on it. This generalizes the characterization of free motion in terms of a vanishing variational derivative. As a numerical example, the method is applied to the damped harmonic oscillator. It is shown that the method follows accurately a change of the amplitude by a factor of 1024 within 20 oscillation periods when taking only 32 steps for one such period. The computed oscillation gets out of phase by an angle of 11.5 degrees over these 20 oscillations. 1

Cited by 0

A multisymplectic variational integrator for the nonlinear Schrödinger equation

Jing‐Bo Chen, Mengzhao Qin · Numerical Methods for Partial Differential Equations · 2002

Abstract The multisymplectic structure for the nonlinear Schrödinger equation is presented. Based on the multisymplectic structure, we derive a nine‐point variational integrator from the discrete variational principle and a six‐point multisymplectic integrator from the Preissman multisymplectic scheme. We show that the two integrators are essentially equivalent. Therefore, we call it a multisymplectic variational integrator. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 523–536, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10021

Cited by 16

Gauge properties of the guiding center variational symplectic integrator

H. Qin and W. Tang J. Squire, H. Qin, W. Tang · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 2012

Recently, variational symplectic algorithms have been developed for the long-time simulation of charged particles in magnetic fields1-3. As a direct consequence of their derivation from a discrete variational principle, these algorithms have very good long-time energy conservation, as well as exactly preserving discrete momenta. We present stability results for these algorithms, focusing on understanding how explicit variational integrators can be designed for this type of system. It is found that for explicit algorithms an instability arises because the discrete symplectic structure does not become the continuous structure in the t → 0 limit. We examine how a generalized gauge transformation can be used to put the Lagrangian in the "antisymmetric discretization gauge," in which the discrete symplectic structure has the correct form, thus eliminating the numerical instability. Finally, it is noted that the variational guiding center algorithms are not electromagnetically gauge invariant. By designing a model discrete Lagrangian, we show that the algorithms are approximately gauge invariant as long as A and are relatively smooth. A gauge invariant discrete Lagrangian is very important in a variational particle-in-cell algorithm where it ensures current continuity and preservation of Gauss's law4.

Cited by 1Open access

Variational integrator graph networks for learning energy-conserving dynamical systems

Shaan Desai, Marios Mattheakis, Stephen Roberts · Physical review. E · 2021

Recent advances show that neural networks embedded with physics-informed priors significantly outperform vanilla neural networks in learning and predicting the long-term dynamics of complex physical systems from noisy data. Despite this success, there has only been a limited study on how to optimally combine physics priors to improve predictive performance. To tackle this problem we unpack and generalize recent innovations into individual inductive bias segments. As such, we are able to systematically investigate all possible combinations of inductive biases of which existing methods are a natural subset. Using this framework we introduce variational integrator graph networks-a novel method that unifies the strengths of existing approaches by combining an energy constraint, high-order symplectic variational integrators, and graph neural networks. We demonstrate, across an extensive ablation, that the proposed unifying framework outperforms existing methods, for data-efficient learning and in predictive accuracy, across both single- and many-body problems studied in the recent literature. We empirically show that the improvements arise because high-order variational integrators combined with a potential energy constraint induce coupled learning of generalized position and momentum updates which can be formalized via the partitioned Runge-Kutta method.

Cited by 1Open access

Variational integrator graph networks for learning energy-conserving dynamical systems

Shaan Desai, Marios Mattheakis, Stephen Roberts · arXiv (Cornell University) · 2021

Recent advances show that neural networks embedded with physics-informed priors significantly outperform vanilla neural networks in learning and predicting the long-term dynamics of complex physical systems from noisy data. Despite this success, there has only been a limited study on how to optimally combine physics priors to improve predictive performance. To tackle this problem we unpack and generalize recent innovations into individual inductive bias segments. As such, we are able to systematically investigate all possible combinations of inductive biases of which existing methods are a natural subset. Using this framework we introduce variational integrator graph networks-a novel method that unifies the strengths of existing approaches by combining an energy constraint, high-order symplectic variational integrators, and graph neural networks. We demonstrate, across an extensive ablation, that the proposed unifying framework outperforms existing methods, for data-efficient learning and in predictive accuracy, across both single- and many-body problems studied in the recent literature. We empirically show that the improvements arise because high-order variational integrators combined with a potential energy constraint induce coupled learning of generalized position and momentum updates which can be formalized via the partitioned Runge-Kutta method.

Cited by 14Open access

Simpson variational integrator for nonlinear systems: a tutorial on the Lagrange top

Juan Antonio Rojas-Quintero, François Dubois, F Jourdan · ArXiv.org · 2025

This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using quadratic finite elements interpolation of the state and Simpson's quadrature, leading to discrete motion equations. The scheme is implicit, symplectic, and fourth-order accurate. The proposed integrator is compared with the implicit midpoint variational integrator on two examples of systems with inseparable Hamiltonians. First, the example of the nonlinear double pendulum illustrates how the method can be applied to multibody systems. The analytical solution of the Lagrange top is then used as a reference to analyze accuracy, convergence, and precision of the numerical method. A reduced Lagrange top system is also proposed and solved with a classical fourth-order method. Its solution is compared with the Simpson solution of the complete system, and the convergence order of the difference between both is consistent with the order of the classical method.

Cited by 0Open access

Variational integrator for fractional Pontryagin's systems. Existence of a discrete fractional Noether's theorem

Loïc Bourdin · arXiv (Cornell University) · 2012

Fractional Pontryagin's systems emerge in the study of a class of fractional optimal control problems but they are not resolvable in most cases. In this paper, we suggest a numerical approach for these fractional systems. Precisely, we construct a variational integrator allowing to preserve at the discrete level their intrinsic variational structure. The variational integrator obtained is then called shifted discrete fractional Pontryagin's system. We provide a solved fractional example in a certain sense. It allows us to test in this paper the convergence of the variational integrator constructed. Finally, we also provide a discrete fractional Noether's theorem giving the existence of an explicit computable discrete constant of motion for shifted discrete fractional Pontryagin's systems admitting a discrete symmetry.

Cited by 0Open access

Variational integrator for constrained mechanical systems with pulsed disturbances and optimal feedback control

Daniel Glaas, Sigrid Leyendecker · PAMM · 2017

Abstract An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we use it to control constrained dynamical systems (differential algebraic equations of Index 3) with pulsed disturbances. To describe their discrete dynamics, a constrained variational integrator [1] is used. Using a discrete version of the Lagrange‐d'Alembert principle yields a forced constrained discrete Euler‐Lagrange equation in a position‐momentum form that depends on the current and future time steps [2]. The desired optimal trajectory ( q opt , p opt ) and according control input u opt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input u R . Adding u opt and u R to the discrete Euler‐Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cited by 0