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Chebyshev spectral variational integrator and applications

Applied Mathematics and Mechanics(English Edition) 2020年第5期41卷 753-768页

作者： Zhonggui YI Baozeng YUE Mingle DENG School of Aerospace Engineering Beijing Institute of TechnologyBeijing 100081China China Academy of Space Technology Beijing 100094China

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.

关键词： geometric numerical method spectral method variational integrator Clenshaw-Curtis quadrature rule barycentric Lagrange interpolation orbital propagation

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variational integratorS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2003年第2期21卷 135-144页

作者： AajuanSun MengzheoQin InstituteofMathematics AcademyofMathematicsandSystemsSciencesChinsesAcademyofSciencesBeijing100080China StateKeyLaboratoryofScientific/EngineeringComputing InstituteofComputationalMathematicsandScientific/EngineeringComputingAcademyofMathematicsandSystemSciencesChineseAcademyofSciencesP.O.Box2719Beijing100080China

We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.

关键词： variational integrator Symplectic mapping

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New way to construct high order Hamiltonian variational integrators

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Applied Mathematics and Mechanics(English Edition) 2016年第8期37卷 1041-1052页

作者： Minghui FU Kelang LU Weihua LI S. V. SHESHENIN School of Engineering Sun Yat-sen University Guangzhou 510275 China Guangdong Provincial Academy of Building Research Group Company Limited Guangzhou 510500 China College of Electromechanical Engineering Guangdong Polytechnic Normal University Guangzhou 510635 China Faculty of Mechanics and Mathematics Lomonosov Moscow State University Moscow 119992 Russia

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

关键词： Hamiltonian system variational integrator symplectic algorithm unconventional Hamilton＇s variational principle nonlinear dynamics

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Hamilton–Pontryagin spectral-collocation methods for the orbit propagation

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Acta Mechanica Sinica 2021年第11期37卷 1696-1713,I0003页

作者： Zhonggui Yi Baozeng Yue Mingle Deng School of Aerospace Engineering Beijing Institute of TechnologyBeijing100081China China Academy of Space Technology Beijing100094China

According to the discrete Hamilton–Pontryagin variational principle,we construct a class of variational integrators in the real vector spaces and extend to the Lie groups for the left-trivialized Lagrangian mechanical systems by employing the spectral-collocation method to discretize the corresponding Lagrangian and kinematic constraints.The constructed framework can be transformed easily to the well-known symplectic partitioned Runge–Kutta methods and the higher order symplectic partitioned Lie Group methods by choosing same interpolation nodes and quadrature points.Two numerical experiments about the orbit propagation of Kepler two-body system and the rigid-body flow propagation of a free rigid body are conducted,respectively.The simulating results reveal that the constructed update schemes can possess simultaneously the excellent exponent convergence rates of spectral methods and the attractive long-term structure-preserving properties of geometric numerical algorithms.

关键词： Hamilton-Pontryagin principle Spectral-collocation method Lie groups variational integrator Orbit propagation

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General techniques for constructing variational integrators

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Frontiers of Mathematics in China 2012年第2期7卷 273-303页

作者： Melvin LEOK Tatiana SHINGEL Department of Mathematics University of California San Diego La Jolla CA 92093 USA

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi＇s solution of the Hamilton- Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators We also illustrate these systematic methods for constructing variational integrators with numerical examples.

关键词： Geometric numerical integration geometric mechanics symplectic integrator variational integrator Lagrangian mechanics

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Implementation Details for the Phase Field Approaches to Fracture

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Journal of Shanghai Jiaotong university(Science) 2018年第1期23卷 166-174页

作者：沈泳星 MOLLAALI Mostafa 李毅环马维馨蒋家皓 State Key Laboratory of Metal Matrix Composites Shanghai Jiao Tong University University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University Department of Mechanical Engineering University of Michigan

Phase field description of fracture is a very promising approach for simulating crack initiation, propagation, merging and branching. This method greatly reduces the implementation complexity, compared with discrete descriptions of cracks. In this work, we provide an overview of phase field models for quasistatic and dynamic cases. Afterward, we present useful vectors and matrices for the implementation of this method in two and three dimensions.

关键词： phase field approach to fracture variational fracture brittle fracture dynamic fracture variational integrator

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