New way to construct high order Hamiltonian variational integrators

doi:10.1007/s10483-016-2116-8

Applied Mathematics and Mechanics (English Edition) ›› 2016, Vol. 37 ›› Issue (8): 1041-1052.doi: https://doi.org/10.1007/s10483-016-2116-8

New way to construct high order Hamiltonian variational integrators

Minghui FU¹, Kelang LU^1,2, Weihua LI³, S. V. SHESHENIN⁴

1. School of Engineering, Sun Yat-sen University, Guangzhou 510275, China;
2. Guangdong Provincial Academy of Building Research Group Company Limited, Guangzhou 510500, China;
3. College of Electromechanical Engineering, Guangdong Polytechnic Normal University, Guangzhou 510635, China;
4. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119992, Russia

收稿日期:2015-10-29 修回日期:2016-04-14 出版日期:2016-08-01 发布日期:2016-08-01
通讯作者: Weihua LI E-mail:lwh927@163.com
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11172334 and 11202247) and the Fundamental Research Funds for the Central Universities (No. 2013390003161292)

New way to construct high order Hamiltonian variational integrators

Minghui FU¹, Kelang LU^1,2, Weihua LI³, S. V. SHESHENIN⁴

1. School of Engineering, Sun Yat-sen University, Guangzhou 510275, China;
2. Guangdong Provincial Academy of Building Research Group Company Limited, Guangzhou 510500, China;
3. College of Electromechanical Engineering, Guangdong Polytechnic Normal University, Guangzhou 510635, China;
4. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119992, Russia

Received:2015-10-29 Revised:2016-04-14 Online:2016-08-01 Published:2016-08-01
Contact: Weihua LI E-mail:lwh927@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11172334 and 11202247) and the Fundamental Research Funds for the Central Universities (No. 2013390003161292)

摘要/Abstract

摘要：

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 momentum 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: nonlinear dynamics, unconventional Hamilton's variational principle, symplectic algorithm, Hamiltonian system, variational integrator

Abstract:

Key words: unconventional Hamilton's variational principle, nonlinear dynamics, symplectic algorithm, Hamiltonian system, variational integrator

中图分类号:

Minghui FU, Kelang LU, Weihua LI, S. V. SHESHENIN. New way to construct high order Hamiltonian variational integrators[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(8): 1041-1052.

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New way to construct high order Hamiltonian variational integrators

New way to construct high order Hamiltonian variational integrators

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