AN UNCONDITIONALLY STABLE DYNAMICAL INTEGRATION ALGORITHM BASED ON HAMEL’S FORMALISM

Gu Wei; Liu Cheng; An Zhipeng; Shi Donghua

doi:10.6052/0459-1879-22-131

Volume 54 Issue 9

Sep. 2022

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Chinese Journal of Theoretical and Applied Mechanics > 2022 > 54(9): 2577-2587. > DOI: 10.6052/0459-1879-22-131 CSTR: 32045.14.0459-1879-22-131

Gu Wei, Liu Cheng, An Zhipeng, Shi Donghua. An unconditionally stable dynamical integration algorithm based on Hamel’s formalism. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 2577-2587. DOI: 10.6052/0459-1879-22-131

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AN UNCONDITIONALLY STABLE DYNAMICAL INTEGRATION ALGORITHM BASED ON HAMEL’S FORMALISM

Gu Wei^*,
Liu Cheng^{†, **},
An Zhipeng^††,
Shi Donghua^{***, †††}

*.
School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China
†.
CAEP Software Center for High Performance Numerical Simulation, Beijing 100088, China
**.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
††.
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
***.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
†††.
Beijing Key Laboratory on MCAACI, Beijing 100081, China

Funds: The project was supported by the (12345678)and (9876543)

Received Date: March 28, 2022
Accepted Date: May 26, 2022
Available Online: May 27, 2022

Graphical Abstract

Abstract

Abstract

Time integration algorithm is a key issue in solving dynamical system. An unconditionally stable Hamel generalized α method is proposed to solve the instability issue arising in the time integration of dynamic equations and to eliminate the pseudo high order harmonics incurred by the spatial discretization of finite element simultaneously. Therefore, the development of numerical integration algorithm to solve the above-mentioned problems has important theoretical and application value. The algorithm proposed in this paper is developed based on the moving frame method and Hamel’s field variational integrators along with the strategy to construct an unconditionally stable Hamel generalized α method. It is shown that a new numerical formalism with higher accuracy can be derived under the same framework of the unconditional stable algorithm established through a special variational formalism and variational integrators. The above-mentioned formalism can be extended from general linear space to Lie group by utilizing the moving frame method and the Lie group formalism of the Hamel generalized α method has been obtained. Both the convergence and stability of the algorithm are discussed, and some numerical examples are presented to verify the conclusion. It is demonstrated by the theoretical analysis that the Hamel generalized α method proposed in the paper is unconditionally stable, second-order accurate and can quickly filter out pseudo high-frequency harmonics. Both conventional and proposed methods have been applied to numerical examples respectively. Comparisons between results of numerical examples show that the aforementioned advantages of the proposed method in terms of accuracy, dissipation and stability are tested and verified. At the same time, it can be developed that new numerical integration algorithms with even higher order accuracy. The scheme can also be proposed, which is suitable for both general linear space and Lie group space. A new way for constructing variational integrators is also obtained in this paper.
- generalized α method,
- Hamel’s field variational integrators,
- Hamel-generalized α method,
- Lie group,
- moving frame

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References (41)

References

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AN UNCONDITIONALLY STABLE DYNAMICAL INTEGRATION ALGORITHM BASED ON HAMEL’S FORMALISM