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An energy‐conserving and decaying time integration method for general nonlinear dynamics
Author(s) -
Zhang H. M.,
Xing Y. F.,
Ji Y.
Publication year - 2019
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6251
Subject(s) - gaussian quadrature , nonlinear system , discretization , quadrature (astronomy) , dissipation , mathematics , numerical integration , finite element method , energy balance , internal energy , mathematical analysis , nyström method , mathematical optimization , integral equation , physics , quantum mechanics , optics , thermodynamics
Summary An energy‐conserving and decaying time integration method is derived from the discretized energy balance equation in this paper. Based on the Gauss‐Legendre quadrature rule, the new method evaluates the mean internal force by collecting the internal forces of several quadrature points in the time interval. Compared with previous conserving methods, the new method avoids the reconstruction of finite element models as used in the energy‐momentum method, and the computation of energy functions as used in the constraint energy method, so it is simple and straightforward, and has no difficulty in implementation for dynamics with general nonlinear resilience including nonlinear damping force and internal force. By employing enough quadrature nodes, this method can preserve system energy accurately for conservative system, and then the unconditional stability is achieved automatically. In addition, the energy‐decaying method is developed by introducing the energy dissipation term to the energy balance equation, and two free parameters controlling the numerical dissipation are determined based on linear spectral analysis. Finally, the performance of the proposed method is checked on several examples and the results are compared with that of the trapezoidal rule.