An accurate predictor-corrector time integration method for structural dynamics

Rezaiee‐Pajand, Mohammad; Karimi-Rad, Mahdi

doi:10.1007/s13296-017-9014-9

Cited by 11 publications

(4 citation statements)

References 33 publications

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“…Effective and accurate numerical simulations for various structural dynamic problems have attracted increasing attention and there are various types of integration algorithms developed based on different strategies. In general, these methods can be divided into explicit [1][2][3][4][5][6][7] and implicit [8][9][10][11][12][13] schemes, and both have their own advantages and disadvantages. This article focuses mainly on the development of implicit algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…4𝛽 + 1)(𝛼 f − 1) − 2𝛽𝛼 f ] 𝜔 2 𝛥t 2 + 4(2𝛾 − 1 + (1 − 3𝛾)𝛼 f )𝜉𝜔𝛥t + 4 − 6𝛼 m 2𝛽(𝛼 f − 1)𝜔 2 𝛥t 2 + 4𝛾(𝛼 f − 1)𝜉𝜔𝛥t + 2(𝛼 m − 1) , 2𝛾 − 2𝛽 − 1 + 2(3𝛽 − 2𝛾)𝛼 f ] 𝜔 2 𝛥t 2 + 4(1 − 𝛾 + (3𝛾 − 2)𝛼 f )𝜉𝜔𝛥t + 2 − 6𝛼 m 2𝛽(𝛼 f − 1)𝜔 2 𝛥t 2 + 4𝛾(𝛼 f − 1)𝜉𝜔𝛥t + 2(𝛼 m − 1) , (D1c) A 0 = (2𝛾 − 2𝛽 − 1)𝛼 f 𝜔 2 𝛥t 2 + 4(1 − 𝛾)𝛼 f 𝜉𝜔𝛥t − 2𝛼 m 2𝛽(𝛼 f − 1)𝜔 2 𝛥t 2 + 4𝛾(𝛼 f − 1)𝜉𝜔𝛥t + 2(𝛼 m − 1) . (D1d)Substituting 𝜆 = exp[(𝜈 + 𝜏) ⋅ 𝛥t] into Equation (D1a) and expanding the result as the series form in terms of 𝛥t yield 3∑ j=0 A j 𝜆 j = 𝜔 2 + 2(𝜈 + 𝜏)𝜉𝜔 + (𝜈 + 𝜏) 2 1 2𝛾(𝛼 f − 1)𝜉𝜔 3 + (𝜈 + 𝜏)(4(𝛼 f − 1)𝛾𝜉 2 − 1 2 (𝛼 m − 1)(2𝛾 − 2𝛼 f + 3))𝜔 2 +(𝜈 + 𝜏) 2 (3 − (2𝛾 + 3)𝛼 m + 2(𝛾 + 𝛼 m − 1)𝛼 f )𝜉𝜔 + (𝜈 + 𝜏) 3 (𝛼 m − 2)(𝛼 m − 1) 𝛼 m ) 2𝛥t 𝛥t 4 + c 5 𝛥t 5 + O(𝛥t6 ).…”

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confidence: 99%

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On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

Lian

et al. 2023

Numerical Meth Engineering

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Hilber and Hughes gave several competitive demands on the implicit methods in 1978. However, there are no such integration methods in the traditional algorithm design. Via imposing subsidiary variables, this article proposes two novel implicit methods to splendidly achieve these competitive demands without increasing computational costs. The two novel methods are self‐starting, unconditionally stable, single‐solve, identically second‐order accurate, controllably dissipative, and non‐overshooting. The novel methods achieve the maximal dissipation of auxiliary variables in the high‐frequency limit, although such the design is not a must. The first novel method uses auxiliary velocity and acceleration variables, whereas the second one employs two auxiliary acceleration variables. After embedding desired numerical characteristics, the novel methods employ four eigenvalues of amplification matrices in the high‐frequency limit as user‐specified parameters to flexibly control numerical high‐frequency dissipation. To reduce the number of use‐specified parameters, this article gives some recommended sub‐families of algorithms, such as optimal low‐frequency dissipation schemes. The article further refines the error analysis to analytically calculate amplitude and phase errors for some implicit methods. The error analysis technique can be applied to almost all direct time integration methods. Comparisons of amplitude and phase errors highlight the superiority of the novel methods over the published algorithms. In addition, some classical measures, such as numerical damping ratios and relative period errors, are analyzed and compared. Numerical examples are solved to show the novel methods' superiority.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

Lian

et al. 2023

Numerical Meth Engineering

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“…Researchers have carried out a lot of research on the application of Runge–Kutta method to the characteristic analysis or response solution of mechanical dynamic system (Zhang and Deng, 2003; Negrut et al ., 2003; Zhang et al ., 2007; Rabiei, 2013; Yin, 2013; Rezaiee-Pajand and Karimi-Rad, 2017; Fang et al ., 2018; Kim, 2019; Tiwari and Pandey, 2020; Egger et al. , 2021), but the relevant research about the problem of calculation efficiency of dynamic system with periodic time-varying meshing stiffness is still rare.…”

Section: Introductionmentioning

confidence: 99%

Improved method for analysing the dynamic response of gear transmission systems

Liu

et al. 2022

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PurposeThe purpose of this paper is to propose an improved method which can shorten the calculation time and improve the calculation efficiency under the premise of ensuring the calculation accuracy for calculating the response of dynamic systems with periodic time-varying characteristics.Design/methodology/approachAn improved method is proposed based on Runge–Kutta method according to the composition characteristics of the state space matrix and the external load vector formed by the reduction of the dynamic equation of the periodic time-varying system. The recursive scheme of the holistic matrix of the system using the Runge–Kutta method is improved to be the sub-block matrix that is divided into the upper and lower parts to reduce the calculation steps and the occupied computer memory.FindingsThe calculation time consumption is reduced to a certain extent about 10–35% by changing the synthesis method of the time-varying matrix of the dynamics system, and the method proposed of paper consumes 43–75% less calculation time in total than the original Runge–Kutta method without affecting the calculation accuracy. When the ode45 command that implements the Runge–Kutta method in the MATLAB software used to solve the system dynamics equation include the time variable which cannot provide its specific analytic function form, so the time variable value corresponding to the solution time needs to be determined by the interpolation method, which causes the calculation efficiency of the ode45 command to be substantially reduced.Originality/valueThe proposed method can be applied to solve dynamic systems with periodic time-varying characteristics, and can consume less calculation time than the original Runge–Kutta method without affecting the calculation accuracy, especially the superiority of the improved method of this paper can be better demonstrated when the degree of freedom of the periodic time-varying dynamics system is greater.

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“…This also emphasizes the need to study the stability of these numerical integration methods in nonlinear structures problems. In Rezalee-Pajand and Karimi-Rad (2017), a new accurate predictor-corrector time integration method is proposed, which is shown superior to the centered differences method, and has second-order accuracy for structures with and without damping. Moreover, it is observed the importance of studying these methods in nonlinear beam problems, which are very common and can be the basic model of several complex systems.…”

Section: Introductionmentioning

confidence: 99%

Application of direct integration methods in the solution of a nonlinear beam problem

Rufato

Enriquez-Remigio

Morais

2019

25th International Congress of Mechanical Engineering

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This work applies different numerical methods involved in the solution of a nonlinear clamped beam problem. The methodology used in the discretization of the dynamic problem is based on the Finite Element Method (FEM), followed by mode superposition, where a localized nonlinearity is applied at the free end of the beam. The solution of the nonlinear problem is performed by five different integration methods. The solution code is implemented in FORTRAN language, validated with ANSYS and the dynamic response and the graphs are obtained with the help of MATLAB R software. The work shows the convergence of the implemented methods with various validation problems.

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An accurate predictor-corrector time integration method for structural dynamics

Cited by 11 publications

References 33 publications

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

Improved method for analysing the dynamic response of gear transmission systems

Application of direct integration methods in the solution of a nonlinear beam problem

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