Higher-order explicit time integration methods for numerical analyses of structural dynamics

Kim, Wooram

doi:10.1590/1679-78255609

Cited by 17 publications

(8 citation statements)

References 30 publications

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“…Inserting Equation (24) into Equation (B.6) and collecting all terms with respect to Ω, yield e 𝜉 = 𝜒Ω 3 + O ( 𝜒Ω 7 ) , Ω ≤ Ω b , (B.7)…”

Section: Discussionmentioning

confidence: 99%

“…Here a two-degree-of-freedom elastic spring-pendulum problem shown in Figure 23 is tested. 24 This problem is described as { mr (2) − m (L 0 + r) ( 𝜃 (1) ) 2 − mg cos 𝜃 + k s r = 0 m𝜃 (2) + m(2r (1) 𝜃 (1) +g sin 𝜃)…”

Section: 32mentioning

confidence: 99%

“…Here a two‐degree‐of‐freedom elastic spring‐pendulum problem shown in Figure 23 is tested 24 . This problem is described as

$$ \left\{\begin{array}{c}m{r}^{(2)}-m\left({L}_0+r\right){\left({\theta}^{(1)}\right)}^2- mg\ \cos \theta +{k}_sr=0\\ {}{m\theta}^{(2)}+\frac{m\left(2{r}^{(1)}{\theta}^{(1)}+g\sin \theta \right)}{L_0+r}=0\end{array}\right.

…”

Section: Numerical Examplesmentioning

confidence: 99%

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A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

Wan

Liu

et al. 2023

Numerical Meth Engineering

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A novel explicit time integration method is formulated with sub‐step strategy and Cubic B‐spline interpolation method. Theoretical and numerical analysis are conducted to obtain optimized algorithm properties including algorithm accuracy, spectral stability and numerical dissipation/dispersion. A demonstrative dispersion analysis for wave propagation is presented to acquire optimal algorithm parameter value for finite element analysis of wave propagation problems. Numerical tests demonstrate that new method show desirable algorithm accuracy and convergence for dynamic problems. Especially, for highly nonlinear problems, the new method can provide very accurate and stable solutions.

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“…Inserting Equation (24) into Equation (B.6) and collecting all terms with respect to Ω, yield e 𝜉 = 𝜒Ω 3 + O ( 𝜒Ω 7 ) , Ω ≤ Ω b , (B.7)…”

Section: Discussionmentioning

confidence: 99%

Section: 32mentioning

confidence: 99%

“…Here a two‐degree‐of‐freedom elastic spring‐pendulum problem shown in Figure 23 is tested 24 . This problem is described as

$$ \left\{\begin{array}{c}m{r}^{(2)}-m\left({L}_0+r\right){\left({\theta}^{(1)}\right)}^2- mg\ \cos \theta +{k}_sr=0\\ {}{m\theta}^{(2)}+\frac{m\left(2{r}^{(1)}{\theta}^{(1)}+g\sin \theta \right)}{L_0+r}=0\end{array}\right.

…”

Section: Numerical Examplesmentioning

confidence: 99%

See 1 more Smart Citation

A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

Wan

Liu

et al. 2023

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…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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“…Since the computational aspect of structural dynamics is heavily dependent on the characteristic of a given problem, it is very difficult to choose a perfect scheme that works nicely for all cases. For example, the long-term analysis of highly nonlinear dynamic problems may require a non-dissipative higher-order accurate time integration scheme to conserve the total energy of the system while minimizing the period error [14,20,21,22,23]. In this case, higher-order accurate explicit time integration schemes may become more efficient than implicit schemes, because matrix factorizations and iterative nonlinear solution finding procedures can be avoided if the mass matrix is diagonal in explicit schemes.…”

Section: Introductionmentioning

confidence: 99%

A critical assessment of two-stage composite time integration schemes with a unified set of time approximations

Kim

Reddy

2021

Lat. Am. j. solids struct.

Self Cite

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A simple unified set of displacement and velocity approximations is presented to express various two-stage composite time integration schemes. Based on the unified approximations, two novel sets of optimized parameters are newly proposed for the implicit composite schemes to enhance the capability of conserving total energy. Two special cases of the unified approximations are also considered to overcome some shortcomings of the existing schemes. Besides, the newly proposed unified set of approximations can include many of the existing composite time integration schemes. To be specific, both implicit and explicit composite schemes can be expressed by using the unified set of approximations. Thus, both implicit and explicit types of composite schemes can be selected from the unified set by simply changing algorithmic parameters. To demonstrate advantageous features of the proposed unified set of approximations, various numerical examples are solved, and results are analyzed.

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Higher-order explicit time integration methods for numerical analyses of structural dynamics

Cited by 17 publications

References 30 publications

A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

Improved method for analysing the dynamic response of gear transmission systems

A critical assessment of two-stage composite time integration schemes with a unified set of time approximations

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