An alternative to the Bathe algorithm

Li, Jinze; Yu, Kaiping

doi:10.1016/j.apm.2018.12.027

Cited by 31 publications

(11 citation statements)

References 28 publications

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“…The single-step direct integration algorithms often advance numerical solutions at the previous instant t n into those in the next instant t n+1 within each time increment until the final simulation time is reached. Analogously to the existing composite two-sub-step implicit schemes, 15,18,20,36 where 𝛾 1 and 𝛾 2 are the splitting ratios of sub-step size. The constructed three-sub-step implicit scheme is described in each sub-interval as follows.…”

Section: General Composite Three-sub-step Implicit Schemementioning

confidence: 99%

“…The three-sub-step implicit method (4) can achieve fourth-order accuracy when the conditions ( 9), (18), and ( 19) are fulfilled as…”

Section: 3mentioning

confidence: 99%

“…Apart from the better robustness than the trapezoidal rule for solving nonlinear problems, the composite two-sub-step Bathe method 11,12 successfully avoids drawbacks that single-step dissipative implicit algorithms suffer from. Later, many researchers proposed and analyzed some novel composite multi-sub-step implicit methods [14][15][16][17][18][19][20][21] to improve accuracy, stability, and dissipation control. The main attention of this study is paid to the composite three-sub-step implicit integration methods, and thus the following only reviews this type of integration algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Zhao

et al. 2023

Numerical Meth Engineering

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SummaryThis paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters ( and ) to control numerical dissipation imposed in the second and third sub‐steps. The parameter adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via . The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.

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Section: General Composite Three-sub-step Implicit Schemementioning

confidence: 99%

“…The three-sub-step implicit method (4) can achieve fourth-order accuracy when the conditions ( 9), (18), and ( 19) are fulfilled as…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Zhao

et al. 2023

Numerical Meth Engineering

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“…Li and Yu believe that data mining can not only bring huge competitive advantages to enterprises and institutions but also create huge economic benefits for the society. Under such temptation, many famous companies in the world have also entered the ranks of data mining, using data mining systems to develop software and tools related to the company's future development [5]. Cui et al argue that the composition of quality standards is largely based on the experience of manufacturers.…”

Section: Literature Reviewmentioning

confidence: 99%

Construction of Data Mining Analysis Model in English Teaching Based on Apriori Association Rule Algorithm

Wang

2022

Mathematical Problems in Engineering

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How to create an English data mining analysis model based on prior association strategy algorithm is learned. First, the basic principles of data mining and organizational strategy are investigated, and the cooperative strategy algorithm in data mining technology is studied. This document makes a comprehensive analysis of the classical Apriori algorithm and studies the extension of organizational strategy and the technique of deleting participation strategy after decision-making. Then, the application of Apriori algorithm in instruction management is analyzed. There is often a lot of information in command management. We need to collect the data, then sort, and review it. Finally, taking the two adults' classes, class (39) and class (40) as the research objects, of which class (39) has 41 people and class (40) has 43 people. The two classes were brought to the board of directors. The results showed that over the course of 4 months, the students scored better on the mixed tests in the lab than those in the control room. The average score of the subjects before the experiment was 17.6, the average score of the subjects after the experiment was 20.1, and the significant P value of the corresponding t-test was p ≤ 0.001 , less than 0.05, indicating significant differences between the two groups of data. After studying data using writing styles in class, students focused more on complex patterns, concepts, technical expressions, and line patterns. Dynamic assessment of student writing is an effort to improve student writing by using interactive assessment techniques.

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“…Later, the composite multi-sub-step implicit algorithms using the non-dissipative trapezoidal rule and more general multi-point backward difference scheme have been constructed and analyzed in the literature [19][20][21][22][23]. Besides, other integration algorithms without adopting the trapezoidal rule in the first sub-step have been successfully developed in [24][25][26][27] and they predict more accurate solutions for solving some dynamical problems. Two and three sub-steps have been widely used to construct the composite multi-sub-step algorithms since these sub-step schemes can be readily derived and analyzed.…”

Section: Introductionmentioning

confidence: 99%

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for second-order hyperbolic problems

Li¹,

Li²,

Yu³

et al. 2022

Preprint

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This paper develops an implicit family of sub-step integration algorithms, which firstly requires identical effective stiffness matrices and third-order consistency within each sub-step. Consequently, the trapezoidal rule has to be employed in the first sub-step and optimal spectral properties are naturally embedded into the proposed algorithms. The analysis reveals at the first time that the constructed s-sub-step implicit schemes with s ≤ 6 can reach sth-order accuracy when achieving dissipation control and unconditional stability simultaneously. Hence, only four cost-optimal high-order implicit algorithms corresponding to three, four, five, and six sub-steps are developed. Unlike some published high-order algorithms, four novel methods do not suffer from order reduction for solving forced vibrations. Moreover, the novel methods overcome the defect that the authors' previous high-order algorithms require an additional solution to obtain accurate accelerations. Linear and nonlinear examples are solved to confirm the numerical performance and superiority of four novel high-order algorithms.Keywords implicit time integration • composite multi-sub-step • dissipation control • high-order accuracy • optimal spectral features

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An alternative to the Bathe algorithm

Cited by 31 publications

References 28 publications

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Construction of Data Mining Analysis Model in English Teaching Based on Apriori Association Rule Algorithm

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for second-order hyperbolic problems

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