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

Li, Jinze; Yu, Kaiping; Zhao, Rui; Fang, Yong

doi:10.1002/nme.7291

Cited by 4 publications

(1 citation statement)

References 41 publications

(367 reference statements)

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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%

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%

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

Self Cite

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Novel Implicit Integration Algorithms With Identical Second‐Order Accuracy and Flexible Dissipation Control for First‐Order Transient Problems

Li,

Cui,

Zhu

et al. 2024

Numerical Meth Engineering

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For the solutions of first‐order transient problems with optimal efficiency, conventional single‐step implicit methodologies, unaided by additional post‐processing techniques, encounter limitations in concurrently attaining identical second‐order accuracy and controllable numerical dissipation. Addressing this challenge, the present study contributes not only a comprehensive analytical framework for formulating implicit integration algorithms but also leverages the auxiliary variable and the composite sub‐step technique to propose two distinct types of implicit integration algorithms. Each type is characterized by self‐initiation, unconditional stability, identical second‐order accuracy, controllable numerical dissipation, and zero‐order overshoots. Recognizing that single‐step implicit methods can be conceptualized as composite single‐sub‐step ones, both algorithm types achieve identical effective stiffness matrices, thereby reducing the computational effort for solving linear problems. The utilization of auxiliary variables endows the proposed single‐step method with numerical attributes akin to established counterparts, while achieving identical second‐order accuracy. Conversely, the adoption of the composite sub‐step technique in the proposed two‐sub‐step methods surpasses the published algorithms by significantly reducing the error constants. This superiority persists when enforcing the same sub‐step sizes and sub‐step dissipation levels. Numerical simulations affirm that the proposed methods consistently outperform existing alternatives without incurring additional computational expenses.

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

Li,

et al. 2024

Arch Appl Mech

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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

Cited by 4 publications

References 41 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

Novel Implicit Integration Algorithms With Identical Second‐Order Accuracy and Flexible Dissipation Control for First‐Order Transient Problems

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

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