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An alternative to the bathe algorithm

APPLIED MATHEMATICAL MODELLING 2019年 69卷 255-272页

作者： Li, Jinze Yu, Kaiping Harbin Inst Technol Dept Astronaut Sci & Mech 92 West Dazhi St Harbin 150001 Heilongjiang Peoples R China

This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the bathe algorithm for solving linear and nonlinear problems;(b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers;(c) The identical effective stiffness matrices can be obtained inside two sub-steps;(d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the bathe algorithm over the dissipative CH-alpha algorithm and the non-dissipative trapezoidal rule. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Sub-steps algorithm bathe algorithm Structural dynamics Implicit integration algorithms

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Enhanced studies on the composite sub-step algorithm for structural dynamics: The bathe-like algorithm

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APPLIED MATHEMATICAL MODELLING 2020年 80卷 33-64页

作者： Li, Jinze Li, Xiangyang Yu, Kaiping Harbin Inst Technol Dept Astronaut Sci & Mech 92 West Dazhi St Harbin 150001 Peoples R China

The bathe algorithm is superior to the trapezoidal rule in solving nonlinear problems involving large deformations and long-time durations. Generally, the parameter gamma = 2 - root 2 is highly recommended due to its optimal numerical properties. This paper further studies this implicit composite sub-step algorithm and thus presents a class of the bathe-like algorithm. It not only gives a novel family of composite algorithms whose numerical properties are the exactly same as the original bathe algorithm with gamma = 2 - root 2. but also provides the generalized alternative to the original bathe algorithm with any gamma. In this study, it has been shown that the bathe-like algorithm, including the original bathe algorithm, can reduce to two common single-step algorithms: the trapezoidal rule and the backward Euler formula. Besides, a new parameter called the algorithmic mode truncation factor is firstly defined to describe the numerical property of the bathe-like algorithm and it can estimate which modes to be damped out. Finally, numerical experiments are provided to show the superiority of the bathe-like algorithm over some existing methods. For example, the novel bathe-like algorithms are superior to the original bathe algorithm when solving the highly nonlinear pendulum. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Sub-step composite algorithm bathe algorithm Structural dynamics Implicit time integration algorithmic mode truncation factor

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A novel family of composite sub-step algorithms with desired numerical dissipations for structural dynamics

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ARCHIVE OF APPLIED MECHANICS 2020年第4期90卷 737-772页

作者： Li, Jinze Yu, Kaiping Harbin Inst Technol Dept Astronaut Sci & Mech 92 West Dazhi St Harbin 150001 Peoples R China

A novel family of composite sub-step algorithms with controllable numerical dissipations is proposed in this paper to obtain reliable numerical responses in structural dynamics. The new scheme is a self-starting, unconditionally stable and second-order-accurate two-sub-step algorithm with the same computational cost as the bathe algorithm. The new algorithm can control continuously numerical dissipations in the high-frequency range in an intuitive way, and the ability of numerical dissipations can range from the non-dissipative trapezoidal rule to the asymptotic annihilating bathe algorithm. Besides, the new algorithm only involves one free parameter and always achieves the identical effective stiffness matrices inside two sub-steps, which is not always achieved in three bathe-type algorithms, to reduce the computational cost in the analysis of linear systems. Some numerical examples are given to show the superiority of the new algorithm over the bathe algorithm and the CH-alpha algorithm.

关键词： Sub-step algorithm bathe algorithm Implicit integration algorithm Controllable numerical dissipation

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A second-order accurate three sub-step composite algorithm for structural dynamics

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APPLIED MATHEMATICAL MODELLING 2020年第Part2期77卷 1391-1412页

作者： Li, Jinze Yu, Kaiping He, Haonan Harbin Inst Technol Dept Astronaut Sci & Mech 92 West Dazhi St Harbin 150001 Heilongjiang Peoples R China

In this paper, a novel three sub-step composite algorithm with desired numerical properties is developed. The proposed method is a self-starting, unconditionally stable and second-order accurate implicit algorithm without overshoot. Particularly, the second-order accuracy in time is achieved in its final form, but it is not required in each sub-step. Its unique algorithmic parameter is analyzed to achieve the unconditional stability and it shares the identical effective stiffness matrix inside three sub-steps to save the computational cost in linear analyses. The same as the bathe algorithm, the proposed algorithm is always L-stable, meaning that the spurious high-frequency modes can be effectively eliminated. Three numerical examples are simulated to illustrate the superiority of the proposed algorithm over some existing implicit algorithms. The first numerical simulation, solving a linear single-degree-of-freedom system, shows less period elongation errors and the second-order accuracy of the present scheme. The second one, a clamped-free bar excited by the end load, shows the ability of effectively damping out the unexpected high frequency modes. The last example solves the nonlinear mass-spring system with variable degree-of-freedoms and illustrates that the composite sub-step algorithm can save more computational cost than the traditional implicit algorithm when the integration step size is selected appropriately. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Three sub-step algorithm bathe algorithm Structural dynamics Implicit integration algorithms

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A novel family of controllably dissipative composite integration algorithms for structural dynamic analysis

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NONLINEAR DYNAMICS 2019年第4期96卷 2475-2507页

作者： Li, Jinze Yu, Kaiping Li, Xiangyang Harbin Inst Technol Dept Astronaut Sci & Mech 92 West Dazhi St Harbin 150001 Heilongjiang Peoples R China

In this paper, a new family of controllably dissipative composite algorithms is developed to obtain reliable numerical response of structural dynamic problems. The proposed algorithm is a self-starting, unconditionally stable and second-order accurate three sub-step composite algorithm. The new method includes two optimal sub-families of algorithms, both of which can control numerical dissipations in the high-frequency range by an intuitive way, and their numerical dissipations can range from the non-dissipative case to the asymptotic annihilating case. Besides, they actually involve only one free parameter and always share the identical effective stiffness matrices inside three sub-step to save the computational cost, which does not hold in some existing sub-step algorithms. Some numerical examples are given to show the superiority of the new algorithm with respect to controllable numerical dissipations and the ability of capturing the free-play nonlinearity.

关键词： Composite algorithm bathe algorithm Structural dynamics Three sub-step algorithm Controllable numerical dissipations

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