On the optimization of n-sub-step composite time integration methods

AbstractA family of n-sub-step composite time integration methods, which employs the trapezoidal rule in the first $$n-1$$ n - 1 sub-steps and a general formula in the last one, is discussed in this paper. A universal approach to optimize the parameters is provided for any cases of $$n\ge 2$$ n ≥ 2 , and two optimal sub-families of the method are given for different purposes. From linear analysis, the first sub-family can achieve nth-order accuracy and unconditional stability with controllable algorithmic dissipation, so it is recommended for high-accuracy purposes. The second sub-family has second-order accuracy, unconditional stability with controllable algorithmic dissipation, and it is designed for heuristic energy-conserving purposes, by preserving as much low-frequency content as possible. Finally, some illustrative examples are solved to check the performance in linear and nonlinear systems.

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Comparisons of Structure-Dependent Explicit Methods for Time Integration

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455414500552 ◽

2015 ◽

Vol 15 (03) ◽

pp. 1450055 ◽

Cited By ~ 11

Author(s):

Shuenn-Yih Chang

Keyword(s):

Time Integration ◽

Low Frequency ◽

Unconditional Stability ◽

Explicit Methods ◽

Explicit Method ◽

Computationally Efficient ◽

Total Response ◽

Time Step ◽

Order Accuracy ◽

Practical Applications

Chang explicit method (CEM)1,2 and CR explicit method3 (CRM) are two structure-dependent explicit methods that have been successfully developed for structural dynamics. The most important property of both integration methods is that they involve no nonlinear iterations in addition to unconditional stability and second-order accuracy. Thus, they are very computationally efficient for solving inertial problems, where the total response is dominated by low frequency modes. However, an unusual overshooting behavior for CR explicit method is identified herein and thus its practical applications might be largely limited although its velocity computing for each time step is much easier than for the CEM.

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A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method

Journal of Applied Mechanics ◽

10.1115/1.2900803 ◽

1993 ◽

Vol 60 (2) ◽

pp. 371-375 ◽

Cited By ~ 1277

Author(s):

J. Chung ◽

G. M. Hulbert

Keyword(s):

Structural Dynamics ◽

Time Integration ◽

Low Frequency ◽

Numerical Dissipation ◽

Integration Algorithm ◽

Integration Methods ◽

New Family ◽

Time Integration Methods ◽

Improved Performance ◽

Dynamics Problems

A new family of time integration algorithms is presented for solving structural dynamics problems. The new method, denoted as the generalized-α method, possesses numerical dissipation that can be controlled by the user. In particular, it is shown that the generalized-α method achieves high-frequency dissipation while minimizing unwanted low-frequency dissipation. Comparisons are given of the generalized-α method with other numerically dissipative time integration methods; these results highlight the improved performance of the new algorithm. The new algorithm can be easily implemented into programs that already include the Newmark and Hilber-Hughes-Taylor-α time integration methods.

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One-Parameter Controlled Non-Dissipative Unconditionally Stable Explicit Structure-Dependent Integration Methods with No Overshoot

Applied Sciences ◽

10.3390/app112412109 ◽

2021 ◽

Vol 11 (24) ◽

pp. 12109

Author(s):

Veerarajan Selvakumar ◽

Shuenn-Yih Chang

Keyword(s):

Time Integration ◽

Unconditional Stability ◽

Second Order ◽

Numerical Dissipation ◽

Implicit Methods ◽

Time Step ◽

Order Accuracy ◽

Structural Dynamic ◽

Integration Methods ◽

New Family

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

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Time Integration Method With Structure-Dependent Difference Equations

Volume 8: Seismic Engineering ◽

10.1115/pvp2013-97044 ◽

2013 ◽

Author(s):

Shuenn-Yih Chang ◽

Chiu-Li Huang

Keyword(s):

High Frequency ◽

Integration Method ◽

Time Integration ◽

Low Frequency ◽

Unconditional Stability ◽

Total Response ◽

Time Step ◽

Order Accuracy ◽

Time Integration Method ◽

Frequency Responses

A novel family of structure-dependent integration method is proposed for time integration. This family method can have the possibility of unconditional stability, second-order accuracy and the explicitness of each time step. Since it can integrate the most important advantage of an implicit method, unconditional stability, and that of an explicit method, the explicitness of each time step, a lot of computational efforts can be saved in solving an inertial type problem, where the total response is dominated by low frequency modes and high frequency responses are of no interest.

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