Time Integration Method With Structure-Dependent Difference Equations

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.

Comparisons of Structure-Dependent Explicit Methods for Time Integration

2015 ◽  
Vol 15 (03) ◽  
pp. 1450055 ◽  
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.

scholarly journals An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

2021 ◽  
Vol 11 (4) ◽  
pp. 1932
Author(s):  
Weixuan Wang ◽  
Qinyan Xing ◽  
Qinghao Yang
Keyword(s):  
High Frequency ◽  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Weak Form ◽  
High Accuracy ◽  
Numerical Dissipation ◽  
Frequency Range ◽  
Time Integration Method ◽  
Numerical Damping

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

A Precise and Stiffly Stable Time Integration Method for Vibration Equations

2001 ◽  
Author(s):  
Takeshi Fujikawa ◽  
Etsujiro Imanishi
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Quadratic Function ◽  
Time Step ◽  
Integration Algorithm ◽  
Time Integration Method ◽  
Time Integration Algorithm ◽  
Time Integration Methods ◽  
Motion Problems

Abstract A method of time integration algorithm is presented for solving stiff vibration and motion problems. It is absolutely stable, numerically dissipative, and much accurate than other dissipative time integration methods. It achieves high-frequency dissipation, while minimizing unwanted low-frequency dissipation. In this method change of acceleration during time step is expressed as quadratic function including some parameters, whose appropriate values are determined through numerical investigation. Two calculation examples are demonstrated to show the usefulness of this method.

Partitioned Integration Method Based on Newmark’s Scheme for Structural Dynamic Problems

2016 ◽  
Vol 16 (01) ◽  
pp. 1640009 ◽  
Author(s):  
Chuanguo Jia ◽  
Zhou Leng ◽  
Yingmin Li ◽  
Hongliu Xia ◽  
Liping Liu
Keyword(s):  
Integration Method ◽  
Computational Cost ◽  
Low Frequency ◽  
State Variables ◽  
Time Step ◽  
Order Accuracy ◽  
Mass System ◽  
Structural Dynamic ◽  
Time Integrators ◽  
Stability And Accuracy

Systems of ordinary differential equations (ODEs) arising from transient structural dynamics very often exhibit high-frequency/low-frequency and linear/nonlinear behaviors of subsets of the state variables. With this in mind, the paper resorts to the use of different time integrators with different time steps for subsystems, which tailors each method and its time step to the solution behaviors of the corresponding subsystem. In detail, a partitioned integration method is introduced which imposes continuity of velocities at the interface to couple arbitrary Newmark schemes with different time steps in different subdomains. It is proved that the velocity continuity of the method is the primal factor of its reduction to first-order accuracy. To maintain second-order accuracy without increasing drift and computational cost, a novel method with the acceleration continuity is proposed whose velocity constraint is also ensured by means of the projection strategy. Both its stability and accuracy properties are examined through numerical analysis of a Single-degree-of-freedom (DoF) split mass system. Finally, numerical validations are conducted on Single- and Two-DoF split mass systems and a four-DoF nonlinear structure showing the feasibility of the proposed method.

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

2020 ◽  
Vol 102 (3) ◽  
pp. 1939-1962
Author(s):  
Huimin Zhang ◽  
Runsen Zhang ◽  
Yufeng Xing ◽  
Pierangelo Masarati
Keyword(s):  
Time Integration ◽  
Low Frequency ◽  
Unconditional Stability ◽  
Frequency Content ◽  
Order Accuracy ◽  
Integration Methods ◽  
Universal Approach ◽  
Energy Conserving ◽  
Time Integration Methods ◽  
Linear And Nonlinear Systems

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.

Modified precise direct time integration method for the transient response analysis of viscoelastic systems using an internal variable model

2019 ◽  
Vol 26 (3-4) ◽  
pp. 161-174
Author(s):  
Taufeeq Ur Rehman Abbasi ◽  
Hui Zheng
Keyword(s):  
State Space ◽  
Integration Method ◽  
Time Integration ◽  
Internal Variable ◽  
New Method ◽  
Computational Time ◽  
Direct Integration ◽  
Computational Accuracy ◽  
Time Step ◽  
Time Integration Method

Engineering systems for different levels of energy dissipation use internal variable models, which may lead to tremendous problems in accurate analysis. This article aims to provide an alternative direct integration method for the analysis of systems involving an anelastic displacement field model. A new state-space formulation built on an augmented set of anelastic variables for asymmetric systems is developed. Then, a precise time integration method based on state-space matrix formulation is proposed by introducing a Legendre–Gauss quadrature. The new integration method in terms of numerical stability and its implementation is discussed. The effect of sensitivity of the selection of the time-step and computational time on the performance of the new method is investigated by using a multi-degree-of-freedom system. The performance of the new method is also evaluated in terms of both computational accuracy and efficiency at higher degrees of freedom by using a continuum system. It is demonstrated that the computational accuracy and efficiency of the new method on large-scale problems are higher than that of the direct integration linear displacement–velocity method.

Research in Numerical Simulation of Lager Multi-Scale Water Conveyance Tunnel Based on Mixed Time Integration Method

2012 ◽  
Vol 203 ◽  
pp. 432-437
Author(s):  
Jun Jie Zhao ◽  
Yan Zhi Yang
Keyword(s):  
Numerical Simulation ◽  
Computational Efficiency ◽  
Integration Method ◽  
Time Integration ◽  
Scale Model ◽  
Integration Time ◽  
Time Step ◽  
Detailed Model ◽  
Multi Scale ◽  
Time Integration Method

The global integration time step of multi-scale model is subject to local detailed model, resulting in lower computational efficiency. Mixed time integration method uses different integration time-step in different scale model, it can effectively avoid the above problem. Rest on a large-scale water tunnel under construction, in order to achieve the synchronization of the global and the local simulation, the paper establishes multi-scale finite element model of the tunnel, and calculate it by mixed time integration method. The final calculation and analysis show that the algorithm can guarantee the computational accuracy of multi-scale numerical simulation, and can effectively improve the computational efficiency, it can also provide references for related tunnel project.

High-Frequency Responses in Relation to the Use of Explicit Time Integration Method

2018 ◽  
Author(s):  
Sungjin Bae
Keyword(s):  
High Frequency ◽  
Time Integration ◽  
Implicit Time ◽  
Spent Fuel ◽  
Time Step ◽  
Explicit Time Integration ◽  
Frequency Responses ◽  
Spent Fuel Storage ◽  
Fuel Storage ◽  
Damping Model

Design of an Independent Spent Fuel Storage Installation (ISFSI) pad requires examining seismic stability and determining the static and dynamic loads of casks in accordance with the provision of 10 CFR 72.212 (Ref. 1). Since spent fuel storage casks are free standing structures on a concrete pad, cask rocking and sliding are expected to occur during an earthquake. NUREG/CR-6865 (Ref. 2) requires that the dynamic analyses consider the effect of soil-and-structure interactions between supporting soil and concrete pad as well as between casks and concrete pad. When casks exhibit rocking or sliding, nonlinear analyses are performed to handle the contact, and explicit time integration analyses with very short period of time step are often used to analyze the dynamic contact behavior of casks. Explicit time integration analyses are more prone to contain high-frequency response than implicit time stepping analyses because of the use of very small time step and the limitation of damping model. A cask analysis case is presented in this paper to illustrate high frequency responses associated with the use of Rayleigh mass proportional damping, which is a commonly used damping model for explicit time integration analyses. A modified mass proportional damping expression is proposed to improve the responses of soil and casks.

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

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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