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.

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

2014 ◽  
Vol 580-583 ◽  
pp. 2996-3002 ◽  
Author(s):  
Chuan Guo Jia ◽  
Ying Min Li ◽  
Hong Liu Xia ◽  
Li Ping Liu
Keyword(s):  
Integration Method ◽  
Low Frequency ◽  
State Variables ◽  
Time Step ◽  
Mass System ◽  
Structural Dynamic ◽  
Time Integrators ◽  
Systems Of Odes ◽  
Stability And Accuracy ◽  
Projection Strategy

Systems of ODEs arising from transient structural dynamics generally exhibit high-frequency/low-frequency and linear/nonlinear behaviours of subsets of state variables. This paper resorts to the use of different time integrators with different time steps for subsystems, to tailor each method and its time step to the solution behaviour of the corresponding subsystem. Anovel partitioned integration method with the acceleration continuity is proposed whose velocity continuity is also ensured by means of the mass projection strategy. Both its stability and accuracy properties are examined through numerical analysis on Single-DoF split mass system.

A Taylor Series Integration Method with Coupling in Structural Dynamics

2012 ◽  
Vol 166-169 ◽  
pp. 9-13
Author(s):  
Ze Ying Guo
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Integration Method ◽  
Damping Ratio ◽  
Time Integration ◽  
Taylor Series Expansion ◽  
Time Step ◽  
Structural Dynamic ◽  
Precise Time Integration ◽  
Stability And Accuracy

Based on the coupled precise time integration method and basic assumptions of constant average acceleration method in Newmark family, implicit series solution of structural dynamic equation is put forward by introducing the Taylor series expansion. Relevant time step integration formulas were designed. Stability and accuracy of the method were analyzed. Stability analyses show that the coupling implicit method is stable when damping ratio is equal to 0, and is conditionally stable when damping ratio are other values. The results show that the accuracy of the algorithm can be controlled by choosing the number of truncation order of Taylor series expansion and is better than that of traditional scheme with the increase of time step. Number examples are given to demonstrate the validity of the proposed method.

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.

A Precise Time Step Integration Method

1994 ◽  
Vol 208 (6) ◽  
pp. 427-430 ◽  
Author(s):  
W X Zhong ◽  
F W Williams
Keyword(s):  
High Precision ◽  
Integration Method ◽  
Linear Time ◽  
Step Size ◽  
Time Step ◽  
Structural Dynamic ◽  
Time Step Size ◽  
Linear Time Invariant ◽  
Wide Range ◽  
Time Invariant

A high-precision numerical time step integration method is proposed for a linear time-invariant structural dynamic system. Its numerical results are almost identical to the precise solution and are almost independent of the time step size for a wide range of step sizes. Numerical examples illustrate this high precision.

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.

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.

Precise Integration Method of Pseudo-Dynamic Testing of Structures

2014 ◽  
Vol 580-583 ◽  
pp. 1574-1580
Author(s):  
Hai Bo Wang ◽  
Rong Liu
Keyword(s):  
Integration Method ◽  
Dynamic Test ◽  
Multistep Method ◽  
The State ◽  
State Matrix ◽  
Time Step ◽  
Order Accuracy ◽  
Central Difference ◽  
Precise Integration ◽  
Precise Integration Method

Based on nonlinear precise integration method, two new numerical integration methods of pseudo-dynamic test of structures are presented. One is explicit predict-correct, four order accuracy and multistep method avoiding calculating the inversion of the state matrix. The other is implicit predict-correct, four order accuracy and multistep method that need calculate the inversion of the state matrix. Since their accuracies are superior to the central difference method by enlarging time step and their stabilities are better, the structural systems of multi-degree of freedom could be well tested and the testing work would be largely reduced. Finally, a pseudo-dynamic test of combined tube structure has been executed with the explicit multi-step method.

scholarly journals A new and inexpensive non-bit-for-bit solution reproducibility test based on time step convergence (TSC1.0)

2017 ◽  
Vol 10 (2) ◽  
pp. 537-552 ◽  
Author(s):  
Hui Wan ◽  
Kai Zhang ◽  
Philip J. Rasch ◽  
Balwinder Singh ◽  
Xingyuan Chen ◽  
...  
Keyword(s):  
Evolution Equations ◽  
Reference Model ◽  
Compiler Optimization ◽  
Test Procedure ◽  
Computational Cost ◽  
Parallel Execution ◽  
State Variables ◽  
Time Step ◽  
Geophysical Model ◽  
Solution Changes

Abstract. A test procedure is proposed for identifying numerically significant solution changes in evolution equations used in atmospheric models. The test issues a fail signal when any code modifications or computing environment changes lead to solution differences that exceed the known time step sensitivity of the reference model. Initial evidence is provided using the Community Atmosphere Model (CAM) version 5.3 that the proposed procedure can be used to distinguish rounding-level solution changes from impacts of compiler optimization or parameter perturbation, which are known to cause substantial differences in the simulated climate. The test is not exhaustive since it does not detect issues associated with diagnostic calculations that do not feedback to the model state variables. Nevertheless, it provides a practical and objective way to assess the significance of solution changes. The short simulation length implies low computational cost. The independence between ensemble members allows for parallel execution of all simulations, thus facilitating fast turnaround. The new method is simple to implement since it does not require any code modifications. We expect that the same methodology can be used for any geophysical model to which the concept of time step  convergence is applicable.

An Improved Structure-Dependent Explicit Method for Structural Dynamics

2008 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang
Keyword(s):  
Low Frequency ◽  
Conditional Stability ◽  
Explicit Method ◽  
Time Step ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Acceleration Method ◽  
The Stability ◽  
Stiffness Softening ◽  
Hardening System

An explicit method is presented herein whose coefficients are determined from the initial structural properties of the analyzed system. Thus, it is structure-dependent. This method has a great stability property when compared to the previously published method [6], which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems while it only has conditional stability for an instantaneous stiffness hardening system. The most important improvement of this method is that it has unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. This implies that a time step may be selected base on accuracy consideration only and the stability problem might be neglected. Hence, many computational efforts can be saved in the step-by-step solution of a general structural dynamic problem, where the response is dominated by the low frequency modes and the high frequency responses are of no great interest, when compared to an explicit method, such as the Newmark explicit method, and an implicit method, such as the constant average acceleration method.

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