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.

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.

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.

scholarly journals Impact of difference between explicit and implicit second-order time integration schemes on isotropic/anisotropic steady incompressible turbulence field

2021 ◽  
Vol 2090 (1) ◽  
pp. 012145
Author(s):  
Ryuma Honda ◽  
Hiroki Suzuki ◽  
Shinsuke Mochizuki
Keyword(s):  
Kinetic Energy ◽  
Energy Conservation ◽  
Integration Method ◽  
Time Integration ◽  
Second Order ◽  
Implicit Time ◽  
Explicit Time Integration ◽  
Time Integration Method ◽  
Implicit Time Integration ◽  
Time Integration Methods

Abstract This study presents the impact of the difference between the implicit and explicit time integration methods on a steady turbulent flow field. In contrast to the explicit time integration method, the implicit time integration method may produce significant kinetic energy conservation error because the widely used spatial difference method for discretizing the governing equations is explicit with respect to time. In this study, the second-order Crank-Nicolson method is used as the implicit time integration method, and the fourth-order Runge-Kutta, second-order Runge-Kutta and second-order Adams-Bashforth methods are used as explicit time integration methods. In the present study, both isotropic and anisotropic steady turbulent fields are analyzed with two values of the Reynolds number. The turbulent kinetic energy in the steady turbulent field is hardly affected by the kinetic energy conservation error. The rms values of static pressure fluctuation are significantly sensitive to the kinetic energy conservation error. These results are examined by varying the time increment value. These results are also discussed by visualizing the large scale turbulent vortex structure.

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.

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

1993 ◽  
Vol 60 (2) ◽  
pp. 371-375 ◽  
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.

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.

Unconditional Stability in Numerical Time Integration Methods

1973 ◽  
Vol 40 (2) ◽  
pp. 417-421 ◽  
Author(s):  
R. D. Krieg
Keyword(s):  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Central Difference ◽  
Integration Methods ◽  
Time Step Size ◽  
Time Integration Method ◽  
Time Integration Methods ◽  
Numerical Time Integration ◽  
Difference Time

Methods of numerical time integration of the equation M¯q¨ + K¯q = f are examined in this paper. A particular class of explicit time integration methods is defined and this class is searched for an unconditionally stable method. The class is found to contain no such method and, furthermore, is found to contain no method with a larger stable time step size than that characterized by the simple central difference time integration method.

Adaptive Time Integration Method for DAEs of Multibody Systems

2012 ◽  
Author(s):  
Jieyu Ding ◽  
Zhenkuan Pan
Keyword(s):  
Multibody Systems ◽  
Integration Method ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Richardson Extrapolation ◽  
Time Interval ◽  
Time Step ◽  
Time Integration Method ◽  
Adaptive Time Integration ◽  
Adaptive Time

An adaptive time integration method is developed for the index-3 differential-algebraic equations (DAEs) of multibody systems to improve the computational efficiency as well as the accuracy of the results. Based on the modified general-α method, the adaptive time integration is presented. At each discrete time interval, the time step size is changed through Richardson extrapolation with definable computation accuracy. A rotary rod slider system is used to validate the presented adaptive time integration. The accuracy and efficiency are determined by the expected order of the accuracy in Richardson extrapolation.

A ROBUST TIME-INTEGRATION ALGORITHM FOR SOLVING NONLINEAR DYNAMIC PROBLEMS WITH LARGE ROTATIONS AND DISPLACEMENTS

2012 ◽  
Vol 12 (06) ◽  
pp. 1250051 ◽  
Author(s):  
SHYH-RONG KUO ◽  
J. D. YAU ◽  
Y. B. YANG
Keyword(s):  
Nonlinear Dynamic ◽  
Finite Difference Methods ◽  
Time Integration ◽  
Nonlinear Dynamic Analysis ◽  
Element Approximation ◽  
Dynamic Problems ◽  
Time Step ◽  
Integration Algorithm ◽  
Current Time ◽  
Time Integration Algorithm

An efficient time-integration algorithm for nonlinear dynamic analysis of structures is presented. By adopting the temporal discretization for time finite element approximation, very large time steps can be used by the algorithm. With an accuracy of fourth order, this technique requires only displacements and velocities to be made available at the start of the current time step for integration in state space. Using the weighted momentum principle, the problem of discontinuity caused by impulsive loads is resolved after time-integration of the applied load in external momentum. Since no knowledge is required of acceleration at the current time step, the errors caused by estimation of acceleration by previous finite-difference methods are circumvented. Moreover, an iterative procedure is included for each time step, involving the three phases of predictor, corrector, and error-checking. The effectiveness and robustness of the proposed algorithm in solving nonlinear dynamic problems is demonstrated in the numerical examples.

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