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 A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

2019 ◽  
Vol 9 (15) ◽  
pp. 3076
Author(s):  
Qinyan Xing ◽  
Qinghao Yang ◽  
Weixuan Wang
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Weak Form ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Structural Dynamic ◽  
Time Integration Method ◽  
Nonlinear Structural ◽  
Nonlinear Dynamic Equations ◽  
Step Algorithm

This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

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 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.

scholarly journals Performance of a Three-Substep Time Integration Method on Structural Nonlinear Seismic Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Jinyue Zhang ◽  
Lei Shi ◽  
Tianhao Liu ◽  
De Zhou ◽  
Weibin Wen
Keyword(s):  
Nonlinear Dynamic ◽  
Integration Method ◽  
Seismic Analysis ◽  
Time Integration ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Dynamic Problems ◽  
Element Analysis ◽  
Time Integration Method ◽  
Period Elongation

In this work, a study of a three substeps’ implicit time integration method called the Wen method for nonlinear finite element analysis is conducted. The calculation procedure of the Wen method for nonlinear analysis is proposed. The basic algorithmic property analysis shows that the Wen method has good performance on numerical dissipation, amplitude decay, and period elongation. Three nonlinear dynamic problems are analyzed by the Wen method and other competitive methods. The result comparison indicates that the Wen method is feasible and efficient in the calculation of nonlinear dynamic problems. Theoretical analysis and numerical simulation illustrate that the Wen method has desirable solution accuracy and can be a good candidate for nonlinear dynamic problems.

The Bathe time integration method revisited for prescribing desired numerical dissipation

2019 ◽  
Vol 212 ◽  
pp. 289-298 ◽  
Author(s):  
Mohammad Mahdi Malakiyeh ◽  
Saeed Shojaee ◽  
Klaus-Jürgen Bathe
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Numerical Dissipation ◽  
Time Integration Method

A Two-Sub-Step Generalized Central Difference Method for General Dynamics

2020 ◽  
Vol 20 (07) ◽  
pp. 2050071
Author(s):  
Yi Ji ◽  
Yufeng Xing
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Numerical Dissipation ◽  
Central Difference ◽  
Single Degree Of Freedom ◽  
Time Integration Method ◽  
General Dynamics ◽  
Central Difference Method ◽  
Spectral Radius Formula

This paper proposes an implicit and unconditionally stable two-sub-step composite time integration method with controllable numerical dissipation for general dynamics called the two-sub-step generalized central difference (TGCD) method. The proposed method is established by performing the generalized central difference scheme in two sub-steps as the nondissipative and dissipative parts to ensure amplitude accuracy and controllable damping, respectively. It is accurate to the second order, with the amount of numerical dissipation controlled exactly by the spectral radius [Formula: see text]. In addition, the related parameters of the proposed method are determined by optimizing the amplitude and phase accuracy of the free vibration of a single degree-of-freedom system. Several representative linear and nonlinear numerical examples are analyzed to demonstrate the advantages of the proposed method in terms of accuracy, stability and efficiency, especially its stability in solving nonlinear problems.

Sign in / Sign up

Export Citation Format

Share Document