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.

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.

Precise Time-Integration Method with Dimensional Expanding for Structural Dynamic Equations

AIAA Journal ◽  
2001 ◽  
Vol 39 (12) ◽  
pp. 2394-2399 ◽  
Author(s):  
Yuanxian Gu ◽  
Biaosong Chen ◽  
Hongwu Zhang ◽  
Zhenqun Guan
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Dynamic Equations ◽  
Structural Dynamic ◽  
Time Integration Method ◽  
Precise Time Integration ◽  
Precise Time

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.

Further Insights Into Time-Integration Method Based on Bernstein Polynomials and Bezier Curve for Structural Dynamics

2019 ◽  
Vol 19 (10) ◽  
pp. 1950113
Author(s):  
Mohammad Mahdi Malakiyeh ◽  
Saeed Shojaee ◽  
Saleh Hamzehei-Javaran ◽  
Behrooz Tadayon
Keyword(s):  
Integration Method ◽  
Bernstein Polynomials ◽  
Time Integration ◽  
Bézier Curve ◽  
Implicit Time ◽  
Bezier Curve ◽  
Structural Dynamic ◽  
Numerical Examples ◽  
Time Integration Method ◽  
Low Dissipation

In [M. M. Malakiyeh, S. Shojaee and S. Hamzehei-Javaran, Development of a direct time integration method based on Bezier curve and 5th-order Berstein basis function, Comput. Struct. 194 (2108) 15–31] an unconditionally stable implicit time-integration method using the Bezier curve was proposed for solving structural dynamic problems. In this study, a new class of the previous algorithm is presented by using the Bernstein polynomials and the Bezier curve as the interpolation functions for solving the equations of motion with the possibility of using large time steps. The spectral radius, period elongation, amplitude decay and overshooting of the present method are investigated and compared with some other methods. To show the high-performance, robustness and validity of this method, five numerical examples are presented. The theoretical analysis and numerical examples show that the proposed method has low dissipation in the lower modes and high dissipation in the higher modes in comparison with the other methods reported in the literature.

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