A new time integration method in structural dynamics using the Taylor series

1998 ◽  
Vol 212 (7) ◽  
pp. 567-575
Author(s):  
S M Wang ◽  
R A Shenoi ◽  
L B Zhao
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Structural Response ◽  
Simultaneous Equation ◽  
Dynamic Responses ◽  
Implicit Methods ◽  
Explicit Integration ◽  
Structural Dynamic ◽  
Order Of Accuracy ◽  
Time Integration Method

The paper presents a new method of time integration for structural dynamic responses. In comparison with well-known methods, it is advantageous in several aspects. It satisfies the governing equations in continuous intervals rather than at discrete time instants (collocation, SSpj) or in average form (weighted, GNpj). It approximates the structural response with user-controllable order of accuracy. It automatically controls the convergence and accuracy so that a correct answer can be assured via auto-adjusted stepping and expansion terms. As far as the accuracy of velocity and acceleration is concerned, the method is much better since rapid convergence can be obtained with ease. Like the explicit integration method, this approach does not demand solution of simultaneous equation sets, yet it can be used with a time increment much larger than that of the implicit methods.

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 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.

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.

scholarly journals Composite Backward Differentiation Formula for the Bidomain Equations

2020 ◽  
Vol 11 ◽  
Author(s):  
Xindan Gao ◽  
Craig S. Henriquez ◽  
Wenjun Ying
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Euler Method ◽  
Implicit Methods ◽  
Differentiation Formula ◽  
Bidomain Equations ◽  
Time Integration Method ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Backward Euler

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

New Explicit Integration Algorithms with Controllable Numerical Dissipation for Structural Dynamics

2018 ◽  
Vol 18 (03) ◽  
pp. 1850044 ◽  
Author(s):  
Xiaoqiong Du ◽  
Dixiong Yang ◽  
Jilei Zhou ◽  
Xiaoliang Yan ◽  
Yongliang Zhao ◽  
...  
Keyword(s):  
Nonlinear Systems ◽  
Time Integration ◽  
Dynamic Responses ◽  
Discrete Control ◽  
Numerical Dissipation ◽  
Explicit Integration ◽  
Structural Dynamic ◽  
New Family ◽  
Recursive Formulas ◽  
Integration Algorithms

This paper presents a new family of explicit time integration algorithms with controllable numerical dissipation for structural dynamic problems by utilizing the discrete control theory. Firstly, the equilibrium equation of the implicit Yu-[Formula: see text] algorithm is adopted, and the recursive formulas of velocity and displacement for the explicit CR algorithm are used in the algorithms. Then, the transfer function and characteristic equation of the algorithms with integration coefficients are obtained by the [Formula: see text] transformation. Furthermore, their integration coefficients are derived according to the poles condition. It was indicated that the proposed algorithms possess the advantages of second-order accuracy, self-starting, and unconditional stability for linear systems and nonlinear systems with softening stiffness. The numerical dissipation of the algorithms is controlled by the spectral radius at infinity [Formula: see text]. It was also shown that the proposed algorithms have the same poles as the Yu-[Formula: see text] algorithm, and thus the same numerical properties. Compared with the implicit Yu-[Formula: see text] algorithm, the proposed algorithms are explicit in terms of both the displacement and velocity formulas. Finally, the effectiveness of the proposed algorithms in reducing the undesired participation of higher modes for solving the dynamic responses of linear and nonlinear systems has been demonstrated.

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