A Novel Single-Step Unconditionally Stable Numerical Integration Scheme With Tunable Algorithmic Dissipation

2020 ◽  
Author(s):  
Huimin Zhang ◽  
Runsen Zhang ◽  
Andrea Zanoni ◽  
Pierangelo Masarati
Keyword(s):  
Time Integration ◽  
Nonlinear Problems ◽  
Single Step ◽  
New Method ◽  
Integration Scheme ◽  
Implicit Methods ◽  
Step Method ◽  
Time Integration Method ◽  
Start Up ◽  
Numerical Integration Scheme

Abstract A novel single-step time integration method is proposed for general dynamic problems. From linear spectral analysis, the new method with optimal parameters has second-order accuracy, unconditional stability, controllable algorithmic dissipation and zero-order overshoot in displacement and velocity. Comparison of the proposed method with several representative implicit methods shows that the new method has higher accuracy than the single-step generalized-α method, and also than the composite P∞-Bathe method when mild algorithmic dissipation is used. Besides, this method is spectrally identical to the linear two-step method, although being easier to use since it does not need additional start-up procedures. Its numerical properties are assessed through numerical examples, and the new method shows competitive performance for both linear and nonlinear problems.

A Novel Sub-Stepping Method with Numerical Dissipation Control for Time Integration of Highly Flexible Structures

2018 ◽  
Vol 10 (10) ◽  
pp. 1850106 ◽  
Author(s):  
Saeed Mohammadzadeh ◽  
Mehdi Ghassemieh
Keyword(s):  
Equilibrium Points ◽  
Flexible Structures ◽  
Time Integration ◽  
Single Step ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Step Method ◽  
Time Step ◽  
Time Increment ◽  
Time Integration Scheme

Sub-stepping time integration methods attempt to march each time step with multiple sub-steps. Generally, for the first sub-step, a single-step method is applied and the following sub-steps are solved using a method that utilizes the data obtained from two or three previous equilibrium points. Despite the robust stability in problems, control of numerical dissipation in sub-stepping schemes is a tough task due to applying different algorithms on a time increment. In order to overcome this insufficiency, a new sub-stepping time integration scheme, which uses two sub-steps in each time increment, is proposed. Newmark and quadratic acceleration methods are applied on the first and second sub-steps, respectively. Both methods utilize constant parameters that enable the control of numerical dissipation in the analysis. For the proposed method, the stability analysis revealed the unconditional stability region for the pertinent parameters. Additionally, the precision investigation disclosed an advantage of the proposed method with the presence of minor period elongation error. Finally, the application of the proposed method is illuminated via several numerical examples. In addition to numerical dissipation control, the proposed method proved to have an outstanding advantage over other methods in solving highly flexible structures more efficiently and more accurately.

An Accurate Explicit Direct Time Integration Method for Computational Structural Dynamics

1999 ◽  
Author(s):  
Bertrand Tchamwa ◽  
Ted Conway ◽  
Christian Wielgosz
Keyword(s):  
Structural Dynamics ◽  
Integration Method ◽  
Time Integration ◽  
Single Step ◽  
New Method ◽  
Phase Portraits ◽  
Structural Dynamic ◽  
Low Frequencies ◽  
Time Integration Method ◽  
Non Linear

Abstract The purpose of this paper is to introduce a new simple explicit single step time integration method with controllable high-frequency dissipation. As opposed to the methods generally used in structural dynamics, with a consistency experimentally chosen of second order, the new method is only first-order-consistent but yields smaller numerical errors in low frequencies and is therefore very efficient for structural dynamic analysis. The new method remains explicit for any structural dynamics problem, even when a non-diagonal damping matrix is used in linear structural dynamics problem or when the non-linear internal force vector is a function of velocities. Convergence and spectral properties of the new algorithm are discussed and compared to those of some well-known algorithms. Furthermore, the validity and efficiency of the new algorithm are shown in a non-linear dynamic example by comparison of phase portraits.

scholarly journals A novel explicit three-sub-step time integration method for wave propagation problems

2022 ◽  
Author(s):  
Huimin Zhang ◽  
Runsen Zhang ◽  
Andrea Zanoni ◽  
Yufeng Xing ◽  
Pierangelo Masarati
Keyword(s):  
Wave Propagation ◽  
Integration Method ◽  
Time Integration ◽  
Second Order ◽  
New Method ◽  
Explicit Methods ◽  
Time Integration Method ◽  
Numerical Performance ◽  
The Stability ◽  
Better Than

AbstractA novel explicit three-sub-step time integration method is proposed. From linear analysis, it is designed to have at least second-order accuracy, tunable stability interval, tunable algorithmic dissipation and no overshooting behaviour. A distinctive feature is that the size of its stability interval can be adjusted to control the properties of the method. With the largest stability interval, the new method has better amplitude accuracy and smaller dispersion error for wave propagation problems, compared with some existing second-order explicit methods, and as the stability interval narrows, it shows improved period accuracy and stronger algorithmic dissipation. By selecting an appropriate stability interval, the proposed method can achieve properties better than or close to existing second-order methods, and by increasing or reducing the stability interval, it can be used with higher efficiency or stronger dissipation. The new method is applied to solve some illustrative wave propagation examples, and its numerical performance is compared with those of several widely used explicit methods.

Impact Analysis of Sandwich Composites

2008 ◽  
Author(s):  
Laura Ferrero ◽  
Ugo Icardi
Keyword(s):  
Impact Analysis ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Implicit Time ◽  
Contact Radius ◽  
Integration Scheme ◽  
Sandwich Composites ◽  
Time Step ◽  
Wide Range ◽  
The Impact

A finite element simulation of impacts on sandwich composites with laminated faces is presented; it is based on a refined multilayered plate model with a high-order zig-zag representation of displacements, which is incorporated through a strain energy updating process. This allows the implementation into existing commercial finite elements codes, preserving their program structure. As customary, the Hertzian law and the Newmark implicit time integration scheme are used for solving the contact problem. The contact radius and the force are computed within each time step by an iterative algorithm which forces the impacted top surface to conform, in the least-squares sense, to the shape of the impactor. Nonlinear strains of von Karman type are used. As appearing by the comparison with experimental results, the present model is able to accurately predict the impact force, the core damage and the damage of face sheets in sandwich composites with foam and or honeycomb core. Moreover, this paper also assesses the accuracy and the range of application of stress based criteria in predicting the onset and evolution of delamination in service. These criteria are widespread by virtue of their low run time and storage costs, although no exhaustive proofs are known weather they are accurate enough for a reasonably wide range of applications. Since where highly iterative solutions are involved (e.g., impact and geometric, or material nonlinear problems) they are the only currently affordable failure models, it appears of primary importance to fill this gap. Aimed to contribute to the knowledge advancement in this field, a comparison is presented with more sophisticate fracture mechanics and progressive delamination models.

Comparison of Implicit Time Integration Schemes for Nonlinear Dynamic Problems

2010 ◽  
Author(s):  
Murat Demiral
Keyword(s):  
Dynamic Equilibrium ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Implicit Time ◽  
Implicit Methods ◽  
Equilibrium Equations ◽  
Integration Schemes ◽  
Implicit Time Integration ◽  
Time Integration Schemes ◽  
The Stability

Implicit time integration schemes are used to obtain stable and accurate transient solutions of nonlinear problems. Methods that are unconditionally stable in linear analysis are sometimes observed to have convergence problems as in the case of solutions obtained with a trapezoidal method. On the other hand, a composite time integration method employing a trapezoidal rule and a three-point backward rule sequentially in two half steps can be used to obtain accurate results and enhance the stability of the system by means of a numerical damping introduced in the formulation. To have a better understanding of the differences in the numerical implementation of the algorithms of these two methods, a mathematical analysis of dynamic equilibrium equations is performed. Several practical problems are studied to compare the implicit methods.

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.

scholarly journals A Two-Step Composite Time Integration Scheme for Vehicle-Track Interaction Analysis considering Contact Separation

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Ya Gao ◽  
Jin Shi ◽  
Chenxu Lu
Keyword(s):  
Degrees Of Freedom ◽  
Interaction Analysis ◽  
Time Integration ◽  
Integration Scheme ◽  
Step Method ◽  
Finite Element Software ◽  
Track System ◽  
Time Integration Scheme ◽  
Contact Relation ◽  
Family Contact

The two-step composite time integration scheme is combined with the linear contact relation between the wheel and rail (termed the two-step method) to solve coupled vehicle-track dynamic problems. First, two coupled vehicle-track models with different degrees-of-freedom are constructed, in which a vehicle-track system is modeled as two subsystems and the interaction between the two subsystems is implemented via kinematic constraints. Then, the two-step method is applied in the models to simulate several cases, and the accuracy and efficiency of this method are discussed in comparison with additional methods of the Newmark family. Contact separation can also be simulated using the same scheme without causing spurious oscillations for large integration steps. The results indicate that track irregularities can cause wheels to momentarily lose contact with the track, causing large impacts between the wheels and rail. Additionally, with the increase of the running velocity, contact separation will occur more frequently. The adoption of the two-step integration method can ensure the accuracy of solutions and significantly increase the computational efficiency; moreover, the matrixes representing the model information will not change during the calculation process, so the substructure data exported from commercial finite element software can be directly adopted.

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 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.

Sign in / Sign up

Export Citation Format

Share Document