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.

scholarly journals Performance of Composite Implicit Time Integration Scheme for Nonlinear Dynamic Analysis

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
William Taylor Matias Silva ◽  
Luciano Mendes Bezerra
Keyword(s):  
Transient Response ◽  
Time Integration ◽  
Nonlinear Dynamic Analysis ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Structural Dynamic ◽  
Implicit Time Integration ◽  
Time Integration Scheme

This paper presents a simple implicit time integration scheme for transient response solution of structures under large deformations and long-time durations. The authors focus on a practical method using implicit time integration scheme applied to structural dynamic analyses in which the widely used Newmark time integration procedure is unstable, and not energy-momentum conserving. In this integration scheme, the time step is divided in two substeps. For too large time steps, the method is stable but shows excessive numerical dissipation. The influence of different substep sizes on the numerical dissipation of the method is studied throughout three practical examples. The method shows good performance and may be considered good for nonlinear transient response of structures.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Ali Akbar Gholampour ◽  
Mehdi Ghassemieh ◽  
Mahdi Karimi-Rad
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Numerical Dispersion ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Similar Time ◽  
Unconditionally Stable ◽  
Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

Studies on the Accuracy Stability and Efficiency of a New Time Integration Scheme for Ballast Modeling Using the Discrete Element Method

2014 ◽  
Author(s):  
G. F. Mathews ◽  
R. L. Mullen ◽  
D. C. Rizos
Keyword(s):  
Particle Interaction ◽  
Time Integration ◽  
Discrete Element ◽  
Critical Discussion ◽  
Implicit Time ◽  
Integration Scheme ◽  
Computationally Efficient ◽  
Time Step ◽  
Central Difference ◽  
Time Integration Scheme

This paper presents the development of a semi-implicit time integration scheme, originally developed for structural dynamics in the 1970’s, and its implementation for use in Discrete Element Methods (DEM) for rigid particle interaction, and interaction of elastic bodies that are modeled as a cluster of rigid interconnected particles. The method is developed in view of ballast modeling that accounts for the flexibility of aggregates and the arbitrary shape and size of granules. The proposed scheme does not require any matrix inversions and is expressed in an incremental form making it appropriate for non-linear problems. The proposed method focuses on improving the efficiency, stability and accuracy of the solutions, as compared to current practice. A critical discussion of the findings of the studies is presented. Extended verification and assessment studies demonstrate that the proposed algorithm is unconditionally stable and accurate even for large time step sizes. It is demonstrated that the proposed method is at least as computationally efficient as the Central Difference Method. Guidelines for the implementation of the method to ballast modeling are discussed.

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 COMPOSITE TIME INTEGRATION SCHEME FOR DYNAMIC ADHESION AND ITS APPLICATION TO GECKO SPATULA PEELING

2014 ◽  
Vol 11 (05) ◽  
pp. 1350104 ◽  
Author(s):  
SACHIN S. GAUTAM ◽  
ROGER A. SAUER
Keyword(s):  
Nonlinear Response ◽  
Time Integration ◽  
Computational Cost ◽  
Integration Scheme ◽  
Time Step ◽  
Integration Schemes ◽  
Newmark Scheme ◽  
Highly Nonlinear ◽  
Time Integration Schemes ◽  
Time Integration Scheme

Simulation of dynamic adhesive peeling problems at small scales has attracted little attention so far. These problems are characterized by a highly nonlinear response. Accurate and stable time integration schemes are required for simulation of dynamic peeling problems. In the present work, a composite time integration scheme is proposed for the simulation of dynamic adhesive peeling problems. It is shown through numerical examples that the proposed scheme remains stable and also has some gain in accuracy. The performance of the scheme is compared with two collocation-based schemes, i.e., Newmark scheme and Bathe composite scheme. It is shown that the proposed scheme and Bathe composite scheme perform equally. However, the proposed scheme adds very little to the computational cost of Newmark scheme. Through a numerical simulation of the peeling of a gecko spatula from a rigid substrate it is shown that the proposed scheme and the Bathe composite scheme are able to simulate the complete peeling process for given time step whereas the Newmark scheme diverges. It is also shown that the maximum pull-off force is within the range reported in the literature.

An Explicit Time Integration Scheme Based on B-Spline Interpolation and Its Application in Wave Propagation Analysis

2017 ◽  
Vol 09 (08) ◽  
pp. 1750115 ◽  
Author(s):  
W. B. Wen ◽  
S. Y. Duan ◽  
Y. Tao ◽  
Jun Liang ◽  
Daining Fang
Keyword(s):  
Wave Propagation ◽  
Hyperbolic Equations ◽  
Spline Interpolation ◽  
Time Integration ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Weighted Residual Method ◽  
Explicit Time Integration ◽  
B Spline ◽  
Time Integration Scheme

An explicit time integration scheme for hyperbolic equations is proposed using B-spline interpolation and weighted residual method. It has simple formulation and calculation procedure. With one adjustable algorithmic parameter, new scheme has higher accuracy when compared with other excellent explicit schemes. New scheme has controllable and also desirable period elongation which is verified by theoretical analysis and numerical simulations. Especially, a demonstrative dispersion analysis coupled with the corresponding wave propagation demonstrate the desirable numerical dissipation property and the effectiveness of the proposed scheme for wave propagation problems.

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.

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