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.

Modeling the damped dynamic behavior of a flexible pendulum

2019 ◽  
Vol 54 (2) ◽  
pp. 116-129 ◽  
Author(s):  
Roberto Ortega ◽  
Geraldine Farías ◽  
Marcela Cruchaga ◽  
Matías Rivero ◽  
Mariano Vázquez ◽  
...  
Keyword(s):  
High Speed ◽  
Time Integration ◽  
Integration Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Rayleigh Damping ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Damping Model

The focus of this work is on the computational modeling of a pendulum made of a hyperelastic material and the corresponding experimental validation with the aim of contributing to the study of a material commonly used in seismic absorber devices. From the proposed dynamics experiment, the motion of the pendulum is recorded using a high-speed camera. The evolution of the pendulum’s positions is recovered using a capturing motion technique by tracking markers. The simulation of the problem is developed in the framework of a parallel multi-physics code. Particular emphasis is placed on the analysis of the Newmark integration scheme and the use of Rayleigh damping model. In particular, the time step size effect is analyzed. A strong time step size dependency is obtained for dissipative time integration schemes, while the Rayleigh damping formulation without time integration dissipation shows time step–independent results when convergence is achieved.

scholarly journals High-Order Time-Integration Schemes with Explicit Time-Splitting Methods

2009 ◽  
Vol 137 (11) ◽  
pp. 4047-4060 ◽  
Author(s):  
Sang-Hun Park ◽  
Tae-Young Lee
Keyword(s):  
Numerical Models ◽  
Time Integration ◽  
High Order ◽  
Integration Scheme ◽  
High Order Schemes ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Time Integration Scheme ◽  
High Order Time ◽  
Fast Waves

Abstract New high-order time-integration schemes for fully elastic models are presented. The new schemes, formulated using the Richardson extrapolation that employs leapfrog-type schemes, can give a good performance for linear model problems and ensure overall stability when they are combined with a forward–backward scheme for fast waves. The new and existing schemes show differences in the order of accuracy. Thus, they can be useful for investigating the impacts of time-integration scheme accuracy on the performance of numerical models. The high-order schemes are found to play an important role in the improvement of high-resolution simulations, according to idealized tests. The new schemes are less efficient than other well-known schemes at moderate spatial resolutions. However, the new schemes can be more efficient than the existing schemes when the resolution becomes very high.

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.

scholarly journals A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

2015 ◽  
Vol 143 (9) ◽  
pp. 3838-3855 ◽  
Author(s):  
Steven Sandbach ◽  
John Thuburn ◽  
Danail Vassilev ◽  
Michael G. Duda
Keyword(s):  
Time Integration ◽  
Standard Test ◽  
Newton Iteration ◽  
Atmospheric Modeling ◽  
Implicit Time ◽  
Integration Scheme ◽  
Baroclinic Wave ◽  
Time Step ◽  
Integration Schemes ◽  
Helmholtz Problem

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

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 Cross Weighted-Residual Time Integration Scheme for Structural Dynamics

2014 ◽  
Vol 14 (06) ◽  
pp. 1450023 ◽  
Author(s):  
Wooram Kim ◽  
Sang-Shin Park ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Linear Time ◽  
Time Integration ◽  
Velocity Fields ◽  
Inner Product ◽  
Integration Scheme ◽  
Time Interval ◽  
Element Discretization ◽  
Integration Schemes ◽  
Time Integration Scheme

In this article, we develop a novel stable time integration scheme for the transient analysis of structural dynamics problems. A second-order (in time) differential operator equation (e.g. obtained after finite element discretization in space) is written as a pair of first-order equations in terms of displacements and velocities. Then the solution is sought by minimizing the inner product of the residuals in the two equations (an unconventional approach) over typical time interval to obtain a symmetric set of algebraic equations involving displacements and velocities at two subsequent intervals. The new time integration scheme is termed the cross weighted-residual (CWR) time integration scheme because each of the two residuals takes the other one as a weight function in the minimization. The CWR time integration scheme is developed by using a uniform linear time approximation of the displacement and velocity fields to yield only a single step time integration scheme, which is comparable to the Newmark family of time integration scheme. A reduced integration technique is used to prevent velocity locking, which is caused by linear approximation of both the displacement and velocity fields. For the verification of the consistency and the stability, the CWR time integration scheme is tested with single-degree as well as multi-degree of freedom problems. The scheme performs extremely well compared with those of the well-known Newmark family of time integration schemes.

scholarly journals Explicit meshfree $${{\varvec{u}}}-{{\varvec{p}}}_\mathbf{\mathrm{w}}$$ solution of the dynamic Biot formulation at large strain

2021 ◽  
Author(s):  
Pedro Navas ◽  
Miguel Molinos ◽  
Miguel M. Stickle ◽  
Diego Manzanal ◽  
Angel Yagüe ◽  
...  
Keyword(s):  
Pore Water Pressure ◽  
Large Strain ◽  
Water Pressure ◽  
Time Integration ◽  
Computational Cost ◽  
Local Maximum ◽  
Integration Scheme ◽  
Time Integration Scheme ◽  
Biot’S Equations ◽  
Predictor Corrector

AbstractIn this paper, an efficient and robust methodology to simulate saturated soils subjected to low-medium frequency dynamic loadings under large deformation regime is presented. The coupling between solid and fluid phases is solved through the dynamic reduced formulation $$u-p_\mathrm{w}$$ u - p w (solid displacement – pore water pressure) of the Biot’s equations. The additional novelty lies in the employment of an explicit two-steps Newmark predictor-corrector time integration scheme that enables accurate solutions of related geomechanical problems at large strain without the usually high computational cost associated with the implicit counterparts. Shape functions based on the elegant Local Maximum Entropy approach, through the Optimal Transportation Meshfree framework, are considered to solve numerically different dynamic problems in fluid saturated porous media.

scholarly journals Acceleration of Runge-Kutta integration schemes

2004 ◽  
Vol 2004 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Phailaung Phohomsiri ◽  
Firdaus E. Udwadia
Keyword(s):  
Computational Cost ◽  
Fixed Time ◽  
Lower Number ◽  
Integration Scheme ◽  
Time Step ◽  
Third Order ◽  
Numerical Examples ◽  
Integration Schemes ◽  
Local Accuracy ◽  
Runge Kutta

A simple accelerated third-order Runge-Kutta-type, fixed time step, integration scheme that uses just two function evaluations per step is developed. Because of the lower number of function evaluations, the scheme proposed herein has a lower computational cost than the standard third-order Runge-Kutta scheme while maintaining the same order of local accuracy. Numerical examples illustrating the computational efficiency and accuracy are presented and the actual speedup when the accelerated algorithm is implemented is also provided.

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.

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