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.

COMPUTATIONAL DYNAMICS OF ORDINARY DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 05 (01) ◽  
pp. 159-174 ◽  
Author(s):  
MAXIM POLIASHENKO ◽  
CYRUS K. AIDUN
Keyword(s):  
Time Integration ◽  
Euler Method ◽  
Point Of View ◽  
Implicit Time ◽  
Time Step ◽  
Navier Stokes Equation ◽  
True Solution ◽  
Integration Schemes ◽  
Backward Euler ◽  
Chaotic Solution

Discrete schemes, used to perform time integration of ODE’s, are expected to exhibit qualitatively ‘true’ dynamics in terms of the solutions and their stability. In past years, it has been discovered that such discretizations may cause spurious steady states and some explicit schemes may produce ‘computational chaos.’ In this study, we show that implicit time integration schemes, such as the backward Euler method, can also produce computationally chaotic solutions. Furthermore, we show that the opposite phenomenon may also take place both for explicit and for implicit schemes: computationally generated ‘spurious order’ may replace the true chaotic solution before the scheme becomes linearly unstable. The numerical solution may become chaotic again as the discretization step is further increased. The spurious computational order and chaos are discussed by solving low-dimensional dynamical systems, as well as a large system of ODE representing the solution to the Navier-Stokes equation. Our results support the point of view that the deviations in the behavior of the computed solution from the true solution has deterministic character with the time step playing the role of an artificial bifurcation parameter.

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.

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.

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.

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.

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.

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.

A New Implicit Integration Scheme for the Simulation of Stiff-Complex Mechanical Systems

2005 ◽  
Author(s):  
Mojtaba Oghbaei ◽  
Kurt S. Anderson
Keyword(s):  
Mechanical Systems ◽  
Implicit Time ◽  
Integration Time ◽  
Integration Scheme ◽  
Algebraic Equations ◽  
Stiff System ◽  
Time Step ◽  
Efficient Manner ◽  
Method Performance ◽  
Integration Schemes

When performing the dynamic simulation of stiff mechanical systems, implicit type integration schemes are usually required to preserve stability. This article presents a new implicit time integrator, which is a particular application of a novel state-time formulation recently developed by the authors in a more general scope. The proposed scheme is constructed with the intent of benefitting from the accuracy and apparent robustness thus far achieved with this algorithm in an integration context. This is realized by first setting up the weighted residual form of the governing equations of the system in a form associated with the application of a time marching integration scheme. The resulting algebraic equations are then solved, minimizing the error of integration time step in a generalized energy sense, allowing one to capture the stiff behavior of solution in an efficient manner. Examples are provided to show the proposed method performance when dealing with a stiff system.

scholarly journals Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

2015 ◽  
Vol 18 (4) ◽  
pp. 1147-1180 ◽  
Author(s):  
David Kamensky ◽  
John A. Evans ◽  
Ming-Chen Hsu
Keyword(s):  
Incompressible Flows ◽  
Fluid Structure Interaction ◽  
Time Integration ◽  
Implicit Time ◽  
Integration Scheme ◽  
Discrete Problem ◽  
Time Step ◽  
Integration Rule ◽  
Fluid Structure ◽  
Structure Interaction

AbstractThe purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

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