On an Improved Time Integration Scheme for Stiff Constitutive Models of Inelastic Deformation

1985 ◽  
Vol 107 (4) ◽  
pp. 282-285 ◽  
Author(s):  
Vinod Banthia ◽  
Subrata Mukherjee
Keyword(s):  
Strain Rate ◽  
Inelastic Deformation ◽  
Time Integration ◽  
Constitutive Models ◽  
Size Control ◽  
Integration Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Step Size Control

For the time-integration of stiff constitutive models of inelastic deformation, the explicit (one step Euler) integration scheme can be used provided the time step size is closely monitored and controlled. The time step size control scheme based on prescribed error bounds is of limited use because it requires an a priori estimate of the maximum nonelastic strain rate for the selection of a proper error bound. In this paper, a new scheme for time-step size control is presented. This scheme automatically scales the time-step size by the maximum nonelastic strain rate. That the new scheme is superior to the old one is evident from the results of the various problems presented here.

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.

Numerical Integration of Some Stiff Constitutive Models of Inelastic Deformation

1980 ◽  
Vol 102 (1) ◽  
pp. 92-96 ◽  
Author(s):  
Virendra Kumar ◽  
Mahesh Morjaria ◽  
Subrata Mukherjee
Keyword(s):  
Numerical Integration ◽  
Inelastic Deformation ◽  
Time Integration ◽  
Constitutive Models ◽  
Integration Scheme ◽  
Uniaxial Deformation ◽  
Time Step ◽  
One Step ◽  
Numerical Time Integration ◽  
Step Control

Several strategies for numerical time-integration of some stiff constitutive models of inelastic deformation are presented in this paper. Numerical results and comparisons are presented for the integration of one such model for the case of uniaxial deformation under various prescribed histories of stress or strain. A simple one step Euler type integration scheme with automatic time-step control, which can be easily adapted to the solution of multiaxial boundary value problems, appears promising.

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 Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

2010 ◽  
Vol 138 (8) ◽  
pp. 3333-3341 ◽  
Author(s):  
Katherine J. Evans ◽  
Mark A. Taylor ◽  
John B. Drake
Keyword(s):  
Shallow Water Equation ◽  
Time Integration ◽  
Atmospheric Model ◽  
Test Cases ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Fully Implicit ◽  
Stability Limits

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Direct Numerical Simulation of the Interaction of an Ultra Short-Pulsed Intense Laser With a H2+ Molecule

2008 ◽  
Author(s):  
Y.-M. Lee ◽  
J.-S. Wu ◽  
T.-F. Jiang ◽  
Y.-S. Chen
Keyword(s):  
Domain Decomposition Method ◽  
Time Integration ◽  
Three Dimensional ◽  
Time Algorithm ◽  
Intense Laser ◽  
Angle Of Incidence ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
H2 Molecule

In this paper, interactions of a linearly polarized ultra short-pulsed intense laser with a single H2+ molecule at various angles of incidence are studied by directly solving the time-dependent three-dimensional Schrodinger equation (TDSE), assuming Born-Oppenheimer approximation. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternative times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using domain decomposition method on distributed memory machines by applying a multi-level graph-partitioning technique. The solver is applied to simulate laser-molecular interaction with test conditions including: laser intensity of 0.5*1014 W/cm2, wavelength of 800 nm, three pulses in time, angle of incidence of 0–90° and inter-nuclear distance of 2 a.u.. Simulation conditions include 4 million hexahedral cells, 90 a.u. long in z direction, and time-step size of 0.005 a.u.. Ionization rates, harmonic spectra and instantaneous distribution of electron densities are then obtained from the solution of the TDSE. Future possible extension of the present method is also outlined at the end of this paper.

scholarly journals Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

2021 ◽  
Author(s):  
Hendrik Ranocha ◽  
Lisandro Dalcin ◽  
Matteo Parsani ◽  
David I. Ketcheson
Keyword(s):  
Fluid Dynamics ◽  
Error Control ◽  
Time Integration ◽  
Size Control ◽  
Spectral Element ◽  
Step Size ◽  
Step Size Control ◽  
Wide Range ◽  
Cfd Applications ◽  
Runge Kutta

AbstractWe develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

Next Generation Safety Analysis Methods for SFRs—(4) Development of a Computational Framework on Fluid-Solid Mixture Flow Simulations for the COMPASS Code

2009 ◽  
Author(s):  
Shuai Zhang ◽  
Koji Morita ◽  
Noriyuki Shirakawa ◽  
Yuichi Yamamoto
Keyword(s):  
Distinct Element ◽  
Integration Scheme ◽  
Step Size ◽  
Time Step ◽  
Mixture Flow ◽  
Computational Framework ◽  
Mps Method ◽  
Time Step Size ◽  
Flow Simulations ◽  
Solid Mixture

The COMPASS code is designed based on the moving particle semi-implicit (MPS) method to simulate various complex mesoscale phenomena relevant to core disruptive accidents (CDAs) of sodium-cooled fast reactors (SFRs). The MPS method, which is a fully Lagrangian method, can be extended for fluid-solid mixture flow simulations in a straightforward approach. In this study, a computational framework for fluid-solid mixture flow simulations was developed for the COMPASS code. In the present framework, the passively moving solid (PMS) model, which is originally proposed to describe the motion of a rigid body in a fluid, used to simulate hydrodynamic interactions between fluid and solids. In addition, mechanical interactions between solids were modeled by the distinct element method (DEM). Since the typical time step size in DEM calculation, which uses an explicit time integration scheme, is much smaller than that in MPS calculation, a multi-time-step algorithm was introduced to couple these two calculations. In order to verify the proposed computational framework for fluid-solid mixture flow simulations, a series of experiments of water-dam break with multiple solid rods was simulated using the COMPASS code. It was found that simulations considering only fluid-solid interactions using the PMS model can not reasonably represent typical behaviors of solid rods observed in the experiments. However, results of simulations taking account of solid-solid interactions using DEM as well as fluid-solid ones were in good agreement with experimental observations. It was demonstrated that the present computational framework enhances the capability of the COMPASS code for mesoscale simulations of fluid-solid mixture flow phenomena relevant to CDAs of SFRs. To improve the computational efficiency for fluid-solid mixture flow simulations, it will be necessary to optimize the time step size used in DEM calculations by adjusting DEM parameters based on additional experiments and numerical tests.

Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem

2011 ◽  
Vol 10 (4) ◽  
pp. 844-866 ◽  
Author(s):  
Jingyan Yue ◽  
Guangwei Yuan
Keyword(s):  
Iterative Method ◽  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Radiation Diffusion ◽  
Reaction Diffusion Systems ◽  
Time Step Size ◽  
Step Control ◽  
Equilibrium Radiation ◽  
Non Equilibrium

AbstractFor a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

Smoothed Profile Method for Particulate Two-Phase Flow

2008 ◽  
Author(s):  
Xian Luo ◽  
Martin R. Maxey ◽  
George E. Karniadakis
Keyword(s):  
Numerical Simulations ◽  
Stokes Equations ◽  
Integration Scheme ◽  
Force Density ◽  
Step Size ◽  
Time Step ◽  
Element Discretization ◽  
Profile Method ◽  
Time Step Size ◽  
Smoothed Profile Method

We re-formulate and demonstrate a new method for particulate flows, the so-called “Smoothed Profile” method (SPM) first proposed in [1]. The method uses a fixed computational mesh, which does not conform to the geometry of the particles. The particles are represented by certain smoothed indicator profiles to construct a smooth body force density term added into the Navier-Stokes equations. The SPM imposes accurately and efficiently the rigid-body constraint inside the particles. In particular, while the original method employs a fully-explicit time-integration scheme, we develop a high-order semi-implicit splitting scheme, which we implement in the context of spectral/hp element discretization. We show that the modeling error of SPM has a non-monotonic dependence on the time step size Δt. The optimum time step size balances the thickness of the Stokes layer and that of the profile interface. Subsequently, we present several numerical simulations, including flow past three-dimensional complex-shaped particles and two interacting microspheres, which are compared against full direct numerical simulations and the force coupling method (FCM).

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

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