Numerical integration of elasto-plasticity coupled to damage using a diagonal implicit Runge–Kutta integration scheme

2012 ◽  
Vol 22 (1) ◽  
pp. 68-94 ◽  
Author(s):  
Eric Borgqvist ◽  
Mathias Wallin
Keyword(s):  
Numerical Integration ◽  
Damage Evolution ◽  
Finite Strain ◽  
Integration Scheme ◽  
Euler Scheme ◽  
Implicit Euler Scheme ◽  
Damage Models ◽  
Integration Schemes ◽  
Evolution Laws ◽  
Runge Kutta

This article is concerned with the numerical integration of finite strain continuum damage models. The numerical sensitivity of two damage evolution laws and two numerical integration schemes are investigated. The two damage models differ in that one of the models includes a threshold such that the damage evolution is suppressed until a certain effective plastic strain is reached. The classical integration scheme based on the implicit Euler scheme is found to suffer from a severe step-length dependence. An alternative integration scheme based on a diagonal implicit Runge--Kutta scheme originally proposed by Ellsiepen ( 1999 ) is investigated. The diagonal implicit Runge--Kutta scheme is applied to the balance of momentum as well as the constitutive evolution equations. When applied to finite strain multiplicative plasticity, the diagonal implicit Runge--Kutta scheme destroys the plastic incompressibility of the underlying continuum evolution laws. Here, the evolution laws are modified such that the incompressibility of the plastic deformation is preserved approximately. The presented numerical examples reveal that a significant increase in accuracy can be obtained at virtually no cost using the diagonal implicit Runge--Kutta scheme. It is also shown that for the model including a discontinuous evolution law, the superiority of the diagonal implicit Runge--Kutta scheme over the implicit Euler scheme is reduced.

scholarly journals Regional and seasonal truncation errors of trajectory calculations using ECMWF high-resolution operational analyses and forecasts

2017 ◽  
Author(s):  
Thomas Rößler ◽  
Olaf Stein ◽  
Yi Heng ◽  
Lars Hoffmann
Keyword(s):  
High Resolution ◽  
Numerical Integration ◽  
Transport Processes ◽  
Integration Scheme ◽  
Free Troposphere ◽  
Time Step ◽  
Truncation Errors ◽  
Trajectory Calculations ◽  
Integration Schemes ◽  
Runge Kutta

Abstract. Lagrangian particle dispersion models (LPDMs) are indispensable tools to study atmospheric transport processes. The accuracy of trajectory calculations, which form an essential part of LPDM simulations, depends on various factors. Here we focus on truncation errors that originate from the use of numerical integration schemes to solve the kinematic equation of motion. The optimization of numerical integration schemes to minimize truncation errors and to maximize computational speed is of great interest regarding the computational efficiency of large-scale LPDM simulations. In this study we analyzed truncation errors of six explicit integration schemes of the Runge Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) model. The simulations were driven by wind fields of the latest operational analysis and forecasts of the European Centre for Medium-range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct domains of the atmosphere, covering the polar regions, the mid-latitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the lower and mid stratosphere. For each domain we performed simulations for the months of January, April, July, and October for the years of 2014 and 2015. In total more than 5000 different transport simulations were performed. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a 4th-order Runge-Kutta integration scheme with a sufficiently fine time step. We assessed the transport deviations with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the truncation errors vary significantly between the different domains and seasons. Especially the differences in altitude stand out. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to generalize, we recommend the 3rd-order Runge Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days time based on ECMWF's high-resolution meteorological data.

scholarly journals Efficient Approximate Techniques for Integrating Stochastic Differential Equations

2006 ◽  
Vol 134 (10) ◽  
pp. 3006-3014 ◽  
Author(s):  
James A. Hansen ◽  
Cecile Penland
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Approximate Solutions ◽  
Integration Scheme ◽  
Time Step ◽  
Integration Schemes ◽  
Runge Kutta

Abstract The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.

Lode angle dependency due to anisotropic damage

2020 ◽  
pp. 105678952094856
Author(s):  
A Mattiello ◽  
R Desmorat
Keyword(s):  
Plastic Strain ◽  
Damage Evolution ◽  
Time Integration ◽  
Anisotropic Damage ◽  
Proportional Loading ◽  
Isotropic Materials ◽  
Damage Models ◽  
Evolution Laws ◽  
Lode Angle ◽  
Fully Coupled

The lode angle dependency introduced by anisotropic damage evolution laws is analyzed in detail for initially isotropic materials. Many rupture criteria are obtained, under the proportional loading assumption, by the time integration of different anisotropic damage evolution laws [Formula: see text] among the three existing families: strain governed, stress governed and plastic strain governed. The cross-analysis of path independent rupture criteria and of anisotropic damage evolution laws finally allows us to improve the Lode angle dependency of (fully coupled) anisotropic damage models.

scholarly journals Is There an Optimal Choice of Configuration Space for Lie Group Integration Schemes Applied to Constrained MBS?

2013 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze
Keyword(s):  
Numerical Integration ◽  
Configuration Space ◽  
Lie Group ◽  
Optimal Choice ◽  
Rigid Bodies ◽  
Integration Scheme ◽  
Integration Schemes ◽  
Numerical Integration Scheme ◽  
Group Integration ◽  
Lie Group Integration

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: SE(3) and SO(3) × ℝ3. Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the SE(3) formulation outperforms the SO(3) × ℝ3 formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the SO(3) × ℝ3 formulation should be used since the SE(3) formulation is numerically more complex, however.

On the Improvement of Deconvolution With Digitized Data Using a Runge-Kutta Integration Scheme

1977 ◽  
Vol 99 (3) ◽  
pp. 190-192
Author(s):  
J. R. Houghton ◽  
M. A. Townsend ◽  
P. F. Packman
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Input Function ◽  
High Order ◽  
Integration Scheme ◽  
Forcing Function ◽  
Pulse Type ◽  
Simple Change ◽  
Runge Kutta

A relatively simple change in the treatment of the input function in numerical integration of high-order differential equations by Runge-Kutta methods provides substantial improvements in accuracy, particularly when the forcing function is in digitized form. The Runge-Kutta-Gill coefficients are modified to incorporate the changes; with pulse-type excitations, improvements on the order of 2 to 50 times greater accuracy are demonstrated.

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.

scholarly journals A direct numerical integration scheme for visco-hyperelastic models using radial return relaxation

2010 ◽  
pp. 129-140
Author(s):  
Stéphane Lejeunes ◽  
Stéphane Méo ◽  
Adnane Boukamel
Keyword(s):  
Numerical Integration ◽  
Taylor Expansion ◽  
Integration Scheme ◽  
Exponential Mapping ◽  
First Order ◽  
Numerical Integration Scheme ◽  
Exponential Solution ◽  
Evolution Laws ◽  
Direct Numerical Integration ◽  
Hyperelastic Models

In this paper, a numerical integration scheme of the evolution laws for viscohyperelastic models is proposed. The starting points of the method are the exponential mapping (Reese et al., 1998) and the radial return (Weber et al., 1990; Simo, 1988). The originality of this work lies in the substitution of a differential tensorial system by a scalar one with two equations and two unknowns and in a first order Taylor expansion of them. In this way an analytical approximated exponential solution is finally obtained.

scholarly journals Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

2018 ◽  
Vol 11 (2) ◽  
pp. 575-592 ◽  
Author(s):  
Thomas Rößler ◽  
Olaf Stein ◽  
Yi Heng ◽  
Paul Baumeister ◽  
Lars Hoffmann
Keyword(s):  
Integration Scheme ◽  
Kutta Method ◽  
Time Step ◽  
Third Order ◽  
Truncation Errors ◽  
The Third ◽  
Trajectory Calculations ◽  
Integration Schemes ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.

scholarly journals An Energy-Work Relationship Integration Scheme for Nonconservative Hamiltonian Systems

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Fu Jingli ◽  
Chen Benyong ◽  
Xie Fengping
Keyword(s):  
Numerical Integration ◽  
Hamiltonian Systems ◽  
Discrete Analog ◽  
Integration Scheme ◽  
Parallel Composition ◽  
Integration Schemes ◽  
Numerical Integration Scheme ◽  
New Energy ◽  
Relation Scheme ◽  
Work Relationship

This letter focuses on studying a new energy-work relationship numerical integration scheme of nonconservative Hamiltonian systems. The signal-stage, multistage, and parallel composition numerical integration schemes are presented for this system. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multistage scheme of order 2 which its order of accuracy is 2n. The connection, which is discrete analog of usual case, between the change of energy and work of nonconservative force is obtained for nonconservative Hamiltonian systems.This letter also shows that the more the stages of the schemes are, the less the error rate of the scheme is for nonconservative Hamiltonian systems. Finally, an applied example is discussed to illustrate these results.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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