scholarly journals The Essential Stability of Local Error Control for Dynamical Systems

1995 ◽  
Vol 32 (6) ◽  
pp. 1940-1971 ◽  
Author(s):  
A. M. Stuart ◽  
A. R. Humphries
Keyword(s):  
Dynamical Systems ◽  
Error Control ◽  
Local Error ◽  
Essential Stability

scholarly journals Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-26 ◽  
Author(s):  
Shaowu Pan ◽  
Karthik Duraisamy
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Polynomial Regression ◽  
Nonlinear Dynamical Systems ◽  
Feedforward Neural Networks ◽  
Data Availability ◽  
Training Data ◽  
Local Error ◽  
Nonlinear Dynamical ◽  
Sparse Polynomial

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of strong correlation between the local error and the maximal singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent.

An Efficient Algorithm for Damage-Coupled Visco-Plastic Fatigue Model

2004 ◽  
Author(s):  
A. H. Zhao ◽  
C. L. Chow
Keyword(s):  
Error Control ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Fatigue Loading ◽  
Material Model ◽  
Tensile Creep ◽  
Local Error ◽  
Single Element ◽  
Numerical Examples ◽  
Uniaxial Test

The paper describes the development of an efficient and robust numerical algorithm for a damage-coupled visco-plastic-fatigue material model. The material chosen for the investigation is a eutectic material, Sn-Pb solder, exhibiting strain-softening behavior. The numerical algorithms employs a modified explicit method with adaptive sub-stepping based on the local error control for which the stress (constitutive) Jacobian explicit solution is derived. The algorithm is implemented in a commercial finite element (FE) code ABAQUS (Version 6.2) via its user-defined material subroutine. The validity of the algorithm is examined with several numerical examples, including (i) single-element simulations for uniaxial test, tensile creep, and fatigue simulations to attain an optimized algorithm, and (ii) two three-dimensional analyses of a miniature specimen under monotonic tensile loading and fatigue loading. The numerical examples illustrate the effectiveness of the modified explicit algorithm in predicting cyclic thermoviscoplastic behavior of a solder material. The algorithm is considered a generalized methodology that can be readily applied characterize thermoviscoplastic behavior and fatigue life of similar materials.

Local error control inSDIRK-methods

1986 ◽  
Vol 26 (1) ◽  
pp. 100-113 ◽  
Author(s):  
Syvert P. Nørsett ◽  
Per G. Thomsen
Keyword(s):  
Error Control ◽  
Local Error

scholarly journals PHASE SPACE ERROR CONTROL WITH VARIABLE TIME-STEPPING ALGORITHMS APPLIED TO THE FORWARD EULER METHOD FOR AUTONOMOUS DYNAMICAL SYSTEMS

2019 ◽  
pp. 16-26
Author(s):  
R. Vigneswaran ◽  
S. Thilaganathan
Keyword(s):  
Fixed Point ◽  
Phase Space ◽  
Dynamical Systems ◽  
Linear System ◽  
Error Control ◽  
Coefficient Matrix ◽  
Euler Method ◽  
Space Error ◽  
Forward Euler ◽  
Forward Euler Method

We consider a phase space stability error control for numerical simulation of dynamical systems. Standard adaptive algorithm used to solve the linear systems perform well during the finite time of integration with fixed initial condition and performs poorly in three areas. To overcome the difficulties faced the Phase Space Error control criterion was introduced. A new error control was introduced by R. Vigneswaran and Tony Humbries which is generalization of the error control first proposed by some other researchers. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In earlier, it was analyzed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. Some theoretical results are obtained and numerical results are given.

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