EQUATION-FREE, EFFECTIVE COMPUTATION FOR DISCRETE SYSTEMS: A TIME STEPPER BASED APPROACH

2005 ◽  
Vol 15 (03) ◽  
pp. 975-996 ◽  
Author(s):  
J. MÖLLER ◽  
O. RUNBORG ◽  
P. G. KEVREKIDIS ◽  
K. LUST ◽  
I. G. KEVREKIDIS
Keyword(s):  
Evolution Equations ◽  
Time Integration ◽  
Discrete Systems ◽  
Computer Assisted ◽  
Discrete Problem ◽  
Arc Length ◽  
Effective Equation ◽  
Discrete Evolution Equations ◽  
Coarse Model ◽  
The Continuum

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

Comparison of Two Time-Integration Algorithms for an Anisotropic Damage Model Coupled with Plasticity

2015 ◽  
Vol 784 ◽  
pp. 292-299 ◽  
Author(s):  
Stephan Wulfinghoff ◽  
Marek Fassin ◽  
Stefanie Reese
Keyword(s):  
Evolution Equations ◽  
Damage Model ◽  
Time Integration ◽  
Three Dimensional ◽  
Anisotropic Damage ◽  
Euler Scheme ◽  
The Finite Element Method ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Integration Algorithms

In this work, two time integration algorithms for the anisotropic damage model proposed by Lemaitre et al. (2000) are compared. Specifically, the standard implicit Euler scheme is compared to an algorithm which implicitly solves the elasto-plastic evolution equations and explicitly computes the damage update. To this end, a three dimensional bending example is solved using the finite element method and the results of the two algorithms are compared for different time step sizes.

scholarly journals Parareal with a learned coarse model for robotic manipulation

2020 ◽  
Vol 23 (1-4) ◽  
Author(s):  
Wisdom Agboh ◽  
Oliver Grainger ◽  
Daniel Ruprecht ◽  
Mehmet Dogar
Keyword(s):  
Time Integration ◽  
Robotic Manipulation ◽  
Multiple Objects ◽  
Initial State ◽  
Link Type ◽  
Planning And Control ◽  
Physics Model ◽  
Coarse Model ◽  
Computational Bottleneck ◽  
And Control

AbstractA key component of many robotics model-based planning and control algorithms is physics predictions, that is, forecasting a sequence of states given an initial state and a sequence of controls. This process is slow and a major computational bottleneck for robotics planning algorithms. Parallel-in-time integration methods can help to leverage parallel computing to accelerate physics predictions and thus planning. The Parareal algorithm iterates between a coarse serial integrator and a fine parallel integrator. A key challenge is to devise a coarse model that is computationally cheap but accurate enough for Parareal to converge quickly. Here, we investigate the use of a deep neural network physics model as a coarse model for Parareal in the context of robotic manipulation. In simulated experiments using the physics engine Mujoco as fine propagator we show that the learned coarse model leads to faster Parareal convergence than a coarse physics-based model. We further show that the learned coarse model allows to apply Parareal to scenarios with multiple objects, where the physics-based coarse model is not applicable. Finally, we conduct experiments on a real robot and show that Parareal predictions are close to real-world physics predictions for robotic pushing of multiple objects. Code (https://doi.org/10.5281/zenodo.3779085) and videos (https://youtu.be/wCh2o1rf-gA) are publicly available.

Numerical Implementation of Continuum Dislocation Dynamics with the Discontinuous-Galerkin Method.

2014 ◽  
Vol 1651 ◽  
Author(s):  
Alireza Ebrahimi ◽  
Mehran Monavari ◽  
Thomas Hochrainer
Keyword(s):  
Discontinuous Galerkin ◽  
Crystal Plasticity ◽  
Evolution Equations ◽  
Total Curvature ◽  
Time Integration ◽  
Three Dimensional ◽  
Conserved Quantity ◽  
Dislocation Dynamics ◽  
Slip Systems ◽  
Continuum Dislocation Dynamics

ABSTRACTIn the current paper we modify the evolution equations of the simplified continuum dislocation dynamics theory presented in [T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, Continuum dislocation dynamics: Towards a physical theory of crystal plasticity. J. Mech. Phys. Solids. (in print)] to account for the nature of the so-called curvature density as a conserved quantity. The derived evolution equations define a dislocation flux based crystal plasticity law, which we present in a fully three-dimensional form. Because the total curvature is a conserved quantity in the theory the time integration of the equations benefit from using conservative numerical schemes. We present a discontinuous Galerkin implementation for integrating the time evolution of the dislocation state and show that this allows simulating the evolution of a single dislocation loop as well as of a distributed loop density on different slip systems.

Exponential Time Integration of Evolution Equations

2010 ◽  
Author(s):  
Alexander Ostermann ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Evolution Equations ◽  
Time Integration ◽  
Exponential Time ◽  
Exponential Time Integration

scholarly journals Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 452
Author(s):  
T. Körpınar ◽  
R. Cem Demirkol ◽  
Z. Körpinar ◽  
V. Asil
Keyword(s):  
Magnetic Flux ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Curvilinear Coordinates ◽  
Arc Length ◽  
Different Types ◽  
Orthogonal Curvilinear Coordinates ◽  
Necessary And Sufficient ◽  
Physical Spacetime ◽  
B Lines

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.

scholarly journals GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems

2018 ◽  
Vol 43 (2) ◽  
pp. 89-100 ◽  
Author(s):  
Hans Christian Öttinger
Keyword(s):  
Continuous Time ◽  
Evolution Equations ◽  
Time Integration ◽  
Mathematical Structure ◽  
Symplectic Integrators ◽  
Structural Elements ◽  
Structure Preserving ◽  
Damped Harmonic Oscillator ◽  
Alternative Approaches ◽  
Non Equilibrium

AbstractThermodynamically admissible evolution equations for non-equilibrium systems are known to possess a distinct mathematical structure. Within the GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) framework of non-equilibrium thermodynamics, which is based on continuous time evolution, we investigate the possibility of preserving all the structural elements in time-discretized equations. Our approach, which follows Moser’s [1] construction of symplectic integrators for Hamiltonian systems, is illustrated for the damped harmonic oscillator. Alternative approaches are sketched.

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