scholarly journals Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190446 ◽  
Author(s):  
Xiaocheng Shang ◽  
Hans Christian Öttinger
Keyword(s):  
Dissipative Systems ◽  
Symplectic Method ◽  
Accuracy Control ◽  
Structure Preserving ◽  
Thermodynamic Structure ◽  
Degeneracy Condition ◽  
Numerical Integrators ◽  
Frame Work ◽  
Irreversible Dynamics ◽  
Damped Systems

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

Temperature Effect on Dynamic Behaviors of Cis-Polyisoprene Chain

2016 ◽  
Vol 08 (01) ◽  
pp. 1650012
Author(s):  
Weipeng Hu ◽  
Zichen Deng ◽  
Hailin Zou ◽  
Tingting Yin
Keyword(s):  
Temperature Effect ◽  
Dynamic Characteristics ◽  
Symplectic Form ◽  
Volume Effect ◽  
Symplectic Method ◽  
Linear Polymers ◽  
Dynamic Behaviors ◽  
Structure Preserving ◽  
Langevin Model ◽  
Novel Structure

A novel structure-preserving method, named as stochastic generalized multi-symplectic method, is proposed to analyze the temperature effect on the dynamic characteristics hided in the motion of the cis-polyisoprene chain in this paper. Ignoring the dynamic backflow and the exhaust volume effect, the motion of the Gaussian chain in linear polymers can be described as the Langevin model, which can be written into the stochastic generalized multi-symplectic form. For this stochastic generalized multi-symplectic form, a box structure-preserving scheme is constructed to simulate the motion of the cis-polyisoprene chain. From the simulation results, the temperature effects on the dynamic behaviors around the glass transition temperature and the viscous flow temperature of cis-polyisoprene are investigated. The structure-preserving method for analyzing the temperature effect on the dynamic characteristics of the cis-polyisoprene chain presented in this paper proposes a new way to study some dynamic characteristics of complex fluid systems.

scholarly journals Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

2019 ◽  
Vol 21 (07) ◽  
pp. 1850064 ◽  
Author(s):  
Guido Gentile ◽  
Alessandro Mazzoccoli ◽  
Faenia Vaia
Keyword(s):  
Potential Energy ◽  
Finite Dimension ◽  
Dissipative Systems ◽  
Diophantine Condition ◽  
Frequency Vector ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Term ◽  
Degeneracy Condition ◽  
The One

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence of dissipation, so that the forcing term is — up to the sign — the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular, we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exists provided the dissipation is strong enough. This extends results previously available in the literature in the one-dimensional case.

scholarly journals Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

2020 ◽  
Vol 42 (1) ◽  
pp. B273-B298
Author(s):  
Zhengdao Chen ◽  
Baranidharan Raman ◽  
Ari Stern
Keyword(s):  
Type Systems ◽  
Structure Preserving ◽  
Numerical Integrators

scholarly journals A symplectic method for structure-preserving modelling of damped acoustic waves

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150105 ◽  
Author(s):  
Xiaofan Li ◽  
Mingwen Lu ◽  
Shaolin Liu ◽  
Shizhong Chen ◽  
Huan Zhang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Numerical Simulations ◽  
Seismic Wave ◽  
Wave Equations ◽  
Superior Performance ◽  
Symplectic Method ◽  
Structure Preserving ◽  
Long Time

In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

GENERALIZED MULTI-SYMPLECTIC METHOD FOR DYNAMIC RESPONSES OF CONTINUOUS BEAM UNDER MOVING LOAD

2013 ◽  
Vol 05 (03) ◽  
pp. 1350033 ◽  
Author(s):  
WEIPENG HU ◽  
ZICHEN DENG ◽  
HUAJIANG OUYANG
Keyword(s):  
Moving Load ◽  
Dynamic Responses ◽  
Viscous Damping ◽  
Symplectic Method ◽  
Continuous Beam ◽  
Local Conservation ◽  
Structure Preserving ◽  
Symplectic Scheme ◽  
Continuous Beams ◽  
Conservation Property

Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multi-symplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths Δt = 0.05 and Δx = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.

Variational Integrators for Dissipative Systems With One Degree of Freedom

2011 ◽  
Author(s):  
Masashi Iura
Keyword(s):  
Mixed Method ◽  
Degree Of Freedom ◽  
Dissipative Systems ◽  
Variational Integrators ◽  
Stationary Condition ◽  
Coupling Term ◽  
D’Alembert Principle ◽  
Discrete Algorithms ◽  
Numerical Integrators ◽  
Integration Techniques

Variational integrators are developed for dissipative systems with one degree of freedom. The dissipation considered herein is of simple Rayleigh dissipation type. The present formulation is based not on the Lagrange-d’Alembert principle, but on Hamilton’s principle. A benefit for using variational integration techniques is stressed in this paper. The discrete algorithms are obtained by a stationary condition of action integral, in which the Lagrangian is directly discretized. Unlike the existing algorithms, a coupling term between mass and dissipation exists in the present algorithms. A mixed method, in which a velocity is independent on a position coordinate, is presented for dissipative systems. In order to investigate an accuracy of numerical integrators, we introduce a new parameter in addition to the energy decay. Numerical examples show that the present variational, integrators are available for not only highly but also weakly dissipative systems.

C. The Continuous Spectrum of Pulsating Variable Stars

1967 ◽  
Vol 28 ◽  
pp. 177-206
Author(s):  
J. B. Oke ◽  
C. A. Whitney
Keyword(s):  
Continuous Spectrum ◽  
Variable Stars ◽  
Velocity Fields ◽  
The Other ◽  
Line Spectrum ◽  
Direct Information ◽  
Thermodynamic Structure ◽  
Diagnostic Interpretation ◽  
Pulsating Variable Stars ◽  
The Continuum

Pecker:The topic to be considered today is the continuous spectrum of certain stars, whose variability we attribute to a pulsation of some part of their structure. Obviously, this continuous spectrum provides a test of the pulsation theory to the extent that the continuum is completely and accurately observed and that we can analyse it to infer the structure of the star producing it. The continuum is one of the two possible spectral observations; the other is the line spectrum. It is obvious that from studies of the continuum alone, we obtain no direct information on the velocity fields in the star. We obtain information only on the thermodynamic structure of the photospheric layers of these stars–the photospheric layers being defined as those from which the observed continuum directly arises. So the problems arising in a study of the continuum are of two general kinds: completeness of observation, and adequacy of diagnostic interpretation. I will make a few comments on these, then turn the meeting over to Oke and Whitney.

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

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