Implicit Integration in Molecular Dynamics Simulation

2008 ◽  
Author(s):  
Nicholas P. Schafer ◽  
Radu Serban ◽  
Dan Negrut
Keyword(s):  
Molecular Dynamics ◽  
Newton Method ◽  
Dynamics Simulation ◽  
Integration Scheme ◽  
Implicit Integration ◽  
Step Size ◽  
Time Step ◽  
Integration Methods ◽  
Size Limitation ◽  
Quasi Newton

Molecular Dynamics (MD) simulation is a versatile methodology that has found many applications in materials science, chemistry and biology. In biology, the models employed range from mixed quantum mechanical and fully atomistic to united atom and continuum mechanical. These systems are evolved in discrete time by solving Newton’s equations of motion at each time step. The numerical methods currently in use limit the step size of a typical all atom simulation to 1 femtosecond. This step size limitation means that many steps need to be taken in order to reach biologically relevant time scales. At each time step, an evaluation of the forces on each atom must be performed resulting in heavy computational loads. This work investigates the use of implicit integration methods in MD. Implicit integration methods have been proven superior to their explicit counterparts in classical mechanical simulation, with which MD has many similarities. Longer time steps reduce the number of force evaluations that must be performed and the corresponding computational load. Herein we present results that compare implicit integration techniques with the current standard for molecular dynamics, the explicit velocity Verlet integration scheme. Total energy conservation is used as a metric for evaluating the dependability of simulations in the microcanonical ensemble. In order to understand the nature of the problem, several long simulations were run and analyzed by performing a Fourier analysis on the position, velocity and acceleration signals. Lastly, several methods for improving the viability of implicit integration methods are considered including replacing the Jacobian used in the Quasi-Newton method with a constant, diagonal mass matrix, evaluating the Jacobian infrequently and finding a better prediction of the system configuration to improve the convergence of the Quasi-Newton method.

A Quantitative Assessment of the Potential of Implicit Integration Methods for Molecular Dynamics Simulation

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Nick Schafer ◽  
Dan Negrut
Keyword(s):  
Molecular Dynamics ◽  
Md Simulation ◽  
Time Integration ◽  
Dynamics Simulation ◽  
Integration Step ◽  
Implicit Integration ◽  
Implicit Methods ◽  
Current Standard ◽  
Step Size ◽  
Computational Overhead

Implicit integration, unencumbered by numerical stability constraints, is attractive in molecular dynamics (MD) simulation due to its presumed ability to advance the simulation at large step sizes. It is not clear what step size values can be expected and if the larger step sizes will compensate for the computational overhead associated with an implicit integration method. The goal of this paper is to answer these questions and thereby assess quantitatively the potential of implicit integration in MD. Two implicit methods (midpoint and Hilber–Hughes–Taylor) are compared with the current standard for MD time integration (explicit velocity Verlet). The implicit algorithms were implemented in a research grade MD code, which used a first-principles interaction potential for biological molecules. The nonlinear systems of equations arising from the use of implicit methods were solved in a quasi-Newton framework. Aspects related to a Newton–Krylov type method are also briefly discussed. Although the energy conservation provided by the implicit methods was good, the integration step size lengths were limited by loss of convergence in the Newton iteration. Moreover, a spectral analysis of the dynamic response indicated that high frequencies present in the velocity and acceleration signals prevent a substantial increase in integration step size lengths. The overhead associated with implicit integration prevents this class of methods from having a decisive impact in MD simulation, a conclusion supported by a series of quantitative analyses summarized in the paper.

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.

Investigation of Fractional Characteristic of Molecular Motion by Molecular Dynamics Simulation

2004 ◽  
Author(s):  
Chao Liu ◽  
Liming Wan ◽  
Xinming Zhang ◽  
Danling Zeng
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Vapor Phase ◽  
Molecular Motion ◽  
Mean Free Path ◽  
Dynamics Simulation ◽  
Fractional Dimension ◽  
Time Step ◽  
Vapor Interface ◽  
Liquid Vapor

Molecular dynamics simulation (MDS) is adopted to investigate the characteristic of fractional motion of molecules in liquid phase, vapor phase and liquid-vapor interface in the paper. Based on the theory of mean free path and Shannon sampling theorem, the way to determine a universal criterion of time step of simulation is presented. It is shown that there exists difference in the regular pattern of molecular motion in the state of liquid and vapor phase. The fractional features are different for different matter states. Under the condition of same temperature, the characteristic fractional number of molecular motion in liquid state is greater than one in vapor state. It is shown that the fractional dimension numbers in the X, Y and Z direction of the liquid-vapor interface are different. This proves that the liquid-vapor interface has anisotropic character.

A Numerical Coning Model

1970 ◽  
Vol 10 (04) ◽  
pp. 418-424 ◽  
Author(s):  
J.P. Letkeman ◽  
R.L. Ridings
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Two Dimensions ◽  
Solution Technique ◽  
Production Rates ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Size Limitation ◽  
Single Well

Abstract The numerical simulation of coning behavior bas been one of the most difficult applications of numerical analysis techniques. Coning simulations have generally exhibited severe saturation instabilities in the vicinity of the well unless time-step sizes were severely restricted. The instabilities were a result of using mobilities based on saturations existing at the beginning of the time step. The time-step size limitation, usually the order of a few minutes, resulted in an excessive amount of computer time required to simulate coning behavior. This paper presents a numerical coning model that exhibits stable saturation and production behavior during cone formation and after breakthrough. Time-step sizes a factor of 100 to 1,000 times as large as those previously possible may be used in the simulation. To ensure stability, both production rates and mobilities are extrapolated production rates and mobilities are extrapolated implicitly to the new time level. The finite-difference equations used in the model are presented together with the technique for incorporating the updated mobilities and rates. Example calculations which indicate the magnitude of the time-truncation errors are included. Various factors which affect coning behavior are discussed. Introduction The usual formulation of numerical simulation models for multiphase flow involves the evaluation of flow coefficient terms at the beginning of a time step and assumes that these terms do not change over the time step. These assumptions are valid only if the values of pressure and saturation in the system do not change significantly over the time step. The design of a finite-difference model to evaluate coning behavior of gas or water in a single well usually results in a model which uses radial coordinates. A two-dimensional single-well model is illustrated in Fig. 1. This type of model will often produce finite-difference blocks with pore volumes less than 1 bbl near the wellbore while producing large blocks with pore volumes greater producing large blocks with pore volumes greater than 1 million bbl near the external radius. If one chooses to use a reasonable time-step size of, say, 1 to 10 days, then normal well rates would result in a flow of several hundred pore volumes per time step through blocks near the wellbore. Therefore the assumption that saturations remain constant, for the purpose of coefficient evaluation, is not valid. Welge and Weber presented a paper on water coning which recognized the limitation of using explicit coefficients and applied an arbitrary limitation on the maximum saturation change over a time step. While this method is workable for a certain class of problems, it is not rigorous and is not generally applicable. In 1968, Coats proposed a method to solve the gas percolation problem which is similar in that it also results from explicit mobilities. This proposal involved adjusting the relative permeability to gas at the beginning of the time step so that an individual block would not be over-depleted of gas during a time step. This method is not conveniently extended to two dimensions nor to coning problems where a block is voided many times during a time step. Blair and Weinaug explored the problems resulting from explicitly determined coefficients and formulated a coning model with implicit mobilities and a solution technique utilizing Newtonian iteration. While this method is rigorous, achieving convergence on certain problems is difficult and, in many cases, time-step size is still severely restricted. In addition to the problems resulting from explicit flow-equation coefficients in coning models, the specification of rates requires attention to ensure that the saturations remain stable in the vicinity of the producing block. SPEJ P. 418

The Recent Developments of Molecular Dynamics Simulation

2013 ◽  
Vol 444-445 ◽  
pp. 1370-1373
Author(s):  
Wen Hai Gai ◽  
Ran Guo ◽  
Yuan Yuan Liu
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Rapid Development ◽  
Equations Of Motion ◽  
Dynamics Simulation ◽  
Impact Factors ◽  
Lagrange Equations ◽  
Time Step ◽  
Basic Principles ◽  
Recent Developments

Based on the development of nanomaterials and the research on performance parameters of materials, molecular dynamics simulation has been rapid development and application. It is widely found that the material's physical, mechanical and other properties are both closely related to its macroscopic state and microstructure [. In order to explore and understand the nature of the material properties we need to analyze various impact factors including macroscopic, mesoscopic and microscopic. This paper describes the basic concepts and methods of molecular dynamics. The contents are comprised of time step, formulas such as Lagrange equations of motion and Hamiltonian equations of motion. The basic principles and recent developments of molecular dynamics were reviewed.

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.

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