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Research Papers

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

[+] Author and Article Information
Nick Schafer

Department of Mechanical Engineering and Materials Science Program, University of Wisconsin, 1513 University Avenue, Madison, WI 53706-1572npschafer@wisc.edu

Dan Negrut1

Department of Mechanical Engineering and Materials Science Program, University of Wisconsin, 1513 University Avenue, Madison, WI 53706-1572negrut@wisc.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(3), 031012 (Jun 15, 2010) (10 pages) doi:10.1115/1.4001392 History: Received August 04, 2009; Revised March 09, 2010; Published June 15, 2010; Online June 15, 2010

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.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Proteins used in numerical experiments: (a) alanin, (b) da, and (c) ubiquitin

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Figure 2

Alanin: total energy versus time step. Δt=10−15 and tend=10−11 s.

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Figure 3

Energy error versus step size, alanin

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Figure 4

Visualization of the HHT Jacobian during a typical time step for alanin. In (a) above, the vertical axis reports the absolute value of a Jacobian Jij entry (the sensitivity). On the horizontal axes are the i and j indices for each Jij entry. The contour plot shows that the diagonal elements dominate the off-diagonal elements. Figure (b) shows the nonzero pattern generated in MATLAB . Although the off-diagonal elements are small, most are not zero because of the long range interactions between atoms.

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Figure 5

Velocity Verlet Fourier transform plots, alanin: (a) Hydrogen position signal, (b) Hydrogen position power spectrum, (c) Oxygen velocity signal, (d) Oxygen velocity power spectrum, (e) Nitrogen acceleration signal, (f) Nitrogen acceleration power spectrum. The step size used was 0.125 fs.

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Figure 6

HHT Fourier transform plots, alanin: (a) Hydrogen position signal, (b) Hydrogen position power spectrum, (c) Oxygen velocity signal, (d) Oxygen velocity power spectrum, (e) Nitrogen acceleration signal, and (f) Nitrogen acceleration power spectrum. The step size used was 0.125 fs.

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Figure 7

Spectral structure of iterative matrix M−1J for alanin at various step sizes: (a) 1 fs, (b) 2 fs, (c) 4 fs, and (d) 8 fs. The horizontal axis gives the real part of each eigenvalue and the vertical axis gives the imaginary part. For all values of the step size, the eigenvalues are purely real and tend to be clustered on the real axis around 1.0.

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