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Title: LINCS: A Linear Constraint Solver for Molecular Simulations


1
LINCS A Linear Constraint Solver for Molecular
Simulations
  • Berk Hess, Hemk Bekker, Herman J.C.Berendsen,
    Johannes G.E.M.Fraaije
  • Journal of Computational Chemistry, 1997

Ankur Dhanik
2
Outline
  • Introduction to Molecular Dynamics
  • Problem description
  • Some solutions
  • LINCS
  • Results

3
Introduction to Molecular Dynamics
  • A classical molecular simulations method
  • Successive configurations of the system are
    generated by integrating Newtons laws of motion
  • Integration algorithms should strive to reduce
    computation and conserve energy
  • Various integration algorithms
  • Standard Verlet algorithm
  • Leap-frog algorithm

4
Introduction to Molecular Dynamics
  • Standard Verlet Algorithm
  • r(t?t) 2r(t) - r (t-?t) ?t2a(t)
  • v(t) r(t?t) - r(t-?t)/2?t
  • Velocities are not directly generated, and are
    one time step behind
  • Leap-frog algorithm
  • r(t?t) r(t) ?tv(t?t/2)
  • v(t?t/2) v(t-?t/2) ?ta(t)
  • v(t) v(t?t/2) v(t-?t/2)/2
  • Velocities are half time step behind

5
Introduction to Molecular Dynamics
  • Choosing the time step
  • One order of magnitude smaller than the shortest
    motion (bond vibrations)
  • Severe restriction as these high frequency
    motions have minimal effect on the overall
    behavior of the system
  • Constrained dynamics

6
Introduction to Molecular Dynamics
System Type of motion present Suggested times step(s)
Atoms translation 10-14
Rigid molecules Translation and rotation 5 X 10-15
Rigid molecules, rigid bonds Translation, rotation, torsion 2 X 10-15
Rigid molecules, flexible bonds Translation, rotation, torsion, vibration 10-15or 5 X 10-16
The different types of motion present in various
systems together with suggested time steps
7
Introduction to Molecular Dynamics
  • In constrained dynamics bonds and angles are
    forced to adopt specific values throughout a
    simulation
  • Constraints are categorized as holonomic and
    non-holonomic.
  • Suppose a particle on the surface of a sphere
  • r2 a2 0 Holonomic
  • r2 a2 gt 0 Non-holonomic
  • In a constrained system
  • Particles are not independent
  • Magnitude of constraint forces are unknown

8
Introduction to Molecular Dynamics
  • Integration algorithms
  • Standard Verlet Algorithm
  • Leap-frog algorithm
  • Choosing time step
  • Constrained dynamics
  • Holonomic and non-holonomic constraints

9
Outline
  • Introduction to Molecular Dynamics
  • Problem description
  • Some solutions
  • LINCS
  • Results

10
Problem Description
  • Design an algorithm for solving constrained
    molecular dynamics, the constraints being
    holonomic
  • The algorithm should strive for following
    features
  • Numerical stability
  • Energy conservation
  • Computational efficiency

11
Some solutions
  • Reset coupled constraints after an unconstrained
    update
  • Non-linear problem
  • SETTLE
  • Solves analytically
  • Very fast, but unsuited for large molecules
  • SHAKE
  • Iterative method
  • Sequentially all the bonds are set to the correct
    length
  • Simple and numerically stable
  • No solutions may be found when displacements are
    large, difficult to parallelize

12
Some solutions
  • EEM
  • The second derivatives of constraint equations
    are set to zero
  • All the constraints are dealt with simultaneously
  • Corrections are necessary to achieve accuracy and
    stability

13
LINCS
  • Does an unconstrained update
  • Sets the projection of the new bonds onto the old
    directions of the bonds to the prescribed lengths
  • Similar to EEM, with some practical differences
  • Implements
  • Efficient solver for the matrix equation
  • A velocity correction that prevents rotational
    lengthening
  • A length correction that improves accuracy and
    stability

14
LINCS
15
LINCS
Newtons equation of motion
(I-TB) is projection matrix which sets the
constrained coordinates to zero. T transforms
motions in the constrained coordinates into
motions in cartesian coordinates
16
LINCS
17
LINCS
  • Numerical errors can accumulate which leads to
    drifts

Velocity correction
Position correction
  • Drawback the projection of the new bonds onto
    the old directions rather than the new bonds are
    set to prescribed lengths

18
LINCS
  • Correction for rotational lengthening
  • To correct rotation of bond i, the projection of
    the bond on the old direction is set to
  • The corrected positions are

19
LINCS
Constrained new position is given by
20
LINCS
  • The first power of An gives the coupling effects
    of neighboring bonds
  • The second power gives the coupling effect over a
    distance of two bonds
  • The inversion through a series expansion makes
    parallelization easy
  • In one timestep, the bonds influence each other
    when they are separated by fewer bonds than
    highest order in expansion

21
LINCS
  • Parallelization
  • Consider a linear-bond constrained molecule of
    100 atoms to be simulated on a dual processor
    computer
  • Uses rotation correction and an expansion to the
    second power of An
  • Because order of expansion is two, bonds
    influence each other over a distance of 6
  • Update of position and call of LINCS algorithm
    must be done for atom 1-56 and 44-100 on
    processors 1 and 2 respectively
  • 1-50 update from processor 1 and 50-100 from
    processor 2

22
Results
23
Results
  • Solves constrained molecular dynamics
  • Numerically stable
  • Conserves energy
  • Three to four times faster than SHAKE
  • Can be easily parallelized
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