Seminar 1: Efficient Algorithms for Molecular Dynamics Simulation

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Seminar 1Efficient Algorithms for Molecular
Dynamics Simulation

• Andrew Noske

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About the Project

• Part of TOMSK (Towards Molecular Structure
  Kinetics)
• Build an engine layer for particle simulations

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About Molecular Simulations

• Real molecules obey quantum laws but can
  approximate with classical laws.
• e.g. Newtons law forcemassacceleration
• Molecular Dynamics (MD) integrates equations of
  motion of atoms each timestep.
• Cannot predict precisely what will happen
  generates statistical prediction.

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Chosen Interaction Model

• Lennard Jones Potential Pair Equation
• I will use this to simulateatoms in a stable
  liquid.
• NOTE Will be easy to overwrite/change
  interaction model add statistical analysis
  functions (e.g. calc Temperature).

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Simulating Liquids

• Can only simulate so many 1000s of particles.
• To simulate bulk liquidI will use Periodic
  Bounding Condition (PBC) ? (boundaries wrap
  around)

Surface particle has less neighbors
Range search on box with PBC
Microscopic droplet (finite particles)
PBC on 2D box
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N-body Problem

• N-body problem all (N) particles in a system
  have pair-wise interaction.
• Solutions
• Brute force approach compare all pairs ? O(N2)
• Better approach approximate distant forces
• Lead to many specific solutions.

Consider ALL
Approximate
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Chosen Solution
Single Range Query

• Chosen approach chose cutoff radius, and ignore
  particles beyond this.
• Involves moving self-spatial join query (many
  range queries) ? has numerous applications
• GIS, Computer graphics, etc.

Ignore
Cutoff radius (Rc)
Spatial join query
Symmetrical attractive/ repulsive forces
O--
NOTEDirection forces!

H

H
Permanent dipole
Argon atom (inert)
Water molecule




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Spatial Data Structure Fixed Grid

• Reviewed many types of structures.
• Fixed grid most effective for uniform particle
  distribution.
• Time to build (placeall pointsinto index)
  O(N)
• cell index atom coordinate / cellLen
• (along each dimension)

Fixed grid
NOTE cells per side (CPS)5
boxLen
Cutoff radius (rc)
cellLen
Cell List technique
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Scientific Process

• Using Visual C console application.
• Am using OO principles.
• Have read Effective C now reading More
  Effective C.
• Testing process
• Run a series of simulations in batch
• Output results to CSV including
• grid parameters,
• clock tics elapsed, ? primary focus
• distance calculates,
• more
• Analyze/graph CSV using Excel.
• Can be time consuming! ?

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Simulation Steps

• Set-up
• 1) Setup grid structure
• 2) Setup atoms in offset lattice
• 3) Assign random velocities.
• Iterate
• 1) Build grid (assign all atoms to cells).
• 2) Build neighbor list (for each atom in each
  cell find neighbors). ? can take 95 of time
• 3) Calculate force and move atoms ? implements
  interaction model (can change).
• 4) Wrap atoms back into cell boundaries.
• 5) Increment timestep.

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Finding Good Neighbours
Love thy neighbour -- the bible
Cell list approach Predetermine which cells are
within rc of each cell
Atom list approach For each atom check which
cells are within rc of atom

• NOTE Volume sphere 52 of its bounding cube

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Optimization Trick Half-sphere Query
Normal approach Search sphere
Faster approach Search upper hemi-sphere
i
NOTE will capture each neighbor twice (once from
each end)
j
i
j
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Smaller Optimization Tricks

• Early elimination if distancebetween atoms along
  anydimension gt rc.
• Dont calculate sqrt
• Lennard-Jones can be doneusing dist2
• NOTE if (dist2 lt cutoffRadius2)
  ? is in range
• Determine optimal of cells per size.
• Is cell length cutoff radius optimal?

j
i
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Idea Using MBRs in cells

• For each cell, maintain a Minimum Bounding
  Rectangle (MBR) around its atoms.
• For any cell JUST tipped by rc, check atom is
  inside rc of MBR before considering atoms
  exhaustively.

This cell is just tipped

• NOTE can also beused inconjunction with sub
  grid

MBR
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Improving Cache Hits through Spatial Locality

• Spatial locality principle objects close to
  referred ones will probably be requested again in
  the future.
• Unsorted atoms ? means many cache misses.

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5
2
3
6
1
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Space Filling Curves

• Space-filling curve A line passing through every
  point in a space, in some order (according to
  some algorithm).
• Resort atom periodically (group by cells order
  using curve).
• Improves CPU performance gt50 in 2D moving point
  query ? worth trying.

Z-ordering
Row-wise
Hilbert curve
Gray curve
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Verlet Neighbour List

• Choose a skin radius greater than rc.
• Build the verlet neighbor list using skin
  radius
• Next few iterations update list check which
  neighbour pairs are inside rc.
• Refinement only rebuild list when sum of 2 max
  displacement (of any 2 atoms) gt skin thickness.

Skin/verlet radius (Rv)
5
Rl
6
7
6'
7'
1
Cut-off sphere
Rc
2
3
Skin
4
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Some Verlet Specific Ideas

• Dont check displacement each iteration check at
  decreasing periods.
• Dont update distances of atoms outside cut-off
  sphere each iteration check more frequently as
  it gets closer.
• Determine optimal skin radius.

Skin/verlet radius (Rv)
5
Rl
6
7
6'
7'
1
Cut-off sphere
Rc
2
3
Skin
4
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Performance vs. Accuracy

• Some techniques that will improve performance,
  but decrease accuracy
• Dont always rebuild/update neighbor list when
  necessary.
• Increasing timestep
• Decreasing cutoff radius
• I can graph these by testing against a control.

small timestep
larger timestep
large rc
smaller rc
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Progress

• Basic grid is implemented
• Thesis started.
• Several graphs obtained.
• Added front end
• Uses MFC (Microsoft Foundation Class) using
  OpenGL.
• Demonstrates building on engine layer.
• Lets me see particles animate ( work out
  problems)
• Code messy
• Is still MUCH to implement test.

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Conclusion

• Molecular dynamics an (approximated)
  simulation of real worldparticle behaviour.
• Periodic Bounding Condition used for bulk
  liquid
• Best existing approach
• Verlet list, using cell list fixed grid to
  build.
• Ideas for improvement
• Minimal half cell list templates.
• MBRs inside cells.
• Space filling curves.
• Thesis may resemble a guide to implementing/optimi
  zing moving spatial join queries.
• Still much to be done.

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Thank YouAny Questions?
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Revised Goal
EXTRA SLIDE

• Implement engine as efficient as possible.
• How it should work
• Pass in minimum parameters
• atoms, offset, max velocity box size(or load
  atom data from file).
• cutoff radius
• Engine should set remaining details to optimal.
  Including
• cells per side,
• verlet radius,
• subgrid? mbr? algorithm? etc.

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Idea Sub Grid
EXTRA SLIDE

• Sub grid adjacency list template guide break
  each cell into (imaginary) sub-grid.
• When considering an atom, check with sub-cell it
  belongs to, that sub-cell will refine which
  adjacent cells to search.
• NOTE
• My primary focus is reducing time per
  iterationbut main memory requirements ( setup
  time) also important.
• Choosing cell length equal to rc is typical.
  But is it optimal?
• No one method/metric is likely to be optimal in
  ALL cases ? too many parameters.