Seminar 1: Efficient Algorithms for Molecular Dynamics Simulation Andrew Noske.

Seminar 1:Seminar 1:

Efficient Algorithms for Efficient Algorithms for Molecular Dynamics Molecular Dynamics

SimulationSimulation

Andrew NoskeAndrew Noske

About the ProjectAbout the Project

Part of Part of TOMSKTOMSK (Towards Molecular (Towards Molecular Structure Kinetics)Structure Kinetics)

Build an “Build an “engine layerengine layer” for particle ” for particle simulationssimulations

About Molecular SimulationsAbout Molecular Simulations

Real molecules obey Real molecules obey quantum lawsquantum laws… but … but can approximate with can approximate with classical lawsclassical laws..

e.g.: e.g.: Newton’s lawNewton’s law: force=mass*acceleration: force=mass*acceleration

Molecular Dynamics (MD)Molecular Dynamics (MD): : integrates equations of motion integrates equations of motion of atoms each of atoms each timesteptimestep.. Cannot predict precisely what will Cannot predict precisely what will

happen: generates statistical prediction.happen: generates statistical prediction.

Chosen Interaction ModelChosen Interaction Model

Lennard Jones Potential Pair EquationLennard Jones Potential Pair Equation::

I will use this to simulateI will use this to simulateatoms in a stable liquidatoms in a stable liquid..

NOTE: Will be easy to overwrite/change NOTE: Will be easy to overwrite/change interaction modelinteraction model & add statistical analysis functions (e.g.: calc & add statistical analysis functions (e.g.: calc Temperature).Temperature).

Simulating LiquidsSimulating Liquids

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PBC on 2D box

Range search on box with PBC

Microscopic droplet(finite particles)

Can only simulate so many 1000’s of particles.Can only simulate so many 1000’s of particles.To simulate To simulate bulk liquidbulk liquid::I will use I will use Periodic Bounding Condition (PBC)Periodic Bounding Condition (PBC) (boundaries wrap around)(boundaries wrap around)

Surface particle has

less neighbors

N-body ProblemN-body Problem

N-body problemN-body problem: all (: all (NN) particles in a system ) particles in a system have pair-wise interaction.have pair-wise interaction.

Solutions:Solutions: Brute force approachBrute force approach

compare all pairs compare all pairs O(N O(N22)) Better approachBetter approach: approximate: approximate

distant forces distant forcesLead to many Lead to many specific solutions.specific solutions.

Consider ALL

Approximate

Chosen SolutionChosen Solution

Chosen approachChosen approach: chose : chose cutoff radiuscutoff radius, and ignore , and ignore particles beyond this.particles beyond this. Involves: Involves: moving self-spatial join querymoving self-spatial join query (many (many range range

queriesqueries) ) has numerous applications: has numerous applications:GIS, Computer graphics, etc.GIS, Computer graphics, etc.

Ignore

O--

H+ H+

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Permanent dipole

Water molecule

Argonatom(inert)

Symmetrical attractive/ repulsive

forces

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– ––

Cutoff radius (Rc)

Spatial join query:

NOTE:Direction forces!

Single Range Query

Spatial Data Structure: Fixed GridSpatial Data Structure: Fixed Grid

Reviewed many types of structures.Reviewed many types of structures.

Fixed gridFixed grid most effective for uniform particle most effective for uniform particle distribution.distribution. Time to Time to buildbuild (place (place

all pointsall pointsinto index) = O(N)into index) = O(N)

cell index = atom coordinate / cellLencell index = atom coordinate / cellLen

(along each dimension)(along each dimension)

Cell List technique

Cutoff radius(rc)

Fixed grid

cellLen

boxLen

rc

NOTE: cells per side (CPS)=5

Scientific ProcessScientific Process

Using Using Visual C++ console Visual C++ console applicationapplication..

Am using OO principles.Am using OO principles. Have read “Effective C++” & now Have read “Effective C++” & now

reading “More Effective C++”.reading “More Effective C++”.

Testing processTesting process:: Run a series of simulations in batch…Run a series of simulations in batch… Output results to Output results to CSVCSV… including:… including:

grid parameters,grid parameters,clock tics elapsed, clock tics elapsed, primary focus primary focusdistance calculates,distance calculates,& more& more

Analyze/graph CSV using Analyze/graph CSV using ExcelExcel.. Can be time consuming! Can be time consuming!

Simulation StepsSimulation Steps

Set-up:Set-up: #1) Setup grid structure#1) Setup grid structure #2) Setup atoms in offset lattice #2) Setup atoms in offset lattice #3) Assign random velocities. #3) Assign random velocities.

Iterate:Iterate: #1) #1) Build gridBuild grid (assign all atoms to cells). (assign all atoms to cells). #2) #2) Build neighbor listBuild neighbor list (for each atom in (for each atom in

each cell: find neighbors). each cell: find neighbors). can take 95% of time can take 95% of time #3) #3) Calculate force and move atomsCalculate force and move atoms implements implements

interaction model (can change).interaction model (can change). #4) Wrap atoms back into cell boundaries.#4) Wrap atoms back into cell boundaries. #5) Increment timestep.#5) Increment timestep.

Finding Good NeighboursFinding Good Neighbours

q

““Love thy neighbour” -- the bibleLove thy neighbour” -- the bible

q

q

Atom list approach:For each atom: check which cells are within

rc of atom

Cell list approach:Predetermine which cells are

within rc of each cell

NOTE: Volume sphere = NOTE: Volume sphere = 52%52% of it’s bounding cube of it’s bounding cube

Optimization Trick: Half-sphere QueryOptimization Trick: Half-sphere Query

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Normal approach:

Search sphere

Faster approach:

Search upper hemi-sphere

NOTE: will capture each

neighbor twice (once from each end)

Smaller Optimization TricksSmaller Optimization Tricks

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Early elimination if distanceEarly elimination if distancebetween atoms along anybetween atoms along anydimension > rc.dimension > rc.

Don’t calculate sqrt:Don’t calculate sqrt: Lennard-Jones can be doneLennard-Jones can be done

using distusing dist22

NOTE: if (NOTE: if (distdist2 2 <= cutoffRadius<= cutoffRadius22) ) is in range is in range

Determine optimal # of cells per size.Determine optimal # of cells per size. Is cell length = cutoff radius optimal?Is cell length = cutoff radius optimal?

Idea: Using MBRs in cellsIdea: Using MBRs in cells

For each cell, maintain a For each cell, maintain a Minimum Bounding Minimum Bounding Rectangle (MBR)Rectangle (MBR) around it’s atoms. around it’s atoms. For any cell JUST tipped by rc, check atom is inside For any cell JUST tipped by rc, check atom is inside

rc of MBR before considering atoms exhaustively.rc of MBR before considering atoms exhaustively.

This cell is just tipped”

MBR NOTE: can NOTE: can

also bealso beused inused inconjunction conjunction with “with “sub sub gridgrid””

Improving Cache Hits through Improving Cache Hits through Spatial LocalitySpatial Locality

Spatial locality principleSpatial locality principle: : objects objects close toclose to referred ones will referred ones will probablyprobably be requested again be requested again in the future.in the future. Unsorted atoms Unsorted atoms means many cache misses. means many cache misses.

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Space Filling CurvesSpace Filling CurvesSpace-filling curveSpace-filling curve: : A line passing through A line passing through every point in a space, in some order (according every point in a space, in some order (according to some algorithm).to some algorithm).

Resort atom Resort atom periodicallyperiodically (group by cells & order using curve). (group by cells & order using curve). Improves CPU performance “>50% in 2D moving point query” Improves CPU performance “>50% in 2D moving point query”

worth trying.worth trying.

Row-wise Hilbert curveGray curve Z-ordering

Verlet Neighbour ListVerlet Neighbour List

Choose a “skin radius” greater than rc.Choose a “skin radius” greater than rc.

Build the “verlet neighbor list” using skin radiusBuild the “verlet neighbor list” using skin radius

Next few iterations: update list; check which neighbour pairs are Next few iterations: update list; check which neighbour pairs are inside rc.inside rc.

Refinement: only rebuild list when: sum of 2 max displacement (of Refinement: only rebuild list when: sum of 2 max displacement (of any 2 atoms) > skin “thickness”.any 2 atoms) > skin “thickness”.

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Cut-off sphere

Skin

Skin/verlet radius (Rv)

2 max displacements since rebuild

Some Verlet Specific IdeasSome Verlet Specific Ideas

Don’t check Don’t check displacementdisplacement each iteration: check at each iteration: check at decreasing periods.decreasing periods.

Don’t update Don’t update distances of atoms outside cut-off spheredistances of atoms outside cut-off sphere each iteration: check more frequently as it gets closer.each iteration: check more frequently as it gets closer.

Determine optimal skin radius.Determine optimal skin radius.

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Cut-off sphere

Skin

Skin/verlet radius (Rv)

2 max displacements since rebuild5

Performance vs. AccuracyPerformance vs. Accuracy

Some techniques that will Some techniques that will improve performance, but improve performance, but decrease “accuracy”:decrease “accuracy”: Don’t always rebuild/update Don’t always rebuild/update

neighbor list when necessary.neighbor list when necessary.

Increasing timestepIncreasing timestep

Decreasing cutoff radiusDecreasing cutoff radius

I can graph these by testing I can graph these by testing against a control.against a control.

larger timestep

small timestep

smaller rc

large rc

ProgressProgress

Basic grid is implementedBasic grid is implemented

Thesis started.Thesis started.

Several graphs obtained.Several graphs obtained.

Added front end:Added front end: Uses Uses MFCMFC (Microsoft Foundation (Microsoft Foundation

Class) using Class) using OpenGLOpenGL.. Demonstrates building on engine layer.Demonstrates building on engine layer. Lets me see particles animate (& work out problems)Lets me see particles animate (& work out problems) Code messyCode messy

Is still Is still MUCHMUCH to implement & test. to implement & test.

ConclusionConclusionMolecular dynamicsMolecular dynamics : an (approximated) : an (approximated) simulation of real worldsimulation of real worldparticle behaviour.particle behaviour.

Periodic Bounding ConditionPeriodic Bounding Condition used for “bulk” liquidused for “bulk” liquid

Best existing approach:Best existing approach: Verlet listVerlet list, ,

using using cell listcell list & & fixed gridfixed grid to build. to build.

Ideas for improvement:Ideas for improvement: Minimal half cell list templates.Minimal half cell list templates. MBRs inside cells.MBRs inside cells. Space filling curves.Space filling curves.

Thesis may resemble a guide to implementing/optimizing Thesis may resemble a guide to implementing/optimizing moving spatial join queriesmoving spatial join queries..

Still much to be done.Still much to be done.

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Thank You…Thank You…

Any Questions?Any Questions?

cubic atom list vs cell list

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atoms=5000 - min atom lisboundary mbrsatoms=5000 - min atom list mbrs

atoms=5000 - min atom list ef f

atoms=5000 - cubic atom list

atoms=5000 - min cell list unloaded

atoms=5000 - min cell list

numAtoms: 1000-5000rc: 0.01-0.25boxLen: 1CPS: 10timeStep: 0.01timeStepsToExe: 20useVerlet: 0

Adjacency List Size Comparison

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Revised GoalRevised Goal

Implement engine as efficient as possible.Implement engine as efficient as possible.How it should work:How it should work: Pass in minimum parameters:Pass in minimum parameters:

# atoms, offset, max velocity & box size# atoms, offset, max velocity & box size(or load atom data from file).(or load atom data from file).cutoff radiuscutoff radius

Engine should set remaining details to Engine should set remaining details to optimal. Including:optimal. Including:

cells per side,cells per side,verlet radius,verlet radius,subgrid? mbr? algorithm? etc.subgrid? mbr? algorithm? etc.

rc

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Idea: Sub GridIdea: Sub Grid

““Sub grid adjacency list template guideSub grid adjacency list template guide”: break each cell ”: break each cell into (imaginary) sub-grid.into (imaginary) sub-grid.

When considering an atom, check with sub-cell it belongs to, When considering an atom, check with sub-cell it belongs to, that sub-cell will refine which adjacent cells to search.that sub-cell will refine which adjacent cells to search.

NOTENOTE:: My primary focus is: My primary focus is: reducing time per iterationreducing time per iteration……

but but main memory requirementsmain memory requirements (& (& setup timesetup time) also important.) also important. Choosing cell length equal Choosing cell length equal

to rc is typical. to rc is typical. But is it optimal?But is it optimal?

No one method/metric is No one method/metric is likely to be optimal in ALL likely to be optimal in ALL cases cases too many too many parameters.parameters.

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