Molecular Dynamics Simulation – Introduction

Scientific Computing II

Molecular Dynamics Simulation – Introduction

Michael Bader – SCCSTechnical University of Munich

Summer 2017

The Simulation Pipeline – Revisitedphenomenon, process etc.

mathematical model?

modelling

numerical algorithm?

numerical treatment

simulation code?

implementation

results to interpret?

visualization

��

HHHHj embedding

statement tool

-

-

-

validation

Michael Bader – SCCS | Scientific Computing II | Molecular Dynamics – Intro | Summer 2017 2

The Seven Dwarfs of HPC – Dwarf # 4

“dwarfs” = key algorithmic kernels in many scientific computingapplications

P. Colella (LBNL), 2004:1. dense linear algebra2. sparse linear algebra3. spectral methods4. N-body methods5. structured grids6. unstructured grids7. Monte Carlo

→ discuss simulation pipeline for molecular dynamics(as example for N-body methods)


Overview: Particle-Oriented Simulation Methods

General Modelling Approach:• “N-body problem”→ compute motion paths of many individual particles

• requires modelling and computation of inter-particle forces• typically leads to ODE for particle positions and velocities

Numerical Aspects:• how to discretize the resulting modelling equations?• efficient time stepping algorithms?

Implementation Aspects:• suitable data structures?• efficient algorithms to compute short- and long-range forces?• parallelisation?


Applications for Micro and Nano Simulations

Flow through a nanotube (where the assumptions of continuum mechanicsare no longer valid)



Protein simulation: human haemoglobin (light blue and purple: alpha chains;red and green: beta chains; yellow, black, and dark blue: docked stabilizers or

potential docking positions for oxygen)Michael Bader – SCCS | Scientific Computing II | Molecular Dynamics – Intro | Summer 2017 6


Material science: hexagonal crystal grid of Bornitrid


HPC Example – Gordon Bell Prize 2005• Gordon-Bell-Prize 2005 (most important annual supercomputing award)• phenomenon studied: solidification processes in Tantalum and Uranium• method: 3D molecular dynamics, up to 524,000,000 atoms simulated• machine: IBM Blue Gene/L, 131,072 processors (world’s #1 in November

2005)• performance: more than 101 TeraFlops (almost 30% of the peak

performance)

(Streitz et al., 2005)


HPC Example – Millennium-XXL Project

(Springel, Angulo, et al., 2010)

• N-body simulation with N = 3 · 1011 “particles”• study gravitational forces

(each “particles” corresp. to ∼ 109 suns)• simulates the generation of galaxy clusters

served to “validate” the cold dark matter modelMichael Bader – SCCS | Scientific Computing II | Molecular Dynamics – Intro | Summer 2017 9

Millennium-XXL Project (2)

Simulation Figures:• N-body simulation with N = 3 · 1011 particles• 10 TB RAM required only to store positions and velocities (single

precision)• entire memory requirements: 29 TB• JuRoPa Supercomputer (Jlich)• computation on 1536 nodes

(each 2x QuadCore, i.e., 12 288 cores)• hybrid parallelisation: MPI plus OpenMP/Posix threads• execution time: 9.3 days; ca. 300 CPU years


HPC Example – Gordon Bell Prize 2010

(Rahimian, . . . , Biros, 2010)

• direct simulation of blood flow• particulate flow simulation (coupled problem)• Stokes flow for blood plasma• red blood cells as immersed, deformable particles


HPC Example – Gordon Bell Prize 2010 (2)

Simulation – HPC-Related Data:• up to 260 Mio blood cells, up to 9 · 1010 unknowns• fast multipole method to compute Stokes flow

(octree-based; octree-level 4–24)• scalability: 327 CPU-GPU nodes on Keeneland cluster,

200,000 AMD cores on Jaguar (ORNL)• 0.7 Petaflops/s sustained performance on Jaguar• extensive use of GEMM routine (matrix multiplication)• runtime: ≈ 1 minute per time step

Article for Supercomputing conference:http://www.cc.gatech.edu/~gbiros/papers/sc10.pdf


http://www.cc.gatech.edu/~gbiros/papers/sc10.pdf

Scales – an Important Issue

• length scales in simulations:• from 10−9m (atoms)• to 1023m (galaxy clusters)

• time scales in simulations:• from 10−15s• to 1017s

• mass scales in simulations:• from 10−24g (atoms)• to 1043g (galaxies)

• obviously impossible to take all scales into acount in an explicit andsimultaneous way

• first molecular dynamics simulations reported in 1957


Laws of Motion

• force on a molecule: ~Fi =∑

j 6=i~Fij

• leads to acceleration (Newton’s 2nd Law):

~̈ri =~Fi

mi=

∑j 6=i~Fij

mi= −

∑j 6=i

∂U(~ri ,~rj )∂|rij |

mi(1)

• system of dN ODE (2nd order)(N: number of molecules, d : dimension),

• reformulated into a system of 2dN 1st-order ODEs:

~pi := mi ~̇ri (2a)

~̇pi = ~Fi (2b)


Example: Hooke’s Law

i j

rij

• “harmonic potential”: Uharm(rij)= 1

2 k(rij − r0

)2

• potential energy of a spring of length r0 when extended or compressed tolength rij

• resulting force:

1D: Fij = −grad U(rij)= −∂U

∂rij= −k

(rij − r0

)2D, 3D: ~Fij = −k

(~rij −~r0

)Michael Bader – SCCS | Scientific Computing II | Molecular Dynamics – Intro | Summer 2017 15

Example: Gravity

• attractive force due to the mass of two bodies (planets, etc.)• gravity potential: Ugrav

(rij)= −g mi mj

rij


1D: Fij = −grad U(rij)= −g

mimj

r2ij


Example: Coulomb Potential

1q

2qr12

+ −

• attractive or repulsive force between charged particles• Coulomb potential: Ucol

(rij)= 1

4πε0

qi qjrij


1D: Fij = −grad U(rij)=

14πε0

qiqj

r2ij


Example – Smoothed Particle Hydrodynamics

“Forces” result from discretisation of a PDE:• approximate functions using kernel functions W :

f (x) ≈∫V

f (r ′)W (|r − r ′|,h)dV ′

• for h→ 0: W → δ (Dirac function)• approximation of derivatives→ integration by parts:

∇f (x) ≈∫V

f (r ′)∇W (|r − r ′|,h)dV ′


Example – Smoothed Particle Hydrodynamics (2)

• approximate integrals at particle positions:

f (ri) ≈N∑

j=1

mj

ρ(rj)f (rj)W (|ri − rj |,h)

• in particular for the density

ρ(ri) ≈N∑

j=1

mjW (|ri − rj |,h)

• similar for derivatives:

∇f (ri) ≈N∑

j=1

mj

ρ(rj)f (rj)∇W (|ri − rj |,h)

• leads to N-body problem (based on Navier-Stokes equations, e.g.)


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Molecular Dynamics Simulation – Introduction

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