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Introduction

Examples

Essentials from. . .

Molecular Dynamics . . .


MD Approximations. . .

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Numerical Methods for. . .

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Algorithmen des WissenschaftlichenRechnens II

1. Molecular DynamicsSimulation

Hans-Joachim Bungartz

1) Molecular Dynamics Simulation

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1.1. Introduction

Remember what Scientific Computing deals with:From phenomena to predictions!

phenomenon, process etc.

mathematical modelc

modelling

numerical algorithmc

numerical treatment

simulation codec

implementation

results to interpretc

visualization

%

rrrrj embedding

statement tool

E

E

E

va

li

da

t

ion

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Overview

modelling aspects of molecular dynamics simulations:

why to leave the classical continuum mechanics point of view?

where appropriate?

which models, i.e. which equations?

numerical aspects of molecular dynamics simulations?

how to discretize the resulting modelling equations?

efficient algorithms?

implementation aspects of molecular dynamics simulations?

suitable data structures?

parallelisation?

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Hierarchy of Models

Different points of view for simulating human beings:

issue level of resolution model basis (e.g.!)global increasein population

countries, regions population dynamics

local increase in

population

villages, individuals population dynamics

man circulations, organs system simulatorblood circulation pump/channels/valves network simulatorheart blood cells continuum mechanicscell macro molecules continuum mechanicsmacromolecules

atoms molecular dynamics

atoms electrons or finer quantum mechanics

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Scales an Important Issue

length scales in simulations:

from 109m (atoms)

to 1023m (galaxy clusters)

time scales in simulations: from 1015s

to 1017s

mass scales in simulations:

from 1024g (atoms)

to 1043g (galaxies)

obviously impossible to take all scales into acount in an explicit

and simultaneous way first molecular dynamics simulations reported in 1957

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Applications for Micro and Nano Simulations

Lab-on-a-chip, used in brewing technology (Siemens)

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Flow through a nanotube (where the assumptions of continuummechanics are no longer valid)

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Protein simulation: actin, important component of muscles (overlayof macromolecular model with electron density obtained by X-ray

crystallography (brown) and simulation (blue))

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Protein simulation: human haemoglobin (light blue and purple:alpha chains; red and green: beta chains; yellow, black, and darkblue: docked stabilizers or potential docking positions for oxygen)

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Material science: hexagonal crystal grid of Bornitrid

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A Prominent Recent Example:

Gordon-Bell-Prize 2005 (most important annual supercomput-ing award)

phenomenon studied: solidification processes in Tantalum andUranium

method: 3D molecular dynamics, up to 524,000,000 atoms sim-ulated

machine: IBM Blue Gene/L, 131,072 processors (worlds #1 inNovember 2005)

performance: more than 101 TeraFlops (almost 30% of thepeak performance)

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1.2. Essentials from Continuum Mechanics

Fluids

fluid: notion covering liquids and gases

liquids: hardly compressible

gases: volume depends on pressure

small resistance to changes of form continuum:

space, continuously filled with mass

homogeneous

subdivision into small fluid voxels with constant physicalproperties is possible

idea valid on micro scale upward (where we consider con-tinuous masses and not discrete particles)

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Lagrange and Euler Formulation

Lagrange formulation:

considers the change of position of material particles

example: velocities of material particles

typical for structural dynamics

Euler formulation:

considers a fixed point in space

example: velocity field, describing the velocities of virtualparticles at fixed positions

typical for fluid mechanics

Arbitrary Lagrangean Eulerian (ALE) formulation:

refers to an arbitrary configuration

example: a particle, moving with its own system of refer-ence through a fluid, having a fixed system of reference

widely used in fluid-structure interactions

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Description of State

consideration of a control volume V0 (Eulerian perspective)

description of the fluids state via

the velocity field v(x, t) and two thermodynamical quanti-ties, typically

the pressure p(x, t) and

the density (x, t)

for incompressible fluids, the density is constant (if there areno chemical reactions)

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Hard Sphere Model

first, atoms are described in a simplified way

as suspended round disks in 2D space, or

as suspended round balls in 3D space

this is called the hard sphere model the fixed (and rather dense) arrangement in a crystal leads to a

solid

for gases, the average distances between particles are verylarge

atoms and molecules of liquids do not have a fixed arrange-ment, but the density compared with gases is higher

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Solids in 2D

D

3D

D

the maximum density in 2D comes with a triangu-lar packing

an ideal crystal without thermal motion provides aperfect 2D grid

the elementary cell for disks of diameter D forms

a rhombus of areaAE = 2 12 D

123D = 1

23D2 0.866D2

in an elementary cell, there are parts that can becombined to a complete disk:AP =

4 D

2 0.785D2

density Cm = APAE =4D2

12

3D2

= 6

3 0.9069

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Solids in 3D

D2D

2D

fill a box with balls of diameter D

periodic arrangement is neither formed automati-cally nor unique

face-centered cubic (fcc) lattice with basis vectorscycl2

2D 2

2D 0

elementary cell: brick,

VE =

2D3

= 2

2D3 2.828D3 VP =

6 1

2+ 8 1

8

6

D3 = 23

D3 2.094D3

Cm =VPVE

=23D3

2

2D3=

6

2 0.7405

VE = volume of elementary cell

VP = volume of atoms

Cm = maximum density ratio

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Fluids

intermediate states for the transition solid-liquid:

plasticity: irreversible deformation of the crystal lattice alonggliding planes

thixotropy: liquid state after destruction of frame-like ag-glomerations of molecules

for C0 0.8 the disks can escape from their positions 2Dliquid without a periodic crystal lattice; average distance of twoparticles: O(D)

gases: smaller densities (C C0) with an average distance ofO( D

C)

C = density ratio of material to reference cell volume

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Scope of Application

number of molecules to be taken into account:

Loschmidt number 2, 687 1019cm3: number of moleculesin 1 cm3 of an ideal gas

Avogadro constant 6.02214151023mol1: number of carbon-

12 atoms in 12g of carbon-12, or number of molecules in 1mol (1 mol, under normal conditions, taking a volume of22.4 litres)

Avogadro number: notion being used in different ways forboth of the above constants, which depend on each other(2, 687 1019cm3 22.413996 103cm3mol1 = 6.0221415 1023mol1)

time steps for numerical simulations are typically in the or-der of femtoseconds (1f s := 1015s)

hence: scope of application is limited to nanoscale simulations(at least for the near future)

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1.3. Molecular Dynamics the Physical Model

Quantum Mechanics a Tour de Force

particle dynamics described by the Schrdinger equation

its solution (state or wave function ) only provides probabilitydistributions for the particles (i.e. nuclei and electrons) positionand momentum

Heisenbergs uncertainty principle: position and momentum cannot be measured with arbitrary accuracy simultaneously

there are discrete values/units (for the energy of bonded elec-trons, e.g.)

in general, no analytical solution available

high dimensional problems: dimensionality corresponds to num-ber of nuclei and electrons

= (R1, . . . , RN, r1, . . . , rK, t) - wave functionR - position of nucleusr - position of electront - time

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hence, numerical solution is possible for rather small systemsonly

therefore, various (simplifying and approximating) approachessuch as density functional method or Hartree-Fock approach(ab-initio Molecular Dynamics, see next slide)

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Classical Molecular Dynamics

Quantum mechanicsapproximation classical Molecular Dynamics

classical Molecular Dynamics is based on Newtons equationsof motion

molecules are modelled as particles; simplest case: point masses

there are interactions between molecules multibody potential functions describe the potential energy of

the system; the velocities of the molecules (kinetic energy) area composition of

the Brownian motion (high velocities, no macroscopic movement),

flow velocity (for fluids)

ab-initio Molecular Dynamics uses quantum mechanical cal-culations to determine the potential hypersurface, apart from

semi-empirical potential functions (cf. Car Parrinello MolecularDynamics (CPMD) methods)

total energy is constant energy conservation

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Fundamental Interactions

Classification of the fundamental in-teractions:

strong nuclear force

electromagnetic force weak nuclear force

gravity

O

rk

ri

rj

interaction potential energy the total potential of N particles is the sum of multibody poten-

tials:

U := 0

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Well-Known Potentials

i j

rij

some potentials from mechanics:

harmonic potential (Hookes law): Uharm (rij) = 12 k (rij r0)2;potential energy of a spring with length r0, stretched/clinchedto a length rij

gravitational potential: Ugrav (rij) = g mimjrij ;potential energy caused by a mass attraction of two bodies(planets, e.g.)

the resulting force isFij = gradU(rij) =

U

rijintegration of the force over the displacement results in the energy or a potential

difference

Newtons 3rd law (actio=reactio):Fij = Fji

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Intermolecular Two-Body Potentials

2

1.5

1

0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

potentialU

distance r

hard sphere potentials

hard sphereSquarewell

Sutherland

2

1.5

1

0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

potentialU

distance r

soft sphere potentials

soft sphereLennardJonesvan der Waals

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Intermolecular Two-Body Potentials

hard sphere potential: UHS(rij) =

rij d0 rij > d

Force: Dirac Funktion

soft sphere potential: USS (rij) = rijn

Square-well potential: USW (rij) =

rij d1 d1 < rij < d20 rij d2

Sutherland potential: USu (rij) =

rij dr6ij

rij > d Lennard Jones potential

van der Waals potential UW (rij) = 46

1rij6

Coulomb potential: UC (rij) = 140qiqjrij

= energy parameter

= length parameter (corresponds to atom diameter, cmp. van der Waals

radius)

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Lennard Jones Potential

e s

e,s

O

ri

rj

rij

Lennard Jones potential: ULJ (rij) =

rij

n

rij

mwith n > m and = 1

n

m n

n

mm

1

nm

continuous and differentiable (C), since rij > 0

LJ 12-6 potential

ULJ (rij) = 4

rij

12

rij

6 m = 6: van der Waals attraction (van der Waals potential) n = 12: Pauli repulsion (softsphere potential): heuristic application: simulation of inert gases (e.g. Argon)

force between 2 molecules:Fij = U(rij)rij = 24rij

2

rij

12 rij

6 very fast fade away short range (m = 6 > 3 = d dimen-

sion)

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LJ Atom-Interaction Parameters

e s

atom [1.38066 1023J]1 [101nm]2

H 8.6 2.81He 10.2 2.28C 51.2 3.35N 37.3 3.31O 61.6 2.95F 52.8 2.83

Ne 47.0 2.72S 183.0 3.52Cl 173.5 3.35Ar 119.8 3.41Br 257.2 3.54Kr 164.0 3.83

= energy parameter = length parameter (cmp. van der Waals radius)

parameter fitting to real world experiments

1 Boltzmann-constant: kB := 1.38066 1023 JK2 101nm = 1010m = 1 (ngstrm)

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Dimensionsless Formulation

using reference values such as , , reduced forms of the equationscan be derived and implemented transformation of the problem

position, distancer :=

1

r (1a)

timet

:=

1

mt

(1b)

velocityv :=

t

v (1c)

potential (atom-interaction parameters are eliminated!): U := U

U

LJ (rij) :=ULJ (rij)

= 4

r

ij26

r

ij23

(1d)

U

kin :=Ukin

=1

mv2

2

=v

2

2t2

(1e)

forceF

ij :=Fij

= 24

2r

ij26

r

ij23

rij

rij2

(1f)

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Multi-Centered Molecules

CA1 CA2

CA

CB1

CB2

CB

FA1B1FA1B2

FA2B1FA2B2

FB1A1FB1A2

FB2A1

FB2A2

FAB

FBA

molecules can be composed with multipleLJ-centers rigid bodies without internal degrees offreedom

additionally: orientation (quarternions), an-gular velocity

additionally: moment of inertia (principalaxes transformation)

calculation of the interactions betweeneach center of one molecule to each centerof the other

resulting force (sum) acts at the center ofgravity, additional calculation of torque

MBS (Multi Body System) point of view: instead of moving multi-centered molecules, there is a holonomically constrained mo-tion of atoms (for a constraint to be holonomic it can be expressible as a function f(r,v,t) = 0)

advantage: better approximation of unsymmetric molecules

there is not necessarily one LJ center for each atom

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Mixtures of Fluids

simulation of various components (molecule types)

modified Lorentz-Berthelot rules for interaction of molecules ofdifferent types

AB :=A + B

2(2a)

AB := aB (2b)with 1e.g. N2 + O2 = 1.007, O2 + CO2 = 0.979 . . .

A A

B B

e ,sA A

e ,sA A

e ,sAB AB

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NVT-Ensemble, Thermostat

statistical (thermodynamics) ensemble: set of possible states a sys-tem might be in

for the simulation of a (canonical) NVT-ensemble, the followingvalues have to be kept constant:

N: number of molecules V: volume

T: temperature

a thermostat regulates and controls the temperature (the kineticenergy), which is fluctuating in a simulation

the kinetic energy is specified by the velocity of the molecules:Ekin =

12

i miv

2i

the temperatur is defined by T = 23NkB

Ekin(N: number of molecules, kB : Boltzmann-constant)

simple method: the isokinetic (velocity) scaling:

vcorr := vact mit =

TrefTact

further methods e.g. Berendsen-, NosHoover-thermostat

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Domain

aa

b

b

Periodic Boundary Conditions (PBC):

modelling an infinite space, built from identical cells domain with torus topology

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Domain

Minimum Image Convention (MIC):

with PBC, each molecule and the associated interactionsexist several times

with MIC, only interactions between the closest represen-tants of a molecule are taken into consideration

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1.4. Molecular Dynamics the Mathematical Model

System of ODE

resulting force acting on a molecule: Fi =

j=i Fij

acceleration of a molecule (Newtons 2nd law):

ir =Fimi

=

j=iFij

mi= j=i

U(ri,rj)

|rij |mi

(3)

system of dN coupled ordinary differential equations of 2nd or-der transferable (as compared to Hamilton formalism) to 2dNcoupled ordinary differential equations of 1st order (N: number ofmolecules, d: dimension), e.g. independent variables q := r and pwith

pi := miri (4a)

pi = Fi (4b)

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Boundary Conditions

Initial Value Problem:position of the molecules and velocities have to be given;initial configuration e.g.:

molecules in crystal lattice (body-/face-centered cell) initial velocity

* random direction

* absolute value dependent of the temperature

(normal distribution or uniform), e.g.32

N kBT =12

Ni=1 mv

2i with vi := v0

v0 :=

3kBTm

resp. v0 :=

3Tt

time discretisation: t := t0 + i t time integration procedure

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1.5. MD Approximations and Discretization

Euler Method

Taylor series expansion of the positions in time:

r(t + t) = r(t) + tr(t) +1

2t2r(t) +

ti

i!r(i)(t) + . . . (5)

(r, r, r(i)

: derivatives)

approximation of (5), neglecting terms of higher order of t, aswell as an analogous formulation of v(t) := r(t) with a(t) :=v(t) = r(t) =

F(t)m

leads to the explicit Euler method:

v(t + t).

= v(t) + t a(t) (6a)

r(t + t).

= r(t) + t v(t) (6b)

implicit Euler method derivatives at the time step end:v(t + t) .= v(t) + t a(t + t) (7a)

r(t + t).

= r(t) + t v(t + t) (7b)

(6a) in (7b) r(t + t) .= r(t) + tv(t) + t2a(t)

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Strmer Verlet Method

the Taylor series expansion in (5) can also be performed fort: (Richardson extrapolation for = 1)

r(t t) = r(t) tr(t) + 12

t2r(t) +(t)i

i!r(i)(t) + . . . (8)

from (5) and (8) the classical Verlet algorithm can be derived:

r(t + t) = 2r(t) r(t t) + t2r(t) + O(t4) 2r(t) r(t t) + t2a(t) (9)

direct calculation of r(t + t) from r(t) and F(t)

the velocity can be estimated with

v(t) = r(t).

=r(t + t)

r(t

t)

2t (10)

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Crank Nicolson Method

explicit approximation (11a) for half step [t, t + t2

] inserted intoimplicit approximation (11b) for half step [t + t2 , t + t] gives forv (11c):

v(t +t

2) = v(t) +

t

2a(t) (11a)

v(t + t) = v(t +t

2 ) +t

2 a(t + t) (11b)

v(t + t) = v(t) +t

2(a(t) + a(t + t)) (11c)

alternative conversion to integral equation

v(t + t) v(t) =t+tt

a() d

numerical integration with trapezoidal rule (11c) generalization with further subdivisions in subintervals leads to

Runge Kutta methods. More force-evaluations necessary!

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Velocity Strmer Verlet Method

The velocity Strmer Verlet method is a composition of a

Taylor series expansion of 2nd order for the positions (5), and a

Crank Nicolson method for the velocities (11c)

r(t + t) = r(t) + tv(t) +t2

2 a(t) (12a)

v(t + t) = v(t) +t

2(a(t) + a(t + t)) (12b)

tt-Dt t+Dt tt-Dt t+Dt tt-Dt t+Dt

r

v

F

tt-Dt t+Dt

Forward Euler r Force calculation Crank-Nicolson v

memory requirements: (3 + 1) 3N(calculation of v(t + t) requires not only v(t), r(t + t) and F(t + t), but also F(t) and

therefore 3 + 1 vector fields)

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Dimensionless Velocity Strmer Verlet Method

insertion of the equations (1) leads to:(r := r, v :=

v, t2 := 2 m

t2, r = 1

mF := 1

m

F)

r(t + t) = r(t) + v(t) +t2

2F(t) (13a)

v(t + t) = v(t) +t2

2

F(t) +t2

2

F(t + t) (13b)

procedure:


r

v

F

tt-Dt t+Dttt-Dt t+Dt

Forward Euler r Forward Euler v Force calculation Backward Euler v

1. calculate new positions (13a),partial velocity update: +t

2

2F(t) in (13b)

2. calculate new forces, accelerations (computationally intensive!)3. calculate new velocities: +t22 F(t + t) in (13b)

memory requirements: 3 3N

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Leapfrog Method

for the Leapfrog method, the velocity calculations are shifted fora half time step with respect to the position calculations:

v(t +t

2

) = v(t

t

2

) + t a(t) (14a)

r(t + t) = r(t) + t v(t +t

2) (14b)


r

v

F

t-Dt/2 t+Dt/2t-Dt/2 t+Dt/2 t-Dt/2 t+Dt/2 t-Dt/2 t+Dt/2

tt-Dt t+Dt

exact arithmetic: Strmer Verlet, Velocity Strmer Verlet and

Leapfrog schemes are equivalent the latter two are more robust with respect to roundoff errors

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Leapfrog Method Thermostat

for the Leapfrog method, the velocity calculations are shifted fora half time step with respect to the position calculations:

v(t +t

2) = v(t t

2) + t a(t)

r(t + t) = r(t) + t v(t +t

2

)

t-Dt t+Dt t+Dt t-Dt t+Dt

r

v

F

tt-Dt t+Dt tt-Dt t+Dt tt-Dt t+Dt tt-Dt t+Dt tt-Dt t+Dt

Thermostat

an intermediate step may be introduced for the thermostat v(t) :=v(t+t2 )+v(tt2 )

2to synchronize the velocity:

vact(t) = v(t t

2 ) +t

2 a(t) (16a)

v(t +t

2) = (2 1)vact(t) + t

2a(t) (16b)

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Multistep, Predictor Corrector Methods

Multistep methods:

results are stored for several time steps, which define a(polynomial) interpolant

use the interpolant (extrapolation) for the integration

initialization with single-step-methods

increased memory requirements caused by storage of dataof previous steps data!

Predictor Corrector methods:

1. use explicit method to determine predictor values for t + t

2. implicit method uses predictor values instead of the un-known ones for t + t

3. increased computational effort!4. quality of the predictor step caused by the complex chaotic

behaviour is often not very good

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Multi-Centered Molecules

for multi-centered molecules, besides position r and velocitiesv, orientations qand angular velocities w have to be also calcu-lated

candidate: explicit or implicit version of the Fincham Leapfrogrotational algorithm

r,v,F using classical Leapfrog method additional orientation q, angular velocity w as well as angu-

lar momentum j

t-Dt/2 t+Dt/2t+Dt/2 t-Dt/2 t+Dt/2

r

v

F,a

tt-Dt t+Dttt-Dt t+Dt tt-Dt t+Dt tt-Dt t+Dt

q

t-Dt/2 t+Dt/2t+Dt/2 t-Dt/2 t+Dt/2j,w

t

t+Dt/2

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Evaluation of Time Integration Methods

accuracy (not of great importance)

stability

conservation

of phase space density (symplectic) of energy

of momentum (especially with PBC (Periodic Boundary Conditions))

reversibility of time

use of resources:

computational effort (number of force evaluations)

maximum time step size

memory usage

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Reversibility of Time

time reversal for a closed system means

a turnaround of the velocities and also momentums; posi-tions at the inversion point stay constant

traverse of a trajectory back in the direction of the origin

demand for symmetry for time integration methods

+ e.g. Verlet method

- e.g. Euler method, Predictor Corrector methods

contradiction with

the H-theorem (increase of entropy, irreversible processes)?(Loschmidt objection)

the second theorem of thermodynamics?

reversibility in theory only for a very short time

Lyapunov instability Kolmogorov entropy

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Lyapunov Instability

Example of a simple system:

stable case:jumping ball on a plane with slightly disturbed initial hori-zontal velocity

linear increase of the disturbance

instable case:jumping ball on a sphere with slightly disturbed initial hor-izontal velocity exponential increase of the disturbance(Lyapunov exponent)

for the instable case, small disturbances result in large changes:chaotic behaviour (butterfly hurricane?)

non-linear differential equations are often dynamically instable

calculation of the trajectories: badly conditioned problem;

a small change of the initial position of a molecule may result ina distance to the comparable original position, after some time,in the magnitude of the whole domain!

there are also conserved quantities for whichnumerical simulations make sense!

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Lyapunov Instability: A Numerical Experiment

setup of 4000 fcc atoms

for a second setup, the position of a singleatom was changed with a displacement of0.001

tracing the movement of the atom for bothsetups, the distance increases for eachstep

colours indicate velocity

2.53

3.54

4.55

5.57.1

7.2

7.3

7.4

7.5

7.6

7.7

3.5

3.6

3.7

3.8

3.9

4

tracing a Molecule (with initial displacement)

Molecule 25, run1

Molecule 25, run2

4.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

2.5 3 3.5 4 4.5 5 5.5

Molecule deviation (with initial displacement)

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Short-Range Potential

choosing m = 6 (negative exponent in the LJ-potential) fast decay of po-tential and force

for each molecule, an influence volume (closed sphere) withcut-off radius rc (Euclidian metrics) can be assumed where ev-

ery molecule outside this influence volume is neglected

ULJ,rc

rij

=

4

rij26 rij23 for rij rc

0 for rij > rc(17a)

Fij,rc

rij

=

24

2

rij26 rij23 rijrij2 for rij rc

0 for rij > rc(17b)

consider only a subgraph of the interaction-graph the anti-symmetric force matrix, related to this graph, is sparse

the complexity of the calculation can be reducedfrom O (N2) to O(N).

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Short-Range Interactions

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0 1 2 3 4 5

U*

r*

dim. red. LennardJones 126 Potential

2.5

2

1.5

1

0.5

0

0.5

0 1 2 3 4 5

F*

r*

dim. red. LennardJones 126 Force

1

2

3

4

5

Fij Force matrix/Interaction-graph- F12 F13 F14 F15

F12 - F23 F24 F25F13 F23 - F34 F35F14 F24 F34 - F45F15 F25 F35 F45 -

fast decay of force contributions with increasing distance

dense force matrix with O(n2

), mostly very small, entries

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Short-Range Interactions

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0 1 2 3 4 5

U*

r*

dim. red. finites LennardJones 126 Potential (rc=2)

2.5

2

1.5

1

0.5

0

0.5

0 1 2 3 4 5

F*

r*

dim. red. finite LennardJones 126 Force (rc=2)

1

2

3

4

5

Fij Force matrix/Interaction-graph- F12 F13 F14 0

F12 - 0 F24 F25F13 0 - F34 0F14 F24 F34 - F45

0 F25 0 F45 -

cut-off radius leads to a reduction of the computational effortsparse force matrix with O(n) entries

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Shifted Potentials

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0 1 2 3 4 5

U*

r*

shifted dim. red. finites LennardJones 126 Potential (rc=2)

2.5

2

1.5

1

0.5

0

0.5

0 1 2 3 4 5

F*

r*

dim. red. finite LennardJones 126 Force (rc=2)

ULJ,rc,shifted

rij

=

ULJ

rij ULJ (rc ) for rij rc

0 for rij > rc

Fij,rc

rij

=

Fij

rij

for rij rc0 for rij > r

c

additionally, constant additive term for the potential continuous potentialreduced error for the overall potential

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Shifted Potentials

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0 1 2 3 4 5

U*

r*

shifted dim. red. finites LennardJones 126 Potential (rc=2)

2.5

2

1.5

1

0.5

0

0.5

0 1 2 3 4 5

F*

r*

shifted dim. red. finite LennardJones 126 Force (rc=2)

ULJ,rc,shifted

rij

=

ULJ

rij ULJ (rc) FLJ (rc) rij rc for rij rc

0 for rij > rc

Fij,rc,shifted

rij

=

Fij

rij FLJ (rc) for rij rc

0 for rij > rc

additionally, constant additive term for the potential

continuous potential additionally, linear additive term for the potential

continuous force

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Cut-Off Corrections

due to the cut-off radius, the calculation of

the potential energy

the pressure

neglects some addends with small absolute values

(small) errors

cut-off correction tries to correct this error

constant density and a homogeneus distribution are a prerequi-site

physical values in the calculated volume can be approximatelyextrapolated

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1.6. MD Implementational Aspects

Verlet Neighbour Lists

rc

rmax

every molecule stores its neigh-bours for a distance rmax > rc

every nupd time steps (dep. on rmax),the lists are updated

the "buffer" has to be larger than thecovered distance of a molecule forthat time:

rmax rc > nupd t vm

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Classical Linked-Cell Algorithm

molecules are ranged in a lattice of cu-bic cells of side length rc

hash table with"geometrically motivated" hashfunction

"Binning" resp. "Bucketing"-techniques from "ComputationalGeometry"

direct volume representation(voxel) of the influence region

runtime: O(n) only half (point symetry) of the neigh-

bour cells are explicitly traversed (New-tons 3rd law)

erase and generate the data structurein each time step

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Variable Linked-Cell Algorithm

lattice might be built up from cellsof side length rc

twith t R+

preferable integer numbers are usedfor the divisor t

N

for t , the examinatedinfluence volume will converge tothe (optimal) sphere

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 1 0

rc/cellwidth

searchvolume/hemispherevolume

t = 1 t = 2 t = 4 t = 3

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Linked-Cell Algorithm Data Structure I

0 1 2 3 4 5 6 7 8 9 10 11

12 23

24 35

108 119

inner zoneboundary zoneHalo

rc

molecule 1

data

nextincell

molecule 2

data

nextincell

molecule 3

data

nextincell

molecule 4

data

nextincell

molecule 5

data

nextincell

molecule x

data

nextincell

molecule N

data

nextincell

cellseq. 1 cellseq. 2cellseq. x cellseq. i

cells are stored as a one-dimensional array (vector)

intrusive list for the cell molecules

list to determine the processing sequence

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Linked-Cell Algorithm Data Structure II

-50 -49 -48 -47 -46

-39 -38 -37 -36 -35 -34 -33

-28 -27 -26 -25 -24 -23 -22 -21 -20

-16 -15 -14 -13 -12 -11 -10 -9 -8

-4 -3 -2 -1

rc

-25 -24 -23

-19 -18 -17 -16

-14 -13 -12 -11 -10

-8 -7 -6 -5 -4

-2 -1

rc

-20

offset mask to determine the neighbours

cache efficiency is influenced by the processing order(temporal locality)

1

2

3

4

5

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1.7. MD Parallelisation

Profiling

!"#$%

&

'&%

()&!*

+,

,-%

,-%

,--%

,--%

+,

'&%

/0

force calculation is dominating

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Shared Memory Parallelisation

each process calculates one part (Np

) of the molecules (cells)

availability of all relevant data (position) because of commonmemory

Shared Memory algorithm: Velocity Strmer Verlet method1. parallel explicit Euler method r,v (half step) for N

pmolecules

2. parallel force calculations for Np

molecules or the respectivecells(force summation critical, respecting Newtons 3rd law: reduction; same with

linked-cell algorithm)

3. parallel implicit Euler method v (half step) for Np

molecules

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Replicated Data I

"Atom Decomposition" Shared Memory parallelisation every node has to store all position data

collective communication for thesynchronization of redundant data

"Atom Decomposition" algorithm: Velocity Strmer Verlet method

1. explicit Euler method r,v (half step) for Np

molecules

2. distribute (gather-to-all) the Np

position data for each PEto all other PEs

3. force calculation for Np

molecules

4. possible distribution of partial forces to the appropriate PEs

5. implicit Euler method for v (half step) for Np

molecules

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Replicated Data II

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S

Force matrix is only virtual and not allocated/set up

costs

calculation: Np

communication partners per PE: p 1 memory requirements: N positions and N

p

forces

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Replicated Data II

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S


F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S


costs

calculation: N2p communication partners per PE: p 1 memory requirements: taking advantage of Newtons 3rd

law needs a vector for N (partial) forces and additional com-

munication

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Force Decomposition I

each process calculates a part of the molecules and the forcematrix

on each node: position data of 2 Np

molecules

communication: distribution of positions and calculated forces "Force Decomposition" algorithm: Velocity Strmer Verlet method

1. explicit Euler method for r,v (half step) for Np

molecules

2. distribution of Np

position data per PE to 2

Np

1

PEs

3. force calculation of a

p Np

sub-matrix

4. distribution of partial forces to

p 1 PEs5. implicit Euler method for v (half step) for N

p

molecules

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Force Decomposition II

1 2 3 4

5 6 8

9 10 11 1213141516

7

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F16

S

1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16


costs

calculation: Np

communication partners per PE: 2

p 1

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Spatial Decomposition

domain is decomposed into subdomains

each processor handles one subdomain

amount of molecules per processor is variable(molecules are moving!)

overlapping buffer regions (halo, rc) have to be synchronized

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Spatial Decomposition

domain is decomposed into subdomains

each processor handles one subdomain

amount of molecules per processor is variable(molecules are moving!)

overlapping buffer regions (halo, rc) have to be synchronized

point-to-point communication, dependent of decomposition molecule movement (flow velocity) communication method: "x-y-z" vs. "direct"

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Domain Decomposition: Cubes Slices (1)

assumption:

homogeneous molecule distribution

subdomains with Np

molecules and volume Ld

p: N Ld

communication size proportional to halo volume full halo

slices:

2 neighbour PEs

halo volume: Ld12 rc = 2Ld rcL bad scaling properties

relatively easy to implement

special case of a cartesian topol-ogy ( cube)

communication

amount: 2p comm./PE: 2 rc

LN

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Domain Decomposition: Cubes Slices (2) assumption:

homogeneous molecule distribution

subdomains with Np

molecules and volume Ld

p: N Ld

communication size proportional to halo volume "complete" halo

cubes: 3d 1 neighbour PEs side length: l = d

Ld

p= Ldp

halo volume (l + 2rc)d ld =di=1

d

i

ldi(2rc)i

d1ld12rc = 2 d ld rcl =2 d Ldp

1d1 rc

L

communication

amount:

3d 1p (direct) or 2d p (x-y-z) comm./PE: 2d rc

LN p

1d1

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1.8. Molecular Dynamics Examples of Nanofluidic

Simulations

1.8.1. Simulating Diffusion

Fluid

a fluid is a continuum (a space continuously filled with mass)without a rigid crystal structure:

liquids: hardly compressible

gases: volume depends on pressure

(looking at isothermal processes)

small resistance to changes of form

length scales of a system have to be large compared to themean free path of the molecules Knudsen number

Kn < 0.01: ideal fluid

0.01 < Kn < 0.1: viscous fluid (Navier Stokes equation)

0.005 < Kn: kinetic (Boltzmann theory)

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the transport of properties in a fluid is caused by

diffusion

advection

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Diffusion

diffusion and thermal conduction are triggered due to the Brow-nian motion

stochastic models of a microscopic model lead to macroscopicmodels

the process is driven for instance through a concentration or atemperature gradient, (i.e. a spatial difference)

00

x

Concentration profiles

Dt=0.1Dt=0.5

Dt=1Dt=2

Dt=10

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MD Diffusion Simulation

t = 1 t = 5 t = 10 t = 15 t = 20

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30 35 40 45 50

Diffusion

t=1t=1t=5

t=10t=15t=20

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Gradient Diffusion

the Ficks equation provides the power density of the materialdiffusion (D: molecular diffusion coefficient [m2s1]):

M = Dgrad (18)

the thermal flow of a thermal conduction equation (k: thermalconductivity [k g m s3 K1]):

W = k gradT (19)

Gradient: multiplication of the Nabla operator with a scalar function(vector):

grad f = f =

x

y

z

T f =

fx

fy

fz

T

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Diffusion Equation

change in the state variable in the control volume denotes thesum or the integral of all flows over the surface

m

t=

t

d =

D grad dn (20)

applying Gaussian integration leads to

t d =

div (D grad)dand to the diffusion equation (parabolic differential equation):

t= D (21)

divergence: multiplication of the Nabla operator with a vector (scalarvalue):

div v = v =

x

y

z

T v = vxx

+vyy

+vzz

Laplace operator: = 2, z.B. f = div gradf = 2fx2

+ 2f

y2+

2fz2

Gaussian integration:

div v d =

v dn

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1.8.2. Nucleation

Nucleation process of supersaturated Argon

t = 10000 t = 125000 t = 230000 t = 360000

nucleation process for an oversaturated Argon vapour at 0.97 Mol/land 80k

the simulation program automatically detects clusters (droplets)

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0

2

4

6

8

10

12

35030025020015010050

numberofclusters

timeste s 103

Clusters in a supersaturated Argon vapor, 80 K, 0.97 Mol/l

f1(x) = 0.00731 x -2.36f2(x) = 0.00774 x -4.09f3(x) = 0.00749 x -4.698

cluster size > 20cluster size > 30cluster size > 40

f1(x)f2(x)f3(x)

counting and grouping clusters of certain size ranges, a statisticcan be generated

the growth of the clusters (slope) is known as nucleation rateand important for macroscopic simulations

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1.9. Numerical Methods for Long-Range Potentials

1.9.1. Introduction

so far: focus on short-range potentials such as Lennard-Jones,e.g.

resulting mutual interactions are restricted to particles insome local neighbourhood

facilitates numerical treatment and algorithmic organization:no quadratic complexity induced by an "each-with-each"behaviour

now: tackle long-range potentials, too

examples: Coulomb or gravitation potential

interactions between remote particles must not be neglected

simple cut-off not possible nevertheless need for approaches that avoid quadratic com-

plexity

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what is long-range?

intuitively: potential function U(r) does not decrease rapidlywith increasing r

formally (one possibility): for d > 2, potentials not decreas-ing faster than rd for increasing r (criterion: integrabilityover IRd)

typical potentials in applications have both a short-range part

(to be dealt with according to the previous sections) and a long-range part, represented as two additive components:

U(r) := Ushort + Ulong

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Two Main Classes

grid-based methods:

decompose Ulong itself into a smooth but long-range and asingular but short-range part

for the latter, use the linked-cell approach again

for the first, there exist special so-called grid-based meth-ods:

* P3M (Particle-ParticleParticle-Mesh) method

* PME (Particle-Mesh-Ewald) method

* SPME (Smooth-Particle-Mesh-Ewald) method)

starting point: PDE-representation of the potential

(x) = 10

(x)

* potential as solution of a potential (Poisson) equation

* efficient solution with standard discretisation techniquessuch as Finite Differences or Finite Elements in case ofsmooth solutions

* hence, feasible for the smooth long-range part and forhomogeneous particle distributions

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hierarchical or tree-based methods:

starting point: integral representation of the potential

(x) =1

40

(y)

1

y xdy

advantageous especially for heterogeneous particle distri-butions (frequent in astrophysics, relevant also for molecu-lar dynamics)

examples:

* panel clustering

* Barnes-Hut method

* (fast) multipole methods

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1.9.2. Tree-Based Methods

based on integral representation of the potential

hierarchical decompositions of the domain of simulation

adaptive approximation of the particle distribution

widespread scheme: octrees

allow for separation of near-field and far-field influences

log-linear or even linear complexity can be obtained

high flexibility with respect to more general potentials (as neededfor special applications, such as biomolecular problems)

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Series Expansion of the Potential

general (integral) representation of the potential:

(x) =

G(x, y)(y)dy

(general kernel G, particle density (y), and domain )

Taylor expansion of the kernel G (if sufficiently smooth apart

from the singularity in x = y) in y around y0:G(x, y) =

j1p

1

j!G0,j(x, y0)(y y0)j + Rp(x, y)

(multi-index j = (j1, j2, j3), j! = j1!j2!j3!, Gk,j(x, y) mixed (k, j)-th derivative (k-th w.r.t. x, j-th w.r.t. y), remainder Rp(x, y))

leads to expansion (and approximation) of the potential:

(x) j1p1

j!Mj(, y0)G0,j(x, y0)

with the so-called moments

Mj(, y0) :=

(y)(y y0)jdy

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Subdivision of the Domain

separation of near-field and far-field for given x:

= near far , near far =

decomposition of the far-field into disjoint, convex subdomains:

far =i

fari

note: this decomposition depends on x, i.e. it is done for eachparticle position x (efficient derivation possible from one hierar-chical tree structure)

each fari has an associated point yi0

how to choose the subdivision?

diam

x yi0:=

supyfari

y yi0x yi0

for some suitable constant 0 < < 1

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resulting approximation for (x):

(x) =

(y)G(x, y)dy

=

near

(y)G(x, y)dy +

far

(y)G(x, y)dy

= near

(y)G(x, y)dy + i

far

i

(y)G(x, y)dy

near(y)G(x, y)dy +

i

j1p

1

j!Mj(

fari , y

i0)G0,j(x, y

i0)

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Error Estimates

error characteristics for one fix particle position x : local relative approximation error for one fari can be shown

to be of order O(p+1)

global relative approximation error (summation over wholefar-field) can be shown to be of order O(p+1)

this clarifies the role of :

allows to control the global approximation error in x

geometric requirement to the far-field subdivision: the closerfari is located to x, the smaller it has to be to fulfil the -condition

hence: a typical "level of detail"

the closer, the higher resolved cf. terrain representation in flight simulators

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Tree Structures

central question: how can we construct all these necessaryseparations of near-fields and far-fields and subdivisions of far-fields in an efficient way?

idea: recursive decomposition of (a square in 2D, a cube in3D without loss of generality) in cells of different size, termi-nating the subdivision process if a cell is either empty or con-tains just one particle

concepts:

kd-tree: alternate subdivision in coordinate direction (x, y,and z), such that the separation produces two subdomainsthat roughly contain the same number of particles each

quadtree (2D) or octree (3D): subdivision into four congruentsubsquares or eight congruent subcubes, respectively

the following algorithms (Barnes-Hut etc.) use the octree ap-proach

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kd-Trees Example

3

4

5

7

6

11

9

8

10

12

13

1 2 3 4 5

6

7

9

8 10 1312

11

2

1

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Quadtrees and Octrees Examples

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Recursive Computation of the Far-Field Subdivision

starting point: create the octree corresponding to the set of par-ticles

each node of the octree represents a subdomain of orone cell

for each cell i, define some yi0 (the centre point or the centre

of gravity of all particles contained, e.g.) for doing the Taylorexpansion for each cell i, let the parameter diam just denote the diam-

eter of the smallest surrounding sphere, e.g.

objective: for each particle position x, use as few cells as pos-sible (i.e. as big cells as possible) for fulfilling the diam- rule

hence: start from root node, checkdiam

x

yi0

,

stop if fulfilled (no need for further subdivision) and proceed ifnot yet fulfilled

note that for each x, we typically get a different subdivision

but note also that all these subdivisions are just subtrees of ourconstructed octree

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Recursive Computation of the Moments

now: use this subdivision for the efficient calculation of the localmoments Mj(fari , y

i0)

direct (numerical) integration or direct summation are not effi-cient

therefore: use hierarchical tree structure to calculate all mo-

ments for all cells in one run and store them crucial property for that:

Mj(1 2, y0) = Mj(1, y0) + Mj(2, y0) ,if the point of expansion y0 is the same

in the (standard) case of different y10 and y20, there are simple

conversion formulas:

Mj(1, y0) =kjj

k

(y0 y0)j

k

Mk(1, y0)

(k j component-wise, multiplicative binomial coefficients)

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this allows for a bottom-up calculation of the moments from theleaves to the root

in the leaves:

if no particle present: zero

if one particle of mass m there in x: m(x yi0)j

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Using these Building Blocks

still to be done for a numerical routine:

how to construct the tree, starting from a given set of particles?

how to store the tree?

how to choose cells and expansion points?

how to determine far-field and near-field?

several algorithmic variants to be discussed in the following

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