parallel implementation of dissipative particle dynamics and application to energetic materials

8/12/2019 Parallel Implementation of Dissipative Particle Dynamics and Application to Energetic Materials

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Parallel Implementation of Dissipative Particle Dynamics:Application to Energetic Materials

1 Introduction

The simulation of various systems of particles provide an alternative for conducting costly

or risky experiments in the eld. However, these simulations often require a trade-off between

accuracy or level of detail and the size of the simulation. For example, a simulation which strives

to maintain electronic detail such as Quantum Monte Carlo can only be used to simulate systems

of 30 or 40 atoms for a picosecond on current computer systems. Molecular Dynamics (MD)

can be used to simulate molecules on an atomistic scale. Atomistic MD follows the classical

laws of physics with atomic movement governed by Newtons second law [1]. In atomistic MD

simulations, only atoms are modeled as opposed to quantum mechanics simulations, which model

atoms and electrons. Millions of atoms can be modeled on nanosecond timescales in atomistic MD.

Another method to run larger simulations with acceptable trade-off in detail is coarse graining.

Methods of coarse graining include representing groups of atoms as beads or considering the

system in terms of elds instead of individual forces. Coarse grained MD would be calculated

much the same way as MD, by considering the conservative forces on each bead via Newtons

second law. Dissipative Particle Dynamics (DPD) improves CG MD by adding dissipative and

random forces [2]. DPD has been prevalent in the simulation of polymers, surfactants and colloids

since its inception, but has recently been used in other applications including energetic materials

[3, 4, 5].

DPD, as originally formulated, samples the canonical ensemble ( i.e. , constant temperature),

but variants of DPD have also been developed that sample the isothermal, isobaric ensemble (con-

stant pressure DPD), the microcanonical ensemble (constant energy DPD), as well as isoenthalpic

conditions (constant enthalpy DPD). The purpose of this research project was to develop parallel

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versions of these DPD variants for a previously written parallel MD code, called CorexMD. Im-

plementation of these parallel DPD variants will allow larger simulations of millions or billions

of particles on longer length (up to microns) and time (up to microseconds) scales. From this

work, researchers can model microstructural voids in energetic materials, which are known to be

important for detonation of explosives. Current experiments also cannot provide information about

materials and their detailed mechanisms so this project will assist in that respect.

2 Dissipative Particle Dynamics

In DPD, particles are dened by mass ( m), position ( r ), and momentum ( p). They interact

through a pairwise force ( F i j ).

F i j = FC i j + F Di j + F

Ri j (1)

In Eq. (1), FC i j is the conservative force, F Di j is the dissipative force, and F

Ri j is the random force,

which are given by Eqs. (2) - (4).

FC i j = duCGijdr ij

r i jr i j

(2)

F Di j = i j D(r i j)(

r i jr i jvi j )

r i jr i j

(3)

F Ri j = i j R(r i j)W i j

r i jr i j

(4)

In Eqs. (2) - (4), r ij is the separation vector between particles i and j, r i j = |r ij | , is the friction

coefcient, is the noise amplitude, vij = pimi

p jm j

, W i j is an independent Wiener process such that

W i j = W ji , and D and R are weighting functions that vanish for r r c where r c is the cut-off

radius [6].

DPD variants exist that conserve temperature (DPD-T), energy (DPD-E), pressure (DPD-P),

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and enthalpy (DPD-H). The equations of motion for each variant are briey discussed below.

2.1 Constant Temperature DPD, DPD-T

In constant-temperature DPD, temperature and momentum are conserved. The following are

the equations of motion that describe DPD-T:

dr i = pimi

dt (5)

dp i = j= i

(FC i j + F Di j + F

Ri j)dt (6)

In order for the system to sample the canonical ensemble ( i.e. constant number of particles,volume, and temperature), DPD-T must obey the following uctuation-dissipation theorem [7]:

2i j = 2 i jk BT (7)

and

D(r ) = [ R(r )]2 (8)

Typically, the weighting functions D(r ) and R(r ) are chosen to be:

D(r ) = [ R(r )]2 = ( 1 r r c

)2 (9)

2.2 Constant Energy DPD, DPD-E

In constant-energy DPD (DPD-E), the total energy of the system is conserved. In order to do

so, the equations of motion for DPD-T are coupled with an internal mesoparticle equation of state,

ui, which is taken to be a sum of two terms that account for internal energy transfer via mechanical

and conductive means. This requires that d ui = dumechi + ducond i [8]. The equations of motion for

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DPD-E then become [2] :

dr i = pimidt (10)

dp i = j= i

FC i j dt i j D(

r ijr i j

vij )r ijr i j

dt + i j Rr ijr i j

dW ij (11)

duvisi = 12 j= i

i j D(r ijr i j

vij )r ijr i j

dt 2i j2

( 1mi

+ 1m j

)( R)2dt i j R(r ijr i j

vij )dW ij (12)

ducond i = j= i

i j ( 1

i

1

j) Dq dt + i j RqdW ij (13)

In Eqs. (10) - (13), and are the mesoscopic thermal conductivity and noise amplitude,

respectively, is the internal temperature, and W ij is an independent Wiener process, such that

W ij = W ji.

The uctuation theorem becomes:

2i j = 2 i j k Bi j (14)

where i j = 12 ( 1i +

1 j ) and with,

D(r ) = [ R(r )]2 (15)

The mesoscopic thermal conductivity and noise amplitude are also related through a uctuation-

dissipation theorem [9]:

2i j = 2k B i j (16)

Dq (r ) = [ Rq(r )]2 (17)

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Similar to DPD-T, the weighting functions are generally assumed to be [2][8]:

Dq (r ) = D(r ) = [ R(r )]2 = [ Rq(r )]2 = ( 1 r r c

)2 (18)

2.3 Constant Pressure DPD, DPD-P

A constant pressure variant of DPD (DPD-P) can be formulated by coupling the equations

of motion for DPD-T with a barostat. This barostat xes pressure upon some imposed pressure

and allows volue to uctuate. Thus, DPD-P conserves pressure, temperature and momentum. For

uniform dilation using a Langevin barostat, the equations of motion for DPD-P[10] are:

dr i = p imi

dt + pW

r idt (19)

dp i = j= i


Ri j )dt (1 +

d N f

) pW

p idt (20)

dlnV = dpW

dt (21)

d p = F dt (22)

In Eqs. (19) - (22), = ln V V 0 , V is the volume, W is a mass parameter, p is a momentum

conjugate, and F = dV (P P0) + d N f ip ip imi p p + pW p. The pressure, P , is calculated from

the virial formula [11],

P = 1dV

(i

p i p imi

+ i

j> i

FC i j r i j ) (23)

In Eq. (23), P0 is the imposed pressure, d is the dimensionality, N f = dN d , p and p are

Langevin barostat parameters, and W p is the Wiener process associated with the piston.

The associated uctuation-dissipation theorem relationships are those from DPD-T, Eqs. (7) -

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(8), along with

2 p = 2 pW k BT (24)

As i j and p go to zero, the expression H = K + U + P0V + p22W should be conserved, where K

is the kinetic energy and U is the potential energy.

2.4 Constant Enthalpy DPD, DPD-H

Constant-enthalpy DPD is a new DPD variant proposed by M. L sal et al. [2]. It combines

the equations of motion for DPD-E with the barostat of DPD-P. As it conserves both energy and

pressure, DPD-H conserves enthalpy ( i.e . a system at constant energy and pressure is at constant

enthalpy). The resulting equations of motion become:

dr i = p imi

dt + pW

r idt (25)

dp i = j= i


Ri j )dt (1 +

d N f

) pW

p idt (26)

dlnV = dpW

dt (27)

d p = F dt (28)

dumechi = 12 j= i

i j D(r ijr i j

vij )2dt 2i j2

( 1mi

+ 1m j

)( R)2dt i j R(r ijr i j

vij )dW ij (29)

ducond i = j= i

i j ( 1i

1 j

) Dq dt + i j RqdW ij (30)

The uctuation-dissipation theorem relationships from DPD-P, Eq. (24), and DPD-E, Eqs. (14)

- (17), dene the necessary relations for DPD-H.

In DPD-H, the total enthalpy is conserved [12]. Thus, H = K + U + U i + P0V + p22W is con-

served.

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2.5 DPD Model

The DPD variants were tested with two types of potential models. The rst was the DPD

uid model, which is commonly used as a potential in DPD. The model denes the conservative

potential as [13]:

uCGi j = a i j r c D(r i j ) (31)

where a i j denes the magnitude of the repulsion between two particles.

DPD does not require that Eq. (31) be used as the potential. Thus, tabulated potentials have

also been implemented into the code which allow for any conservative potential and force to be

used within the DPD framework. As an example, simulations have been performed using a density

dependent tabulated potential for RDX, which was tted through a technique called force matching

by Izvekov et al . [14].

3 FORTRAN90 and CoreXMD

FORTRAN90 is a general-purpose, procedural programming language that is especially suited

to numeric computation and scientic computing. Because of its capabilities, it is used for pro-gramming the DPD variants into CoreXMD.

CoreXMD is a software package designed for performing particle simulations over multiple

processors [15]. In CoreXMD, simulations follow a typical procedure within the code involving

initialization, iteration and destruction. During the iteration step, CoreXMD splits up pairs of

particles onto different domains on different processors such that the forces and energy of those

pairs of particles can be calculated simultaneously and therefore, parallel speed-up occurs. The

goal of this project was to implement variants of DPD such that the parallel abilities of CoreXMD

were used effeciently.

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4 Specics of Implementation of DPD Variants into CoreXMD

The fundamental steps required when performing simulations in CoreXMD involve the creation

and initialization of the system variables then the iteration for each time step in which forces are

calculated and nally the destruction of the simulation variables.

The creation and initialization of the simulation variables dene the simulation. Various pa-

rameters are read in from input which dene the unit cell, periodic boundaries, the initial positions

and velocities, as well as other parameters which dene the variant of DPD to be used ( i.e. , DPD-T,

DPD-E, DPD-P, DPD-H).

The iterative step is the most expensive step of the simulation as it requires calculating the

forces for each pair of particles as well as updating particle velocities and positions through iter-

ation of the pair lists. The iterative step from DPD as implemented requires a Shardlow Splitting

Algorithm (SSA) [16] and a two step velocity Verlet algorithm to update the positions and veloc-

ities [11]. Between the two velocity Verlet steps, the conservative forces for that time step are

calculated.

The Shardlow Splitting Algorithm method updates the velocity based on the dissipative and

random forces via integration of stochastic differential equations, while the velocity Verlet algo-

rithm updates the velocities via integration of ordinary differential equations. In DPD-E and DPD-

H, the Shardlow Splitting Algorithm also calculates the mechanical and conductive energies and

the internal temperature. The Shardlow Splitting Algorithm was implemented in the serial code as

it allows much larger time steps to be taken compared to velocity Verlet alone [16]. However, in

the following results section it is shown that it may be incompatible with domain decomposition,

which is fundamental to the parallel processing capabilities of CoreXMD. The nal step, the de-

struction of the simulation variables, is used for memory optimization, with the memory of various

variables and arrays being released.

DPD-P was implemented into CoreXMD by adding a stochastic Langevin barostat to the DPD-

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T code. The CoreXMD DPD-E was implemented by the addition of the internal particle energy

comprising the conductive and mechanical energies, which account for some internal degrees of

freedom lost due to coarse graining. DPD-H was implemented through the addition of the Langevin

barostat from DPD-P into the DPD-E code. DPD-T was not implemented as a part of this project

as it was previously coded into CoreXMD.

5 Results

The success of the implementation of the DPD variants into the parallel code was measured

by two criteria: how well the results conserved the quantity stated in the variant ( i.e. , constant

temperature, pressure, energy, or enthalpy) and how well the parallel code scaled. This paper will

rst discuss the former by comparing results from CoreXMD codes to those from serial versions

of the codes.

5.1 DPD-E Results

In Figure 1, the total energy calculated from 1-million step DPD-E simulations is shown for the

DPD uid potential. For the serial code, very little energy drift occurs as evidenced by the black,

horizontal line. All of the CorexMD runs show energy drift, with the magnitude of the energy drift

increasing with the number of processors used. For the serial version of the code, the percentage

drift over 1-million steps is nearly 0, while in the 1 processor CorexMD runs it is > 0.01 % increas-

ing to > 0.05 % for 16 processors. In Appendix B1, the conductive, mechanical, congurational,

and kinetic contributions to the total energy are shown. When compared to the serial code, all have

similar standard deviations in their uctuations except total energy in CoreXMD. This suggests that

there is an incompatability between the splitting algorithm employed and domain decomposition.

Since the percentage drift is increasing with the number of processors, the error could be due

to the use of the Shardlow Splitting Algorithm. The Shardlow Splitting Algorithm relies on a

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sequential update over the pairs of particles, which is fundamentally known to be incompatible with

parallelization via domain decomposition. This is due to domain decomposition utilizing loops

over pairs on different processors non-sequentially. Despite the drift, it is noted that the percentage

drift is small over 1 million steps, with the total energy varying only in the fourth decimal place in

Figure 1 in the CoreXMD results. Generally for velocity Verlet and other integration algorithms

for microcanonical molecular dynamics simulations, the energy should be constant within 0.01

% over the course of the simulation [1], and the results for DPD-E nearly meet this requirement.

Thus, even with the current energy drift, the results may be suitable for many applications.

Figure 1. A DPD-E simulation of 10,125 molecules over 1,000,000 timesteps using the DPD uid potentialfrom Eq. 31. Note that the serial run displays constant energy, but the CoreXMD parallel runs showconsiderable drift as the number of processors increases. While the drift looks signicant, it is important tonote that the drift is on the order of 10 4 which is quite minute.

5.2 DPD-P Results

Figure 2 displays the pressure from a DPD-P simulation of the DPD uid potential. The uctu-

ations of the CoreXMD results on 1, 8 and 16 processors compare well with that of the serial code

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be in agreement between the serial and CorexMD implementations of DPD-H, with the enthalpy

uctuations being one or two orders of magnitude larger than that of the serial code.

Figure 3. DPD-H simulation of 10,125 molecules over 1,000,000 teimsteps using the DPD uid potentialfrom Eq. 31. Note that because the DPD-H was built off the DPD-E, there is still a drift between the en-thalpy of the CoreXMD parallel runs and the constant serial runs. It is important to note that these drifts aresignicantly close in magnitude to the serial run because DPD-H employs the same stochastic barostat usedin DPD-P.

5.4 Error Suppression

A technique to suppress energy and enthalpy drift in the DPD-E and DPD-H methods, respec-

tively, was proposed by M. L sal et al [2]. This was implemented into the CorexMD DPD-E and

DPD-H variants. Using error suppression, the mechanical energy is adjusted such that any energyor enthalpy drift is corrected, ensuring the total energy or enthalpy to be perfectly constant. It is

advised not to use this option until the cause of the DPD-E and DPD-H drift is further investigated.

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6 Scalability

To measure the success of the parallel implementation of the DPD variants, the scalability of

the codes was investigated. The DPD-P simulation runs are used to demonstrate the scalability and

speed up from the serial counterpart. In Tables 1 - 3, the run times and parallel speedups are shown

for 9,216, 124,416 and 1,152,000 RDX particles for DPD-P as implemented into CorexMD with

density dependent tabulated potentials. Parallel speedups are shown in relation to 8 processors, the

number of cores on 1 node on the supercomputer employed. As the number of particles increases

the parallel speedup increases (Figure 4). This is the expected behavior as particle dynamic codes

generally scale better for larger systems due to the implementation of domain decomposition. For

effective utilization of processors, the correct choice of the number of processors and nodes based

on the number of particles is important. A scalability graph (Figure 4) is important to fully quantify

the efciency of ones parallel code. Generally, the aim is to distribute 1000 to 2000 particles per

processor. Thus, for 1.1 million particles using 2000 particles per processor, the scalability should

be efcient to 550 processors. However the scalability in Figure 4 is shown to drop off signicantly

with a parallel efciency of 75 % on 32 processors for 1.1 million particles, then decreasing to 37

% on 128 processors. Further optimization of the code should improve the scalability, allowing

for larger numbers of processors and nodes to be utilized. However, the parallelization has dra-

matically increased the speed at which results can be obtained. For example, for 124,416 particles

using 8 processors, CoreXMD was 293 % faster than the serial DPD-P code.

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Table 1. This table shows the parallel speedup of the DPD-P CoreXMD run for 9,216 beads using the RDXpotential. All speedup values are relative to the run time on 8 processors because that is how many

processors are present on each node.

CoreXMD DPD-P Speedup of 9,216 RDX beads

# of Processors Run Time Parallel Speedup

1 0:21:27 N/A

8 0:03:29 8

16 0:02:19 12.02

Table 2. This table shows the parallel speedup of the DPD-P CoreXMD run for 124,416 RDX beads. Allspeedup values are relative to the run time on 8 processors because that is how many processors are presenton each node. It is important to note that as the number of processors increases, the speedup increases, but

in decreasing amounts. This is due to the additional processors not used to each ones fullest potential.CoreXMD DPD-P Speedup of 124,416 RDX beads


1 4:11:37 N/A

8 0:36:53 8

16 0:21:49 13.52

32 0:14:13 20.75

64 0:11:05 26.62

128 0:09:26 32.48

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Table 3. This table shows the parallel speedup of the DPD-P CoreXMD run for 1,152,000 RDX beads. Allspeedup values are relative to the run time on 8 processors because that is how many processors are presenton each node. It is important to note that as the number of processors increases, the speedup increases, but

in decreasing amounts. This is due to the additional processors not used to each ones fullest potential

CoreXMD DPD-P Speedup of 1,152,000 RDX beads


1 N/A N/A

8 1:16:02 8

16 0:43:55 13.85

32 0:25:32 23.82

64 0:16:48 36.20

128 0:12:43 47.83

Fig 4. This is the scalability of DPD-P as implemented into CoreXMD for the Izvekov et al. [17] densitydependent model of RDX. Note that as the number of molecules increases, the speedup improves; thisis due to increased efciency in the use of the processors full potential. The 45 degree line is perfectscalability, essentially for some increase in the number of processors, the program will speed up by thatamount. This line is the objective so the white space between the current scalability lines and the 45 degreeline demonstrates room for improvement.

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7 Summary

Constant temperature (DPD-T), pressure (DPD-P), energy (DPD-E) and enthalpy (DPD-H)

variants of dissipative particle dynamics have been implemented into ARLs CorexMD code. DPD-

T and DPD-P conserved temperature and pressure, respectively, in CorexMD as well as their serial

counterparts. However, the DPD-E and DPD-H implementations of CorexMD showed larger en-

ergy and enthalpy drift, respectively, compared to their serial counterparts. This was attributed to

the known problems of utilizing the Shardlow Splitting Algorithm [16] with domain decomposi-

tion. In light of that, the energy and enthalpy drifts were relatively small, a maximum of < 0.06

% over 1 million steps on 16 processors. An error suppression algorithm was implemented which

can eliminate this error but was advised not to be used until the source of the error was investigated

further.

Density dependent tabulated potentials were implemented which allow for any potential to be

used for the conservative force, not just the DPD uid potential. This allowed 1.1 million RDX

molecules to be simulated within the DPD-P code for the rst time.

The scalability of the DPD variant implementations shows that further improvement to the

codes are needed. This can be accomplished through removing numerous multiple variables and

transforming them into particle data or type variables. There is overhead from the initial coding

and removing these overheads would provide a signicant improvement toward scalability and

speedup. For 1.1 million RDX molecules the DPD-P code was shown to be 75 % efcient on 32

processors and was 293 % faster on 8 processors compared to the serial code. The effort presented

here provides a suitable starting point for future improvements and optimization of the code.

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A DPD Variant Results

A.1 Constant Energy DPD, DPD-E

Table B1-1. Table containing components in DPD-E for a DPD-E simulation using DPD uid potentialfrom Eq. 31. Format is: average standard deviations

CoreXMD

Component Serial Serial 8 Processors 16 Processors

Umech/eV 0.776 0.000374 0.776 0.000372 0.776 0.000441 0.776 0.000457

Ucond/eV 0.776 0.0 0.775 0.0 0.775 0.0 0.775 0.0

Uconf/eV 0.117 0.000202 0.117 0.000195 0.117 0.000206 0.117 0.000203

Ukin/eV 0.0382 0.000308 0.0382 0.00031 0.0382 0.000316 0.0382 0.00029

Utot/eV 1.707 3.52 E -9 1.707 0.0000638 1.707 0.000175 1.707 0.000266

Tkin/K 295.484 2.381 295.292 2.399 295.226 2.442 295.229 2.241

Press/bar 1537.571 2.302 1537.427 2.284 1537.41 2.241 1537.418 2.27

A.2 Constant Pressure DPD, DPD-P

Table B2-1. Table containing components in DPD-P for DPD-P simulation using the uid potential fromEq. 31. Format is: average standard deviations

CoreXMD

Component Serial Serial 8 Processors 16 Processors

Tkin/K 299.987 2.403 299.649 2.323 299.487 2.411 299.536 2.42

Press/bar 1536.241 8.108 1537.073 7.972 1537.363 7.745 1536.801 8.37

Uconf/eV 0.117 0.000486 0.117 0.000493 0.117 0.000473 0.117 0.000514

Dens 0.00471 1.19 E -05 0.00471 1.14 E-05 0.00471 1.11 E -05 0.00471 1.22 E -05

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A.3 Constant Enthalpy DPD, DPD-H

Table B3-1. Table containing components of DPD-H from a DPD-H simulation using the DPD uidpotential from Eq. 31. Format is: average standard deviations

CoreXMDComponent Serial Serial 8 Processors 16 Processors

Umech/eV 0.776 0.000364 0.776 0.00042 0.776 0.00048 0.776 0.000461

Ucond/eV 0.776 0.0 0.776 0.0 0.776 0.0 0.777 0.0

Uconf/eV 0.117 0.000506 0.117 0.000472 0.117 0.000508 0.117 0.000489

Ukin/eV 0.0382 0.000325 0.0382 0.000333 0.0382 0.000345 0.0381 0.000306

Utot/eV 1.707 0.00052 1.707 0.000498 1.707 0.000545 1.707 0.000559

Tkin/K 295.579 2.325 295.289 2.574 295.172 2.671 295.945 2.366

Press/bar 1536.3995 8.2 1536.948 7.935 1537.667 8.318 1537.396 8.205

Enthalpy/eV 1.911 6.60 E-6 1.91 0.000217 1.911 0.000158 1.911 0.000221

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parallel implementation of dissipative particle dynamics and application to energetic materials

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