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CEE 618 Scientific Parallel Computing (Lecture 12)Dissipative Hydrodynamics (DHD)
Albert S. Kim
Department of Civil and Environmental EngineeringUniversity of Hawai‘i at Manoa
2540 Dole Street, Holmes 383, Honolulu, Hawaii 96822
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Particle Dynamics
Outline
1 Particle Dynamics
Introduction
Brownian Dynamics
Stokesian Dynamics
Lab work and Project
2 Raster3D
Visualizing Spheres
2 / 26
Particle Dynamics Introduction
What is Particle Dynamics?
A study of motion of multiple particles,
influenced by forces and torques
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Particle Dynamics Introduction
What is the force?
FORCE
A push or pull that can cause an object with mass to accelerate
Newton’s second law:
F = ma
Acceleration:
a =dv
dt=
d2r
dt2
ENERGY
A scalar physical quantity that is a property of objects and
systems which is conserved by nature
The ability to do work:
E = −∫
r2
r1
F · dr
only if F = F(r).
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Particle Dynamics Introduction
Statistical Mechanical Approaches
1 Nano-scale (10−9 m)
MD (Molecular Dynamics) = Deterministic simulation of solving
Newton’s second law for ion species
2 Nano to Micro-scale (10−6 m)
BD (Brownian Dynamics) = Updated simulation protocol of MD for
ions in a fluid medium, but more applied to volumeless (point)
colloidal/nano-particles: Random Forces/Torques
DPD (Dissipative Particle Dynamics) = Simulation method for
Brownian motion of multiple particles using (approximate) pair-wise
hydrodynamics.
3 Nano to Meso-scale (10−3 m)
SD (Stokesian Dynamics) = Accurate simulation method for
micro-hydrodynamics of spherical particles
DHD = General simulation method for micro-hydrodynamics of
Brownian and non-Brownian particles
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Particle Dynamics Brownian Dynamics
Brownian Dynamics: Langevin’s Equation
The Langevin equations for the system of N Brownian particles:
for particle i interacting with j’s
pi = mivi = Fi (r) +∑
j
(−) ξijvj +∑
j
αijfj
1 Molecular Dynamics for conservative forces/torques2 Stokesian Dynamics for hydrodynamic forces/torques3 Dissipative Particle Dynamics for stochastic forces/torques
* On the average hydrodynamic ≈ stochastic
pi = mivi is the momentum,
ξij is the hydrodynamic friction tensor,
Fi is the sum of inter-particle and external forces, and∑
j αijfj represents the randomly fluctuating force exerted on a
particle by the surrounding fluid: negligible if particles are much
bigger than 1.0 µm.
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Particle Dynamics Brownian Dynamics
Properties of Random Fluctuating Force, fi
1 Time average is zero:
〈fi〉 = 0 (1)
2 Independently exerted on i and j particle of different positions
(i.e., ri and rj) and at different times (i.e., t and t′)
〈fi (t) fj(
t′)
〉 = 2δijδ(
t− t′)
(2)
3 δ is the Dirac-delta function:
δij = 0 if i 6= j; and δij = 1 if i = j;
δ (t− t′) = 0 if t 6= t′; and δ (t− t′) = 1 if t = t′.
4 Related to the friction coefficient
ξij =1
kBT
∑
k
αikαjk (3)
indicating α ∼√ξ.
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Particle Dynamics Brownian Dynamics
Brownian Dynamics
Integration of the Langevin equation gives the time evolution equation:
ri (t+∆t) = ri (t) +∑
j
Dij (t)
kBT· Fj ∆t+ (∇ ·D)∆t+∆rGi (4)
where the components of ∆rGi are random displacements selected
from 3N variate Gaussian distribution with zero means and
covariance matrix
〈∆rGi 〉 = 0 and 〈∆rGi ∆rGj 〉 = 2Dij∆t (5)
The Oseen tensor (crude approximation) is given by
Dij =kBT
6πηa1, for i = j (6a)
=kBT
8πηrij
(
1+rijrij
r2ij
)
, for i 6= j (6b)
and one calculates ∇ ·D = 0. If Fj ≈ 0, the random motion is
dominant in multi-particle dynamics: ∆rGi ∝√∆t.
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Particle Dynamics Brownian Dynamics
Brownian Dynamics (BD)
Langevin equation1 with inter-particle (conservative) forces fP ,
drag forces fH = −ξv, and random Brownian forces fB
mdv
dt= fP + fH + fB (t) (7a)
fH = −ξv (7b)
〈fB(t)〉 = 0 (7c)
〈fB(0) · fB(t)〉 = 6ξkBTδ (t) (7d)
1Ermak and McCammon, J. Chem. Phys. 69 (1978) 1352-1360; Langevin,
C. R. Acad. Sci. (Paris) 146 (1908) 530-5339 / 26
Particle Dynamics Brownian Dynamics
e.g., a falling body in liquid with x(0) = 0 & v(0) = 0
ma = −mg − βv + fB (t)
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Particle Dynamics Stokesian Dynamics
Stokesian Dynamics: Langevin’s Equation
The Langevin equations for the system of N force-free, non-Brownian
particles
pi = mivi = −∑
j
ξij (vj − U) ≡ FH
FH is the hydrodynamic forces/torques,
pi = mivi is the momentum, and
ξij is the hydrodynamic friction tensor.
If particles are at rest,
U = M∞ · FH (8)
FH = R∞ ·U (9)
R∞ = (M∞)−1 (10)
where U is the translational/rotational velocity vector, and M∞ and
R∞ are the grand mobility and grand resistance matrixes, respectively.
R∞ is dependent on particle positions and calculated as an inverse
matrix of M∞.11 / 26
Particle Dynamics Stokesian Dynamics
Stokesian Dynamics (SD)
Particles translate and rotate in a fluid field of
V = U∞ + r ×Ω∞ +E∞ : r
where U∞ is the uni-directional flow; and the vorticity Ω∞ and rate of
strain E∞ are represented as
Ω∞ = 1
2∇× V (r)
E∞
ij =1
2
(
∂Vi
∂xj+
∂Vj
∂xi
)
= 1
2(∂jVi + ∂iVj) = Eji
respectively. If no shear, E∞ = 0
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Particle Dynamics Stokesian Dynamics
Directions of force/torque: Fx, Fy, Fz, Tx, Ty, Tz
Exerted on each particle with upflow, U = +1 (↑).
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Particle Dynamics Lab work and Project
Lab work
SD code code for hydrodynamic force/torque calculation is in
/opt/cee618s13/class12/hasonjee/
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Raster3D
Outline
1 Particle Dynamics
Introduction
Brownian Dynamics
Stokesian Dynamics
Lab work and Project
2 Raster3D
Visualizing Spheres
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Raster3D Visualizing Spheres
Raster3D
http://skuld.bmsc.washington.edu/raster3d/
1 Raster3D is a set of tools for generating high quality raster images
of proteins or other molecules.2 The core program renders spheres, triangles, cylinders, and
quadric surfaces with specular highlighting, Phong shading, and
shadowing.22 / 26
Raster3D Visualizing Spheres
Example 1
1 Copy all the files from
/opt/cee618s13/class12/raster3d/example1/
to your own directory.2 Type and enter: qsub⊔raster_ex1.pbs3 This pbs script will execute example1h.script and generate an
image file, example1h.tff
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Raster3D Visualizing Spheres
Sphere configuration: 6× 6× 6 array
Under ‘/mnt/home/albertsk/UHTraining/cee618-sp2012/class09/DHD’1 In “sHsnj_obsd_fts_64.f”
To rotate image change Euler angles of alpha0, beta0, and
gamma0.
To change the distance between the center and your eyes, control
distance “sHsnj_obsd_fts_64.f”.
2 “Raster3Dspheres.f” is included in the main code
“sHsnj_obsd_fts_64.f”.3 There will be three output files from this serial run:
1 “sForceFTS.dat” stores force/torque calculation data.2 “sCoordXYZ.dat” includes (x, y, z) coordinates of Np particles.3 “sCoordXYZ.r3d” contains Raster3D format coordinate data,
translated to the center of mass.
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Raster3D Visualizing Spheres
How to generate an image
1 Copy all the files in /opt/cee618s13/class12/dhd-raster3d/ to your
own directory.2 Execute
$ make
$ make⊔run3 Then, a file like “sCoordXYZ.tff” will be generated.4 Download the .tff file and view it.
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Raster3D Visualizing Spheres
Raster file: sCoordXYZ.r3d, x, y, z, a, and 3 more
1 Example of material properties and file indirection
2 80 64 tiles in x,y
3 8 8 pixels (x,y) per tile
4 4 3x3 virtual pixels -> 2x2 pixels
5 0 0.1 0 background colour
6 T cast shadows
7 25 Phong power
8 0.15 secondary light contribution
9 0.05 ambient light contribution
10 0.25 specular reflection component
11 4.0 eye position
12 1 1 1 main light source position
13 0.578E+00 -0.259E+00 0.483E+00 0.000E+00
14 0.224E+00 0.966E+00 0.129E+00 0.000E+00
15 -0.500E+00 0.000E+00 0.866E+00 0.000E+00
16 0.000E+00 0.000E+00 0.000E+00 0.900E+02
17 3 mixed objects
18 *19 *20 *21 # Draw a bunch of spheres
22 #
23 #
24 #
25 @orange.r3d
26 2
27 -.241800E+02 -.241800E+02 -.241800E+02 0.100000E+01 0.100000E+01 0.100000E+01 0.100000E+01
28 @green.r3d
29 2
30 -.806000E+01 -.241800E+02 -.241800E+02 0.100000E+01 0.100000E+01 0.100000E+01 0.100000E+01
31 @blue.r3d
32 2
33 0.806000E+01 -.241800E+02 -.241800E+02 0.100000E+01 0.100000E+01 0.100000E+01 0.100000E+01
34 @red.r3d
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