Clutter Reduction in Rendering of Particle (Atom) Trajectories with
Adaptive Position Merging
Bidur Bohara
1
and Bijaya B. Karki
1-3
1
School of Electrical Engineering and Computer Science, Louisiana State University, Baton Rouge, U.S.A.
2
Department of Geology and Geophysics, Louisiana State University, Baton Rouge, U.S.A.
3
Center for Computation and Technology, Louisiana State University, Baton Rouge, U.S.A.
Keywords: Trajectory Rendering, Clutter Reduction, Position Merging, Atomistic (Molecular) Visualization
Abstract: Visualization of position-time series data from molecular dynamics simulations of a material has to render
atomic trajectories, and relevant structural and dynamical information. Clutter/occlusion associated with
overlapping trajectories becomes serious even for moderate data sizes. We present an adaptive hierarchical
scheme for merging multiple positions along trajectories to significantly reduce the number of points/line
segments used for rendering. Our approach finds positions lying within a space window (cut-off distance)
from a reference position and merges them into a single position. The window is then moved in time order
with merging performed at each successive location. All original positions are thus processed to a reduced
number of new (merged) positions, which are further merged with the same or a different cut-off to obtain
even fewer positions. This hierarchical merging may continue several levels deep. Moreover, merging can
be performed subject to constraint of information, which is displayed (color-coded) along individual
trajectories. Both the trajectory geometry and underlying atomic structure become increasingly visible after
merging so the nature and extent of atomic arrangements and movements can be better assessed.
1 INTRODUCTION
Molecular dynamics (MD) simulations of real
material systems produce massive amounts of time-
varying positional data for constituent atoms or
molecules (Allen and Tidesley, 1987); (Peng et al.
2011). Visualization of such correlated data allows
us to gain insight into the structural and dynamical
behaviour of the material system under
consideration. Different approaches depending on
the nature of the atomic systems studied have
previously been used to visualize the scattered MD
data (see the Section 2). Two approaches for a rapid
navigation through the data are animation and
trajectories (Bhattarai and Karki 2009); (Li, 2005);
(Humphrey et al., 1996). Particle (atom) trajectories
allow a complete representation of position-time
series by rendering positions of all atoms at all-time
steps as points or line segments, so full simulation
information is contained in a single display. As
shown in Figure 1 (left), the main problem is that
trajectory rendering becomes too crowded because
there are simply many trajectories that are long,
distributed in 3D space, and overlapped with each
other. Note that the trajectories shown are unfolded
so they become continuous curves extending in the
space both inside and outside the simulation
supercell.
The challenge is how we can render the atomic
trajectories with a minimum clutter while preserving
the dynamical and structural information contained
in the data. The degree of trajectory crowdedness is
expected to increase with the size of the data, which
depend on the number of atoms and the number of
time steps. Though the clutter cannot be completely
avoided, it may be reduced to an acceptable level via
repetitive processing and analysis of the data.
Deciding whether to accept or reject the processed
trajectories may require a domain specific
knowledge. Various factors such as the geometrical
shape, spatial confinement, and other physical
properties of the trajectory are relevant.
Furthermore, the information about the geometric
arrangement of constituent atoms (i.e., atomic
structures) also needs to be rendered together with
the trajectories.
In this paper, we report an improvement in
rendering of the atomic trajectories from MD
264
Bohara B. and B. Karki B..
Clutter Reduction in Rendering of Particle (Atom) Trajectories with Adaptive Position Merging.
DOI: 10.5220/0004720102640271
In Proceedings of the 9th International Conference on Computer Graphics Theory and Applications (GRAPP-2014), pages 264-271
ISBN: 978-989-758-002-4
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
Figure 1: Visualization of molecular dynamics simulation data: atom trajectories and structure are rendered together. As
multiple positions along the trajectories are merged with our adaptive hierarchical merging (described in section 3), the
structure (consisting of Si-O polyhedra, H-O bonds, and floating Mg atoms) becomes increasingly visible (left to right).
simulations. We overcome the clutter/occlusion
problem associated with the trajectories by position
merging, encode useful information along the
trajectories, and superimpose the atomic structure.
While our approach should work for any MD data
set, here we deal with atomic systems that are
routinely simulated using the first-principles
molecular dynamics (FPMD) method based on the
fundamental equations of quantum mechanics. Two
atomic systems considered are hydrous silicate
(MgSiO
3
+H
2
O) and silica (SiO
2
) liquids (e.g., Karki
et al., 2010). The first system consists of 84 atoms in
a supercell of length of 10.11Å and simulation run
of 60,000 time steps (with each time step being 1
femtosecond). There are about 5 millions of data
points in total. The second system consists of 72
atoms in a supercell of length of 10.32Å and
simulation run of 1.2 million time steps. There are
over 80 million data points in total. We consider
these data sets to be of moderate size in the sense
that the classical MD simulations based on pairwise
potential functions can handle very large systems
even exceeding one billion atoms (Nakano et al.
2007).
2 RELATED WORK
Atomistic (molecular) visualization is one of the
most widely studied applications of computer
graphics and visualization (Bhattarai and Karki
2009); (Li, 2005); (Humphrey et al., 1996); (Grottel
et al., 2010); (Stone et al., 2011); (Zhang et al.,
2007). To gain insight into MD and FPMD
simulations, visualization has previously been
exploited in various ways depending on the nature of
the atomic systems studied. Few common examples
include VMD (Humphrey et al., 1996), Molscript
(Kraulis, 1991), XcrysDen (Kokalj, 1999), Atomeye
(Li, 2003), etc. In addition to rendering the atomic
configurations in various forms, they also support
animation and trajectories. To the best of our
knowledge, not much work was previously done in
manipulating the atomic trajectories.
Trajectory rendering is also relevant for other
movement-related data such as traffic network
consisting of roads and trails (Buchin et al., 2011).
Removing selected trajectories or portions of those
trajectories, and line simplification of highly curved
trajectories can help reduce the clutter to some
extent (Borcan, 2012). It is relevant to point out that
the pathline or trajectory technique is widely used in
vector visualization (Shi et al., 2007) with a lot of
efforts made on correct illumination and 3D
rendering of lines (Zockler et al., 1996), and
encoding the relevant information (Jones et al.,
2007). In the case of MD simulations, the input
position-time series directly represent the trajectory
data so trajectory rendering displays the spatio-
temporal behaviour of atomic system. Few ideas on
constraining trajectory rendering in space and time
have recently been explored (Bohara and Karki,
2012). Here, we present an improvement in
atomistic visualization by using adaptive
hierarchical (repeated) position merging based on
spatial proximity for trajectory rendering and by
superimposing instantaneous atomic structures.
3 POSITION MERGING ALONG
TRAJECTORY
Constituent atoms of a simulated material system are
expected to show non-uniform movements. This
ClutterReductioninRenderingofParticle(Atom)TrajectorieswithAdaptivePositionMerging
265
(a) (b) (c) (d)
Figure 2: Space proximity based two-level position merging along trajectory. Spheres represent space windows (cutoffs).
An original trajectory before merging (a), and the same trajectory after first merging (b), second merging (c) and smoothing
(d). Time is encoded along the trajectory (green to red).
means that the atoms spend longer time in certain
regions in 3D space than they do in other regions.
Positions in slow regions where atoms are perhaps
making oscillating (going back and forth) motions
form short and crowded portions of the trajectories.
A “proximity window” can be introduced based on
the assumption of such confined atomic movements
or localities so that positions lying in the window
can be merged together. This window can be
modeled as temporal proximity or spatial proximity.
It is important that the 3D geometry of the trajectory
be preserved during clutter reduction through
merging of multiple positions. This also means that
smaller amount of positional information is copied
to the video memory for rendering.
3.1 Basic Merging Algorithm
We first consider the temporal proximity, which
involves positions occupied by the concerned atom
over a finite time interval. A time window is defined
in terms of the number of successive MD steps, e.g.,
every 100 steps form a window. For each window,
we find a representative point by calculating the
mean (centroid) of all positions occupied at time
steps belonging to that window. Single position is
thus generated per window so the number of
windows (N
STEP
/N
TW
) gives the number of positions
to be rendered, where N
STEP
is the total number of
steps and N
TW
is the size of the time window. Line
segments are drawn between two successive mean
positions to render the trajectories. The length of the
line segments can vary significantly since the atoms
move at different speeds at different time. Generally,
N
TW
needs to be very large in order to reduce the
clutter in highly confined regions. This means that
line segments in extended regions (where the atoms
are moving fast without reversing directions) can be
unusually long so the time-window based merging
may not work well for the entire trajectory length.
Perhaps a more appealing approach is to apply
spatial proximity, which considers positions located
in the close vicinity of each other irrespective of
time when these positions are occupied by the
concerned particle. We define spatial proximity in
terms of a finite space window (cut-off distance) so
that all positions lying within such window are
considered for merging. The idea is to scan all time
steps to pick up positions whose distances measured
from the reference position are shorter than the cut-
off distance, and merge them together. The locality
being approximated is thus a spherical region of
radius equal to the cut-off distance (Figure 2).
Our position-merging algorithm for each
trajectory works as follows:
Step 1: Set the space window at the starting
(reference) position of an atom trajectory.
Step 2: Scan subsequent time steps to pick up
positions lying inside the window until an outside
position is encountered for the first time.
Step 3: Merge/map all positions found in the step 2
to single position.
Step 4: Start at the first position not yet picked up
(new reference position), and do the steps 2 and 3.
Step 5: Move the window in the time order with
merging performed at each successive location
(repeating the step 4) until the end.
Thus, all positions are processed to generate a
reduced set of new (merged) positions. If successive
positions are well separated because either the cut-
off distance is too small or atoms move too fast,
merging is not effective at all and every window
contains single but original position. On the other
hand, if atoms have covered small distances because
either atom are slowly moving/oscillating or the cut-
off distance is too large, an arbitrarily large number
of positions can be found in the window under
GRAPP2014-InternationalConferenceonComputerGraphicsTheoryandApplications
266
consideration to map to a single position. Note that
processing atomic positions in the time order
ensures the correct geometry of the trajectories while
merging nearby positions as much as possible.
3.2 Adaptive Hierarchical Merging
After the first run of position merging using a cutoff
distance, the clutter though it might have already
reduced considerably is still not at an acceptable
level and merged positions are still too close. So it
may be desirable to apply additional merging with
the same or larger cutoff. The next level merging
then considers all new positions derived in the
previous merging by placing a space window (of the
same or bigger size) at the first merged position.
Figure 2 illustrates two-level merging. As in the
original merging, all positions lying within the
current window at each successive location are
picked up and merged. It is likely that the positions
even after the second merging are still closely
spaced, and/or again it may be desirable to apply an
even larger cut-off. A third/higher level merging is
then performed for the positions, which have
undergone merging more than once. This adaptive
hierarchical processing will continue until the
number of repeatedly merged positions does not
decrease anymore. This situation arises relatively
fast if the cut-off distance is constant from the one
level to the next. However, if the cut-off distance is
increased progressively, the merging process may
continue over many levels. The user has to specify
when to stop either by not increasing the cut-off
further or by visually assessing the crowdedness and
correctness of the trajectories being increasingly
approximated with multilevel position merging.
Merging is expected to be sensitive to
information that we want to display (color code)
along the trajectories. Examples of useful
information include the distance travelled by atom,
time spent, coordination states, bond events, merge
count, mean squared error, etc. Such information is
computed during merging process from original set
of positions and appropriately mapped to the merged
positions. Information extraction for coordination
states involves the positions of nearby atoms.
Atomic coordination ሺܥ
ఈఉ
is the number of the
nearest neighbour atoms of the same species (
) or
other species (
) around a given atom of species
.
Coordination is of multivariate type as it can take
multiple values for a given window. Our approach to
impose a coordination constraint while merging is
that successive positions within a space window are
merged only if they are in same coordination state.
A position with different coordination state than the
previous positions will start a new space window for
merging next sequence of positions of the same
coordination state. It is expected to result in more
crowded trajectories than original merging.
3.3 Trajectory Rendering
To visualize the particle (atom) trajectories, we
render a series of connected line segments for each
trajectory by using the original positions (no
merging) or new (merged) positions at different
levels of merging. The user may choose to display
single trajectory for a selected atom or a subset of
trajectories corresponding to an atomic species or a
group of atoms (e.g., SiO
4
unit), or trajectories for
all constituent atoms.
Successive positions generated by the final
hierarchical level merging are well separated (by
more than the cut-off distance) so trajectories are no
longer smooth and can be visually distracting. This
becomes more serious when larger cut-offs are used.
We perform a piece-wise smoothing of line
segments by generating cubic and quadratic Bézier
curves. A subset of merged positions along the
trajectory is used as control points in the Bézier
equation to generate the local curves. Additional
information about the material system in question
can be extracted on the fly and encoded along the
trajectories using a color map.
4 RESULTS AND DISCUSSION
How much the trajectories are cluttered/crowded
depends on the size of the data and the nature of the
atomic movements. Here, we present our case
studies by visualizing complete position-time series
of two material systems (silicate and silica liquids)
mentioned earlier.
4.1 Effects of Space Window Size
For a given data set, the size of space window (cut-
off) applied controls the degree of clutter reduction
to a great extent. Larger the cut-off, more positions
merged along the trajectories and less crowdedness
(Figure 3). However, too wide space windows can
cover multiple localities and new positions produced
by merging process may be too few to approximate
the trajectory geometry to any acceptable level
(Figure 3, right). There is a trade-off between cut-off
(clutter measure) and the trajectory geometry.
The choice of the space cut-off requires domain
ClutterReductioninRenderingofParticle(Atom)TrajectorieswithAdaptivePositionMerging
267
Figure 3: Effects of the space window (cutoff) size on the trajectory geometry. One H trajectory (top row) and one Si
trajectory (bottom row) are processed using four different cutoffs. On increasing cutoff from left to right, the trajectories
become simpler with local features disappearing more and more. The final trajectory (right) in each case fails to preserve
the overall geometry. The color encodes time information along the original and processed trajectories.
specific knowledge and some pre-processing of a
given position-time series data (such as radial
distribution function and mean square
displacement). Our adaptive approach allows the
user to adjust the space cut-off by considering a few
factors such as the number of trajectories, the shape
and extent of trajectories, the desired level of clutter
reduction, and the degree of trajectory
approximation. For instance, if a single trajectory
(one selected atom) or a few trajectories (atoms
forming a structural unit or cluster) are visualized, a
small cut-off may reduce the associated clutter to an
acceptable level. However, if many (corresponding
to all atoms of one species) or entire trajectories are
rendered, relatively large cut-offs are needed.
4.2 Visualization of Trajectories
A detailed visualization of trajectories of hydrous
silicate (MgSiO
3
) and pure silica (SiO
2
) liquids
shows us that the atomic movements involve two
types of motion: extended portions of the trajectories
represent rapid continuous motion whereas the
crowded portions represent confined (oscillating)
motion. The atoms jump from one confined region
to another in relatively short time interval.
For the hydrous system, each trajectory of each
species (H, Mg, Si, and O) is obtained by rendering
60,000 successive positions occupied by the
corresponding atom. Here we display 16 H
trajectories obtained by processing about one million
positions in the total (Figure 4). They start from the
respective initial atomic positions all lying within
the supercell and they eventually extend beyond the
super-cell reaching as 3 or 4 times the supercell
length. Note that H atoms are the most mobile
species. It is difficult to trace these trajectories
except in the outermost regions. Also, any atomic
structures such as initial configurations shown by
atomic spheres can hardly be seen. The H
trajectories are approximated using our adaptive
position merging approach with a cut-off 1.5Å and
five levels of merging (Figure 4). This process
suppresses local wiggling features and reduces the
number of positions along the trajectories.
Individual trajectories can be seen almost
everywhere and all atomic spheres are visible.
During visualization process, the user can explore
the complete trajectory geometry with the aid of
operations like rotation, translation, scaling, and
highlighting.
The atoms in the silica liquid are very slow
covering relatively small distances even over much
longer simulation duration. In other words, the
atoms are trapped locally most of the time. The Si
trajectories obtained by rendering one million
positions per atom are highly confined to narrow
regions and they extend very little outside the
supercell (Figure 5). Different trajectories tend to
form distinct regions displayed with different colors
but these regions overlap among each other. It is
difficult to assess the shape and extent of these
trajectories because the geometric structures are
almost invisible. Our adaptive hierarchical position
merging approach is expected to be useful.
4.3 Performance Analysis
The effectiveness of position merging along the
trajectories can be assessed by counting the number
GRAPP2014-InternationalConferenceonComputerGraphicsTheoryandApplications
268
Figure 4: Trajectories of 16 H atoms (each colored differently) in hydrous silicate liquid after hierarchical merging.
Figure 5: Trajectories of 24 Si atoms (each colored differently) in silica liquid after hierarchical merging.
of final (merged) positions used for trajectory
rendering, examining the geometries (shape and
extent) of individual trajectories, and having a
measure (qualitative) of clutter reduction in
visualization of trajectories and atomic structures.
Key factors relevant for this assessment include
space window size (cut-off distance), hierarchical
merging level, and also imposed information
constraint. Here, we present a performance analysis
considering two cases: weak merging (with a narrow
space window) and strong merging (with a wide
space window) for each species trajectory for both
material systems.
As shown in Table 1, the effects of window size
(or cut-off) are substantial. For H trajectories,
merging with a cut-off of 0.8Å involves 5
hierarchical levels and yields 16,450 final points
compared to about 1 million original positions. The
visibility of the trajectories improves considerably
and some atomic spheres start to appear. When a
larger cut-off of 1.5Å is used (again with 5
hierarchical levels) the number of final positions
drops to 3,170 – a reduction by a factor of 300 with
respect the original positions. The individual
trajectories and underlying atomic structure become
visible. For O trajectories, the number of final
positions drops from about 9,000 (weak merging) to
1,770 (strong merging), compared to 2.6 million
original positions. Similarly, the number of final
positions drops by a factor of 4 and 6 between the
two cases of merging for Mg and Si trajectories,
respectively. A relatively larger effect is seen for Si
trajectories because most of them are more confined.
Position merging along atomic trajectories for
silica liquid is much more sensitive to window size
than that for hydrous silicate liquid. A weak merging
along Si trajectories using a cut-off of 0.5Å saturates
at the 8
th
level and generates around 57 thousands
positions, compared to the original 24 million
positions. The number of reduced (final) positions
drops to around 4 thousands with somewhat larger
cut-off (0.7Å). The overall reduction in the number
of positions needed for trajectory rendering is by a
factor of about 6,200. Thus, a much smaller set of
processed positions, which represents only 0.016 %
of the original positional data are rendered thereby
dramatically reducing clutter in rendering of Si
trajectories. Since O atoms cover longer distances,
we use larger cut-offs of 0.8Å (weak merging) and
1.2Å (strong merging). The weak merging yields
ClutterReductioninRenderingofParticle(Atom)TrajectorieswithAdaptivePositionMerging
269
Table 1: Number of atomic positions, original and after merging, of individual atom species in hydrous silicate liquid (top
part), and silica liquid (bottom part) for weak merging (small cut-offs) and strong merging (large cut-offs). Number of
positions after first level merging at corresponding cut-off is presented inside the parentheses. The last column represents
the number of positions for a constrained merging with large cut-off.
Original Weak Merging Strong Merging
Cutoff Levels Positions Cutoff Levels Positions Constrained merging
H 960,000 0.8 5 16,450
(24,710)
1.5 5 3,170
(7,160)
15,630
(16,830)
Mg 720,000 0.5 4 6,818
(7,975)
1.0 4 1,585
(2,520)
14,000
(14.015)
Si 720,000 0.5 4 2,920
(4,540)
1.0 3 510
(740)
1,975
(2.115)
O 2,640,000 0.8 5 8,945
(14,125)
1.5 4 1,770
(2,615)
5,480
(6,170)
Si 24 millions 0.5 8 57,420
(507,490)
0.7 5 3,865
(12,055)
51,530
(75,110)
O 48 millions 0.8 8 93,880
(989,570)
1.2 6 3,150
(12,290)
-
Figure 6: Trajectories of H atoms with color-coded coordination information with respect to O atom (red: singly
coordinated, yellow double coordinated), before (left) and after (right) constrained merging with cut-off of 1.50Å.
little less than one hundred thousand positions, and
the strong merging case yields much fewer positions
(3,000 positions) for reduced trajectory, which
represent about 0.0066 % of the original positional
data and results in a much improved visibility.
The number of hierarchical levels needed to
saturate merging varies somewhat depending on the
space cut-off and atomic species. This number spans
the range of 3 to 5 for two cases (weak and strong
merging) considered for all species of hydrous
silicate. The number is higher (5 to 8 levels) for Si
and O trajectories of silica. Hierarchical merging
tends to continue deeper for the cases of highly
confined trajectories. The number of new (merged)
positions along trajectories decreases as merging
goes deeper the merger hierarchy. As shown in
Table 1, the numbers of positions differ significantly
between the first level and the final level of
hierarchical merging in both weak and strong
merging.
Finally, we consider the sensitivity of position
merging to a coordination information constraint
(Figure 6). As shown in Table 1, fewer levels are
needed to saturate merging in the hierarchy, and the
number of final positions generated by merging
along H trajectories, with H-O coordination as
constraint, increases from around 3,000 (for original
merging) to above 15,000 for a cut-off of 1.5Å.
Similar increases in the number of final positions
were found for other cases.
5 CONCLUSIONS
We have presented an improvement in atomistic
visualization of position-time series data obtained
from molecular dynamics simulations of real
GRAPP2014-InternationalConferenceonComputerGraphicsTheoryandApplications
270
materials by rendering both the atom (particle)
trajectories and instantaneous structures. To reduce
the clutter caused by the crowded trajectories, we
perform adaptive hierarchical merging of multiple
positions along the trajectories. According to our
approach, the multiple positions to be merged at
each level are picked up using a proximity window,
which is defined in terms of space window (distance
cutoff). Our analysis shows that the number of
effective positions needed to render the trajectories
decreases dramatically under merging (with/out
information constraint) and the processed (reduced)
trajectories show significantly reduced clutter. We
can further enhance the visualization process by
encoding additional information (time, 3D position,
coordination number, and merge count) along the
trajectories. Improved trajectories allow us to better
assess the nature and extent of the corresponding
atomic movements. In particular, they suggest that
atoms move via discrete jumps (hopping-like
motion) in addition to continuous forward motion.
More importantly, the underlying atomic structures
become visible with all trajectories rendered. While
moderate-size data sets containing several millions
of data points (atomic positions) were considered in
this study, we anticipate to extend the proposed
position merging to larger data sets produced by
large-scale molecular dynamics simulations.
ACKNOWLEDGEMENTS
This work is supported in part by a grant from
National Science Foundation (EAR 1118869).
REFERENCES
Allen M. P., Tidesley D. J., 1987. Computer Simulation of
Liquids, Oxford University Press.
Bhattarai D., Karki B. B.,
2009. Atomistic visualization:
Space-time multiresolution integration of data analysis
and rendering. Journal of Molecular Graphics and
Modelling, 27: 951-68.
Bohara B., Karki B. B., Rendering particle trajectories
with color-coded information for atomistic
visualization, 12
th
IASTED Int’l Conf. on
Visualization, Imaging and Image Processing
(VIIP’12), 2012, 782-049: 35-42.
Borcan M.,
2012. Improving Visualization of Trajectories
by Dataset Reduction and Line Simplification. Master
Thesis 2012, Utrecht University.
Buchin K., Demšar U., Slingsby A., Willems E., 2011.
Results of the break-out group: Visualization,
Dagstuhl SeminarProceedings.
Grottel S., Reina G., Dachsbacher C., Ertl T., 2010.
Coherent culling and shading for large molecular
dynamics visualization, Eruographics/IEEE-VGTC
Symp. on Visualization, 29, 3.
Humphrey W., Dalke A., Schulten K., 1996. VMD –
Visual molecular dynamics, Journal of Molecular
Graphics 14 (1996) 33-38.
Jones C., Ma K.L., Ethier S., Lee W.L., 2007. An
integrated visual exploration approach to particle data
analysis. Tech. Rep. CSE-2007-20, University of
California at Davis.
Karki B.B., Bhattarai D., Mookherjee M., Stixrude L.,
2010. Visualization-based analysis of structural and
dynamical properties of simulated hydrous silicate
melt. Physics and Chemistry of Minerals 37:103-117.
Kokalj A.,
1999. XCrySDen – a new program for
displaying crystalline structures and electron densities,
Journal of Molecular Graphics and Modelling 17
(1999) 176-179.
Kraulis P.J., 1991. MOLSCRIPT: A program to produce
both detailed and schematic plots of protein structures,
Journal of Applied Crystallography 24 (1991) 946-
950.
Li J., 2003. Atomeye: an efficient atomistic configuration
viewer, Modelling and Simulation in Materials
Science and Engineering 11 (2003) 173-177.
L
I J., 2005. Atomistic visualization, in: S. Yip (Ed),
Handbook Materials Modeling, Springer, pp. 1051-
1068.
Nakano A., Kalia R. K., Nomura K., Sharma A.,
Vashishta P., Shimojo F., Duin A. C. T., Goddard W.
A., Biswas R., Srivastava D.,2007. A divide-and-
conquer/cellular-decomposition framework for
million-to-billion atom simulations of chemical
reactions, Computational Materials Science, 38:642-
652.
Peng L., Kunaseth M., Durson H., Nomura K., Wang R.,
Kalia R. K., Nakano A., Vashishta P., 2011.
Exploiting hierarchical parallelisms for molecular
dynamics simulation on multicore clusters, Journal of
Supercomputing 57: 20-33.
Shi K., Theisel H., Hauser H., Weinkauf T., Matkovic K.,
Hege H. C., Seidel H.P., 2007. Path line attributes – an
information visualization approach to analyzing the
dynamic behavior of 3D time-dependent flow fields.
Topology-Based Methods in Visualization.
Stone E.J., Vandivort K.L., Schulten K.,
2011. Immersive
out of core visualization of large-size and long-
timescale molecular dynamics trajectories. Lecture
Notes in Computer Science, 6939:1-12.
Zhang C., Callaghan S., Jordan T., Kalia R.K., Nakano A.,
Vashishta P.,
2007. Paraviz: a spatially decomposed
parallel visualization algorithm using hierarchical
visibility ordering. International Journal of
Computational Science 1: 407.
Zockler M., Stalling D., Hege H.C., 1996. Interactive
visualization of 3D-vector fields using illuminated
streamlines. Proc. IEEE Visualization’96, San
Francisco, pp. 107-113.
ClutterReductioninRenderingofParticle(Atom)TrajectorieswithAdaptivePositionMerging
271