doi:10.21311/001.39.4.22 A KMC-Based Multi-Scale …tjfeonline.com/admin/archive/2218.05.20161463541303.pdf · A KMC-Based Multi-Scale Evolution Simulation Method of Dust in Virtual

Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 4, 176 - 187, 2016

176

doi:10.21311/001.39.4.22

A KMC-Based Multi-Scale Evolution Simulation Method of Dust in

Virtual Campus

Xiaomei Hu, Yubin Wang, Junjun Hu

Shanghai Key Laboratory of Intelligent Manufacturing and Robotics, School of Mechatronic Engineering and

Automation, Shanghai University, Shanghai200072, China

Abstract

Dust evolution simulation provides an important way to analyze the impact of dust on the environment. Since

wind speed plays a key role in dust evolution, multi-scale simulation is applied to simulate the wind speed in

order to improve the accuracy of wind speed simulation and reduce the amount of calculation. Through k-means clustering, the virtual campus can be divided by grids with different scales and multi-scale simulation of wind

speed can be realized. KMC-based multi-scale simulation model of dust evolution in virtual campus is proposed.

Experimental results show that KMC-based multi-scale dust evolution simulation model can improve the

accuracy of the simulation and reduce the computational cost at the same time, which proves the effectiveness

of dust evolution simulation system based on multi-scale modeling.

Key words:Multi-scale Modeling, Dust Evolution Simulation, Kinetic Monte Carlo, Dust accumulation.

1. INTRODUCTION

Nowadays, with rapid economic development and sharp growth of urban population, dust floating in the air

has become part of the urban pollution and does harm to human beings seriously. Moreover, dust can do harm to the ecological system covertly and potentially over a long period. For example, dust can also take the pollutants

into urban water system via runoff caused by rainfall, declining the water quality and damaging the city aquatic

ecosystem. Because the contents of heavy metal elements in dust are much higher than the soil background

values, these heavy metal elements can cause serious damage to the nervous system, the respiratory system and

the endocrine system of the human beings (Chang,et al.,2007;Zheng, Li and Huang, et al., 2009;Shi,Chen and

Xu,et al., 2008;Zhang,Chen and Xu,et al.,2006;Al-Khashman,2004). As a result, research on the dust evolution

has great significances on environment quality and health protection.

The evolution process of dust includes sediment, diffusion and re-suspension(Gao andLi,2011).

Nevertheless, it is quite difficult to study the evolution process of dust by experiment equipment currently

because of the extreme complication of the evolution process. Therefore, simulation has become one of the most

important methods to research the dust evolution process (Da, Yang, Li and Lu,2005;Xi and Jiang,2002;Ye, Wang and Li,2006). A lot of methods can be used in the simulation, such as first principal method (FP),

molecular dynamic method (MD), Monte Carlo method (MC), Finite element method (FEM) and so on(Peng,

Lu and Qi, et al., 2003). The Kinetic Monte Carlo (KMC) related to the MC, has the advantage of simulating in

a long period. In addition, KMC is a stochastic process, which makes it fit for evolution simulation of the dust

particles (Okin,2003;Yao, Hao andHu,2003;Sun, Qian and Zhang,2011;Peng and Yuan,2008;Meixner, Kunert

and Scholl,2003;Wang,2003).

In the KMC-based dust evolution simulation, wind speed plays a key role. However, it is hard to get the

real wind speed affecting the movement of dust. Therefore CFD software is used to simulate the wind speed in

the virtual area (Hu, Wang, Fan, Xu and Chang,2014;Wang, Hu, Xu and Fan,2013). Firstly, the virtual area is

divided by 3D grid; Secondly, the wind speed of each vertex of 3D grid can be gained by CFD software; Finally,

the wind speed of the other positions can be calculated according to the wind speed of each vertex of the grid by interpolation. In order to improve the accuracy of wind speed simulation, thin grid should be adopted, which

will increase the computation cost. In order to decrease the computation cost, thick grid should be adopted,

which will reduce the accuracy of wind speed simulation. In order to balance the accuracy of wind speed

simulation and the amount of simulation computation, a multi-scale simulation method is used to divide the grid.

Multi-scale simulation has been widely used in many fields in recent years. It is a balance solution between

accuracy and computational cost. The construction method of multi-scale computational model can be divided

into two kinds: scale separation and scale coupling (Rudd and Broughton,2000). The former focuses on

analyzing different parts of the object using different scales, and the latter focuses on finding the connection

between the macro and micro. In this paper, multi-scale simulation model based on scale separation is

considered. That is, the model with appropriate scale is selected according to the intensity of wind speed change

and realizes the cooperation among models with different scales.


177

In this paper, Shanghai University is chosen as the simulation district. The 3D model of the university can

be gained by 3D graphics software. A unit grid clustering algorithm is proposed and the virtual campus is

divided by multi-scale grid. The area with relatively severe wind speed change is divided by thin grid, and the

rest of the area is divided by thick grid. The wind speed of each vertex of thin grid and thick grid in virtual

campus can be calculated. Based on the numerical simulation results of wind speed, multi-scale KMC evolution

simulation of dust in virtual campus is realized. Finally, the visualization of dust evolution can be obtained by OpenGL.

2. A KMC-BASEDMULTI-SCALESIMULATIONOFDUSTEVOLUTIONINVIRTUALCAMPUS

2.1. Multi-scale grid division of virtual campus

Shanghai University is chosen as the simulation area and modeled by 3D graphics software. The campus is

120 meters in width and 470 meters in length. The chosen simulation space is 120meter 470meter 6meter

considering that dust mainly moves in the height of 0 to 5 meter.

The basic idea of multi-scale grid division is to divide the areas where wind speed changes intensely by

thin grid and divide the areas where wind speed changes slightly by thick grid. In this way, the precision of wind

speed model can be ensured and the computational cost can be decreased. The specific steps are as follows:

(1) Divide the virtual campus by Fluent using unit grid, that is, 1meter 1meter 1meter grid; (2) Set wind speed change threshold. When the change of wind speed in unit grid does not exceed the threshold,

unit grids can be merged together with different scales through k-means clustering.

According to two steps above, the virtual campus can be divided by multi-scale grid.There are eight

vertexes in each unit grid, shown in Figure 1. The wind speed of each vertex in unit grid has three components

in , ,x y z directions respectively. The wind speed [ ]V j of vertex j can be expressed as ( [ ], [ ], [ ])x y zV j V j V j , and

the position of vertex j can be expressed as ( [ ], [ ], [ ])x y zP j P j P j .

The intensity of wind speed change in unit grid can be defined as:

8

1

1 2 3 4 5 6 7 8

x x x x x x x x

x

x

j

V V V V V V V Vf

V j

(1)

8

1

1 3 2 4 5 7 6 8

y y y y y y y y

y

y

j

V V V V V V V Vf

V j

(2)

8

1

1 5 2 6 3 7 4 8

z z z z z z z z

z

z

j

V V V V V V V Vf

V j

(3)

1 2

3 4

5

7 8

6

x

z

y

Figure 1.The vertex number of unit grid

xf , yf , zf are the intensity value of wind speed change in unit grid in directions respectively. Given that

the wind speed change threshold is 0.2, the wind speed change can be considered severe if the intensity value of

wind speed change exceeds 0.2 in any direction.

If the following conditions are satisfied in unit grid:


178

0.2

0.2

0.2

x

y

z

f

f

f

(4)

the wind speed change in unit grid can be considered slight and this kind of unit grids should be merged together

through clustering. The wind speed change of the rest of the unit grids are considered severe and the unit grids

are left unprocessed.

According to the unit grid clustering criterion in virtual campus, unit grid k-means clustering algorithm in

virtual campus is adopted to merge the unit grids with slight wind speed change and close wind speed to form

much larger grids. K-means clustering algorithm is an adaptive search algorithm, and its basic idea is to adjust

the clustering center repeatedly by iterations and finally divide theindividuals in dataset into K clusters to

minimize the sum of all the Euclidean distances from each individual to the clustering center of the cluster the

individual belongs to(Kanungo and Mount,2004;Wu,2012;Sabo and Scitovski,2013).

Wind speed of unit grid is the average of wind speed of eight vertexes in unit grid. Since wind speed of

each vertex in unit grid has three components, so wind speed can be written as _ _ _

( , , )ix iy izV V V , which is described

as: _ _ _ _

8

_1

8

_1

8

_1

( , , )

[ ]

8

[ ]

8

[ ]

8

i ix iy iz

x

jix

y

jiy

z

jiz

V V V V

V j

V

V j

V

V j

V

(5)

The goal of k-means clustering is to divide all the unit grids into K clusters according to wind speed to

form the cluster set { 1,2,..., } kT t k K . The clustering center of the cluster kt is k

. According to the

definition of wind speed attribute in unit grid, the clustering center also has three components accordingly:

( , , ) k kx ky kz(6)

In k-means clustering, initial values should be set to the clustering centers. According to the initial

clustering centers, the Euclidean distance from unit grid to the clustering center is calculated as:

_ _ _2 2 2( , ) ( ) ( ) ( ) i k ix kx iy ky iz kzD c V V V (7)

By calculating the Euclidean distance from unit grid to the clustering center of each cluster, unit grid can

be divided into the cluster with the shortest Euclidean distance.

The sum of all the Euclidean distances from each unit grid to the clustering center of the cluster the unit grid

belongs to is defined as:

1 1

( ) ( , )

K n

i i k

k i

S c D c (8)

n is the number of all the clustering unit grids. And

1,

0,

i k

i

i k

if c t

if c t(9)

After the initial division of all the unit grids, the average value of the wind speed attributes of all the unit

grids that the cluster kt contains can be considered the new clustering center of the cluster kt .The new clustering

center of the cluster is described as:


179

_

1

_

1

_

1

1,

0,

n

i ix

ikx

k

n

i iy

iky

k

n

i iz

ikz

k

i k

i

i k

V

n

V

n

V

n

if c t

if c t

(10)

In (10), kt is the number of unit grids in the cluster.According to the new clustering centers, ( )S c can be

recalculated:

(1)If the recalculated value is equal to the original one, the clustering comes to an end;

(2) Otherwise, the unit grids are re-divided according to the new clustering centers to regain the clustering

centers and ( )S c .

The k-means clustering ID of unit grids represents the results of k-means clustering. The finalID of unit

grids is the final results of grid area expansion considering grid spatial adjacency. The process of the k-means

clustering algorithm of unit grids in virtual campus is as follows:

Step1. Divide the virtual campus by unit grid;

Step2. Select the first unit grid and read the wind speed of its eight vertexes. Calculate the intensity value of

wind speed change in each direction and decide whether the wind speed change of the unit grid is severe;

Step3. Dispose of all the unit grids as Step2 in order;

Step4. The k-means clustering is applied in all the unit grids with slight wind speed change and obtain the k-

means clustering ID of each unit grid;

Step5. Select the first unit grid and set its finalID as 1; Step6. Dispose all the unit grids in order. For each unit grid, if its k-means clustering ID is the same as that of its

adjacent unit grid, the finalID of the unit grid is equal to that of the adjacent unit grid; otherwise, the finalID of

the unit grid is one more than the maximum value of the finalIDs of all the processed unit grids;

Step7. All the unit grids get their finalID and the k-means clustering ends.

After the k-means clustering, all the unit grids with the same finalID make up a large grid, and the multi-scale

grid division of virtual campus is complete. The virtual campus is divided by the unit grids with severe wind

speed change and the clustering non-unit grids.

2.2. KMC-based multi-scale simulation of dust evolution

The dust particles move in the simulation process randomly, and three kinds of events occur in the process

of movement: sediment, diffusion and re-suspension. (1) dust diffusion

Considering drag force, gravitational setting, Saffmanlift force and turbulent diffusions in numerical calculation,

motion equations of the particle can be written as:

( )( )

y p

D

p

gdvF v

dt(11)

DF u v is the drag force per unit particle mass and

2

18

24

D

D

P P

C ReF

d(12)

representswind speed, v is particle speed, is air density, p is the density of particles, pd is

particle diameter, is the molecular viscosity of the fluid, yg is the gravitational acceleration, DC is drag

coefficient and eR is Reynolds number.

In the KMC-based simulation, the particles’ motion can be expressed as follows:

( ) ( ) ( 1)

( ) ( ) ( 1)

( ) ( ) ( 1)

x

y

z

X t V t X t

Y t V t Y t

Z t V t Z t

(14)


180

' ( )xV t , ' ( )yV t ,and ' ( )zV t is the particle acceleration in each direction at t time step, and they can be

calculated by the following formula: '

'

'

( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )

x Dx

y P

y Dy

P

z Dz

V t F u v c t

gV t F u v c t

V t F u v c t

(15)

( )DxF u v , ( )DyF u v , ( )DzF u v mean the effectiveness of wind field to dust particles. In (16),

D x

Dx

D y

Dy

D z

Dz

F uF

u

F uF

u

F uF

u

(16)

xu ,yu and

zu are the components of wind speed in three directions. ( )c t is a random factor, and it is

defined as:

2 2 2

max

( ) ( ) ( )( )( )

max ( )

x y zV t V t V tv tc t

v v t(17)

So the particles diffuse in the virtual campus environment according to the formulas above.

(2) dust sediment

When particles fulfill the following condition:

'

0 ( ) 5

v ( ) 0

z t mm

t(18)

they can be taken as sediment on the earth in the simulation model. ( )z t isthe vertical height of the particles.

'v ( )t is the particle acceleration.

(3) dust re-suspension

Particles can be re-suspension when the wind speed increases to critical friction velocity. The equation can

be shown as follows:

u ( ) / Pf PA d g (19)

In (19), fu is the critical friction velocity, p

is particle density, pd is the diameter of the particle, g is

the acceleration of gravity, is the air density, A is a random coefficient with values between 0.16 and 0.21.

From the KMC-based simulation of dust evolution, wind speed is a key parameter. In the process of KMC-

based multi-scale simulation of dust evolution, wind speed affecting on dust particle is calculated according to

the position of dust:

(1) If the position where particle locates is in the non-unit grids, the wind speed is approximately equal to the

clustering center of k-means clustering in this grid;

(2) If the position where particle locates is in the unit grid, the wind speed can be obtained by tri-linear

interpolation.

Tri-linear interpolation is the method of using linear interpolation in the tensor product grid of 3D discrete

sampling data(Zhang, Guan and Qin,2002; Yi , et al.,2006;Zhang and Han, et al.,2013).

In Figure 1, the wind speed of the position ( , , )x y zP P P can be obtained as follows:

Firstly, interpolate in the Z-axis:

1

2

1

2

[1]( [5] ) [5]( [1])

[3]( [5] ) [7]( [1])

[2]( [5] ) [6]( [1])

[4]( [5] ) [8]( [1])

z z z z

z z z z

z z z z

z z z z

i V P P V P P

i V P P V P P

j V P P V P P

j V P P V P P

(20)

Then, interpolate in the Y-axis:

1 1 2

2 1 2

( [3] ) ( [1])

( [3] ) ( [1])

y y y y

y y y y

w i P P i P P

w j P P j P P(21)


181

Finally, interpolate in the X-axis:

1 2( [2] ) ( [1]) x x x xu w P P w P P (22)

The wind speed prediction of the position ( , , )x y zP P P can be obtained according to the formula (20),(21),(22)

and the result can be taken into equation(15).

3.THERESULTSANDDISCUSSION

To prove the effectiveness of the multi-scale simulation model, seven collection points in the virtual

campus are chosen. Among them, five places are on the ground: location 1 is influenced mostly by buildings

while less by the transportation; location 2 is located on the campus road which is close to the large lawn, so it is

less affected by building but greatly affected by the flow of people; the location 4, similar to location 3, is both

affected by buildings and the flow of people and the flow of car influences the location 5 most. The location 6

and 7 are highly close to location 1, but their heights from the ground are 6m and 3m respectively. The 3D

model of virtual campus is shown in Figure 2. The tools we use to collect dust include lint brush, fine mesh,

storage bags, precision balance, marking pen. We gather dust with lint brush to storage bags at each location at

10am every morning, and then weigh every bag with precision balance respectively and record the weight of

dust.

Figure 2. 3D model of virtual campus

Shanghai University is modeled by 3D graphics software, and Fluent is used to simulate the wind field of

Shanghai University. In Fluent software, the grids within the buildings are automatically eliminated. The initial

input wind speed is the northeast wind because the northeast wind is the most frequent wind in the area.

According to the wind field simulation result from Fluent, the wind speed of eight vertexes of unit grid in

virtual campus can be obtained. The data file of wind speed is imported into the dust KMC evolution simulation

system developed by C language. At the beginning of the simulation phase, 108 particles are born in a stochastic

location in the virtual campus. The speed of dust particles can be obtained by the wind speed of eight vertexes in the grid. Then KMC method is used to calculate the velocity and the trajectory of each particle.

In dust KMC evolution simulation system, the virtual campus is firstly divided by the unit grid, that is,

1meter 1meter 1meter grid. The number of the clusters of k-means clustering (k) is set as 9. According to the

initial clustering centers, the unit grids can be divided into different clusters and the unit grids in the same

cluster may not be adjacent. The initial clustering centers are shown in Table 1.

Table 1. The initial clustering centers of k-means clustering The clusters of k-means

clustering kx ky kz

1 -5 -0.6 -2

2 -4 -0.5 -1.5

3 -3 -0.4 -1

4 -2 -0.2 0

5 -1 0 1

6 0 0.2 2

7 1 0.4 3

8 2 0.5 3.5

9 3 0.6 4


182

After 67 times of iteration, the clustering centers of k-means clustering remain the same, and the sum of all

the Euclidean distances from each unit grid to the clustering center of the cluster the unit grid belongs to remains

unchanged. The results of k-means clustering can be obtained.

The final clustering centers and the number of unit grids each cluster contains are shown in Table 2.

Table 2. The clustering results of k-means clustering

The clusters of k-

means clustering kx

ky kz

The number of

unit grids each

cluster contains

1 -8.475657 -0.000711 2.343375 9967

2 -5.467401 0.005255 -0.514765 17893

3 -3.072794 0.002147 -0.064701 54578

4 -3.558064 -0.003680 1.894925 23792

5 -1.535472 0.002284 0.094976 44178

6 -1.042483 0.002974 2.399580 20669

7 -5.798894 0.003674 3.436155 11763

8 0.055294 -0.003875 -0.024551 46690

9 1.601643 -0.000310 -0.277950 2838

The virtual campus is divided into multi-scale grid by disposing all the unit grids. For each unit grid, if its

adjacent unit grid has the same k-means clustering ID as it, the finalIDs of the two unit grids are the same too.

Otherwise, the unit grid is assigned a new finalID. After the final clustering, 844 non-unit grids can be obtained

and visualized in Figure 3.

Figure 3. The multi-scale division of virtual campus

After the division of virtual campus by multi-scale grid, the wind speed in non-unit grid is approximately

equal to the clustering center of k-means clustering in this grid. Due to using approximation instead of

interpolation in calculating wind speed, the computational cost of simulation can be reduced. In order to validate

the reliability and effectiveness of multi-scale modeling, the virtual campus is divided by the thick 10meter

10meter 3meter grid and the thin 1meter 1meter 1meter grid. The wind speed in two grids can be obtained

by interpolation. The simulation results of three models are compared as follows.

3.1. The comparison of computational cost

Take the dust accumulation in 10 minutes for example, the computational cost of three models is shown in

Table 3.

Table3. The computational cost of different models

Thick grid model Thin grid model Multi-scale model

Calculation time 27min 32min 15min

As shown in Table 4, multi-scale simulation model has the shortest calculation time, and the calculation

time of thick grid based simulation model is shorter than that of thin grid based simulation model. In multi-scale

model, wind speed is decided by the attributes of the multi-scale grid, and the results are obtained without

interpolation. However, interpolation is used in thick grid model and thin grid model. Besides thin grid model

takes more computational costs in data importing and storage.


183

3.2. Comparison of wind speed simulation error

Because wind speed is the most important parameter in the dust evolution simulation, wind speed

simulation error will directly affect the accuracy of dust evolution simulation. Compared with thin grid model,

multi-scale model and thick grid model both have the errors. is the simulation error of multi-scale model, and' is the interpolation error of the thick grid model. The definitions of the errors are as follows:

~

_1

1

n

i

ii

V

V

n(23)

_1

'

1

n

i

ii

V

V

n(24)

In (23) and (24),

iV is the clustering wind speed of the gridic ,

iV is the wind speed of the gridic obtained in

thick grid model by interpolation.

The error comparison of multi-scale model and thick grid model is shown in Table 4.

Table 4. The error comparison of multi-scale model and thick grid model

simulation error of multi-scale model( ) interpolation error of the thick grid model( ' )

0.280959 0.645181

According to the definition, the smaller the error is, the more accurate the simulation model is. From Table

4, multi-scale model has a relatively smaller error, so it is more accurate than the thick grid model. Actually, the

thick grids often intersect with the buildings and the interpolation results hardly reflect the subtle wind speed

change.From the analysis above, multi-scale model has low computational cost and small simulation error of wind speed.

3.3. The comparison of simulation models and experimental results

To validate the accuracy of multi-scale model further, the experimental data of dust accumulation during

non-rain period of 7 days is compared with the simulation data of dust accumulation in different models. Table 5

shows the average weight of dust accumulation during the different non-rain periods. The experimental data of

dust accumulation is compared with the simulation data of dust accumulation in different models, shown in

Figure 4. Figure 4(a) to (g) show that the dust accumulation becomes heavier and heavier with the increase of

non-rain period in seven locations, which proves the effectiveness of the models.

Table 5. The average weight of dust accumulation in different locations

Non-rain period The average weight of dust accumulation in seven locations

L1 L2 L3 L4 L5 L6 L7

1day 0.034 1.355 1.364 1.378 2.031 0.022 0.047

2days 0.059 2.905 2.592 2.406 3.232 0.037 0.086

3days 0.095 4.104 3.788 3.405 4.298 0.049 0.117

4days 0.124 5.556 4.801 4.157 5.444 0.065 0.144

5days 0.163 6.882 6.173 5.388 6.745 0.083 0.169

6days 0.197 8.146 7.752 6.537 7.965 0.109 0.200

7days 0.231 8.961 8.801 7.394 9.200 0.123 0.228


184

(a) Relationship of dust accumulation and non-rainperiod at location-1

(b) Relationship of dust accumulation and non-rain period at location-2

(c) Relationship of dust accumulation and non-rain period at location-3

(d) Relationship of dust accumulation and non-rain period at location-4


185

(e) Relationship of dust accumulation and non-rain period at location-5

(f) Relationship of dust accumulation and non-rain period at location-6

(g) Relationship of dust accumulation and non-rain period at location-7

Figure 4. Comparison of experimental results and simulation results(From left to right,histograms are:

experimental result, thin grid model, thick grid model and multi-scale model).

In Figure 4，the dust accumulation in thin grid model is closest to the experimental data, so thin grid

model has the highest precision and the best reliability. The dust accumulation in thick grid model usually has

large difference with the experimental data and is usually much less than the experimental data. Therefore, thick

grid model has the lowest precision. The dust accumulation of multi-scale model is less close to the

experimental data than that of thin grid model and closer to the experimental data than that of thick grid model,

so the accuracy of multi-scale model is relatively modest.

From the comparison of computational cost, wind speed simulation error and experimental results, multi-

scale simulation has a better performance because it can make a tradeoff between simulation accuracy and computation cost, which proves the effectiveness of KMC-based multi-scale simulation of dust evolution.


186

4. CONCLUSIONS

Wind speed is a key factor to simulate dust evolution. In order to improve the accuracy of wind speed

simulation and reduce the computational cost at the same time, multi-scale grids are generated by K-means

clustering algorithm. Based on multi-scale grids, a KMC-based multi-scale simulation of dust evolution is

proposed. The experimental results show that multi-scale simulation has the lowest computational cost among the three kind models, and smaller simulation error of wind speed comparing to thick grid model. Compared

with the experimental result, the multi-scale model has more accurate performance than thick grid model, which

shows the effectiveness and reliability of multi-scale simulation of dust KMC evolution.

Acknowledgements

This work was supported in part by NSFC (Project No. 41101454).

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