Research Article Kinetic-Monte-Carlo-Based Parallel ...Research Article Kinetic-Monte-Carlo-Based Parallel Evolution Simulation Algorithm of Dust Particles XiaomeiHu,ZhifengXu,HongxiaCai,andJunjunHu

Research ArticleKinetic-Monte-Carlo-Based Parallel EvolutionSimulation Algorithm of Dust Particles

Xiaomei Hu Zhifeng Xu Hongxia Cai and Junjun Hu

Shanghai Key Laboratory of Mechanical Automation and Robotics School of Mechatronic Engineering and AutomationShanghai University Mailbox 232 No 149 Yanchang Road Shanghai 200072 China

Correspondence should be addressed to Hongxia Cai hx cai163com

Received 24 September 2013 Accepted 24 December 2013 Published 21 January 2014

Academic Editor Bernard J Geurts

Copyright copy 2014 Xiaomei Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The evolution simulation of dust particles provides an important way to analyze the impact of dust on the environment KMC-based parallel algorithm is proposed to simulate the evolution of dust particles In the parallel evolution simulation algorithm ofdust particles data distribution way and communication optimizing strategy are raised to balance the load of every process andreduce the communication expense among processes The experimental results show that the simulation of diffusion sedimentand resuspension of dust particles in virtual campus is realized and the simulation time is shortened by parallel algorithm whichmakes up for the shortage of serial computing and makes the simulation of large-scale virtual environment possible

1 Introduction

Many ecological environmental problems have emerged dur-ing the process of urbanization Sedimentation of lots ofsurface dust in cities caused by transportation is one of themDust has close relation with particulates in the atmosphere[1] Ecological system can be hurt by dust covertly over a longperiod while it is human body that dust can do most directand greatest harm to Dust (especially suspended particulatematter with aerodynamic diameters less than 10 120583m) has beenone of themost serious air pollutants inChina formany yearsThe surface dust can be resuspension under some certaincircumstances and the contaminant will make bad influenceon our body It is shown by some researches that Pb canretard childrenrsquos intellectual development and weaken theirintelligence while these acknowledged prisoners includingCu Cd Cr Zn As and Hg are in a position to change humanbeingsrsquo nervous and respiratory system A lot of problemsare caused by the pollution of dust such as laze respiratorydisease and lung cancer [2] As a result research on thesurface dust is not only a crucial aspect of environmentevaluation but has also great significance on human healthwhen the urbanization is accelerating

The evolution process of surface dust includes sedimentdiffusion and resuspension Nevertheless it is quite difficultto study the evolution process of surface dust by experimentequipment currently because of the extreme complicationof the evolution process Therefore simulation has becomeone of the most important means to research the dustevolution process The current simulation research includesthe diffusion of dust around buildings in cities and therelationship between dust deposition and wind power [3ndash7] Dust diffusion is one of the hot topics of the dustevolution Dust diffusion is a complex dynamic process Itrepresents a series of particles diffusion process in the windsuch as particles deposition resuspension and transit [8] Anumber of investigations have been carried out for predictingturbulent diffusion around buildings using wind tunnel tests[9 10] Numerical methods for simulating flow and diffusionfields have developed rapidly Particle deposition in verticalsquare ventilation duct flows by two different numericalmodels has been studied [11 12] The effects of particlediameter dimensionless relaxation time flow direction andair speed in vertical upward and downward square duct flowon particle deposition velocities are discussed lift and gravity

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 839726 11 pageshttpdxdoiorg1011552014839726

2 Journal of Applied Mathematics

have also been taken into account Ali has investigated a time-dependent partial differential equation governing the trans-port of heavy dust into the atmosphere Dust concentrationis expressed in terms of a series of confluent hyper geometricfunctions [13] Sun et al investigated the microparticle depo-sition and distribution by employing the Eulerian approachwith Reynolds stress turbulent model and a Lagrangian tra-jectorymethod [14]With the rapid development in computertechnology computational fluid dynamics (CFD) methodhas matured to simulate the ventilation performance andcontaminant dispersion and transfer in buildings Saha etal have used CFD to assess the effect of wind tunnel sizeson air velocity and concentration boundary layers and onammonia emission estimation [15] Tominaga et al have usedCFD to predict the air diffusion around a construction [1617] Besides these Schneider et al proposed a semiempiricaltwo-compartment constant parameter model [8] Roney andWhite compared the near-surface wind-tunnel fugitive dustconcentration profiles arising from soil surfaces beds withthose calculated numerically [18] However most of thestudies paid attention to simulate the diffusion process ofparticles The whole dust evolution process is seldom traced

A lot ofmethods can be used in the simulation of dynamicprocess such as first principal method (FP) moleculardynamic method (MD) Monte Carlo method (MC) andfinite elementmethod (FEM) [19ndash22] Among thesemethodsMCmethod has been well used in simulation of air pollutiondispersion and river water pollution Three decades agoMonte Carlo (MC) method was used to study the dynamicprocess The basic idea of the MCmethod is that the solutionof the problem is equivalent to hypothetical statistical modelparameters using random number and the parameters areestimated by the statistical model of a sample [23] Okinproposed MC method to simulate airborne dust diffusionmodel [24] Yao et al usedMCmethod in simulation of emis-sion height effects on building [25] Tian presented the MCsimulation of complex terrain effect on dust diffusion [26]Dejun GU puts forward a model for the convective boundarylayer of the line source diffusion along MC method Xu et alsimulate gas dispersion based on Monte Carlo which couldsatisfy the requirement of the long-distance pipeline disastersemergency decision making [27] The object researched bykinetic Monte Carlo (KMC) method is unbalanced or arelaxation process Time evolution correct or not is the keyfactor in the simulation process simulation time step mustbe the system real time step so KMC method is an effectivemethod of studying on the kinetic behavior Sun et al appliedKMC analysis in environmental risk assessment of a chlorinerelease accident [28] Peng and Yuan simulated gas-solidflow behavior in desulphurization tower based on KMC [29]Because dust have a similar dynamic property with water andair pollutant KMC method has widely been used in waterflow simulation and atmospheric diffusion field The kineticMonte Carlo (KMC) related to the MC has the advantage ofsimulating in a long period In addition KMC is a stochasticprocess which makes it fit for evolution simulation of thesurface dust particles In the current conditions it is hard tosimulate the KMC evolution of dust particles in a large-scaleurban environment by one computer Therefore the parallel

5

4

1

2

3

Figure 1 3D model of Shanghai University campus

computing method can be used to shorten the simulationtime [30ndash33]

Virtual reality (VR) has the property of intuition andinteractive which makes up the defeat of computer simu-lation and makes the results of simulation take on effect of3D stereo display [34ndash38]This paper will adopt visual realitytechnology (VRT) to visualize the results of dust evolution Avirtual Shanghai University campus is rendered by OpenGLand a parallel simulation system of dust evolution in thevirtual campus is created in order to study the impact ofdifferent parameters (such as nonrain period and wind)on dust evolution By comparison with experimental datathe validity of the model is verified The research aims atproviding theory and quantitative reference for dust particlesevolution

2 Kinetic Monte Carlo Simulation ofSurface Dust Evolution

21The 3DModeling of Virtual Campus Shanghai Universitycampus is simplified and the 3D model is shown in Figure 1

The campus is 120 meter in width and 470 meter inlength Since most of surface dust works in the height of 0to 5 meters the virtual campus is in the scale of 120m lowast

470m lowast 6m To verify the validity of the KMC simulationmethod five positions are chosen as the sampling point andsigned as 1 2 3 4 and 5 shown in Figure 1 Location 1 isgreatly influenced by the buildings while the transportationhas less impact on it Location 2 is located on the campus roadwhich is near the greensward The flow of people has greaterimpact on it than the buildings Location 3 and Location 4are moderately influenced by the buildings and the flow ofpeople Location 5 is on a road where traffic is quite large andslightly impacted by buildings By collecting and analyzingthe weight of dust at the 5 positions and comparing themwith the KMC simulation result the initial parameters of dustevolution simulation are defined to suit the real situation

Journal of Applied Mathematics 3

22 KMC Simulation Modeling of Surface Dust Particle inVirtual Campus The surface dust modeling is dynamic andthe dust particles in the simulation system have three periodsincluding ldquoemergingrdquo ldquomovingrdquo and ldquovanishingrdquo With thepassage of time some of the existing particles are vanishingand the new ones are emerging The survival particles movein the simulation process randomly and three kinds of eventsoccur in the process of movement sediment diffusion andresuspension Therefore the KMC simulation modeling ofdust particles consists of five events

(1) Dust particles merge and join in the simulation sys-tem with attributes provided

(2) Dust particles diffuse in the air and update theirattributes

(3) Certain dust particles sediment on the ground andupdate their attributes

(4) Some particles on the ground are resuspension andupdate their attributes

(5) Dust particles exceeding the life cycle are deleted fromthe simulation system

221The Initialization of Dust Particles It is assumed that 108dust particles are released into the virtual campus environ-ment Every dust particle has six initial attributes (1) initialposition (2) initial wind speed (3) initial size (4) initialcolor (5) shape (6) life cycle

The initial positions of dust particles are uniformly dis-tributed These particles have the same size shape and colorTheir density is 1800 kgm3 The initial speed of each particleis the wind speed of its position which can be obtained bynumerical simulation of wind field in the virtual campusCommon Computing Fluid Dynamics (CFD) software suchas Fluent CFX and Phonics Star-CD can simulate the windfield

Assuming air is incompressible viscous fluid the typeof flow is turbulent density is regarded as constant Con-trol equation includes continuity and momentum equationRelated equations are as follows [39]

120597119906119894

120597119909119894

= 0

120597119906119894

120597119905+120597 (119906119894119906119895)

120597119909119895

= minus1

120588

120597119875

120597119909119894

+120597

120597119909119895

(V120597119906119894

120597119909119895

)

(1)

119909119894and 119909

119895(119894 119895 = 1 2 3) present the distance in each

coordinate (119909 119910 119911) 119906119894and 119906119895present the velocity component

in each coordinate (119909 119910 119911) 119875 is air pressure 120588 is air densityand 119905 is time

To solve the strongly swirling flow problem in the numer-ical simulation of wind field in the virtual campus 119896-120576modelis used to produce certain distortion

119896 =3

2(119906ref119879119894)

2 120576 = 119862

34

120583

11989632

ℓ ℓ = 007119871 (2)

119906ref denotes inflow point velocity 119871 denotes equivalentlength

Gy Gx

Gz

Figure 2 Grid division in virtual campus

Besides in order to obtain the accurate simulation resultsof wind field some improvements are made

(1)There are two types of grid during Shanghai Universityvirtual campus space grid division including triangle andhybrid grid Triangle grid has strong boundary adaptabilityand hybrid grid can save compute time In order to simplymodel campus local area and plane grid is used [40]

The simulation area contains two parts building intervalarea and external wind field area shown in Figure 2 Thegrid has been divided into unstructured grid and structuredgrid separately In order to improve the accuracy of calcula-tion the building interval area is encrypted Especially theexternal wind field uses quadrilateral structure grid and theinternal wind field uses triangle unstructured grid

(2) There are three main boundaries in the simulationof wind field flow inlet boundary flow outlet boundaryand solid boundary In order to simplify and optimize theboundary conditions specific setting is listed in Table 1

(3) Before the simulation it is necessary to make sureof its validity and judge its convergence The experimentalresult shows that when iteration is around 1800 times mostkinds of iterative curves are close to our setting number underthe residuals 1119890 minus 06 As shown in Figure 3 the wind fieldsimulation result in this condition has good convergency andfidelity

According to themeteorological record the average windspeed 32ms is set and wind direction is northeastThe windspeed in virtual campus is shown in Figure 4

CFD postprocessing software such as Ensight Tecplotand FieldView can deal with the 3D grid dataMeanwhile thespatial coordinates of dust particles are written on the text in agrid data formWith the help of CFD wind speed at each gridnode is available and thewind speed anywhere on campus canbe drawn by interpolation


Table 1 Boundary conditionsBoundary Condition FormulaSky No slip wall boundary 119906 = V = 0Inlet condition Velocity inlet boundary ] = 32ms

Ground building Free-slip boundary V = 0 120597119906

120597119910= 0

Outlet condition Pressure-type boundary 120597119875

120597119909= 0 Δ120588 = 23556 pa

1e + 02

1e + 00

1e minus 02

1e minus 04

1e minus 06

1e minus 08

1e minus 10

1e minus 12

1e minus 14

1e minus 16

1e minus 18

0 2 4 6 8 10 12 14 16 18 20

times102

Iterations

ResidualsContinuityx-velocityy-velocity

Energyk

Epsilon

Figure 3 Residual curve simulated by Fluent

265e + 01

252e + 01

239e + 01

226e + 01

212e + 01

199e + 01

186e + 01

173e + 01

159e + 01

146e + 01

133e + 01

119e + 01

106e + 01

929e + 00

796e + 00

664e + 00

531e + 00

396e + 00

266e + 00

133e + 00

132e minus 03

Figure 4 Profile of wind speed

222 Dust Particlesrsquo KMC Movement Once the particlesemerge they begin to move when they get the initial attrib-utes Their move attributes can be derived by the initial onesconsidering sediment diffusion and resuspension incidentsrespectively

(1) The Diffusion of Surface Dust Particles Considering dragforce gravitational setting Saffman lift force and turbulentdiffusions in the process of computation motion equationsof the particle can be written as [12]

119889119906119901

119889119905= 119865119863(119906 minus 119906

119901) +

119892119909(120588119901minus 120588)

120588119901

(3)

119865119863(119906 minus 119906

119901) is the drag force per unit particle mass

119865119863=18120583

1205881199011198892119901

119862119863Re24

(4)

119906 represents wind speed 119906119901is particle speed 120588 is air density

120588119901is the density of particles 119889

119901is particle diameter 120583 is the

molecular viscosity of the fluid119862119863is drag coefficient and Re

is Reynolds numberThis event mainly deals with dust particles on airflow

field namely particle moves on the MC lattice mentionedabove If particles move with no memory and have equalprobability to each direction the particlersquos motion can betaken as Monte-Carlo motion [26]

The trajectory of particles is shown as follows

119883119894= V119905119905 + 1198830119894

(119894 = 1 2 3) (5)

119894 is the coordinate direction of 119909 119910 119911The motion of a particle is defined as

119906 (119905 + Δ119905) = 119906 (119905) + 1199061015840(119905)

V (119905 + Δ119905) = V (119905) + V1015840 (119905)

119908 (119905 + Δ119905) = 119908 (119905) + 1199081015840(119905)

(6)

119906(119905) V(119905) and 119908(119905) are the mean values in this time step1199061015840(119905) V1015840(119905) and 1199081015840(119905) are compounded by two parts

relevant part and stochastic part

1199061015840(119905) = 119865

119863119909(119906 minus 119906

119901) 119888

V1015840 (119905) = 119865119863119910(119906 minus 119906

119901) 119888 +

119892119910(120588119901minus 120588)

120588119901

1199081015840(119905) = 119865

119863119911(119906 minus 119906

119901) 119888

(7)

119865119863119909(119906minus119906119901) 119865119863119910(119906minus119906119901) and 119865

119863119911(119906minus119906119901) are the effectiveness

of wind to the particles 119888 is the probability to any of thedirection 119888 = VVmax


Virtual campus

6 processes

(a) Lattice division

Virtual campus

5 processes

(b) Sheet division

Figure 5 KMC division of virtual campus

So the particles diffuse in the virtual campus environmentaccording to the formulas above

(2)The Sediment of Surface Dust ParticlesWhen the particlesfulfill the following condition

0 lt 119911119895(119905) lt 5mm

V1015840 (119905) lt 0(8)

they can be taken as sediment on the surface of the earth inthis model At this time V(119905 + Δ119905) = 0

(3) The Resuspension of Surface Dust Particles Manyresearches based on Reynolds stress have taken that the par-ticles may be resuspended when the wind speed is increasingto critical friction velocity The equation can be shown asfollows

119906119891= 119860radic

(120588119904minus 120588) 119889119892

120588 (9)

119906119891is the critical friction velocity 120588

119904is particle density 119889 is the

diameter of the particle 119892 is the acceleration of gravity 120588 isthe density of wind and119860 is a random coefficient with valuesbetween 016 and 021

When the particle is resuspended

119911119895(119905) = 5mm (10)

And the particle velocity is equal to the wind speed

223 The Vanish of Surface Dust Particles The particles havelife cycle once they emerge in the virtual environment Theycan be deemed as vanished when they move out of theboundary of the virtual campus or theymove into the interiorof the buildings

3 KMC-Based Parallel Simulation ofDust Evolution

The premise of KMC-based simulation of dust evolution isthat the surface dust location can be described by a point inthe virtual campusThus it is hard for a computer to complete

the simulation task with the increase of virtual environmentor dust particles At this time the virtual environment canbe divided into several small subspaces and particles inthe subspaces are assigned to multiprocessors to simulateconcurrently

31 The Way of Data Distribution Since the evolution ofsurface dust particles is a random process the virtual campusspace should be divided into continuous space and everyspace contains the same number of dust particles Eachprocess simulates the evolution of dust particles in a subspacewhich can effectively ensure load balancing of each process[41ndash43]

When virtual environment is divided into subspacesthe dust particles in regional boundary of a subspace maymove to another subspace which causes the communicationbetween processes When the particles are in the regionalboundary two adjacent regions need to communicate andconfirm where particles are and their specific locations Inorder to divide the virtual campus block data distribution canbe achieved by two ways lattice and sheet divisions as shownin Figure 5

The division in Figure 5(a)makes each region be requiredto communicate with at least three adjacent regions whichhave the same boundary while the division in Figure 5(b)makes each region be required to communicate with at mosttwo adjacent regions which have the same boundary So thedivision in Figure 5(b) will reduce the traffic and it is used inthe division of the virtual campus

32 Communication Optimization Strategy Particlesrsquo sed-iment diffusion and resuspension should be taken intoaccount when the dust evolution based on KMC is used Tooptimize the communication strategy and reduce the trafficbetween processors the data storage space is divided into 2parts Local Store and Neighbor Copy Local Store keeps thedata of particles in the local subspace while Neighbor Copykeeps the data of particles in the regional boundary of otherneighboring subspace

When the sediment and resuspension processes are simu-lated the attributes of particles in Local Store need renewingand the particles which meet the condition sediment on


(1) Begin(2) Initialization MPI Define the number of processes and simulation time 119905(3) Master process reads the data structure of dust particles(4) Master process distributes the data to Local Store of each sub-processors according tothe data distribution way of sheet division and the simulation timer 119894 = 0(5) if 119894 lt= 119905 go to (6) otherwise go to (10)(6) Each sub-processor executes sediment of dust particles(7) Each sub-processor executes re-suspension of dust particles(8) Each sub-processor executes diffusion of dust particles(9) Each sub-processor updates the value of simulation timer 119894 go to (5)(10) Each sub-processor sends data structure of dust particles to the master process(11) The master processor collects the data sent by each sub-processor and writes to the file Output(12) End

Algorithm 1 Frame of KMC-based dust evolution parallel simulation algorithm

(1) Begin(2) Each processor updates the position of each particle in the Local Store(3) If 119860particle(119895) sdot pos sdot 119911 lt 5mmampamp119860particle(119895) sdot vel sdot 119911 lt 0 go to (4) Otherwise go to (5)(4) Particle 119895 is sediment on the ground 119860particle(119895) sdot pos sdot 119911 = 0(5) If all particles in Local Store are finished scanning go to (6) Otherwise go to (3)(6) End

Algorithm 2 The sediment simulation algorithm of dust particles

(1) Begin(2) Each processor updates the position of each particle in the Local Store(3) If 119860particle(119895) sdot pos sdot 119911 = 0 go to (5) Otherwise go to (5)(4) If the wind speed in the location of particle 119895 is greater than the critical friction velocityParticle 119895 is re-suspended and 119860particle(119895) sdot pos sdot 119911 = 5mm(5) If all particles in Local Store are finished scanning go to (6) Otherwise go to (3)(6) End

Algorithm 3 The re-suspension simulation algorithm of dust particles

the ground or re-suspend in the air So the processors donot need to communicate with each other and reduce thecommunication frequency

In the diffusion process particles in the regional bound-ary update their attributes both in Local Store and NeighborCopy However particles which are not in the regionalboundary only update their attributes in Local Store whichwill reduce the communication frequency and traffic betweenprocessors

According to the description above each process not onlystores the data of particles in the local subspace but also storesthe data of particles in the neighborhood space In Figure 6the virtual environment is divided into three processors anddashed areas contain the Neighbor Copy space of process 2because it needs to communicate with processes 1 and 3 inthe diffusion process

Data structure of particle in the communication isdescribed as follows

119860particle = (pos vel size color shape lifecyle) (11)

Processor 2

Processor 3

Processor 1

Figure 6 Dust data storage model in three processes

where pos and vel are the particlersquos location and velocity in thevirtual campus at this time size color shape and lifecycle ofeach particle in the Neighbor Copy have the same value

33 KMC-Based Dust Evolution Parallel Simulation Algo-rithm Algorithm 1 is the frame of KMC-based dust evolu-tion parallel simulation algorithm Three main processes ofdust evolution are shown in Algorithms 2 3 and 4


(1) Begin(2) Each processor sends the update data of particles in the border to other processors thenreceives the update data of particles from other processors and copies them in Neighbor Copy(3) Calculate the diffusion probability and diffusion direction of the particle j in Local Store(4) Execute the diffusion of particles update 119860particle(119895) sdot pos mark particles which enter other Neighbor Copy(5) If all particles in Local Store are finished scanning go to (6) Otherwise go to (3)(6) Communicate with other processor to update the data of particles in Neighbor Copy(7) End

Algorithm 4 The diffusion simulation algorithm of dust particles

(1) Begin(2) Create a transparent border of virtual campus(3) Render a yellow ground(4) Render the buildings in the virtual campus(5) Set the attributes of dust particles including their size color and shape(6) Read the coordinates of dust particles from file Output calculate NParts(7) If the particles are in the area of sampling points 119871

119894

calculate the accumulation amount of dust particles 119878(119871119894)

(8) Render the particles according to the value of NParts(9) Output the accumulation amount of dust particles in five sampling points(10) End

Algorithm 5 The visualization algorithm of dust particlesrsquo KMC evolution

4 Visualization on Surface Dust Evolution

The coordinates of particles obtained from the concurrentcalculation are recorded in the text and OpenGL makes thedust evolution visible To see the dust evolution clearly theboundaries of virtual campus are drawn as transparent

In the visualization on surface dust evolution the dustparticles are drawn pro rata because of the large amount andthe coefficient scale119870 is 10minus3

119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)

Suppose that the total particle amount in output at 119905moment is 119878119906119898119863119873(119905) Then the visible amount of dustparticles is119873119875119886119903119905119904(119905) at the simulation of 119905moment

The visualization algorithm of dust particlesrsquo KMC evo-lution is shown in Algorithm 5 And Figure 7 shows thevisualization result It is clear to see the evolution of dustparticles in the virtual campus

5 Results and Analysis

In order to evaluate the effectiveness of KMC-based parallelsimulation algorithm of dust evolution in virtual campusenvironment the experiment is designed as follows

Dust in five collection areas of campus is collected eachnonrain day during four months The weight of dust isgained by a delicate electronic balance and recorded Atthe same time the weather condition like wind scalerainy day and nonrain period is marked According to therecords the northeast wind is the most frequent wind during

Figure 7 The visualization result of dust evolution in virtualcampus

the experimental period So the following analysis is basedon the condition of northeast wind The record of dustaccumulation in the northeast wind is shown in Table 2

Figure 8 compares the experimental results and simula-tion results based on KMC serial and parallel algorithm ofdust evolution by the effect of different nonrain periods infive collection areas

Figures 8(a) to 8(e) show that the dust fall accumulationbecame heavier and heavier with the increase of the nonrainperiod which proves the effectiveness of simulation algo-rithm From Figure 8 serial and parallel simulations had the


Table 2 The record of dust accumulation in different locations

Nonrain period (day) Location 1 Location 2 Location 3 Location 4 Location 51 0406 1095 2095 0745 01711 0037 1378 0782 0215 06631 0098 1790 3964 1846 02851 0347 2524 1208 2558 07051 0166 1251 0920 1974 05201 0234 2372 1272 2705 09841 0336 1128 2449 1612 20561 0282 0877 0920 0614 05511 0112 0654 0346 1115 15541 0193 0511 0628 1016 20631 0205 0960 0922 0838 09242 0056 1226 2198 0985 02212 0296 1865 1088 1084 05472 0294 1384 0887 2015 05522 0259 0597 0611 1224 18662 0180 0599 0528 1054 15862 0181 0497 0314 0215 14703 0264 1426 1149 0446 06233 0226 1495 0808 0791 07533 0184 1064 0839 0439 37404 0045 1804 2656 1575 02194 0055 1452 0484 0895 10984 0214 3577 1203 1388 03154 0214 0833 0804 1034 14014 0195 0804 0510 0964 08544 0073 1506 1565 0921 12864 0156 0691 1476 1366 14815 0082 1533 1194 0271 01485 0323 0602 1958 0978 06945 0194 0399 1430 1138 26306 0049 2416 2467 1561 04536 0116 0385 0348 0859 18067 0080 0804 1070 0999 05757 0150 0482 0602 0471 1404

Table 3 Comparison study of results of parallel calculation indifferent processors

The number of processes 119878119875

119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433

same accumulation amount of dust particles in five pointswhich shows the accuracy of parallel simulation algorithm

To evaluate the validity of parallel simulation algorithmits acceleration ratio and efficiency are calculated and resultsare shown in Table 3

The parallel acceleration ratio is defined as

119878119875=119879119878

119879119875

(13)

where 119879119878is the time used by serial algorithm and 119879

119875is the

time used by parallel algorithm in 119875 processesFrom Table 2 the value of the acceleration ratio is small

because the algorithm is related to the text operation whilethe acceleration ratio increases with the number of theprocesses and the computation time reduces evidently Thisindicates that the parallel algorithm on dust evolution canpromote the efficiency although KMC evolution algorithmneeds lots of boundary exams on the particles which makesthe simulation of large-scale virtual environment possible


0

02

04

06

08

1

12

14

16

1 2 3 4 5 6 7Nonrain period

Accu

mul

atio

n of

dus

t

Experimental dataSerial simulation dataParallel simulation data

(a) Relationship of nonrain period and dust fall at Location 1

0

2

4

6

8

10


Accu

mul

atio

n of

dus

t


(b) Relationship of nonrain period and dust fall at Location 2

0

2

4

6

8

10


Accu

mul

atio

n of

dus

t


(c) Relationship of nonrain period and dust fall at Location 3

0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t


(d) Relationship of nonrain period and dust fall at Location 4

0

2

4

6

8

10


Accu

mul

atio

n of

dus

t


(e) Relationship of nonrain period and dust fall at Location 5

Figure 8 Comparison of experimental and simulation results of dust fall accumulation


6 Conclusion

It is efficient to use the parallel algorithm to simulate theKMC evolution of surface dust particles in large-scale virtualenvironment A parallel simulation algorithm of particlesrsquoKMC evolution is proposed It is useful to balance the loadof every process and reduce the communication expenseamong processes with the help of data distribution way ofsheet division and communication optimizing strategy Theexperiment results show that simulation operation time isshortened enormously the acceleration ratio is easy to getand the parallel efficiency is promoted due to the reasonableprocess numbers in the parallel simulation algorithm whichalso compensates the disability of single computer With the3D visible simulation result researchers can have a goodunderstanding of the segmentation diffusion and resuspen-sion of dust particles and analyze their movement disciplineto lay a theoretical foundation for the dust prevention

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by NSFC (Project no41101454) the Grand Science amp Technology Program Shang-hai China (no 13111101300) and Industrial Innovation GrandProjects (no 07CH-008)

References

[1] J Chang M Liu L-J Hou S-Y Xu X Lin and S BalloldquoConcept pollution character and environmental effect ofurban surface dustrdquo Chinese Journal of Applied Ecology vol 18no 5 pp 1153ndash1158 2007

[2] N Li Effect of Haze on Respiratory Healthy in GuangzhouWuhan University Of Technology 2009

[3] L-L Da T-W Yang Y-Y Li andX-T Lu ldquoAccelerating volumerendering of 3D datasets based on PC hardwarerdquo Journal ofSystem Simulation vol 17 no 10 pp 2422ndash2425 2005

[4] D-P Xi and W-P Jiang ldquoResearch and application of three-dimensional visibility based on digital maprdquo Earth Science vol27 no 3 pp 278ndash284 2002

[5] C Ye J-S Wang and X I Li ldquoField measurement and numer-ical simulation for pollutant dispersion from vehicular exhaustin street canyonrdquo Environmental Chemistry vol 25 pp 364ndash366 2006

[6] A Zhang L Zhang and J Zhou ldquoNumerical simulation ofwindenvironment around two adjacent buildingsrdquoChinese Journal ofComputational Mechanics vol 20 no 5 pp 553ndash558 2003

[7] T L Chan G Dong C W Leung C S Cheung and W THung ldquoValidation of a two-dimensional pollutant dispersionmodel in an isolated street canyonrdquo Atmospheric Environmentvol 36 no 5 pp 861ndash872 2002

[8] T Schneider J Kildes and N O Breum ldquoA two compartmentmodel for determining the contribution of sources surface

deposition and resuspension to air and surface dust concentra-tion levels in occupied roomsrdquo Building and Environment vol34 no 5 pp 583ndash595 1999

[9] R Yao Q Qiao and X Yu ldquoWind tunnel simulation of flowand dispersion around complex buildingsrdquoRadiation ProtectionBulletin vol 22 pp 1ndash6 2002

[10] D-P Guo Q-D Qiao and R-T Yao ldquoExamining the k-120576(RNG)model and LES of flow feature and turbulence dispersionaround a building by means of wind tunnel testsrdquo Journal ofExperiments in Fluid Mechanics vol 25 no 5 pp 55ndash63 2011

[11] J Zhang and A Li ldquoStudy on particle deposition in verticalsquare ventilation duct flows by different modelsrdquo EnergyConversion andManagement vol 49 no 5 pp 1008ndash1018 2008

[12] R Gao and A Li ldquoModeling deposition of particles in verticalsquare ventilation duct flowsrdquo Building and Environment vol46 no 1 pp 245ndash252 2011

[13] I Ali S L Kalla and H G Khajah ldquoA time dependent modelfor the transport of heavy pollutants from ground-level aerialsourcesrdquo Applied Mathematics and Computation vol 105 no 1pp 91ndash99 1999

[14] K Sun L Lu andH Jiang ldquoA numerical study of bend-inducedparticle deposition in and behind duct bendsrdquo Building andEnvironment vol 52 pp 77ndash87 2012

[15] C K Saha W Wu G Zhang and B Bjerg ldquoAssessing effect ofwind tunnel sizes on air velocity and concentration boundarylayers and on ammonia emission estimation using computa-tional fluid dynamics (CFD)rdquo Computers and Electronics inAgriculture vol 78 no 1 pp 49ndash60 2011

[16] Y Tominaga SMurakami andAMochida ldquoCFDprediction ofgaseous diffusion around a cubic model using a dynamic mixedSGSmodel based on composite grid techniquerdquo Journal ofWindEngineering and Industrial Aerodynamics vol 67-68 pp 827ndash841 1997

[17] W Nie W-M Cheng G Zhou and Y Yao ldquoThe numericalsimulation on the regularity of dust dispersion in whole-rockmechanized excavation face with different air draft amountrdquoProcedia Engineering vol 26 pp 961ndash971 2011

[18] J A Roney and B RWhite ldquoComparison of a two-dimensionalnumerical dust transport model with experimental dust emis-sions from soil surfaces in a wind tunnelrdquoAtmospheric Environ-ment vol 44 no 4 pp 512ndash522 2010

[19] P Song J Lu Q Hu M Zhao and B Yang ldquoApplication anddevelopment of computer simulation in thin film depositionrdquoMaterials Review vol 17 pp 154ndash157 2003

[20] S Razmyan and F Hosseinzadeh Lotfi ldquoAn application ofMonte-Carlo-based sensitivity analysis on the overlap in dis-criminant analysisrdquo Journal of Applied Mathematics vol 2012Article ID 315868 14 pages 2012

[21] Y X Jie H N Yuan and H D Zhou ldquoBending momentcalculations for piles based on the finite element methodrdquoJournal of Applied Mathematics vol 2013 Article ID 784583 19pages 2013

[22] L Zhang and Z Chen ldquoA stabilized mixed finite elementmethod for single-phase compressible flowrdquo Journal of AppliedMathematics vol 2011 Article ID 129724 16 pages 2011

[23] W Zhu G Hu X Hu L Hongbo and W Zhang ldquoVisualsimulation of GaInP thin film growthrdquo Simulation ModellingPractice andTheory vol 18 no 1 pp 87ndash99 2010

[24] G Okin ldquoThe Role of Spatial Variability in Wind Erosion andDust Emissionrdquo Geophysical Research Abstracts 12583 2003


[25] R Yao H Hao and E Hu ldquoComparison of two kinds of atmo-spheric dispersionmodel chains in rodosrdquoRadiation Protectionvol 23 pp 146ndash155 2003

[26] R Tian ldquoMonte-Carlo model simulates the influence of com-plex terrain on diffusionrdquo Scientia Atmospherica Sinica vol 18pp 37ndash42 1994

[27] K Xu H G He and Y C Zhu ldquoStudy on dispersion simulationof long-distant pipeline leaked gas based on Monte-CarlordquoJournal of Safety Science and Technology vol 8 pp 18ndash23 2012

[28] Y Sun Y Qian and Y Zhang ldquoApplication of Monte Carloanalysis in environmental risk assessment of a chlorine releaseaccidentrdquoActa Scientiae Circumstantiae vol 31 no 11 pp 2570ndash2577 2011

[29] Z-B Peng and Z-L Yuan ldquoNumerical simulation of gas-solid flow behaviours in desulfurization tower based on MonteCarlordquo Proceedings of the Chinese Society of Electrical Engineer-ing vol 28 no 14 pp 6ndash14 2008

[30] S Tanaka T Nishide and K Sakurai ldquoEfficient implementationfor QUAD stream cipher with GPUsrdquo Computer Science andInformation Systems vol 10 no 2 pp 897ndash911 2013

[31] Y Shang G Lu and L Shang ldquoParallel processing on block-based Gauss-Jordan algorithm for desktop gridrdquo Computer Sci-ence and Information Systems vol 8 no 3 pp 739ndash759 2011

[32] FH Pereira and S I Nabeta ldquoA parallel wavelet-based algebraicmultigrid black-box solver and preconditionerrdquo Journal ofApplied Mathematics vol 2012 Article ID 894074 15 pages2012

[33] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[34] Z Wenhua F Xiong H Guihua S Yupeng and X Hu VirtualReality Technology and Application Intellectual Property PressBeijing China 2007

[35] Y-M Chen J-S Bao Y Jin C-C Xu and Y-C Yang ldquoKeytechnology study and application of engineering analysis datarsquosimmersive visualizationrdquo Journal of System Simulation vol 16no 10 pp 2309ndash2312 2004

[36] A Attenberger and K Buchenrieder ldquoModeling and visual-ization of classification-based control schemes for upper limbprosthesesrdquo Computer Science and Information Systems vol 10no 1 pp 349ndash367 2013

[37] B Jin-Song J Ye M Deng-Zhe and Y Jun-Qi ldquoImmersivescientific visualization with realist geometryrdquo Journal of SystemSimulation vol 15 pp 653ndash655 2003

[38] S Sang J Zhao H Wu S Chen and Q An ldquoModeling andsimulation of a spherical mobile robotrdquo Computer Science andInformation Systems vol 7 no 1 pp 51ndash62 2010

[39] P F Li M Y Xu and F FWang FLUENTGAMBIT ICEMCFDTecplot Beijing Institute of Technology Press Beijing China2005

[40] J F Xie ldquoA comparative study of various numerical simulationapproaches to wind environment within urban green spacerdquoAgriculture Network Information vol 26 no 07 pp 18ndash21 2011

[41] K Lu and X Lin ldquoImplementing load balance in MPI parallelprogramrdquo Microcomputer Information vol 05X pp 226ndash2272007

[42] L Lu ldquoResearch on parallel program design strategyrdquo ElectronicComputers vol 141 no 6 pp 2ndash8 1999

[43] B Zhou J Shen and Q Peng ldquoCommunication scheme ofparallel clustering algorithm for PCs clusterrdquo Computer Engi-neering vol 30 no 7 pp 20ndash21 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


have also been taken into account Ali has investigated a time-dependent partial differential equation governing the trans-port of heavy dust into the atmosphere Dust concentrationis expressed in terms of a series of confluent hyper geometricfunctions [13] Sun et al investigated the microparticle depo-sition and distribution by employing the Eulerian approachwith Reynolds stress turbulent model and a Lagrangian tra-jectorymethod [14]With the rapid development in computertechnology computational fluid dynamics (CFD) methodhas matured to simulate the ventilation performance andcontaminant dispersion and transfer in buildings Saha etal have used CFD to assess the effect of wind tunnel sizeson air velocity and concentration boundary layers and onammonia emission estimation [15] Tominaga et al have usedCFD to predict the air diffusion around a construction [1617] Besides these Schneider et al proposed a semiempiricaltwo-compartment constant parameter model [8] Roney andWhite compared the near-surface wind-tunnel fugitive dustconcentration profiles arising from soil surfaces beds withthose calculated numerically [18] However most of thestudies paid attention to simulate the diffusion process ofparticles The whole dust evolution process is seldom traced

A lot ofmethods can be used in the simulation of dynamicprocess such as first principal method (FP) moleculardynamic method (MD) Monte Carlo method (MC) andfinite elementmethod (FEM) [19ndash22] Among thesemethodsMCmethod has been well used in simulation of air pollutiondispersion and river water pollution Three decades agoMonte Carlo (MC) method was used to study the dynamicprocess The basic idea of the MCmethod is that the solutionof the problem is equivalent to hypothetical statistical modelparameters using random number and the parameters areestimated by the statistical model of a sample [23] Okinproposed MC method to simulate airborne dust diffusionmodel [24] Yao et al usedMCmethod in simulation of emis-sion height effects on building [25] Tian presented the MCsimulation of complex terrain effect on dust diffusion [26]Dejun GU puts forward a model for the convective boundarylayer of the line source diffusion along MC method Xu et alsimulate gas dispersion based on Monte Carlo which couldsatisfy the requirement of the long-distance pipeline disastersemergency decision making [27] The object researched bykinetic Monte Carlo (KMC) method is unbalanced or arelaxation process Time evolution correct or not is the keyfactor in the simulation process simulation time step mustbe the system real time step so KMC method is an effectivemethod of studying on the kinetic behavior Sun et al appliedKMC analysis in environmental risk assessment of a chlorinerelease accident [28] Peng and Yuan simulated gas-solidflow behavior in desulphurization tower based on KMC [29]Because dust have a similar dynamic property with water andair pollutant KMC method has widely been used in waterflow simulation and atmospheric diffusion field The kineticMonte Carlo (KMC) related to the MC has the advantage ofsimulating in a long period In addition KMC is a stochasticprocess which makes it fit for evolution simulation of thesurface dust particles In the current conditions it is hard tosimulate the KMC evolution of dust particles in a large-scaleurban environment by one computer Therefore the parallel

5

4

1

2

3

Figure 1 3D model of Shanghai University campus

computing method can be used to shorten the simulationtime [30ndash33]

Virtual reality (VR) has the property of intuition andinteractive which makes up the defeat of computer simu-lation and makes the results of simulation take on effect of3D stereo display [34ndash38]This paper will adopt visual realitytechnology (VRT) to visualize the results of dust evolution Avirtual Shanghai University campus is rendered by OpenGLand a parallel simulation system of dust evolution in thevirtual campus is created in order to study the impact ofdifferent parameters (such as nonrain period and wind)on dust evolution By comparison with experimental datathe validity of the model is verified The research aims atproviding theory and quantitative reference for dust particlesevolution

2 Kinetic Monte Carlo Simulation ofSurface Dust Evolution

21The 3DModeling of Virtual Campus Shanghai Universitycampus is simplified and the 3D model is shown in Figure 1

The campus is 120 meter in width and 470 meter inlength Since most of surface dust works in the height of 0to 5 meters the virtual campus is in the scale of 120m lowast

470m lowast 6m To verify the validity of the KMC simulationmethod five positions are chosen as the sampling point andsigned as 1 2 3 4 and 5 shown in Figure 1 Location 1 isgreatly influenced by the buildings while the transportationhas less impact on it Location 2 is located on the campus roadwhich is near the greensward The flow of people has greaterimpact on it than the buildings Location 3 and Location 4are moderately influenced by the buildings and the flow ofpeople Location 5 is on a road where traffic is quite large andslightly impacted by buildings By collecting and analyzingthe weight of dust at the 5 positions and comparing themwith the KMC simulation result the initial parameters of dustevolution simulation are defined to suit the real situation











120597119906119894

120597119909119894

= 0

120597119906119894

120597119905+120597 (119906119894119906119895)

120597119909119895

= minus1

120588

120597119875

120597119909119894

+120597

120597119909119895

(V120597119906119894

120597119909119895

)

(1)

119909119894and 119909





119896 =3

2(119906ref119879119894)

2 120576 = 119862

34

120583

11989632

ℓ ℓ = 007119871 (2)


Gy Gx

Gz












120597119910= 0


120597119909= 0 Δ120588 = 23556 pa

1e + 02

1e + 00

1e minus 02

1e minus 04

1e minus 06

1e minus 08

1e minus 10

1e minus 12

1e minus 14

1e minus 16

1e minus 18

0 2 4 6 8 10 12 14 16 18 20

times102

Iterations


Energyk

Epsilon


265e + 01

252e + 01

239e + 01

226e + 01

212e + 01

199e + 01

186e + 01

173e + 01

159e + 01

146e + 01

133e + 01

119e + 01

106e + 01

929e + 00

796e + 00

664e + 00

531e + 00

396e + 00

266e + 00

133e + 00

132e minus 03




119889119906119901

119889119905= 119865119863(119906 minus 119906

119901) +

119892119909(120588119901minus 120588)

120588119901

(3)

119865119863(119906 minus 119906


119865119863=18120583

1205881199011198892119901

119862119863Re24

(4)








119883119894= V119905119905 + 1198830119894

(119894 = 1 2 3) (5)


119906 (119905 + Δ119905) = 119906 (119905) + 1199061015840(119905)

V (119905 + Δ119905) = V (119905) + V1015840 (119905)

119908 (119905 + Δ119905) = 119908 (119905) + 1199081015840(119905)

(6)



1199061015840(119905) = 119865

119863119909(119906 minus 119906

119901) 119888

V1015840 (119905) = 119865119863119910(119906 minus 119906

119901) 119888 +

119892119910(120588119901minus 120588)

120588119901

1199081015840(119905) = 119865

119863119911(119906 minus 119906

119901) 119888

(7)

119865119863119909(119906minus119906119901) 119865119863119910(119906minus119906119901) and 119865




Virtual campus

6 processes


Virtual campus

5 processes

(b) Sheet division




0 lt 119911119895(119905) lt 5mm

V1015840 (119905) lt 0(8)



119906119891= 119860radic

(120588119904minus 120588) 119889119892

120588 (9)





119911119895(119905) = 5mm (10)























Processor 2

Processor 3

Processor 1








119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

04

06

08

1

12

14

16


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















120597119906119894

120597119909119894

= 0

120597119906119894

120597119905+120597 (119906119894119906119895)

120597119909119895

= minus1

120588

120597119875

120597119909119894

+120597

120597119909119895

(V120597119906119894

120597119909119895

)

(1)

119909119894and 119909





119896 =3

2(119906ref119879119894)

2 120576 = 119862

34

120583

11989632

ℓ ℓ = 007119871 (2)


Gy Gx

Gz












120597119910= 0


120597119909= 0 Δ120588 = 23556 pa

1e + 02

1e + 00

1e minus 02

1e minus 04

1e minus 06

1e minus 08

1e minus 10

1e minus 12

1e minus 14

1e minus 16

1e minus 18

0 2 4 6 8 10 12 14 16 18 20

times102

Iterations


Energyk

Epsilon


265e + 01

252e + 01

239e + 01

226e + 01

212e + 01

199e + 01

186e + 01

173e + 01

159e + 01

146e + 01

133e + 01

119e + 01

106e + 01

929e + 00

796e + 00

664e + 00

531e + 00

396e + 00

266e + 00

133e + 00

132e minus 03




119889119906119901

119889119905= 119865119863(119906 minus 119906

119901) +

119892119909(120588119901minus 120588)

120588119901

(3)

119865119863(119906 minus 119906


119865119863=18120583

1205881199011198892119901

119862119863Re24

(4)








119883119894= V119905119905 + 1198830119894

(119894 = 1 2 3) (5)


119906 (119905 + Δ119905) = 119906 (119905) + 1199061015840(119905)

V (119905 + Δ119905) = V (119905) + V1015840 (119905)

119908 (119905 + Δ119905) = 119908 (119905) + 1199081015840(119905)

(6)



1199061015840(119905) = 119865

119863119909(119906 minus 119906

119901) 119888

V1015840 (119905) = 119865119863119910(119906 minus 119906

119901) 119888 +

119892119910(120588119901minus 120588)

120588119901

1199081015840(119905) = 119865

119863119911(119906 minus 119906

119901) 119888

(7)

119865119863119909(119906minus119906119901) 119865119863119910(119906minus119906119901) and 119865




Virtual campus

6 processes


Virtual campus

5 processes

(b) Sheet division




0 lt 119911119895(119905) lt 5mm

V1015840 (119905) lt 0(8)



119906119891= 119860radic

(120588119904minus 120588) 119889119892

120588 (9)





119911119895(119905) = 5mm (10)























Processor 2

Processor 3

Processor 1








119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

04

06

08

1

12

14

16


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597119910= 0


120597119909= 0 Δ120588 = 23556 pa

1e + 02

1e + 00

1e minus 02

1e minus 04

1e minus 06

1e minus 08

1e minus 10

1e minus 12

1e minus 14

1e minus 16

1e minus 18

0 2 4 6 8 10 12 14 16 18 20

times102

Iterations


Energyk

Epsilon


265e + 01

252e + 01

239e + 01

226e + 01

212e + 01

199e + 01

186e + 01

173e + 01

159e + 01

146e + 01

133e + 01

119e + 01

106e + 01

929e + 00

796e + 00

664e + 00

531e + 00

396e + 00

266e + 00

133e + 00

132e minus 03




119889119906119901

119889119905= 119865119863(119906 minus 119906

119901) +

119892119909(120588119901minus 120588)

120588119901

(3)

119865119863(119906 minus 119906


119865119863=18120583

1205881199011198892119901

119862119863Re24

(4)








119883119894= V119905119905 + 1198830119894

(119894 = 1 2 3) (5)


119906 (119905 + Δ119905) = 119906 (119905) + 1199061015840(119905)

V (119905 + Δ119905) = V (119905) + V1015840 (119905)

119908 (119905 + Δ119905) = 119908 (119905) + 1199081015840(119905)

(6)



1199061015840(119905) = 119865

119863119909(119906 minus 119906

119901) 119888

V1015840 (119905) = 119865119863119910(119906 minus 119906

119901) 119888 +

119892119910(120588119901minus 120588)

120588119901

1199081015840(119905) = 119865

119863119911(119906 minus 119906

119901) 119888

(7)

119865119863119909(119906minus119906119901) 119865119863119910(119906minus119906119901) and 119865




Virtual campus

6 processes


Virtual campus

5 processes

(b) Sheet division




0 lt 119911119895(119905) lt 5mm

V1015840 (119905) lt 0(8)



119906119891= 119860radic

(120588119904minus 120588) 119889119892

120588 (9)





119911119895(119905) = 5mm (10)























Processor 2

Processor 3

Processor 1








119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

04

06

08

1

12

14

16


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Virtual campus

6 processes


Virtual campus

5 processes

(b) Sheet division




0 lt 119911119895(119905) lt 5mm

V1015840 (119905) lt 0(8)



119906119891= 119860radic

(120588119904minus 120588) 119889119892

120588 (9)





119911119895(119905) = 5mm (10)























Processor 2

Processor 3

Processor 1








119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

04

06

08

1

12

14

16


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Processor 2

Processor 3

Processor 1








119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

04

06

08

1

12

14

16


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t



0

2

4

6

8

1

1

2 3

3

4 5

5

6 7

7

Nonrain period

Accu

mul

atio

n of

dus

t



0

2

4

6

8

10


Accu

mul

atio

n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894







119873119875119886119903119905119904 (119905) = 119878119906119898119863119873 (119905) lowast 119870 (12)















119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




0

02

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Accu

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0

2

4

6

8

10


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mul

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n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119879119875hour

1 1 26312 149 17664 191 13778 287 91716 608 433




119878119875=119879119878

119879119875

(13)


119875is the




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0

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0

2

4

6

8

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Accu

mul

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n of

dus

t





6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

14

16


Accu

mul

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n of

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0

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4

6

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mul

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n of

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6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Conclusion




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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