Research Article Electromagnetic Computation and ...downloads.hindawi.com/journals/mpe/2014/942106.pdf · Research Article Electromagnetic Computation and Visualization of Transmission

Research ArticleElectromagnetic Computation and Visualization ofTransmission Particle Model and Its Simulation Based on GPU

Yingnian Wu12 Lin Zhang1 and Lan Mu1

1 School of Automation Science and Electrical Engineering Beihang University Beijing 100191 China2 Beijing Key Laboratory of High Dynamic Navigation Technology University of Beijing Information Science amp TechnologyBeijing 100192 China

Correspondence should be addressed to Yingnian Wu wuyingnian126com

Received 9 March 2014 Revised 4 July 2014 Accepted 18 July 2014 Published 7 August 2014

Academic Editor Efstratios Tzirtzilakis

Copyright copy 2014 Yingnian Wu 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

Electromagnetic calculation plays an important role in both military and civic fields Some methods and models proposed forcalculation of electromagnetic wave propagation in a large range bring heavy burden in CPU computation and also require hugeamount of memory Using the GPU to accelerate computation and visualization can reduce the computational burden on the CPUBased on forward ray-tracing method a transmission particle model (TPM) for calculating electromagnetic field is presented tocombine the particle methodThemovement of a particle obeys the principle of the propagation of electromagnetic wave and thenthe particle distribution density in space reflects the electromagnetic distribution status The algorithm with particle transmissionmovement reflection and diffraction is described in detail Since the particles in TPMare completely independent it is very suitablefor the parallel computing based on GPU Deduction verification of TPM with the electric dipole antenna as the transmissionsource is conducted to prove that the particle movement itself represents the variation of electromagnetic field intensity caused bydiffusion Finally the simulation comparisons are made against the forward and backward ray-tracing methods The simulationresults verified the effectiveness of the proposed method

1 Introduction

Electromagnetic calculation plays an important role in bothmilitary and civic fields It is realized that simulation of anelectromagnetic environment is useful and can give muchinformation in communication systems design and radartesting It is of great significance to calculate the distributionstatus of the electromagnetic field

Some methods and models are proposed for calculationof indoor and outdoor wave propagation that is methodsbased on the numerical solution of Maxwellrsquos equations suchas finite-difference time-domain method (FDTD) [1] andparabolic equation method (PE) [2ndash6] methods based ongeometrical optics principle such as ray-tracing (RT) [7ndash10]method and so forth

To calculate electromagnetic wave in a large range FDTDmethod brings heavy burden in calculation and also requireshuge amount of memory [11ndash14] PE method performs wellfor complex terrain but has significant limitation in the

calculation of propagation angle Generally speaking RTmethod is applicable to the electromagnetic calculation in alarge range

Ray-tracing method is based on geometrical optics (GO)principle which decides reflection or refraction throughsimulating the propagation route of rays [15ndash17] It alsointroduces diffraction ray in the case of obstacles Accordingto GO theory improved geometry theory of diffraction(GTD) [18] and uniform theory of diffraction (UTD) [19ndash21]are established Ray-tracking is classified into forward ray-tracking and backward ray-tracking [22 23] Backward ray-tracking is generally used because forward ray-tracking haslarger error But backward ray-tracking has huge calculationwhen calculating space field because every point in the spacehas to be recalculated

Particle method is used to simulate the macrostate basedon a huge amount of individual particle which has a processof generation motion and vanishing

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 942106 9 pageshttpdxdoiorg1011552014942106

2 Mathematical Problems in Engineering

Based on forward ray-tracing method and particlemethod a Transmission Particle Model for calculating elec-tromagnetic distribution status in space is presented inthis paper Using the GPU to accelerate computation andvisualization can reduce the computational burden on theCPU The movement of a particle obeys the principle of thepropagation of electromagnetic wave and then the particledistribution density in space reflects the distribution statusof electromagnetic distribution status Since the particles inTPM are completely independent it is very suitable for theparallel computing based on GPU Updated GPU particledata is stored in display memory so the data does notneed to be transferred back to the memory when makingvisualization of the particle and it can directly draw theparticle movement after transmission

In this paper we present Transmission ParticleModel anduse GPU based on CUDA realize particles transmit updatesdemise and other operations and achieve particle modelcomputation and synchronization visualization

2 Description of TPM

The basic idea of TPM is that particles are generated andtransmitted intermittently in the origin of electromagneticfield and the electromagnetic distribution status in space isobtained through calculating the movement of each particle

So far there are many methods for electromagneticcalculation such as finite-difference time-domain method(FDTD) parabolic equation method (PE) and ray-tracingmethod (RT) To calculate electromagnetic wave in a largerange FDTDmethod brings heavy burden in calculation andalso requires huge amount of memory PE method performswell for complex terrain but has significant limitation inthe calculation of propagation angle Generally speaking RTmethod is applicable to the electromagnetic calculation in alarge range

21 Particle Transmission From the generation in the originof electromagnetic field the particle transmission densitymainly depends on antennarsquos angle factor 119891(120579 120593) 120579 and 120601 arethe angle components in sphere coordinates

Take the origin of electromagnetic field as a vertexpartition the space with equal interval in all directions Thesampling intervals for 120579 and 120593 are Δ120579 and Δ120593 respectivelyusually we have Δ120579 = Δ120593

Then we obtain

120579119894= 119894Δ120579 119894 = 1 2 3 int( 120587

Δ120579)

120593119895= 119895Δ120593 119895 = 1 2 3 int( 2120587

Δ120593)

(1)

Denote 119899119894119899119895as the particle numbers in one transmission

in the region (1205791015840 1205931015840) | 1205791015840 isin [120579119894minus Δ1205792 120579

119894+ Δ1205792] 1205931015840 isin

[120593119895minusΔ1205932 120593

119895+Δ1205932] whose maximum value is119873

11198732(119899119894119899119895

is the product of 119899119894and 119873

11198732the product of 119899

1198951198731and 119873

2

s(k)

s(k)998400s(k minus 1)

r(k

r(k)

+ 1)

r(k minus 1)

R0

120575

Figure 1 The distance between the particles

1198731 and 119873

2are given constants if Δ120579 = Δ120593 then we have

1198731= 1198732 119899119894= 119899119895)

119899119894=

119891 (120579119894 120593119895)

119891max (120579 120593)1198731

119899119895=

119891 (120579119894 120593119895)

119891max (120579 120593)1198732

(2)

The initial coordinates are (1198770 120579119894119895119901

120593119894119895119902

) initial speed Vis (1198770120575 120579119894119895119901

120593119894119895119902

) initial distance 119903 = 1198770 the initial interval

angle with other adjacent particles is 120575 asymp (12)(Δ120579119899119894+

Δ120593119899119895) the initial life time is 119905 = 0 and the initial particle

status is active where

120579119894119895119901

= 120579119894minusΔ120579

2+119901Δ120579

119899119894

119901 = 1 2 3 119899119894

120593119894119895119902

= 120593119895minusΔ120593

2+119902Δ120593

119899119895

119902 = 1 2 3 119899119895

(3)

1198770is the distance between the initial position of particle

transmission and the electromagnetic transmission sourcewhich is supposed to be the origin of coordinate

22 Particle Movement The movement of particle isdynamic that is with the elapse of time the currentproperties such as position coordinates speed V movementdistance 119903 angle 120575 life time 119905 and state (active or dead) willaffect the properties of the particles in the particle list

In Figure 1 119903(119896) is the total movement distance of theparticle 119904(119896) is the distance between particles in the samebatch and 119904(119896)1015840 is the distance particle of two consecutivebatches within time interval Δ119879 Particles are consecutivelytransmitted with equal time interval Δ119879

Mathematical Problems in Engineering 3

Since 120575 is very small at initial time we have

119903 (0) = 1198770

119904(0)1015840 ≐ 120575119903 (0) = 1205751198770

119904 (0) = 2 sin(1205752) 119903 (0) = 2 sin(120575

2)1198770

(4)

At time 119896Δ119879 we have

119903 (119896) = 119903 (0) +119896minus1

sum119894=0

119904 (119894) = (120575 + 1)1198961198770

119904(119896)1015840 = 2 sin(120575

2) 119903 (119896) = 2 sin(120575

2) (120575 + 1)

1198961198770

119904 (119896) = 120575119903 (119896) = 120575(120575 + 1)1198961198770

119904 (119896 minus 1) = 120575119903 (119896 minus 1) = 120575(120575 + 1)119896minus11198770

(5)

Since 120575 is very small we have 119904(119896 minus 1) asymp 119904(119896)1015840 asymp119904(119896) Consequently particles are approximately in uniformdistribution in space particle density in (119903 120579 120593) can representthe energy of the electromagnetic energy In the next sectionwe will give the expression between particle density andPoynting vector mean value for electric dipole

The movement speed of particle is

V (119896) =119904 (119896)

Δ119879=120575(120575 + 1)1198961198770

Δ119879 (6)

Correspondingly we have

V (119905) =1

Δ119879(120575 + 1)

119905Δ1198791198770ln (120575 + 1) (7)

23 Particle Reflection When electromagnetic particlescome to the interface of two media they reflect accordingto reflection theory some of them may vanish owing toreflection and the probability of vanishing is decided byreflection factor 119877

The liveability after reflection is

119875reflection = 119877 (8)

Then the vanishing probability is

119875death = 1 minus 119875reflection (9)

If the electric field intensity vector is parallel to incidenceplane (parallel polarization) the reflection factor of uniformplane wave in [21] is

R=radic12057621205761minus sin2120579

1minus (12057621205761) cos 120579

1

radic12057621205761minus sin2120579

1+ (12057621205761) cos 120579

1

(10)

If the electric field intensity vector is vertical to incidenceplane (vertical polarization) the reflection factor of uniformplane wave in [10] is

Rperp=cos 1205791minus radic12057621205761minus sin2120579

1

cos 1205791+ radic12057621205761minus sin2120579

1

(11)

After reflection some particles vanish The transmissioninterval angle between survived particles and their adjacentparticles has been changed to 1205751015840

1205751015840 =120575

119877 (12)

24 Particle Diffraction In Geometry Optics there are threekinds of diffractions that is edge diffraction spire diffrac-tion and surface diffraction Since the attenuation of spirediffraction and surface diffraction is very fast we can ignorethe two kinds of diffractions and only take edge diffractioninto account

When the particle diffraction occurs the position of theparticle needs to be modified according to the diffractionprinciple of electromagnetic wave And further the deathprobability of the particle is calculated based on the diffrac-tion factor which can be used to decide the death of theparticle

Given a distance 119897 there is particle diffraction within therange of 119897 around the edge of terrain After the diffractionthe particle is located on the surface of a cone whose vertexis the diffraction point the axis is the tangent line of thecontour around the diffraction point and the angle betweenthe generatrix and the axis is equal to the angle between theincidence particle and the tangent line

The liveability after diffraction is

119875diffraction = 119863 (13)

Then the vanishing probability after diffraction is

119875death = 1 minus 119875diffraction (14)

Similar to reflection the transmission interval anglebetween the survived particles after diffraction that is 120575 willbe changed to 1205751015840

1205751015840 =120575

119863 (15)

3 Model Verification

Electric dipole is the most simple electromagnetic transmis-sion antenna This paper is focused on the analysis of theelectromagnetic field generated by electric dipole

31 Electric Dipole In unbounded ideal media electromag-netic wave propagates without reflection refraction scat-tering and absorption there is only energy loss caused bydiffusion In calculation of electromagnetic transmission ina large area the average power density of close field (119903 ≪ 120582)is near zero that is no electromagnetic power outputs in theclose field Thus we only need to do calculations for the far-reaching field (119903 ≫ 120582)


In the far-reaching electromagnetic field the electricfield vector and magnetic field vector can be expressedapproximately in [10] as follows

E = 119895IΔ1198971198962 sin 12057941205871205961205760119903

119890minus119895119896119903e120579

H = 119895IΔ119897119896 sin 1205794120587120596119903

119890minus119895119896119903e120601

(16)

I is the virtual value of current Δ119897 is the length ofelectric dipole 119896 is the number of waves 119896 = 2120587120582 (120582 isthe wavelength of electromagnetic wave) 120596 is the angularfrequency and 120576

0is the dielectric constant in free space

Poynting vector mean value of any point in the space is

S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903 (17)

The radiation field generated by electric dipole is not onlyrelevant to the distance from field point to the source but alsorelevant to the 120579 and 120601 in spherical coordinates

By (24) and (25) can be seen electric field strength andmagnetic field strength of electric dipole changes with theangle 119902 changing which can be expressed as a function119891(120579 120601) called antennarsquos angle factor

The antennarsquos angle factor of electric dipole is

119891 (120579 120601) = sin 120579 (18)

According to function119891(120579 120601) we can get the angle factorfigure of electric dipole which is shown in Figure 2

32 Verification Deduction of Electromagnetic TPM Con-sider the coordinate origin is the transmission source withsampling intervals Δ120579 and Δ120601 Δ120579 = Δ120601 then after longenough time the particle density at a point 119875(119903

119901 120579119901 120593119901) is 120588119899

within the range (1205791015840 1205931015840) | 1205791015840 isin [119894119901Δ120579 minus Δ1205792 119894

119901Δ120579 + Δ1205792]

1205931015840 isin [119895119901Δ120593 minus Δ1205932 119895

119901Δ120593 + Δ1205932] Then the number of

transmitted particles at the centre of the above-mentionedarea is 119899

119894119901119899119895119901 where

119894119901= round(

120579119901

Δ120579) (19)

119895119901= round(

120593119901

Δ120593) (20)

119899119894119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198731 (21)

119899119895119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198732 (22)

The distance range of these particles to the transmissionsource is 119903

119901(119896)

119903119901 (119896) = (120575119901 + 1)

119896

1198770 (23)

05

0

minus05

x

z

yminus1 minus1

00

1

1

Figure 2 Angle factor figure of electric dipole

The distance between the transmission source and theparticles transmitted at the previous time instant is

119903119901 (119896 minus 1) = (120575119901 + 1)

119896minus1

1198770 (24)

where

120575119901=1

2(Δ120579

119899119894119901

+Δ120593

119899119895119901

) (25)

Since these particles are uniformly distributed in localarea then we have (1199031015840 1205791015840 1205931015840) | 1199031015840 isin (119903

119901(119896minus1) 119903

119901(119896+1)] 1205791015840 isin

[119894119901Δ120579minusΔ1205792 119894

119901Δ120579+Δ1205792]1205931015840 isin [119895

119901Δ120593minusΔ1205932 119895

119901Δ120593+Δ1205932]

The volume of the area is

119881119901= 119881119901 (119896 + 1) minus 119881119901 (119896 minus 1) (26)

where

119881119901 (119896 + 1) =

4

31205871199033119901(119896 + 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896) (120575119901 + 1)

3

119881119901 (119896 minus 1) =

4

31205871199033119901(119896 minus 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896)

1

(120575119901+ 1)3

(27)

Substituting (27) into (26) we get

119881119901=2Δ120579Δ1205931199033

119901(119896)

3120587((120575119901+ 1)3

minus1

(120575119901+ 1)3)

asymp4Δ120579Δ1205931199033

119901(119896) 120575119901

120587

(28)

Then the density of particles is

120588119899=2119899119894119901119899119895119901

119881119901

=120587119899119894119901119899119895119901

2Δ120579Δ1205931199033119901(119896) 120575119901

(29)


Considering (21) (22) and (25) then we obtain

120588119899=12058711987311198732(119891(119894119901Δ120579 119895119901Δ120593)119891max(120579 120593))

3

Δ120579Δ1205931199033119901(119896) (Δ1205791198731 + Δ1205931198732)

(30)

Since Δ120579 = Δ120593 then 119899119901= 119899119894119901= 119899119895119901 and119873 = 119873

1= 1198732

Further we have

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(119894119901Δ120579 119895119901Δ120593)

119903119901(119896)

)

3

(31)

On account of 120579119901asymp 119894119901Δ120579 120593119901asymp 119895119901Δ120593 119903119901asymp 119903119901(119896) then

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(120579119901 120593119901)

119903119901

)

3

(32)

In order to express Poynting vector mean value by thedensity of particles and since the antennarsquos angle factor ofelectric dipole is

119891 (120579 120601) = sin 120579 (33)

we have

12058823119899

= (120587

2)23 1198732

1198912max (120579 120593) Δ1205792(sin 120579119903

)2

(34)

Then we can get

(sin 120579119903

)2

= 12058823119899(2

120587)231198912max (120579 120593) Δ120579

2

1198732 (35)

Substituting this into (17) we obtain

S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903= 120578(

119868Δ119897

2120582)2

(sin 120579119903

)2

e119903

= 120578(119868Δ119897

2120582)2

(2

120587)231198912max (120579 120593) Δ120579

2

119873212058823119899

e119903= 11987112058823119899

e119903

(36)

where

119871 = 120578(119868Δ119897

2120582)2 (2120587)231198912max (120579 120593) Δ120579

2

1198732 (37)

Consider 119871 is a constant and then we know that theelectromagnetic status in uniform space can be calculatedthrough TPM Poynting vector mean value of any point inthe space is identical to the particle density of that point

4 Computation and SynchronizationVisualization Based on GPU

41 GPU Acceleration Technology In recent years due to thelimit of the integrated circuit components we encounteredbottlenecks when we want to enhance the central process-ing unit (CPU) clock frequency on existing infrastructureWhen upgrading the performance of a single processor core

becomes difficult multicore CPU was launched and peopleare turning to the Graphics Processing Units (GPU) In thepast GPU was primarily used for graphics rendering anddisplay but with the improvements of GPU performance andfunctional structure GPU is not only used for graphics pro-cessing and rendering but also starts to become a powerfulassistant for CPU computing

The current mainstream GPU has the following char-acteristic features (1) supporting IEEE 32-bit floating-pointarithmetic and output (2) providing flexible programma-bility dynamic branching and loop control functions (3)supporting multiple rendering avoiding multiple data trans-fer between the CPU and GPU and (4) supporting tex-ture access function It can store data to the texture easilyand get index access (5) It supports output data to thetexture features which can improve data output speed andreduce the overhead of data readback Using the GPUto accelerate computation and rendering can reduce thecomputational burden on the CPU free CPU from theheavy computing tasks and meet the requirements of othercomputing

As the inventor of the GPU NVIDIA launched acommon computing programming environment CUDAcompletely shielding the underlying graphics API whichmakes GPU programming almost transparent for developersand lays the foundation for wide use of GPU commoncomputing

42 TPM Computation Based on CUDA The process for thecalculation of electromagnetic distribution status based onCUDAwith the particle transmissionmodel can be describedas follows (Figure 3)

Step 1 Transmit particles initiate each of the particles andput the transmitted particles to the particle list

Step 2 Update the properties of the particles in the list for thenext time instant

Step 3 Judge if the particle is below the ground If soreflection or diffraction is needed Next the choice betweenreflection and diffraction is made based on whether theparticle is on the edge of spire of the terrain

Step 4 If reflection is needed modify the position of theparticle according to the reflection principle of electromag-netic wave And further calculate the death probability of theparticle based on the reflection factor and decide the death ofthe particle based on the vanishing probability If diffractionis needed the process is similar to that of particle reflection

Step 5 Judge if the particle is outside the boundary and theneliminate the outside particles and the death particles fromthe list

Step 6 If the number of the listed particles is not stable (iethe particles within the calculation area have not reachedequilibrium) then return to Steps 1ndash5


Start

Transmit particles initiate each of the particles

Put the transmitted particles to the

Update the properties of the particles in the list for the next time instant

Is the particle below

Is the particle on the edgeof spire of the terrain

the ground

No

No

No

No

No

Yes

Yes

Yes

Reflectioncalculate the death

probability

Is the particleIs the particle outside the boundary

Diffraction calculatethe death probability

dead

Yes

End

Write to

Count the number of

Is the number in the

Eliminate the deathparticles

The particle ismarked as death

list stable

particles within each grid

the file

Yes

particle list

Figure 3 The process for the calculation of electromagnetic based on CUDA


Step 7 If the number of the particles in the list is stablepartition the space into grids and then count and output thenumber of particles within each grid

43 Computation and Visualization of Particle Model ofSynchronization Since the particle in Transmission ParticleModel is completely independent TPM is very suitable forthe parallel computing based on CUDA Updated GPUparticle data is stored in display memory so the data doesnot need to be transferred back to the memory when makingvisualization of the particle It can directly draw the particlemovement after transmission

In order to achieve the results we use CUDA to computeand draw directly on the display memory As we needto transfer data between OpenGL and CUDA we shouldestablish buffer zones between these two groups ofAPI for theposition coordinates and speedWe can get the actual addressof the device memory through the kernel function Firstlycall cudaGraphicsMapResources() function to tell CUDAruntime mapping the OpenGL shared memory and thencall cudaGraphicsResourceGetMappedPointer() to request apointer to the mapped resourceThus the OpenGL can sharewith CUDA buffer

5 Simulation and Analysis

51 Comparison of TPM with RT Electromagnetic particlemodel is based on ray-tracing method but it is ratherdifferent from ray-tracing In ray-tracing method the spacedistances between transmission rays are equal we calculatethe initial density on different directions and need to cal-culate the variation of the density of electromagnetic fieldcaused by diffusion in space As to electromagnetic particlemodel the ray is replaced by particles the energy of eachparticle is a constant the intensity of electromagnetic fieldis represented by the density of particles the transmissiondensity on different directions depends on the angle factor ofthe antenna and the movement of particle accords with theattenuation of diffusion in space Consequently the particlemovement itself represents the variation of electromagneticfield intensity caused by diffusion

To calculate the electromagnetic field intensity at a pointforward ray-tracing algorithmadopts themethodof receivingsphere to define the receiving radius and count the numbersof rays in the scope of the receiving sphere But the definitionof receiving sphere may bring in significant errors Considerthere is only direct transmission in an infinite space onereceiving sphere should receive only one ray The receivingradius equals the distance between two rays around the pointIf the receiving sphere is defined larger than the distancethere could be two rays in the receiving sphere if the receivingsphere is defined smaller than the distance there could beno rays in the receiving sphere That is the reason for errorsin the definition of the receiving sphere The energy of eachparticle in electromagnetic particle model is less than that ofa ray in ray-tracing method thus yielding the smaller errorOf course in this way the deficiency of significant predictionerrors in forward ray-tracing method can be overcome

Comparedwith backward ray-tracing TPM can calculatethe electromagnetic distribution status in the whole area atone time but the backward ray-tracing needs to calculateeach point in the area once again In other words TPMneeds less calculation than backward ray-tracing to getelectromagnetic distribution status

To verify the effectiveness of the proposed TPM wemade a comparison between the proposed method andthe ray-tracing method with a simplified terrain and thetransmission source is an electric dipole antenna in themiddle of the area

In order to compare TPM with ray-tracing methodwe choose several points to calculate the propagation lossaccording to the ray-tracingmethod and the particlemethodrespectively The height for calculation is 100m Becausebackward ray-tracing is commonly used the result is calcu-lated by backward ray-tracing as standard The results areshown in Figure 4 The beginning of the curve is not drawnbecause this part is under the ground There are errors in thecalculations using the forward ray-tracing method We cansee that the power is bigger than standard near 95m becausethe receiving sphere is defined larger in Figure 4(a) and thereceiving sphere is defined lower near 78m in Figure 4(b)The calculation results of TPM and backward ray-tracing arealmost the same thus TPM is effective for electromagneticcalculation and can overcome the deficiency of errors inforward ray-tracing method

52 Comparison of Transmission Particle Model and WirelessInSite Wireless InSite is complex electromagnetic environ-ment predictive analysis simulation commercial softwareThe software is based on UTDGTD theory and mainlyuses ray-tracingmethod combined with the improved FDTDmethod to calculate the electromagnetic propagation Thecalculation model includes urban canyon model fast 3Dcity models full three-dimensional model the vertical planemodel the free-space model urban canyon FDTD modelFDTD model with sliding windows HATA model COST-HATA model and WI real-time models By combiningelectricity field with the antenna pattern it calculates the pathloss time of arrival angle of arrival of the electromagneticfield and the electromagnetic trend distribution

To further verify the feasibility of the model of Trans-mission Particle Model we calculated the electromagneticdistribution in the same terrain using particle transmissionsimulation model and Wireless InSite and displayed theelectromagnetic attenuation trend extracting surfaces 5meterfar from ground Figure 5(a) shows the calculation results ofWireless InSite which set 20 lowast 20 receivers and each colourblock represents the energy for one receiver Figure 5(b)shows the calculation of Transmission Particle Model Thespace is divided into 100 lowast 100 lowast 100 grids Figure 5(b) is theextracted 100 lowast 100 grids 5 meters from the ground consti-tuting a surface As we can see from the comparison of thetwo figures the electromagnetic attenuation of TransmissionParticle Model and Wireless InSite is basically in the samesituation which verified the reliability of the TransmissionParticle Model


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

Forward RTBackward RTTPM

50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)

Figure 4 Compare TPM with ray-tracing method

minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90

(b) Transmission Particle Model

Figure 5 Visualization comparing TPM with Wireless InSite software (dBm)

The calculation time of TPM is less than the WI usingRT method The calculation time comparison is shown inFigure 6

When Wireless InSite software needs to calculate theelectromagnetic field strength of a point it uses the methodof setting the receiver and calculates the received electro-magnetic energy for the receiver If we want to calculate theelectromagnetic field distribution trend of the entire spacewe need to provide intensive receiver in the space at equalintervals Transmission Particle Model can calculate electro-magnetic distribution trend in the entire space of the regionIn the calculation of the electromagnetic distribution trendin the entire space the calculation amount of TransmissionParticle Model is less than that of Wireless InSite software

6 Conclusion

Based on ray-tracing method and the particle method thispaper constructs TPM for electromagnetic field calculation

The details including the initiation of the particles updatingthe properties of the particles the reflection and diffractioncalculation are also presented Since the particles in TPM arecompletely independent we used parallel computing basedon GPU for the computation Updated GPU particle datais stored in display memory when making visualization ofthe particle and it can directly draw the particle move-ment after transmission Deduction verification of TPM isconducted based on the electromagnetic field generated byelectric dipole Finally the simulation comparisons are madeagainst the ray-tracing method and the commercial softwareWI The simulation results verified the effectiveness of theproposed method In summary TPM is an effective toolfor electromagnetic calculation such as the electromagneticdistribution status in the space with complex terrain

Conflict of Interests

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


6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)

Figure 6 Calculation time comparing TPM with Wireless InSite(RT)

Acknowledgments

This work is supported by Scientific Research CommonProgram of Beijing Municipal Commission of Education(KM201411232007) Open Research Project of the BeijingKey Laboratory of High Dynamic Navigation Technology(Grant no HDN2014007) and the National Natural ScienceFoundation of China (Grant no 61031001) Beijing NaturalScience Foundation (4142017)

References

[1] J W Schuster and R J Luebbers ldquoComparison of GTD andFDTD predictions for UHF radio wave propagation in a simpleoutdoor urban environmentrdquo in Proceedings of the IEEE Anten-nas and Propagation Society International Symposium pp 2022ndash2024 Digest Montreal Canada July 1997

[2] C Kuifu and Z Senwe ldquoDFT accelaration and its applicationin non-stationary random vibration calculationrdquo Journal ofVibration Engineering vol 12 pp 91ndash96 1999 (Chinese)

[3] M F Levy Parabolic Equation Methods for ElectromagneticWave Propagation Institution of Engineering and TechnologyPress London UK 2000

[4] Z Senwen and C Kuifu ldquoThe monment equation method formulti-degree of freedom system with strongly nonlinear coup-ling and parametric excitationrdquo Acta Mechnica Sinica vol 25pp 362ndash368 1993 (Chinese)

[5] S Amiri and S M Hosseini ldquoSecond-order method for solving2D nonlinear parabolic differential equations based on ADImethodrdquo International Journal of Modeling Simulation andScientific Computing vol 1 no 1 pp 133ndash146 2010

[6] V P Tanana ldquoAn order-optimal method of solving an inverseproblem for a parabolic equationrdquo Journal of Applied andIndustrial Mathematics vol 3 pp 395ndash400 2009

[7] Y Cocheril and R Vauzelle ldquoA new ray-tracing basedwave propagation model including rough surfaces scatteringrdquoProgress in Electromagnetics Research vol 75 pp 357ndash381 2007

[8] B R Epstein and D L Rhodes ldquoGPU-accelerated ray tracingfor electromagnetic propagation analysisrdquo in Proceedings ofthe IEEE International Conference on Wireless InformationTechnology and Systems (ICWITS rsquo10) pp 1ndash4 September 2010

[9] P G Brown and C C Constantinou ldquoInvestigations on theprediction of radiowave propagation in urban microcell envi-ronments using ray-rdquo IEE Proceedings Microwaves Antennasand Propagation vol 143 no 1 p 36 1996

[10] T E Athanaileas G E Athanasiadou G V Tsoulos and DI Kaklamani ldquoParallel radio-wave propagation modeling withimage-based ray tracing techniquesrdquo Parallel Computing vol36 no 12 pp 679ndash695 2010

[11] S Aono M Unno and H Asai ldquoAlternating-direction explicitFDTD method for three-dimensional full-wave simulationrdquo inProceedings of the 60th Electronic Components and TechnologyConference (ECTC rsquo10) pp 375ndash380 June 2010

[12] N Endoh T Tsuchiya and S Matsumoto ldquoAnalysis of acousticcharacteristics of aero ultrasonic sensor calculated by 3D-FDTDrdquo in Forum Acusticum Budapest 2005 4th EuropeanCongress on Acustics Scientific Society for Optics BudapestHungary 2005

[13] J E Bendz H G Fernandes and M K Zuffo ldquoFast andaccurate finite-difference time-domain method for large three-dimensional simulationsrdquo in Proceedings of the 19th IASTEDInternational Conference on Modelling and Simulation pp 20ndash25 Quebec City Canada May 2008

[14] F Yalcin Yamaner and A Bozkurt ldquoP5E-4 3D transient analysisof ultrasound propagation using finite difference time domainmethod and its experimental verificationrdquo in IEEE UltrasonicsSymposium pp 2295ndash2298 2007

[15] G E Athanasiadou and A R Nix ldquoA novel 3-D indoor ray-tracing propagation model the path generator and evaluationof narrow-band and wide-band predictionsrdquo IEEE Transactionson Vehicular Technology vol 49 no 4 pp 1152ndash1168 2000

[16] E G Papkelis I Psarros I C Ouranos et al ldquoA radio-coverageprediction model in wireless communication systems based onphysical optics and the physical theory of diffractionrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 156ndash1652007

[17] A Escobar L Lozano H Cadavid and M F Catedra ldquoA newray-tracing acceleration technique for radio propagationrdquo inProceeding of the IEEE Antennas and Propagation Society Inter-national Symposium (APSURSI 10) pp 1ndash4 Toronto CanadaJuly 2010

[18] J B Keller ldquoGeometrical theory of diffractionrdquo Journal of theOptical Society of America vol 52 pp 116ndash130 1962

[19] R G Kouyoumjian and P H Pathak ldquoA uniform geometricaltheory of diffraction for an edge in a perfectly conductingsurfacerdquo Proceedings of the IEEE vol 62 no 11 pp 1448ndash14611974

[20] M B Tabakcioglu andA Kara ldquoComparison of improved slopeuniform theory of diffraction with some geometrical optic andphysical optic methods for multiple building diffractionsrdquo Elec-tromagnetics vol 29 no 4 pp 303ndash320 2009

[21] G Z Ni Engineering Principle of Electromagnetic Field HigherEducation Press Beijing China 2009

[22] Y Corre and Y Lostanlen ldquoThree-dimensional urban EMwave propagation model for radio network planning andoptimization over large areasrdquo IEEE Transactions on VehicularTechnology vol 58 no 7 pp 3112ndash3123 2009

[23] S Priebe M Jacob C Jastrow T Kleine-Ostmann T Schraderand T Kurner ldquoA comparison of indoor channel measurementsand ray tracing simulations at 300GHzrdquo in Proceedings ofthe 35th International Conference on Infrared Millimeter andTerahertz Waves (IRMMW-THz rsquo10) pp 1ndash2 September 2010

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


Based on forward ray-tracing method and particlemethod a Transmission Particle Model for calculating elec-tromagnetic distribution status in space is presented inthis paper Using the GPU to accelerate computation andvisualization can reduce the computational burden on theCPU The movement of a particle obeys the principle of thepropagation of electromagnetic wave and then the particledistribution density in space reflects the distribution statusof electromagnetic distribution status Since the particles inTPM are completely independent it is very suitable for theparallel computing based on GPU Updated GPU particledata is stored in display memory so the data does notneed to be transferred back to the memory when makingvisualization of the particle and it can directly draw theparticle movement after transmission

In this paper we present Transmission ParticleModel anduse GPU based on CUDA realize particles transmit updatesdemise and other operations and achieve particle modelcomputation and synchronization visualization

2 Description of TPM

The basic idea of TPM is that particles are generated andtransmitted intermittently in the origin of electromagneticfield and the electromagnetic distribution status in space isobtained through calculating the movement of each particle

So far there are many methods for electromagneticcalculation such as finite-difference time-domain method(FDTD) parabolic equation method (PE) and ray-tracingmethod (RT) To calculate electromagnetic wave in a largerange FDTDmethod brings heavy burden in calculation andalso requires huge amount of memory PE method performswell for complex terrain but has significant limitation inthe calculation of propagation angle Generally speaking RTmethod is applicable to the electromagnetic calculation in alarge range

21 Particle Transmission From the generation in the originof electromagnetic field the particle transmission densitymainly depends on antennarsquos angle factor 119891(120579 120593) 120579 and 120601 arethe angle components in sphere coordinates

Take the origin of electromagnetic field as a vertexpartition the space with equal interval in all directions Thesampling intervals for 120579 and 120593 are Δ120579 and Δ120593 respectivelyusually we have Δ120579 = Δ120593

Then we obtain

120579119894= 119894Δ120579 119894 = 1 2 3 int( 120587

Δ120579)

120593119895= 119895Δ120593 119895 = 1 2 3 int( 2120587

Δ120593)

(1)

Denote 119899119894119899119895as the particle numbers in one transmission

in the region (1205791015840 1205931015840) | 1205791015840 isin [120579119894minus Δ1205792 120579

119894+ Δ1205792] 1205931015840 isin

[120593119895minusΔ1205932 120593

119895+Δ1205932] whose maximum value is119873

11198732(119899119894119899119895

is the product of 119899119894and 119873

11198732the product of 119899

1198951198731and 119873

2

s(k)

s(k)998400s(k minus 1)

r(k

r(k)

+ 1)

r(k minus 1)

R0

120575

Figure 1 The distance between the particles

1198731 and 119873

2are given constants if Δ120579 = Δ120593 then we have

1198731= 1198732 119899119894= 119899119895)

119899119894=

119891 (120579119894 120593119895)

119891max (120579 120593)1198731

119899119895=

119891 (120579119894 120593119895)

119891max (120579 120593)1198732

(2)

The initial coordinates are (1198770 120579119894119895119901

120593119894119895119902

) initial speed Vis (1198770120575 120579119894119895119901

120593119894119895119902

) initial distance 119903 = 1198770 the initial interval

angle with other adjacent particles is 120575 asymp (12)(Δ120579119899119894+

Δ120593119899119895) the initial life time is 119905 = 0 and the initial particle

status is active where

120579119894119895119901

= 120579119894minusΔ120579

2+119901Δ120579

119899119894

119901 = 1 2 3 119899119894

120593119894119895119902

= 120593119895minusΔ120593

2+119902Δ120593

119899119895

119902 = 1 2 3 119899119895

(3)

1198770is the distance between the initial position of particle

transmission and the electromagnetic transmission sourcewhich is supposed to be the origin of coordinate

22 Particle Movement The movement of particle isdynamic that is with the elapse of time the currentproperties such as position coordinates speed V movementdistance 119903 angle 120575 life time 119905 and state (active or dead) willaffect the properties of the particles in the particle list

In Figure 1 119903(119896) is the total movement distance of theparticle 119904(119896) is the distance between particles in the samebatch and 119904(119896)1015840 is the distance particle of two consecutivebatches within time interval Δ119879 Particles are consecutivelytransmitted with equal time interval Δ119879



119903 (0) = 1198770

119904(0)1015840 ≐ 120575119903 (0) = 1205751198770

119904 (0) = 2 sin(1205752) 119903 (0) = 2 sin(120575

2)1198770

(4)


119903 (119896) = 119903 (0) +119896minus1

sum119894=0

119904 (119894) = (120575 + 1)1198961198770

119904(119896)1015840 = 2 sin(120575

2) 119903 (119896) = 2 sin(120575

2) (120575 + 1)

1198961198770

119904 (119896) = 120575119903 (119896) = 120575(120575 + 1)1198961198770


(5)



V (119896) =119904 (119896)

Δ119879=120575(120575 + 1)1198961198770

Δ119879 (6)


V (119905) =1

Δ119879(120575 + 1)

119905Δ1198791198770ln (120575 + 1) (7)



119875reflection = 119877 (8)




R=radic12057621205761minus sin2120579

1minus (12057621205761) cos 120579

1

radic12057621205761minus sin2120579

1+ (12057621205761) cos 120579

1

(10)



1


1

(11)


1205751015840 =120575

119877 (12)





119875diffraction = 119863 (13)




1205751015840 =120575

119863 (15)






E = 119895IΔ1198971198962 sin 12057941205871205961205760119903

119890minus119895119896119903e120579

H = 119895IΔ119897119896 sin 1205794120587120596119903

119890minus119895119896119903e120601

(16)




S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903 (17)




119891 (120579 120601) = sin 120579 (18)



119901 120579119901 120593119901) is 120588119899


119901Δ120579 + Δ1205792]

1205931015840 isin [119895119901Δ120593 minus Δ1205932 119895



119894119901119899119895119901 where

119894119901= round(

120579119901

Δ120579) (19)

119895119901= round(

120593119901

Δ120593) (20)

119899119894119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198731 (21)

119899119895119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198732 (22)


119901(119896)

119903119901 (119896) = (120575119901 + 1)

119896

1198770 (23)

05

0

minus05

x

z

yminus1 minus1

00

1

1



119903119901 (119896 minus 1) = (120575119901 + 1)

119896minus1

1198770 (24)

where

120575119901=1

2(Δ120579

119899119894119901

+Δ120593

119899119895119901

) (25)


119901(119896minus1) 119903

119901(119896+1)] 1205791015840 isin

[119894119901Δ120579minusΔ1205792 119894

119901Δ120579+Δ1205792]1205931015840 isin [119895

119901Δ120593minusΔ1205932 119895

119901Δ120593+Δ1205932]


119881119901= 119881119901 (119896 + 1) minus 119881119901 (119896 minus 1) (26)

where

119881119901 (119896 + 1) =

4

31205871199033119901(119896 + 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896) (120575119901 + 1)

3

119881119901 (119896 minus 1) =

4

31205871199033119901(119896 minus 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896)

1

(120575119901+ 1)3

(27)


119881119901=2Δ120579Δ1205931199033

119901(119896)

3120587((120575119901+ 1)3

minus1

(120575119901+ 1)3)

asymp4Δ120579Δ1205931199033

119901(119896) 120575119901

120587

(28)


120588119899=2119899119894119901119899119895119901

119881119901

=120587119899119894119901119899119895119901

2Δ120579Δ1205931199033119901(119896) 120575119901

(29)



120588119899=12058711987311198732(119891(119894119901Δ120579 119895119901Δ120593)119891max(120579 120593))

3

Δ120579Δ1205931199033119901(119896) (Δ1205791198731 + Δ1205931198732)

(30)

Since Δ120579 = Δ120593 then 119899119901= 119899119894119901= 119899119895119901 and119873 = 119873

1= 1198732

Further we have

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(119894119901Δ120579 119895119901Δ120593)

119903119901(119896)

)

3

(31)


120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(120579119901 120593119901)

119903119901

)

3

(32)


119891 (120579 120601) = sin 120579 (33)

we have

12058823119899

= (120587

2)23 1198732

1198912max (120579 120593) Δ1205792(sin 120579119903

)2

(34)

Then we can get

(sin 120579119903

)2

= 12058823119899(2

120587)231198912max (120579 120593) Δ120579

2

1198732 (35)


S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903= 120578(

119868Δ119897

2120582)2

(sin 120579119903

)2

e119903

= 120578(119868Δ119897

2120582)2

(2

120587)231198912max (120579 120593) Δ120579

2

119873212058823119899

e119903= 11987112058823119899

e119903

(36)

where

119871 = 120578(119868Δ119897

2120582)2 (2120587)231198912max (120579 120593) Δ120579

2

1198732 (37)















Start






the ground

No

No

No

No

No

Yes

Yes

Yes


probability



dead

Yes

End

Write to

Count the number of




list stable


the file

Yes

particle list















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903 (0) = 1198770

119904(0)1015840 ≐ 120575119903 (0) = 1205751198770

119904 (0) = 2 sin(1205752) 119903 (0) = 2 sin(120575

2)1198770

(4)


119903 (119896) = 119903 (0) +119896minus1

sum119894=0

119904 (119894) = (120575 + 1)1198961198770

119904(119896)1015840 = 2 sin(120575

2) 119903 (119896) = 2 sin(120575

2) (120575 + 1)

1198961198770

119904 (119896) = 120575119903 (119896) = 120575(120575 + 1)1198961198770


(5)



V (119896) =119904 (119896)

Δ119879=120575(120575 + 1)1198961198770

Δ119879 (6)


V (119905) =1

Δ119879(120575 + 1)

119905Δ1198791198770ln (120575 + 1) (7)



119875reflection = 119877 (8)




R=radic12057621205761minus sin2120579

1minus (12057621205761) cos 120579

1

radic12057621205761minus sin2120579

1+ (12057621205761) cos 120579

1

(10)



1


1

(11)


1205751015840 =120575

119877 (12)





119875diffraction = 119863 (13)




1205751015840 =120575

119863 (15)






E = 119895IΔ1198971198962 sin 12057941205871205961205760119903

119890minus119895119896119903e120579

H = 119895IΔ119897119896 sin 1205794120587120596119903

119890minus119895119896119903e120601

(16)




S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903 (17)




119891 (120579 120601) = sin 120579 (18)



119901 120579119901 120593119901) is 120588119899


119901Δ120579 + Δ1205792]

1205931015840 isin [119895119901Δ120593 minus Δ1205932 119895



119894119901119899119895119901 where

119894119901= round(

120579119901

Δ120579) (19)

119895119901= round(

120593119901

Δ120593) (20)

119899119894119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198731 (21)

119899119895119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198732 (22)


119901(119896)

119903119901 (119896) = (120575119901 + 1)

119896

1198770 (23)

05

0

minus05

x

z

yminus1 minus1

00

1

1



119903119901 (119896 minus 1) = (120575119901 + 1)

119896minus1

1198770 (24)

where

120575119901=1

2(Δ120579

119899119894119901

+Δ120593

119899119895119901

) (25)


119901(119896minus1) 119903

119901(119896+1)] 1205791015840 isin

[119894119901Δ120579minusΔ1205792 119894

119901Δ120579+Δ1205792]1205931015840 isin [119895

119901Δ120593minusΔ1205932 119895

119901Δ120593+Δ1205932]


119881119901= 119881119901 (119896 + 1) minus 119881119901 (119896 minus 1) (26)

where

119881119901 (119896 + 1) =

4

31205871199033119901(119896 + 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896) (120575119901 + 1)

3

119881119901 (119896 minus 1) =

4

31205871199033119901(119896 minus 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896)

1

(120575119901+ 1)3

(27)


119881119901=2Δ120579Δ1205931199033

119901(119896)

3120587((120575119901+ 1)3

minus1

(120575119901+ 1)3)

asymp4Δ120579Δ1205931199033

119901(119896) 120575119901

120587

(28)


120588119899=2119899119894119901119899119895119901

119881119901

=120587119899119894119901119899119895119901

2Δ120579Δ1205931199033119901(119896) 120575119901

(29)



120588119899=12058711987311198732(119891(119894119901Δ120579 119895119901Δ120593)119891max(120579 120593))

3

Δ120579Δ1205931199033119901(119896) (Δ1205791198731 + Δ1205931198732)

(30)

Since Δ120579 = Δ120593 then 119899119901= 119899119894119901= 119899119895119901 and119873 = 119873

1= 1198732

Further we have

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(119894119901Δ120579 119895119901Δ120593)

119903119901(119896)

)

3

(31)


120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(120579119901 120593119901)

119903119901

)

3

(32)


119891 (120579 120601) = sin 120579 (33)

we have

12058823119899

= (120587

2)23 1198732

1198912max (120579 120593) Δ1205792(sin 120579119903

)2

(34)

Then we can get

(sin 120579119903

)2

= 12058823119899(2

120587)231198912max (120579 120593) Δ120579

2

1198732 (35)


S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903= 120578(

119868Δ119897

2120582)2

(sin 120579119903

)2

e119903

= 120578(119868Δ119897

2120582)2

(2

120587)231198912max (120579 120593) Δ120579

2

119873212058823119899

e119903= 11987112058823119899

e119903

(36)

where

119871 = 120578(119868Δ119897

2120582)2 (2120587)231198912max (120579 120593) Δ120579

2

1198732 (37)















Start






the ground

No

No

No

No

No

Yes

Yes

Yes


probability



dead

Yes

End

Write to

Count the number of




list stable


the file

Yes

particle list















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E = 119895IΔ1198971198962 sin 12057941205871205961205760119903

119890minus119895119896119903e120579

H = 119895IΔ119897119896 sin 1205794120587120596119903

119890minus119895119896119903e120601

(16)




S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903 (17)




119891 (120579 120601) = sin 120579 (18)



119901 120579119901 120593119901) is 120588119899


119901Δ120579 + Δ1205792]

1205931015840 isin [119895119901Δ120593 minus Δ1205932 119895



119894119901119899119895119901 where

119894119901= round(

120579119901

Δ120579) (19)

119895119901= round(

120593119901

Δ120593) (20)

119899119894119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198731 (21)

119899119895119901=119891 (119894119901Δ120579 119895119901Δ120593)

119891max (120579 120593)1198732 (22)


119901(119896)

119903119901 (119896) = (120575119901 + 1)

119896

1198770 (23)

05

0

minus05

x

z

yminus1 minus1

00

1

1



119903119901 (119896 minus 1) = (120575119901 + 1)

119896minus1

1198770 (24)

where

120575119901=1

2(Δ120579

119899119894119901

+Δ120593

119899119895119901

) (25)


119901(119896minus1) 119903

119901(119896+1)] 1205791015840 isin

[119894119901Δ120579minusΔ1205792 119894

119901Δ120579+Δ1205792]1205931015840 isin [119895

119901Δ120593minusΔ1205932 119895

119901Δ120593+Δ1205932]


119881119901= 119881119901 (119896 + 1) minus 119881119901 (119896 minus 1) (26)

where

119881119901 (119896 + 1) =

4

31205871199033119901(119896 + 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896) (120575119901 + 1)

3

119881119901 (119896 minus 1) =

4

31205871199033119901(119896 minus 1) (

Δ120579

120587)(

Δ120593

2120587)

=2Δ120579Δ120593

31205871199033119901(119896)

1

(120575119901+ 1)3

(27)


119881119901=2Δ120579Δ1205931199033

119901(119896)

3120587((120575119901+ 1)3

minus1

(120575119901+ 1)3)

asymp4Δ120579Δ1205931199033

119901(119896) 120575119901

120587

(28)


120588119899=2119899119894119901119899119895119901

119881119901

=120587119899119894119901119899119895119901

2Δ120579Δ1205931199033119901(119896) 120575119901

(29)



120588119899=12058711987311198732(119891(119894119901Δ120579 119895119901Δ120593)119891max(120579 120593))

3

Δ120579Δ1205931199033119901(119896) (Δ1205791198731 + Δ1205931198732)

(30)

Since Δ120579 = Δ120593 then 119899119901= 119899119894119901= 119899119895119901 and119873 = 119873

1= 1198732

Further we have

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(119894119901Δ120579 119895119901Δ120593)

119903119901(119896)

)

3

(31)


120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(120579119901 120593119901)

119903119901

)

3

(32)


119891 (120579 120601) = sin 120579 (33)

we have

12058823119899

= (120587

2)23 1198732

1198912max (120579 120593) Δ1205792(sin 120579119903

)2

(34)

Then we can get

(sin 120579119903

)2

= 12058823119899(2

120587)231198912max (120579 120593) Δ120579

2

1198732 (35)


S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903= 120578(

119868Δ119897

2120582)2

(sin 120579119903

)2

e119903

= 120578(119868Δ119897

2120582)2

(2

120587)231198912max (120579 120593) Δ120579

2

119873212058823119899

e119903= 11987112058823119899

e119903

(36)

where

119871 = 120578(119868Δ119897

2120582)2 (2120587)231198912max (120579 120593) Δ120579

2

1198732 (37)















Start






the ground

No

No

No

No

No

Yes

Yes

Yes


probability



dead

Yes

End

Write to

Count the number of




list stable


the file

Yes

particle list















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588119899=12058711987311198732(119891(119894119901Δ120579 119895119901Δ120593)119891max(120579 120593))

3

Δ120579Δ1205931199033119901(119896) (Δ1205791198731 + Δ1205931198732)

(30)

Since Δ120579 = Δ120593 then 119899119901= 119899119894119901= 119899119895119901 and119873 = 119873

1= 1198732

Further we have

120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(119894119901Δ120579 119895119901Δ120593)

119903119901(119896)

)

3

(31)


120588119899=

1205871198733

21198913max (120579 120593) Δ1205793(119891(120579119901 120593119901)

119903119901

)

3

(32)


119891 (120579 120601) = sin 120579 (33)

we have

12058823119899

= (120587

2)23 1198732

1198912max (120579 120593) Δ1205792(sin 120579119903

)2

(34)

Then we can get

(sin 120579119903

)2

= 12058823119899(2

120587)231198912max (120579 120593) Δ120579

2

1198732 (35)


S119886V = 120578(

119868Δ119897

2120582119903)2

sin2120579e119903= 120578(

119868Δ119897

2120582)2

(sin 120579119903

)2

e119903

= 120578(119868Δ119897

2120582)2

(2

120587)231198912max (120579 120593) Δ120579

2

119873212058823119899

e119903= 11987112058823119899

e119903

(36)

where

119871 = 120578(119868Δ119897

2120582)2 (2120587)231198912max (120579 120593) Δ120579

2

1198732 (37)















Start






the ground

No

No

No

No

No

Yes

Yes

Yes


probability



dead

Yes

End

Write to

Count the number of




list stable


the file

Yes

particle list















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start






the ground

No

No

No

No

No

Yes

Yes

Yes


probability



dead

Yes

End

Write to

Count the number of




list stable


the file

Yes

particle list















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100Height (m)

Pow

er (d

Bm)


50

0

minus50

minus100

minus150

(a)


0 20 40 60 80 100Height (m)

Pow

er (d

Bm)

50

0

minus50

minus100

minus150

(b)


minus60

minus65

minus70

minus75

minus80

minus85

minus90

(a) Wireless InSite

minus60

minus65

minus70

minus75

minus80

minus85

minus90





6 Conclusion






6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6000

5000

4000

3000

2000

1000

0TPM

Calc

ulat

ion

time (

s)

WI (RT)


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Electromagnetic Computation and ...downloads.hindawi.com/journals/mpe/2014/942106.pdf · Research Article Electromagnetic Computation and Visualization of Transmission