parallel age solver of multidimensional pde modeling for thermal control of laser beam on...

8/13/2019 Parallel AGE solver of multidimensional PDE modeling for Thermal Control of Laser Beam on Cylindrical Glass

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9th International Conference on Fracture & Strength of SolidsJune 9-13, 2013, Jeju, Korea

Parallel AGE solver of multidimensional PDE modeling for Thermal

Control of Laser Beam on Cylindrical Glass Norma Alias1, Md. Rajibul Islam2,*

1

Ibnu Sina Institute, University Technology Malaysia, Johor, Malaysia2Department of Mathematical Sciences, University Technology Malaysia, Johor, Malaysia

Abstract: The commands of this up-to-date century customer products denote that the glass platesused in their exhibits can no longer be cut by old conventional techniques. Luckily, laser based cuttingevades the vital restrictions of those methods. It can be developed without excessively rising productexpenses, because this simulation will use to remove the past problems. This paper presents a novelmathematical modeling based on multidimensional parabolic equations for cylindrical coordinate ofglass that will be used for simulating the thermal control of the laser glass interaction. Three methodshave been used for the simulation purposes in the Alternating Group Explicit (AGE) method classwhich compromises two variances that are Brian and Douglas variants, and Red Black Gauss Seidelmethod is chosen as a benchmark. This model simulates a large scope of laser beam propagation byimplementing these three methods in a parallel computing platform. So the high computational cost ofthat complete electromagnetic nature of a laser beam is simulated accurately and produced convergenceresults. Precise computation of laser beam passage ensures pragmatic heat generation pattern in thetarget material. The comparison of numerical simulation will be analyzed by conducting numericalanalysis and parallel performance measurement, in terms of speedup, execution time, effectiveness,efficiency, temporal performance and granularity.Keywords : Alternating Group Explicit Method (AGE); Gauss Seidel Red Black; Partial DifferentialEquation (PDE); Parallel Computing; Glass Cutting; Laser.

1. IntroductionVarious forms of the conventional method for cutting glass has been used, such as a mechanicalsnapping force to disseminate the crack completely through the glass by scribing the surface of theglass with a sharp, hard tool (usually a diamond [1] or carbide wheel). A chopper bar, has beenimplementing in the automated systems to perform such parting, which moves down on the glass [2].Unfortunately, this automated system has certain weaknesses for very thin substrates. Specifically, themechanical force of the cutting tool produces microcracks in the material. The following breaking stepdefers small chips and debris, along with an edge that is not necessarily perpendicular to the glasssurface. Moreover, mechanical cutting leaves momentous mechanical stress in the finished edge. Infact, it becomes complicated to utilize mechanical cutting at all with substrates that are below 1mm inthickness because the glass is so easily broken. Therefore, it may be required to grind or polish the cutsurface of the glass after the original cut, to avoid further cracking or breaking. Furthermore, a post- process cleaning step may be essential to eliminate debris that could obstruct with successive methods,such as circuit formation [2].Researchers have developed some glass cutting methods such as abrasive waterjet and laser. Lasercutting of glass has many advantages, it is a noncontact process acquiescent to computer control and isable to construct composite shapes. Two laser shaping methods, termed controlled fracture and scribingare used on a variety of ceramic substrates in the electronic industry [3].Channels of microcracks are formed using a Nd: YAG laser in a different method of laser cutting of plate glass. The channel of microcracks is then developed from pulse to pulse in the direction insideand ended on the face side. As a result the discharge of this stress becomes the formation of big regularcrack along the set of channels [4].Our proposed method of laser glass cutting simulation addresses quite a lot of these limitations. First ofall, it is a non-contact procedure that absolutely removes the difficulty of chipping and microcracking.Also, this cutting method produces will improve the laser glass interaction with more accurate andsmooth because of the use of parallel computing strategy. When force is concerned to the center of aglass panel, any crack generally begins at the edge, which is in fact a significant difficulty. Therefore, proposed method of glass cutting through laser can resist two to three times as much force asconventional technique such as mechanically cut glass that will be shown in numerical analysis by theCOMSOL multiphysics visualization as a proof.

_______________________________*Corresponding author: E-mail: [email protected]; Tel: +6-016-6458243.
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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The key purpose of this study is to develop a new variance algorithm in the class of the AGE method inorder to solve the thermal control process in multidimensional laser glass model problem and tocompare the numerical results and parallel performance evaluations of the Alternating Group ExplicitMethod (AGE) with Douglas-Rachford variant (DOUGLAS) [5-6] and AGE method with Brian variant(BRIAN). Red Black Gauss Seidel is chosen as the standard method for the comparison. We have presented the three dimensional cylinder coordinate system of parabolic equation to determine the

temperature distribution of the laser glass interaction problem. In this model, we have described parallel AGE algorithms implementation on the high performance computing platform.

2. Problem Statement of Laser beam on cylindrical glassTemperature prediction is a wide variety of science and engineering problems, which are governed by partial different equations of the parabolic type. Instead of testing them each time in order to determinewhether the equation is parabolic or not, it is convenient to introduce a sufficiently general parabolicdifferential equation from which numerous other parabolic equation of transport phenomena can beobtained as special cases. In the special case of temperature propagation in an isotropic andhomogeneous medium in the three dimensional space of cylindrical coordinate of parabolic type, theequation is given as,

k q

zU U

r r U

r r U

t U

a+

+

+

+

=

2

2

2

2

2

2

2

2 111

(1)

With the initial conditionT(r,,z,t)| r =0 =T 0 (2)

With T 0 is initial temperature where temperature of glass cylinder before laser process and boundarycondition

T(r,,z,t)| s1=0 = T 1T(r,,z,t)| s2=0 = T 2

With T 1 and T 2 are temperature of inner-surface and outer-surface of glass cylinder. S 1 and S 2 arerepresenting area of inner-surface and outer-surface of glass cylinder.q is rate of heat flow,a is amaterial-specific quantity depending on the thermal conductivity, the density and the heat capacity. The heating due to laser is treated as body heat source. We investigated the moving laser with constant power. Some of the assumptions have been taken into considerations are, (i) material properties areassumed to be constant, (ii) the electromagnetic of the laser beam is not simulated, (iii) the effect ofelectromagnetic wavelength is not explicitly modeled, (iv) the effect of complex refractive index ofglass is modeled using an absorption and reflection coefficient, (v) the simulation does not involvemodeling phase change. The modeling geometry only includes the glass slab.

Figure 1: Schematic illustration of laser scribing

Figure 2: Temperature distribution after laser heating on the glass slab. a) 2D slice plot, b) 3D slice plot

Glass slab

Cooled area (Heat flux=h (Text-T)

Heated area

Direction of glass motion

Laser beam nozzle

(a) (b)


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Figure 3: Cylinder with prescribed angle and radius.

Except the top surface all the other boundaries are assumed to be thermally insulated. The heat flux onthe top surface simulates convective cooling.Figure 1, presents schematic design of laser scribing and figure 2 shows temperature distribution afterlaser heating on a cylindrical glass slab (2D and 3D). Here, figure 3 clearly shows a hot spot where thelaser beam is located at a specific time. The reflection and absorption coefficients are assumed to beconstants. The planar surface of the glass slab incident to the laser beam is assumed to be aligned withthe xy-plane of the global coordinate system. The top planar surface is aligned with z=0 . Hence theeffect of absorption ( Ac) can be simulated by the termexp (-Ac*abs(z)) . The center of the beam can beeasily shifted by changing x 0 and y 0 . The beam width and astigmatism can be easily controlled by thestandard deviation parameters; x and y. This model investigates the transient heating of a glass slabwhen an ancient laser beam shines upon it for a given time.

Table 1: Changing parameters of laser beam simulationsParameter Variable MeasurementSize of Grid , r, z 100, 50, 50Interval increase (mm) , r, z 3.6, 0.2, 0.1Internal/external cylinder residual (mm) r 1, r 2 48/53Density (kg/m3) 2440Hc (W/m2K) H 0Heat conductivity (J/kg.K) K at 300K 0.88Heat capacity C at 300K 840Initial temperature T0 650Environmental temperature T1 50Power of laser 0 100Radial of laser 0.1Speed of laser (rev/sec) w 1Lambda of laser 0.1Time increase t (sec) t 0.001End time (sec) T 1

The laser glass cylindrical coordinate of heat equation has used for this experiment. An isotropic ofthermal conductivity,k x = 163 W/(moC) , k y =163 W/(moC), andk z = 16 W/(moC), density, p = 2330kg/m3, temperature distribution,a= 0 W/m3, radius of trajectory circle =2 cm, angular velocity, = 10rad/s and prescribed heat capacity,C p = 703 J/kg.k have used in the equation (8) that is the laser glasscylindrical coordinate of heat equation for this experiment. The value of laser beam power of thecylinder is taken 50[W]. COMSOL 4.2 Multiphysics software [19] has been used for the visualizationof 3D simulation results and the simulation results are shown in figure 3.

3. Discretization of cylindrical coordinate system and sphereIn equation (1), specifically,a=k/C p wherek is the thermal conductivity,C p is the capacity, and thedensity. Value ofk depends on material used in experiment. Formulas of resistance-capacitance used asan exact solution on parabolic equation coordinate cylinder system.Thermal capacitanceC m = p mc m V m And thermal resistance R mn .The energy ratio of V m toward time interval t is given in equation (3),

= + (3)With time step l and equation shows temperatures change1+lmT , capacitance cm, resistance R mn andQ v source have to estimate every each time step increase.Solution of equation at 1+lmT is given

r


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+1= (1 1) + ( + ) (4) Stability condition of equation solution is coefficient where is cannot have a negative value that

1 1 0 (5) Therefore, stability of criterium appraised for each node and time-step and follow inequality is

(6)The finite difference of second order of partial differential can be obtained by Taylor [7-8] seriesexpansions and second order 3 points differential approximation at the time(p+1). The discretizationare given as the following,

22

)1(

.,1

)1(

,,

)1(

,,12

2

)(2

r Or r

U uuu p

k ji

p

k ji

p

k ji +

+=

+

+

++

22

)1(

.,1

)1(

,,

)1(

,,1

2

2

)(2

r OU uuu

p

k ji

p

k ji

p

k ji +

+=

+++

++

22

)1(

.,1

)1(

,,

)1(

,,1

2

2

)(2

r O z z

U uuu p

k ji

p

k ji

p

k ji +

+=

+

+

++

(7)

Obtained a equation (1) to partial differential with weighted parameter, at the time(p+1) and(p) asthe following,

uu p

k ji zr r

p

k ji zr r r r r r z

U U

r r

U

r r

U )(,,

22

2

2)1(

,,

22

2

2

2

2

2

2

2

2

2

2

)11

()(1()11

(11

+++++++=

+

+

+

+

(8)

Equation (8) is dealing with the real physical problems on three dimensional cylindrical coordinate of parabolic type to predict the temperature distribution on the laser glass interaction model.Finite difference approximation for parabolic equations with cylindrical and spherical coordinates can be expressed as follows, , = 1() 2+ , +1+ (1 ) 1() 2+ . + . + , i=1, 2, 3, ., m= 1 and . += 0 for the equation of cylindrical coordinates and= 2 and . + 0 for

the spherical coordinates. Form a standard three-point formula for parabolic equation; we get one-dimensional polar coordinates as follows,

1, +1+ (1 + 2 ) , +1 +1, +1 = (1 ) 1, + [ (1 2 (1 ) ] , +1 (1 ) +1, i=1, 2, 3, .m

4. Numerical methods4.1 AGE with DOUGLAS methodAGE class with Douglas Rachford variation (DOUGLAS) has been published by Evans & Sahimi [5-6]to solve parabolic problem of three dimensional. The domain decomposition technique(mmm) can be distributed to processor P directly at every time step [9]. The parallelization of DOUGLAS methodis using domain decomposition technique or non-overlap subdomain in block system(22) for everytime step. Matrix A is been split into consistent symmetric and positive definite matrices G1, G2, G3,G4, G5, G6 and the calculation of these matrices will be simplified into four equations as follows,

( )][1][ ][22][ ][11)( ][21)( ][1111)

61(

][1 2 xy N xy

N xy

p xy

p xy

p

xy guF uF u E u DC u ++++= +

(9)

( )][)( ][1][ ][1][ ][12)( ][2)( ][1)( ][1111)

61

(

][ 2)()( xyk N xyk

N xyk

N xyk

p xyk

p xyk

p xyk

p

xyk guF uuF u Duu E C u ++++++= +++

(10)

( )][)( ][1][ ][1][ ][12)( ][1)( ][1)( ][1111)

61

(

][ 2)()( xyk N xyk

N xyk

N xyk

p xyk

p xyk

p xyk

p

xyk guF uuF u Duu E C u ++++++= +++

(11)

( )][)( ][12)( ][1)( ][1)( ][1111)

61

(

][ 2)( xym N

xym N

xym p

xym p

xym

p

xym guF uF u Du E C u ++++= +

(12)Thus, based on (9) to (12), we can derive the calculation using AGE Douglas as (13),

][)()

65

(

][

)(

][1

1

1

)1(

],[

++ ++= p

xz

p

xz

p

z x ruuGrI Gu

(13)


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4.2 AGE with BRIAN methodBRIAN method is exploited the concepts of fractional strategy [10], linear interpolation on Brianvariant and the acceleration parameter,r . Experiments are implemented in both sequential and paralleliterative methods. The formula of BRIAN method for three-dimensional problem leads the method asthe following,

(r I + G1) )7/1(

][

+n

xyu = (r I (G

1 +G

2 +G

3 +G

4 +G

5 +G

6)) )(

][

n

xyu + f

= ((r I +G1) A) )(

][n

xyu + f ,

(r I +G2) )7/2(][

+n xyu = r

)7/1(][

+n xyu + G2

)(][

n xyu

(r I +G3) )7/3(][

+n xyu = r

)7/2(][

+n xyu + G3

)(][

n xyu

(r I +G4) )7/4(][

+n xyu = r

)7/3(][

+n xyu + G4

)(][

n xyu

(r I +G5))7/5(

][+n

xyu = r )7/4(][

+n xyu + G5

)(][

n xyu

(r I +G6))7/6(

][+n

xyu = r )7/5(][

+n xyu + G6

)(][

n xyu

)1(

][

+n

xyu = )(

][

n

xyu + 2( ))(

][

)7/6(

][

n

xy

n

xy uu +

(14)

It has been simplified into(22) matrix block form and the calculation of matrix by AGE Brianmethod for cylindrical coordinate system as follows,

( )][1][ ][2,1][ ][1,1)( ][2,1)( ][1,11,1)

71

(

][1 xy N

xyi N xyi

n xyi

n xyii

n

xy gu H uF u E u DC u ++++= +

( )][][ ][,1][ ][1,1][ ][1,1)( ][,2)( ][1,1)( ][1,11,2)

71

(

][ xyk N

xyk i N

xyk i N

xyk in xyk i

n xyk i

n xyk ii

n

xyk guF u H u H u Du E u E C u ++++++= +++

( )][][ ][,1][ ][1,1][ ][1,1)( ][,1)( ][1,1)( ][1,11,1)

71

(

][ )( xyk N

xyk i N

xyk i N

xyk in xyk i

n xyk i

n xyk ii

n

xyk guF u H u H u Du E u E C u ++++++= +++

( )][][ ][,1][ ][1,1)( ][,1)( ][1,111)

71

(

][ xym N

xymi N

xymin

xymin

xymi

n

xym guF u H u Du E C u ++++= +

4.3 Gauss-Seidel Red Black (GSRB)Parallel algorithm for GSRB method is based on domain decomposition for each odd subdomaindecomposition, R and even subdomain decomposition, H . Local convergence is tested by slave processors before global convergence tested. After that is followed by communication between R and H for update the convergence on the latest grid generation,Grid calculation at R

mk jick

ug

u f ueud uhubud

guubduuhuau

i N

k ji

N k ji

N k ji

N k ji

N k jii

N k ji

N k jii

pk ji

pk jii

pk ji

pk jii

pk jii

pk ji

,...,3,2,1,,

][1,,

][,1,

][,,1

][1,,

][,1,

][,,1

][,,

)(1,,

)(,,1

)(1,,

)(,1,

)(,,1

)1(,,

=++

++++++

=

+

++

+++

where =k ji ,, odd numberGrid calculation at H

mk jick

ug

u f ueud uhubud

guubduuhuau

i N

k ji

N k ji

N k ji

N k ji

N k jii

N k ji

N k jii

pk ji

pk jii

pk ji

pk jii

pk jii

pk ji

,...,3,2,1,,

][1,,

][,1,

][,,1

][1,,

][,1,

][,,1

][,,

)(1,,

)(,,1

)(1,,

)(,1,

)(,,1

)1(,,

=++

++++++

=

+

++

+++

where =k ji ,, even number


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5. Experiment Results and Discussion5.1 Numerical AnalysisTable 2 shows a comparison of the performance of one dimensional AGE and GSRB sequential classin solving cylindrical model and spherical coordinate system. The numerical results based onsequential algorithms of BRIAN, DOUGLAS and GSRB are displayed in Table 3. These tables providethe absolute errors of the discretization of one dimensional AGE cylindrical coordinates with GSRBand 3-dimensional BRIAN, DOUGLAS and GSRB methods using(70 70 70) and(100100100) sizes of matrices. We presented our 3D comparison results of BRIAN, DOUGLAS and GSRB methodsin our previous [16-18] articles. But in this article, we compared one dimensional sequential and parallel results. Moreover, we presented the COMSOL [19] visualization of the simulation results sothat researchers can understand the implementation and contribution of this study. Our analysis showsthat the rate of convergence of sequential 1D AGE is better than GSRB. That mean one dimensionalAGE works more better than GSRB in solving problem of cylindrical model and 3D BRIAN is even better than DOUGLAS and GSRB methods. That denotes 3D BRIAN method will be efficient forsolving laser glass interaction problem. The degree of accuracy of BRIAN method is higher thanDOUGLAS and GSRB methods. This proved by root means square error (rmse) and maximum error(r.maks) generated by BRIAN method have a lowest value for the both sizes of matrixes.

Table 2: Sequential performance comparison of the one dimensional AGE cylindrical coordinatest= 4.8225E-11, x=1.3889E -6, t=9.6450E -13, =0.5, =1.0E -15, and 1=1

rmse=root mean square error, |r|=absolute error, r.maks=maximum error, ave_rmse=average of rmse,and, r =size of grid,

m AGE GSRBAGE 3 2 1 3 2 1 Iters 69 70 157 470 501 951rmse 4.1596E-7 4.1595E-7 3.0538E-2 4.1596E-7 4.1595E-7 3.0536E-2 |r| 1.1031E-8 6.7501E-13 3.4051E-5 4.1373E-13 1.6855E-13 1.3877E-5 Ave_rmse 5.7715E-4 5.7715E-4 1.5986E-1 5.7715E-4 5.7715E-4 1.5987E-1 r_maks 9.5689E-4 9.5689E-4 2.4043E-1 9.5689E-4 9.5686E-4 2.4043E-1 r 0.7 0.5 0.7 - - -

Table 3: Numerical results of the sequential three dimensional BRIAN, DOUGLAS and GSRBmethods for two difference sizes of matrixes.

rmse=root mean square error, |r|=absolute error, r.maks=maximum error, ave_rmse=average of rmse, r=size of grid,t = time increase t (sec) and , r, z = interval increase (mm)

m 707070 100100100AGE BRIAN DOUGLAS GSRB BRIAN DOUGLAS GSRBt lari(s) 38.9550 41.4770 45.7150 116.4450 122.8850 154.6420Iters 10 20 70 10 20 70rmse 7.533E-4 7.536E-4 6.087E-3 8.689E-4 8.690E-4 3.919E-3 |r| 3.592E-6 1.796E-6 4.568E-6 1.785E-6 8.924E-7 4.688E-6 Ave_rmse 1.075E-5 1.076E-5 1.198E-5 1.438E-5 1.439E-5 1.739E-5 r_maks 2.451E-2 2.451E-2 3.614E-1 2.913E-2 2.913E-2 3.495E-1 r 5.000E-1 5.000E-1 5.000E-1 5.000E-1 5.000E-1 5.000E-1 t 1.449E-6 1.449E-6 1.449E-6 1.010E-6 1.010E-6 1.010E-6 9.106E-2 9.106E-2 9.106E-2 6.347E-2 6.347E-2 6.347E-2 r 1.449E-4 1.449E-4 1.449E-4 1.010E-4 1.010E-4 1.010E-4 z 1.449E-5 1.449E-5 1.449E-5 1.010E-5 1.010E-5 1.010E-5 er 7.000E-1 7.000E-1 7.000E-1 7.000E-1 7.000E-1 7.000E-1 Q 1.000E-1 1.000E-1 1.000E-1 1.000E-1 1.000E-1 1.000E-1 q 1.000E-1 1.000E-1 1.000E-1 1.000E-1 1.000E-1 1.000E-1

1 1 1 1 1 1Level 10 10 10 10 10 10 1.000E-5 1.000E-5 1.000E-5 1.000E-5 1.000E-5 1.000E-5

5.2 Computational and Communicational Cost

mmethod

Communication cost

Coefficients

BRIAN

DOUGLAS

GSRB

012L( mm )t data + 6L(t stratu p + t idle )12L( mm )t data + 6L(t stratup + t idle )

12L(2

mm )t data + 6L(t stratu p + t idle )

L = number of iterations

mmethod

Multiplications Additions

Coefficients 7 10

Level time p

mm 22 3030

pmm 22 2015

BRIAN p

mm 22 4)1(8 +

pmm 22 10)1(13 +

DOUGLAS p

mm 22 8)1(12 +

pmm 22 10)1(13 +

GSRB pmm 22 14)1(15 +

pmm 22 12)1(13 +

Table 5: Computational Complexity for parallel strategiesof BRIAN, DOUGLAS and GSRB methods Table 4: Communication cost for parallel strategies

of BRIAN, DOUGLAS and GSRB methods


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Table 4 is presented the communication cost for parallel strategies of BRIAN, DOUGLAS and GSRBmethods. Here also its shown that communication costs of BRIAN are lower than DOUGLAS andGRSB methods. Table 5 shows the parallel computational complexity for BRIAN, DOUGLAS andGRSB methods and here its shown that computational complexity of BRIAN are lower thanDOUGLAS and GRSB methods. Parallel performance for speedup, efficiency, temporal performanceand effectiveness versus number of the processors were plotted in Figure 4-13. The parallel execution

time and computation time are decreasing with the increasing processor, p. The GSRB has the lowestrate of convergence and BRIAN method has highest rate of convergence and adaptable forimplementing on three-dimensional coordinate cylinder system.Table 6 shows Analysis on computational and communication cost of one-dimensional parallel BRIAN,DOUGLAS and GSRB methods and table 7 shows the comparison on computational andcommunication cost for different 3-dimensional method and size of matrices. For(100100100) sizesof matrices, BRIAN has better computation to communication ratio compared to DOUGLAS andGSRB. Increasing number of processors significantly reduces the ratio but all the methods thatexperimentally performed, represent the ability in maintaining the condition where time forcomputation is always more than time consumed for communication. This reflects the beneficial abilityof the blends of methods used with parallel algorithm that had been developed [12-13]. Thecomputation to communication ratio became higher for larger size of matrices(140140140) especially when BRIAN method was being used. As more problems need to be solved, each methodresults in higher time consumed for computation rather than communication. The ratio betweencomputation and communication is known as granularity. High granularity reflects that computationalcost dominating the overall execution time. However, too high granularity will lead to loss parallelismcharacteristics where the algorithm developed involved large size of data passing between processors.Thus, best combination of parallel algorithm and method being used will lead to better parallel performance evaluation where there is balance between computation and communication cost.t para is the parallel execution time and its divided into two parts, computational time that is,t comp andcommunication time that is,t comm . In the parallel algorithm,t comp is the time to compute the arithmeticoperations such as addition operations and multiplications. As all the processors doing the operation atthe similar speed, calculation for thet comm is depending upon the size of the message. The cost ofcommunication comes from the two major phases in sending a message: the start-up phase and the datatransmission phase [10]. The total time to send K units of data for a given system can be written as,

t comm = t startup + Kt data + t idle (15)Table 6: Analysis on computational and communication cost of one-dimensional parallel BRIAN,DOUGLAS and GSRB methods

M p parallel comp ratio comm comm1 idle parallel comp ratio comm Comm1 idle

BRIAN DOUGLAS5 11.00 9.600 6.9 1.400 0.885 0.515 11.57 10.06 6.7 1.506 0.885 0.621% 87.3 12.7 8.1 4.7 87.0 13.0 7.7 5.410 5.848 4.724 4.2 1.124 0.885 0.239 6.243 5.006 4.1 1.237 0.885 0.352% 80.8 19.2 15.1 4.1 80.2 19.8 14.2 5.615 4.347 3.183 2.7 1.164 0.885 0.279 4.708 3.338 2.4 1.371 0.885 0.486% 73.2 26.8 20.4 6.4 70.9 29.1 18.8 10.320 3.617 2.387 1.9 1.230 0.885 0.345 3.982 2.503 1.7 1.479 0.885 0.594% 66.0 34.0 24.5 9.5 62.9 37.1 22.2 14.9

GSRB GSRB5 44.10 30.99 2.4 13.11 2.12 10.99 44.10 30.99 2.4 13.11 2.12 10.99% 70.3 29.7 4.8 24.9 70.3 29.7 4.8 24.910 24.88 16.04 1.8 8.831 2.124 6.707 24.88 16.04 1.8 8.831 2.124 6.707% 64.5 35.5 8.5 27.0 64.5 35.5 8.5 27.015 17.19 10.33 1.5 6.856 2.124 4.732 17.19 10.33 1.5 6.856 2.124 4.732% 60.1 39.9 12.4 27.5 60.1 39.9 12.4 27.520 13.90 7.822 1.3 6.078 2.124 3.954 13.90 7.822 1.3 6.078 2.124 3.954% 56.3 43.7 15.3 28.4 56.3 43.7 15.3 28.4

Wheret comm is time needed to communicate a message ofK bytes,t startup is sometimes referred as thenetwork latency time.t startup is also referred to time to send a message with no data. It includes time to pack the message at source and unpack the message at the destination and to start a point-to-pointcommunication.The t data is time to transmit units of information. It is also the transmission time to send one bytes ofdata. Thet startup andt data are assumed as constants and measured in bits/ sec.t idle is the time for messagelatency and time to wait for all the processors to complete the works. The evaluation of thesecommunication costs via simple codes that time the send/recv messages.The research focus on,t para = time for parallel execution andt comm1 = t data + t startup .

where, and dependents onm and L.


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Heret comm1 is the Communication time 1 which is obtained from the subtraction of idle time fromcommunication time.Communication cost for parallel processing is,

Lmt data + L(t startup + t idle ) (16)where,m= units of data that sending across processor

L= number of step overall the execution.Table 7: Analysis on computational and communication cost of parallel BRIAN, DOUGLAS and

GSRB methods for two difference sizes of matrixes M p

100100100 140140140 parallel comp ratio comm comm1 idle parallel comp ratio comm Comm1 idle

BRIAN BRIAN5 68.04 43.27 1.75 24.77 8.84 15.93 174.5 130.2 2.94 44.34 16.57 27.77% 63.6 36.4 13.0 23.4 74.6 25.4 9.5 15.910 39.16 22.13 1.3 17.03 8.843 8.182 97.54 65.08 2 32.46 16.57 15.89% 56.5 43.5 22.6 20.9 66.7 33.3 17.0 16.315 27.16 14.92 1.22 12.24 8.843 3.394 73.16 43.39 1.46 29.77 16.57 13.2% 54.9 45.1 32.6 12.5 59.3 40.7 22.7 18.120 22.3 11.67 1.1 10.63 8.843 1.79 59.83 32.34 1.18 27.49 16.57 10.92% 52.3 47.7 39.7 8.0 54.1 46.0 27.7 18.3

DOUGLAS DOUGLAS5 75.1 44.21 1.43 30.89 9.612 21.28 196.8 142.1 2.6 54.7 18.07 36.63% 58.9 41.1 12.8 28.3 72.2 27.8 9.2 18.610 43.29 23.61 1.2 19.68 9.612 10.07 111.9 71.04 1.74 40.91 18.07 22.83% 54.5 45.5 22.2 23.3 63.5 36.5 16.2 20.415 29.1 15.24 1.1 13.86 9.612 4.25 83.7 47.36 1.3 36.34 18.07 18.26% 52.4 47.6 33.0 14.6 56.6 43.4 21.6 21.820 24.09 12.15 1.02 11.94 9.612 2.325 68.38 35.52 1.08 32.86 18.07 14.78% 50.5 49.6 39.9 9.7 52.0 48.1 26.4 21.6

GSRB GSRB5 99.9 58.59 1.42 41.31 29.43 11.87 193 118.8 1.38 81.25 49.14 32.11% 58.7 41.4 29.5 11.9 57.9 42.1 25.5 16.610 73.01 29.29 0.67 43.71 29.43 14.27 136.2 55.89 0.7 80.34 49.14 31.21% 40.1 59.9 40.3 19.6 41.0 59.0 36.1 22.915 69.11 19.73 0.4 49.27 29.43 19.84 121 36.66 0.43 84.29 49.14 35.15% 28.6 71.4 42.7 28.8 30.3 69.7 40.6 29.120 63.99 15.15 0.31 48.85 29.43 19.42 110.7 27.15 0.32 83.55 49.14 34.41% 23.7 76.3 46.0 30.3 24.5 75.5 44.4 31.1

5.3 Parallel Performance EvaluationIn order to present the effectiveness of the parallel strategies of DOUGLAS and BRIAN methods, wehave implemented the methods on grid computing platform based on the development of a number ofhomogenous PC cluster systems at difference locations, which contains of 38 Intel Pentium IV CPUs,each with a storage of 20GB, 512 RAM and speed of 1.6MHz, supported by PVM software and CProgramme. This software supports sending and receiving data activities between processors [14].Speed up, effectiveness, efficiency, and temporal performance are used to measure the performances of parallel strategies. Here, the execution time isT 1 on one processor, the execution time isT p on p processors and L p is the unit of work done per micro second.

Speed-up ratio, S p = T 1/T p (17)Efficiency, E p = S p /p (18)Effectiveness, F p = S p /C p (19)

Temporal performance, L p = T p -1 (20)

E x e c u t i o n

T i m e

Numb. Of Processor

E f f i c i e n c y

E f f e c t

i v e n e s s

Figure 4: The execution time versus number of processors using (70X70X70) sizes of matrices

Figure 5: The Speed up versus number of processorsusing (70X70X70) sizes of matrices

Numb. Of Processor

Numb. Of Processor

Figure 6: The efficiency versus number of processors using (70X70X70) sizes of matrices

Numb. Of Processor

Figure 7: The effectiveness versus number of processors using (70X70X70) sizes of matrices

Numb. Of ProcessorFigure 8: The temporal performance versus number of processors using (70X70X70) sizes of matrices Figure 9: The execution time versus number of processors for 1 dimensional problem

E x e c u t i o n

T i m e

Numb. Of Processor

T e m p o r a

l P e r

f o r m a n c e

S p e e

d

u p


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Based on Figure 4, execution time for BRIAN method has lowest value than DOUGLAS and GSRB.Execution time is decreasing with the increasing p processors. Figure 5 shows the highest speedup forBRIAN method. Efficiency for BRIAN, DOUGLAS and GSRB are decreasing with the increasing p based on Figure 6. BRIAN method has the higher efficiency than DOUGLAS and GSRB methods.Based on Figure 7, effectiveness for GSRB method are increasing very slowly than BRIAN andDOUGLAS methods where data decomposition is run asynchronously and concurrently at every timelevel with limited communication cost and allows inconsistencies to load balancing. This shows thatGSRB methods are not effective in solving parabolic 3 dimension model problem than BRIAN andDOUGLAS methods.Figure 8 shows that temporal performance for BRIAN method is better than DOUGLAS and GSRB.Figure 9-13 show comparison for performance analysis one dimensional AGE BRIAN and AGE

DOUGLAS. Parallel performances of 1D AGE BRIAN are better than others for speedup andefficiency versus number of processors. 1D AGE BRIAN has better temporary performance andeffectiveness as well than AGE DOUGLAS and GSRB because execution time for small size ofmatrices is smaller than large size of matrices.

6. ConclusionThe stable and highly accurate BRIAN algorithm is found to be well suited in parallel implementationin this study where the data decomposition runs asynchronously and concurrently at every time level.The BRIAN sweeps involve tri-diagonal systems, which require the solution of(22) block systems.Existing parallel strategies could not be fully exploited to solve such systems. The comparison resultsof AGE class of methods presented the best way to choose parallel algorithm for the thermal profile ofmultidimensional cylindrical laser glass interaction that can solve precisely the computation of laser beam passage ensuring pragmatic heat generation pattern in the target material.BRIAN method on the cylindrical coordinate model of laser glass interaction proved that this methodhas a convergence value better than DOUGLAS and GSRB. Comparative results of numericalanalysis and parallel performance evaluations that came out through this study showed that when thesize of dimension is increase, speed-up and efficiency of AGE Brian and AGE Douglas variants alsoincrease. The factors are when communication cost between processors and idle time increase, the levelequilibrium of data storage decrease. BRIAN method, that is alternative method for AGE is proved intheoretically and experimentally in this study for multidimensional problems. Hence, we can declarethat our proposed method will able to produce the best solutions for laser glass interaction problem andthe thermal profile had been successfully visualized by COMSOL multiphysics simulation in sectiontwo (see Figure 1 - 3).The communication frequency, the size of messages and the cluster platforms influences thecommunication time. Consequently, it can be accomplished that the efficiency, speedup ratio, andeffectiveness of the parallel algorithm can be affected by computational complexity and

Figure 10: The Speed up versus number of processorsfor 1 dimensional problem

Numb. Of Processor Numb. Of Processor Numb. Of Processor

E f f i c i e n c y

Figure 11: The efficiency versus number of processorsfor 1 dimensional problem

Figure 12: The effectiveness versus number o f processors for 1 dimensional problem

T e m p o r a

l P e r f o r m a n c e

Numb. Of Processor

Figure 13: The temporal performance versus numberof processors for 1 dimensional problem

E f f e c

t i v e n e s s

S p e e

d u p


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communication cost. The fully multidimensional primal-mixed finite element approach in steady-stateand thermal control analysis of the solid body is examined.

AcknowledgementThe authors wish to express their gratitude and indebtedness to the Technology University of Malaysiaand the Malaysian government for providing the moral and financial support under the e-Science

Research grants (Grant No. 72919) forth-successful completion of this project.References

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parallel age solver of multidimensional pde modeling for thermal control of laser beam on...

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