2011 Aplicado a Radiacion

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Numerical simulations of a coupled radiativeconductive heat transfer model

using a modified Monte Carlo method

Andrey E. Kovtanyuk a,b, Nikolai D. Botkin c,, Karl-Heinz Hoffmann c

a Institute of Appl. Math. FEB RAS, Radio St. 7, 690041 Vladivostok, Russiab Far Eastern Federal University, Sukhanova St. 8, 690950 Vladivostok, Russiac Technische Universitt Mnchen, Zentrum Mathematik, Boltzmannstr. 3, D-85747 Garching b. Mnchen, Germany

a r t i c l e i n f o

Article history:

Received 1 March 2011

Received in revised form 18 October 2011

Accepted 24 October 2011

Available online 22 November 2011

Keywords:

Radiative heat transfer

Conductive heat transfer

Monte Carlo method

Diffusion approximation

a b s t r a c t

Radiativeconductive heat transfer in a medium bounded by two reflecting and radiating plane surfaces

is considered. This process is described by a nonlinear system of two differential equations: an equation

of the radiative heat transfer and an equation of the conductive heat exchange. The problem is character-

ized by anisotropic scattering of the medium and by specularly and diffusely reflecting boundaries. For

the computation of solutions of this problem, two approaches based on iterative techniques are consid-

ered. First, a recursivealgorithm based on some modification of theMonteCarlomethod is proposed. Sec-

ond, the diffusion approximation of the radiative transfer equation is utilized. Numerical comparisons of

the approaches proposed are given in the case of isotropic scattering.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The study of the coupled heat transfer [13] where the radiative

and conductive contributions are simultaneously taken into ac-

count is important for many engineering applications. So, Andre

and Degiovanni[4,5], Banoczi and Kelley[6], and Klar and Siedow

[7]have studied the thermal properties of some semi-transparent

and insulating materials in the context of a coupled radiativecon-

ductive model. The mathematical treatment of this nonlinear mod-

el is studied in[811]. In[8], Siewert and Thomas use the simple

iteration method and a computationally stable version of the PNapproximation. In work[9],Siewert has applied the Newton itera-

tion method instead of the simple iteration procedure. This allows

the author to calculate some numerical examples which are not

feasible using the simple iteration method (compare [8]). Kelley

has provided existence and uniqueness theorems for the consid-ered problem in the case of isotropic scattering and non-reflecting

boundaries [10]. An analytical version of the discrete-ordinates

method along with Hermites cubic splines and Newtons method

to solve a class of coupled nonlinear radiationconduction heat

transfer problems in a solid cylinder is proposed in [11]. The algo-

rithm is implemented to establish high-quality results for various

data sets which include some difficult cases.

In our paper, some iterative algorithm for solving this problem

is considered. For the calculation of solutions of the radiative trans-

fer equation, two ways are used. The first approach proposed by

the authors utilizes a recursive algorithm based on some modifica-

tion of the Monte Carlo method. This algorithm suits for the appli-

cation of parallel calculations, and hence it can provide a good

accuracy within a reasonable computing time. The second ap-

proach uses the diffusion approximation of the radiative transfer

equation. It is shown that using this approximation gives a good

description of the solution behavior. A numerical comparison of

the approaches proposed is done in the case of isotropic scattering

and reflecting boundaries. The calculations are implemented on a

computer cluster of the Technical University of Munich using the

technology of parallel computing supported by the application pro-

gramming interface OpenMP.

2. Problem formulation

Let us consider the coupled radiativeconductive heat transfer

problem which is formulated as in[8,9]. The equation of the radi-

ation transfer for a homogenous layer is written in the normalized

form as

lIss;l Is;l x2

Z 11pl;l0Is;l0dl0 1xH4s; 1

whereI(s,l) is the normalized density of the radiation flux at thepoint s e[0,s0] in the direction which angle cosine with the positive

0017-9310/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2011.10.045

Corresponding author.

E-mail addresses: [email protected] (A.E. Kovtanyuk), [email protected]

(N.D. Botkin),[email protected](K.-H. Hoffmann).

International Journal of Heat and Mass Transfer 55 (2012) 649654

Contents lists available atSciVerse ScienceDirect

International Journal of Heat and Mass Transfer

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direction of the axiss isl e[1,1];x < 1 the albedo of single scat-tering;p(l,l0) the phase function;H(s) the normalize temperature.Note that the case of non absorbing media (x = 1) is excluded fromthe consideration as unrealistic. Introduce the following sets for the

definition of boundary conditions:

C f0g 0;1 [ fs0g 1; 0;C f0g 1;0 [ fs0g 0;1:We supply Eq.(1)with the boundary conditions

In;l hn BIn;l; n;l 2C; 2

where the functionh and the operator B are defined by

h0 :e1H41; Bf0;l :

qs1I0;l 2qd1Z 10

I0;l0l0dl0; l >0;

hs0 :e2H42; Bfs0;l :

qs2Is0;l 2qd2Z 10

Is0;l0l0dl0; l< 0:

Here, H1 and H2 are the normalized temperatures on the bound-

aries;qsi andqdi the coefficients of specular and diffuse reflections,

respectively; ei 1 qsi qdi the emissivity coefficients for the

boundary surfaces. It is assumed that e1, e2> 0, which provides the

estimate kBk < 1 (see Section 3). Note that the first summand onthe right-hand side of the definition of the operator B describes

the contribution of the specular reflection, whereas the second

one describes the contribution of the diffuse reflection.

The equation of the conductive heat transfer is written as

H00s 12Nc

Z 11Is;lldl

0; 3

andNcis the conduction-to-radiation parameter[8]. For Eq.(3), we

set the following boundary conditions:

H0 H1; Hs0 H2: 4For finding the solution of system(1)(4), we will use a simple

iteration method with parameter. According to that, choose an

initial approximation of the temperature H(s) (for example, thelinear approximation which corresponds to zero value of the

right-hand side of (3)) and denote it as Hh0is. Then, substituteHh0is into(1) instead of the function H(s), find the solution ofthe problem(1) and (2), and denote it as Ih1is;l. Then, find thesolution of the problem (3) and (4) under the given function

Ih1is;l and denote it as ~Hh1is. Choose a small positive real aand set Hh1is a ~Hh1is 1 aHh0is to be the next approxi-mation ofH(s). Then, putHh1isinstead of the function H(s) intoEq. (1), find the next approximation Ih2is;l, and so on. Thus, inthejth step, we use the functions Hhj1isand ~Hhjisto determinethe next approximation of the function H(s) by the followingformula:

Hhj

is a~

Hhj

is 1 aHhj

1

is: 5The main complexity in the numerical realization of this itera-

tive method is finding the solution of the radiative transfer Eq.

(1). For its treatment, we will mainly use a recursive algorithm

based on the Monte Carlo method. As alternative, we will construct

a diffusion approximation of Eq. (1)(P1 approximation). We will

compare the results of these approaches with the numerical data

from[8,9].

3. Solvability of the radiative transfer equation

Let us consider the problem (1) and (2). We assume that the

function H(s) is nonnegative, and H(s) eCb(0, s0), where Cb(X) isthe Banach space of functions bounded and continuous on Xwith

the normkukCbX supx2X

juxj. Also, letpl;l0 2CbX X, where

X 1; 0 [ 0; 1, and

1

2

Z 11pl;l0dl01:

Note that the operatorB : CbC ! CbC

is linear, bounded, non-

negative, and

kBk maxi

qsi qdi< 1:

DenoteX 0; s0 1; 0 [ 0; 1. We define a class D(X) wheresolutions Iof the problem(1) and (2)are sought.

A function I(s, l) belongs to D(X), if the following properties hold:

(1) I(s, l) is absolutely continuous in se

(0, s0] for alll > 0, andabsolutely continuous ins e[0, s0) for alll < 0;

Nomenclature

A an integral operatorB operator of reflectionCb Banach space of bounded and continuous functionsD a functional classI normalized density of the radiation flux

Ihji radiation flux in the jth step of the iterative procedureIn radiation flux in thenth step of the recursive procedureh input radiation fluxL a linear operatorM number of recursive trajectoriesN number of summands of the truncated Neumann seriesNc conduction-to-radiation parameterp phase functionS an integral operatorT operator of the Neumann seriesX the set of optical and angular variables

Greek symbolsa iteration parameterC a set used in the definition of boundary conditions

C+ a set used in the definition of boundary conditionsH1 normalized temperature on the left boundaryH2 normalized temperature on the right boundaryH normalized temperatureHhji temperature in the jth step of the iterative procedure

e1 emissivity coefficient of the left boundarye2 emissivity coefficient of the right boundaryx albedo of single scatteringl angular variables optical depth (point of the layer)s0 optical thickness of the layern boundary pointqd1 coefficient of diffuse reflection of the left boundary

qd2 coefficient of diffuse reflection of the right boundary

qs1 coefficient of specular reflection of the left boundary

qs2 coefficient of specular reflection of the right boundary

/0 diffuse approximation of the average flux

650 A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654


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(2)lIs0(s, l) + I(s, l) eCb(X);(3) I(s, l) eCb(C

) .

For further purposes, we introduce the following function

nl 0; l 2 0;1;s0; l 2 1;0:

The differential expression LIs;l lIss;l Is;l defines alinear operator L: D(X)? Cb(X). In the space D(X), we introducethe norm

kukDmaxfkukCbC; kLukCbXgand notice that the inclusion D(X) Cb(X) holds.

The expressions

Aus;l 1l

Z snlexp s s

0

l

us0;lds0;

Sus;l x2

Z 11pl;l0us;l0dl0;

TIs;l BInl;lexp s

n

ll ASIs;l 6

define linear operators A: Cb(X)?D(X), S: Cb(X)? Cb(X), and

T: D(X)?D(X).

According to[12], the following statements hold:

Theorem 1. A function I is a solution of the problem (1) and (2),iff it

is a solution of the operator equation

Is;l I0s;l TIs;l;

I0s;l exp snll

hnl 1xAH4s;l; 7

in the class DX.

Theorem 2. Assuming that the inequalities ||B|| < 1 and x < 1 hold,there exists a unique solution of the problem (1) and (2) (or of the

integralEq.(7))that can be found in the form of the Neumann series

Is;l X1k0

TkI0s;l 8

converging in the norm of CbX.

Remember that kBk < 1 in our case, and therefore the condi-

tions ofTheorem 2are satisfied.

4. Recursive algorithm based on the Monte Carlo method

Let us consider the iterative algorithm described in Section 2.

For computing a solution to the problem (1) and (2)corresponding

to a given function H(s), we propose a recursive algorithm basedon the Monte Carlo method. According toTheorem 2, there exists

a unique solution of the problems (1) and (2) that can be found

in the form of the Neumann series (8). The Monte Carlo method

is appropriate for computing the finite sums

INs;l XNn0

TnI0s;l: 9

To implement the computation, rewrite(9)as the following recur-

rence relation:

Ins;l TIn1s;l I0s;l; n1;2; . . . ;N:

Let us consider a structure of the operatorT(see Eq.(6)). It containstwo summands: the first one describes reflection effects, the second

one describes the contribution of scattering effects. Consider the

second summand in more detail. Applying simple transformations,

we rewrite it in the form

Js;l : ASIs;l x2 1exp sn

l

Z s

n Z 1

1

exps s0=ll1

exp

s

n

=l

pl;l0Is0;l0dl0ds0;

10where n= n(l). According to the Monte Carlo method, we canapproximate the integral in this expression as the mean value of a

random sequence defined by the random variables s0 andl0 distrib-uted over the intervals (n, s) and (1, 1) with the densities

exps s0=ll1expsn=l ; and

1

2pl;l0; 11

respectively. Therefore, the integral (10) is being approximated

with the following finite sum:

Js;l xM

1 exp s nll

XM

k

1

Izk;lk:

Here,zk,lk,k = 1, 2, . . . , Mare independent realizations of the ran-dom variables s0 andl0 distributed over the intervals (n, s) and(1, 1) with the densities (11). Hence, we can approximate the

functions In(s, l),n = 1, 2, . . . , N, as follows:

Ins;l Ins;l 1M

XMk1sn;ks;l; I0s;l I0s;l; 12

sn;ks;l BIn1nl;lexp snll

x 1 exp snll

In1zk;lk I0s;l: 13

Thus, the finite sum (9)can be calculated using the recurrence rela-

tions(12) and (13).For computing the second summand of the right-hand side of

recursive relation (13), we have to generate points zk and lk,k= 1, . . . , M, distributed on the intervals (n(l), s) and (1, 1),respectively, with the densities given by (11), respectively. The

points zk are defined as follows:

zks lln1 ak1exps n=l;whereak are independent realizations of a random variable uni-formly distributed on the interval (0, 1). The generation of the angu-

lar values,lk, is governed by the phase function p(l, l0). Thus,

lk2ak1in the case of isotropic scattering, and

lkl11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 l2 4lakq

in the case where p(l, l0) = 1 + ll0 (binomial scattering law, see[13]).

For computing the term BIn1 appearing in the first summand of

the right-hand side of(13), the angular variable involving in the

definition of the specular part of the operatorB (see the definition

ofB and the remarks to its structure) remains deterministic and

equals tol. The diffusive part ofB is being computed using ran-dom values generated as

l0k sgnlffiffiffiffiffiak

p ;

whereak are independent realizations of a random variable uni-formly distributed on the interval (0, 1) .

It should be noted that the above recursive algorithm based onthe Monte Carlo method is suitable for the utilization of parallel

A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654 651


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computing technologies. There are two basic ways for the parallel-

ization of the computing process. First, the calculation of the func-

tionIat each point of the layer is performed by a separate thread.

Second, the generation of each recursive trajectory of the Monte

Carlo method is performed by a separate thread. Thus, the algo-

rithm proposed is aimed to multiprocessor systems and moderne

grid computing.

Note, that the method of discrete ordinates, which is rather

popular in the one-dimensional case, shows a restricted paralleliz-

ability so that it is hardly applicable in three dimensions.

5. Implementations of the iterative method

In this section, different approaches to the numerical solution of

the coupled problem(1)(4)are considered. Section5.1proposes

two kinds of recursive relations based of the Monte Carlo method.

The first one is valid in the general case of anisotropic scattering,

and the second one is applicable to the case of isotropic scattering

only. In Section5.2, a model based on the diffusion approximation

of the radiation transfer equation is derived in the case of isotropic

scattering.

5.1. Recursive relations based on the Monte Carlo method for the

coupled heat transfer problem

For more stable numerical implementation of the iterative pro-

cedure described in Section2, we express the solution of the prob-

lem(3) and (4)for a given function I. After integrating Eq.(3), we

obtain

Hs 12Nc

Z s0

Z 11If;lldldfC1sC2; 14

Here the constantsC1 andC2are defined from the boundary condi-

tions(4):

C1s10 H2H1 1

2Nc

Z s0

0

Z 1

1If;lldldf

; C2H1:

Assume that the approximation Hhj1is of the temperature is al-ready obtained in the (j1)th step of the iterative procedure of

Section 2. On the basis of(5) and (14), using the Monte Carlo meth-

od, we obtain the following approximation of the temperature in

thejth step:

Hhjis 1 aHhj1is a sMNc

XMk1

lkIhjiNzk;lk C1s C2

!;

15

where the random variables zk andlk are uniformly distributedover the intervals (0, s) and (1, 1), respectively. The computationof the I

hjiNs;l is implemented on the base of formulas(12) and

(13)assuming that the temperature is equal to Hhj1is. The con-stant C1 is also computed on the base of the Monte Carlo method.

Thus, the approximationHhjis of the function Hhjis is computedby formulas(12), (13), and (15).

The next approach, which is considered under assumptions of

isotropic scattering, is based on different analytical representations

of solutions of the coupled heat transfer problem(1)(4). Accord-

ing to[10], we obtain from Eq. (1):

Z 11Is;lldl

021x H4s 1

2

Z 11Is;ldl

:

After substitution of this expression into (3)and the integration, weobtain

Hs 1 xNc

Z s0

Z f0

H4x 12

Z 11Ix;ldl

dxdfC1s C2:

16The constantsC1 andC2 are determined from the boundary condi-

tions(4)as follows:

C1s

10 H

2H

11

x

NcZ s00

Z f0

H4

x

1

2Z 1

1Ix;ldl

dxdf ;C2H1:On the basis of(5) and (16), using the Monte Carlo method, we ob-

tain the following approximation of the temperature in thejth step

of the iterative procedure:

Hhjis 1 aHhj1is

a s1 xMNc

XMk1xk H

hj1izk 4 IhjiNzk;lk C1sC2

!;

17

wherexk,zk,lkare numerical realizations of random variables uni-

formly distributed on the intervals (0, s), (0,xk) and (1, 1), respec-tively. The computation ofI

hjiNs;l is implemented on the base of

formulas(12) and (13) assuming that the temperature is equal to

Hhj1is. The constantC1is also computed on the base of the MonteCarlo method. Thus, the approximation Hhjis of the functionHhjis is computed using formulas(12), (13), and (17).

Thus, two kinds of recursive relations based on the Monte Carlo

method are proposed in this subsection. The proposed approaches

allow us to avoid the instability that occurs due to differentiating

in the right-hand side of Eq. (3).

5.2. Diffusion approximation for the coupled heat transfer problem

Now we consider an approach based on the diffusion approxi-

mation (also named P1 approximation, see [14]). Note that thismethod is mostly applicable in the case of isotropic scattering.

We represent the function I(s, l) by the two first summands inthe Fourier expansion in Legendre polynomials:

Is;l /0s l/1s: 18It gives us the following approximation of Eq.(1):

/000s 31x/0s 31xH4s; 19There are different approaches to the derivation of the boundary

conditions for the diffusion approximation(19). Exemplary discus-

sions of this issue can be found in book [14]. In the present work, we

choose the following two ways. In the first way, we substitute the

expansion (18) into the boundary conditions (2) instead of the func-

tion I(s, l) and integrate(2) over all incoming directions l of thelayer. This yields

e1/00 1

2 1 qs1

4

3qd1

/000 e1H41; 20

e2/0s0 1

2 1 qs2

4

3qd2

/00s0 e2H42: 21

In the second way, we use the Marshak boundary conditions[15]. It

gives

e1/00 2

31 qs1qd1/000 e1H41; 22

e2/0s0 2

3 1 qs2qd2/00s0 e2H

42: 23



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With new notations, Eq. (3)is rewritten as follows:

H00s r/000s; r 1

3Nc: 24

Note that the function /0s is interpreted here as the functionI(s, l) averaged over all directionsl .

From Eq.(24),we obtain

Hs r/0s C1sC2: 25The constantsC1 andC2 are defined from the boundary conditions

(4)as follows:

C1s10H2H1r/0s0 /00; C2H1r/00:Thus, the coupled problem (1)(4) is reduced to the coupled system

of Eqs.(19), (20), (21), and (25) (or(19), (22), (23), and (25)).

In the next section, we present the results of numerical experi-

ments based on the above proposed approaches.

6. Numerical experiments

Numerical experiments are carried out for two problems con-

sidered by Siewert and Thomas[8]and Siewert[9]where the sim-ple iteration procedure [8] and Newtons iteration method [9]

combined with a computationally stable version of thePNapprox-

imation have been used. In both cases, the following values of

parameters are taken:x 0:9, s0= 3, H1= 1, qs1 0:1, qd1 0:2,

e1 0:7, H2= 0.5,qs2 0:3, qd2 0:1, and e2 = 0.6. The difference

between the two considered problems consists in the value of

the conduction-to-radiation parameter Nc. The calculations are

implemented forNcequals 0.05 and 0.00001. The last value ofNccorresponds to the case of high temperatures.

Fig. 1 presents the following approximations of the temperature

H(s) (Nc= 0.05): first, computed on the basis of the Monte Carlorecursive algorithm(12), (13), and (15); second, computed on the

basis of the diffusion approximation (19), (20), (21), and (25);

and third, obtained by Siewert and Thomas[8]. For the implemen-tation of the Monte Carlo method, the following values are taken:

the number of the summands of the Neumann series, N= 14; the

number of the generated trajectories, M= 10000. The diffusion

approximation is implemented with the Maple 9.5. For both ap-

proaches, 20 steps of the iterative algorithm are used. The param-

eter a of the iteration method described in Section 2 is chosen to beequal 0.5. It is seen all approximations are close enough to each

other. We do not demonstrate results for the diffusion approxima-

tion with the Marshak boundary conditions, because the corre-

sponding plots are visually indistinguishable from those for the

boundary conditions(20) and (21).

Fig. 2 presents the following approximations of the temperature

H(s) for Nc = 0.00001 (this corresponds to higher temperaturescompared with the previous case): first, computed on the basis

of the Monte Carlo recursive algorithm (12), (13), and (15); second,

computed on the basis of the diffusion approximations (19), (20),

(21), and (25); third, computed on the basis of the diffusion

approximation corresponding to (19), (22), (23), and (25); and

fourth, obtained by Siewert [9]. In the implementation of the

numerical method, 500 steps of the iterative procedure are used.

The parameter a of the iteration method is chosen to be equal0.0001. It is seen that the deviation of the temperature curves is

more essential than in Fig. 1. Nevertheless, the diffusion approxi-

mation describes the behavior of the temperature properly. Thus,

it can be successfully applied to various heat transfer problems

which are not require obtaining very high accuracy.

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Normalizedtemperature

Optical thickness

Fig. 1. Results of numerical simulation for Nc= 0.05: the iterative algorithm based

on the Monte Carlo method after 20 iteration steps (solid curve); the diffusion

approximation based on the boundary conditions (20) and (21) (dashed curve); anddata from[8](squares).

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Normalizedtemperature

Optical thickness

Fig. 2. Results of numerical simulation for Nc= 0.00001: the iterative algorithm

based on the Monte Carlo method after 500 iteration steps (solid curve); the

diffusion approximation based on the boundary conditions(20) and (21) (dashed

curve); the diffusion approximation based on the Marshak boundary conditions

(dotted curve); and data from[9](squares).

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Normalizedtempera

ture

Optical thickness

Fig. 3. Numerical experiments, Nc= 0.00001, demonstrating a convergence of the

iterative procedure based on the Monte Carlo method. The plots correspond to 50

steps (dotted curve), 150 steps (dashed curve), and 500 steps (solid curve) of theiterative procedure.

A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654 653


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Fig. 3shows numerical experiments that demonstrate the con-

vergence of the iterative procedure based on the Monte Carlo

method whenNc= 0.00001. The plots correspond to 50 steps, 150

steps and 500 steps of the iterative procedure.

Fig. 4 shows numerical experiments (Nc= 0.00001) that demon-

strate an instability of the iterative procedure based on the Monte

Carlo method. This instability occurs in the case of insufficient

number of the trajectories, M= 2000. The plots correspond to 300

and 900 steps of the iterative procedure. A similar effect is ob-

served in the case of utilizing the diffusion approximation when

few decimal places were used in the computation.

The presented calculations are implemented on a computer

cluster of the Technical University of Munich using the technology

of parallel computing supported by the application programminginterface OpenMP.

7. Conclusion

This paper proposes a modified Monte Carlo algorithm for the

numerical treatment of nonlinear coupled radiativeconductive

heat transfer problems. Compared with PN approximations, the

algorithm proposed allows us to obtain more precise results, be-

cause it deals with the exact model, whereas PN approximations

utilize simplified equations. Compared with the method of discrete

ordinates, the modified Monte Carlo algorithm is well appropriate

for parallelization, because trajectories can be randomly generated

independently on each other, and additionally parallelization over

points of the layer in which the normalized temperature is calcu-

lated can easily be implemented. The potential of parallelization

can be recognized from the second test example given in this pa-

per. Here, the computation of the temperature in each of 20 points

is based on 104 randomly generated trajectories. Therefore, there

are 2 105 independently computable blocks. Thus, the develop-

ment of multiprocessor systems will provide the permanently

growing speedup of the modified Monte Carlo algorithm so that

it expects to show a good performance in complicated cases, in

particular, for thee dimensional problems.

Acknowledgements

This publication was supported in part by the German Aca-

demic Exchange Service (DAAD); German Research Society (DFG),

SPP 1253; Award No. KSA-C0069/UK-C0020, made by King Abdul-

lah University of Science and Technology (KAUST); and Ministry of

Education and Science of Russian Federation (state contracts

14.740.11.0289, 14.740.11.1000, 16.740.11.0456, 07.514.11.4013).

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0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Normalize

dtemperature

Optical thickness

Fig. 4. Numerical experiments, Nc = 0.00001, demonstrating instability of the

iterative procedure based on Monte Carlo method. This instability occurs in the

case of insufficient number of trajectories. The plots correspond to 300 steps (solid

curve) and 900 steps (dashed curve) of the iterative procedure.


2011 Aplicado a Radiacion

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