a consistent interpolation function for the solution of radiative transfer on triangular meshes....

A CONSISTENT INTERPOLATION FUNCTIONFOR THE SOLUTION OF RADIATIVE TRANSFERON TRIANGULAR MESHES. I—COMPREHENSIVEFORMULATION

Daniel R. Rousse1 and Fatmir Asllanaj21Department of Mechanical Engineering, Ecole de Technologie Superieure,Montreal, Quebec, Canada2LEMTA, Nancy-Universite, CNRS, Nancy, France

This article presents a first-order skewed upwinding procedure for application to discretization

numerical methods in the context of radiative transfer involving gray participating media. This

scheme: (1) yields fast convergence of the algorithm; (2) inherently precludes the possibility of

computing negative coefficients to the discretized algebraic equations; (3) reduces false scatter-

ing (diffusion); (4) is relatively insensitive to grid orientation; and (5) produces solutions

completely free from undesirable oscillations. Theses attributes render the scheme attractive,

especially in the context of combined modes of heat transfer and fluid flow problems for which

computational time is a major concern. The suggested scheme has been validated by application

to several basic test problems discussed in a companion article.

1. INTRODUCTION

One of the key issues of the numerical methods that solve the radiative transferequation (RTE) is the closure relation or spatial interpolation function for thediscrete dependent variable intensity over the elements (or over control volumes)of the computational mesh [1]. This is then relevant for discrete ordinates methods(DOMs), finite-volume methods (FVMs), control-volume finite-element methods(CVFEMs), and finite-element methods (FEMs).

Here the discussion pertains to such discretization methods and is notconcerned with traditional zonal, Monte Carlo, ray tracing, discrete transfermethods [2], or the more recently acknowledged collapsed dimension [3–5] and directcollocation meshless methods [6, 7].

Received 15 July 2010; accepted 23 October 2010.

The authors gratefully acknowledge Rodolphe Vaillon (CNRS researcher, CETHIL, INSA de

Lyon), Guillaume Gautier (Eng, PSA Group), and Mathieu Francoeur (Researcher, University of

Kentucky) for their participation, collaboration, and=or comments during these research works. The

authors are also grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC).

Thanks go to Alan Wright for helping in the preparation of this manuscript.

Address correspondence to Daniel R. Rousse, Department of Mechanical Engineering, Ecole de

Technologie Superieure, Montreal, Quebec, Canada. E-mail: [email protected]

Numerical Heat Transfer, Part B, 59: 97–115, 2011

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790.2010.541352

97

A first interpolation scheme was introduced by Carlson and Lathrop [8] in thecontext of neutron transport problems: It is the diamond difference scheme. This schemewas the first to be implemented by Hyde and Truelove [9] and later by Fiveland [10] intheir pioneering radiative heat transfer work. It has since been used extensively.However, as for the central difference scheme used in the context of convection-diffusionand fluid flow problems, the diamond scheme can produce negative coefficients (elemen-tal negative intensities) in the discretized algebraic equations that approximate the orig-inal RTE. This procedure may lead to spatially oscillating, physically unrealisticdistributions of the transported entity (intensity). Fix-up procedures have been proposedin attempts to solve this problem. All of them result in schemes that require morecomputational time per iteration than their original counterpart.

Decades later, Coelho and Aelenei [11] tested several high-order schemes,namely, the MINMOD, CLAM, MUSCL, and SMART schemes. They found thatthe use of high-order schemes is only effective if the angular refinement yieldslow-angular discretization error as well. In this respect, the study of Chenget al. [12] proposes a genuine method (called DRESDOR) that accounts forradiation in 6,658 discrete directions in the hemispheric space to model the angulardiscretization appropriately.

Chai and co-workers [13], after testing several procedures, finallyrecommended relying on the upwind scheme (US). In that article, the upwind scheme(also called step scheme), the diamond scheme, the positive scheme proposed byLathrop [14], and the positive intensity conditions suggested by Fiveland andJessee [15] were discussed in the context of the discrete ordinates method. The articlealso discussed the relative merits of variable-weight schemes proposed by Jamaluddinand Smith [16] but indicated that, in general, schemes based on weighting factor lessthan unity could lead to physically unrealistic solutions. Liu et al. [17] showed that thecriterion for an unconditionally stable scheme is that this weighting function should

NOMENCLATURE

fþ, f�, w weighting functions

Gmn geometric quantity, sr

~JJ/ flux of / (radiative) per unit

solid angle, W=m2 � srM midpoints of element edges~nn unit vector normal to a surface

N number of panels defining a

control volume; node of an

element

O centroid of an element

p control volume surface (panel)

SI source of radiative intensity

S/ rate of volumetric generation

of // Dependent variable

Subscripts

b blackbody

B boundary

l element edge

m discrete direction

n control-volume surface

(panel)

P node of reference

r radiation or reference

point

W integer function: W(n)¼ n,

8n¼ 1, 2, 3; and W(n)¼ n� 3,

8n¼ 4, 5

Superscripts

m discrete direction0 incoming direction! vectorial quantity

Other symbols: ASME, JHT 1999, 121(4),

770–773.

98 D. R. ROUSSE AND F. ASLLANAJ

be superior to 2=3. Berour and co-workers [18] proposed a review of several ofthese differencing scheme efficiencies for the case of strong opacities in purelyabsorbing media.

One of the alternative recommended by Chai and co-workers [13] was to tracethe downstream intensities at an integration point p to an upstream referencelocation r where the intensity can be computed or is known in terms of nodal values.This is the basis of the original CVFEM proposed by Rousse and Baliga [19] andRousse [1, 20]. Although use of unidirectional upwinding removes the potentialfor spatial oscillations, such a procedure is burdened with excessive false diffusion(false scattering).

The smearing here is numerically (not physically) analogous to what is referredto as false diffusion in the context of fluid flow and convective heat transfer [21]. Inthis respect, numerical smearing can be referred to as false scattering, which is anumerical redistribution of energy rather than a physical phenomenon. Hence, theauthors acknowledge the need for a scheme that models the skewness of radiativetransport with improved accuracy. This has been done over a period of some 10years, as exemplified by the articles of Chai et al. [13], Tan et al. [22], and Coelho [23].

In the article of Jessee and Fiveland [24], the authors addressed the issue ofspatial discretization in the context of numerical smearing. In that article, theauthors stated that the upwind scheme is computationally inexpensive butfirst-order-accurate, thus leading to poor modeling and numerical scattering, andthat the diamond scheme modifications, required by the presence of negativecoefficients, are not entirely satisfactory. To overcome these drawbacks, the authorsinvestigated bounded high-resolution schemes. They concluded that a boundedexponential or higher-order scheme, with built-in flux limiters in which skewness isaccounted for, would be promising. However, the nonlinearity of the proposedhigh-resolution schemes often necessitates an increase in computational time. Itwas reported that these high-resolution schemes may be competitive with the stan-dard upwind scheme but for significantly scattering media or reflective boundariesand when the problem may allow the analyst to neglect other heat transfer modes.

Liu et al. [17] came to similar conclusions in their analyses of the conventionaldifference schemes and the SMART scheme. However, the authors recommendedthe use of the central difference scheme even though this scheme may produce spu-rious oscillations, because it yielded integral quantities (flux, incident radiant energy)almost as accurate as the SMART scheme.

The avenue taken here is somewhat different; instead of using a high-orderscheme, a first-order skew upwind scheme is proposed. It will be shown that theproposed scheme can be viewed as a skewed upwind scheme that ensures positivecoefficient to the discretized algebraic equations while limiting the connectivity toa single finite element.

In the context of convection-diffusion and fluid flow problems, false diffusioncan be substantially reduced through the use of a skewed upwind scheme such as thatproposed by Raithby [25]. However, this procedure has the potential for developingspatial oscillations in the solution through the computation of negative coefficientsin the discretized equations. Leonard [26] proposed a quadratic upstream inter-polation scheme that considerably reduces the problem. However, his scheme doesnot entirely circumvent the calculation of negative coefficients. Hassan et al. [27]

RADIATIVE TRANSFER ON TRIANGULAR MESHES. I—FORMULATION 99

proposed a procedure that solves the problem by restricting the range of theupstream weighting factor such that the influence coefficients cannot becomenegative. In some sense, this procedure, based on a mathematical restriction onthe upstream weighting factor, is akin to a similar ideas developed for the intensityof radiation calculation with the diamond scheme [14].

Conventional Galerkin finite-element methods for convection-diffusionproblems experience similar difficulties [28], and although several upwind typeschemes have been suggested [29–32], they all more or less suffer from falsediffusion.

Clearly, in the context of convective transport, a broad range of numericalprocedures have been proposed in attempts to reduce false diffusion and=or elimin-ate the negative coefficients problem. Among these, the skewed positive influencecoefficient upwinding procedure proposed by Schneider and Raw [33], modified bySaabas [34] and by Rousse [20], was found to hold the premise of a suitable interp-olation function for radiation intensity in the context of problems involving radiativeheat transfer in participating media.

The present article is concerned with the progressive development of aninterpolation function for radiative intensity that does account for the directionalityof the radiant energy propagation through a skewed approach, while simultaneouslyprecluding the possibility of negative coefficients.

2. CVFEM FORMULATION

The solution of the radiative transfer equation (RTE) by a CVFEM requiresthe discretizations of both spatial and angular domains. The angular discretizationof the RTE, which is solved along M discrete directions, leads to the solution ofM sets of algebraic discretization equations [1].

2.1. Radiative Transfer Equation

For the sake of completeness, the multidimensional propagation of radiation ingray-diffuse enclosures filled with gray participating media is recalled here; it can bedescribed by the following equation:

~rr � ~XXIðs; ~XXÞ� �

¼ �bIðs; ~XXÞ þ SI ðs; ~XXÞ ð1Þ

where SI, the source function for radiant intensity, is given by

SI ðs; ~XXÞ ¼ jIbðs; ~XXÞ þr4p

Z4pIðs; ~XX0ÞUð~XX0; ~XXÞ dx0 ð2Þ

and the radiative boundary condition at a point B on a gray-diffuse surface is

IBð~XXÞ ¼ eBIbBðTBÞ þð1� eBÞ

p

Zð~XX0 �~nnBÞ<0

~XX0 �~nnB�� IBð~XX0Þ dx0 ð3Þ


In the formulation of CVFEMs, it is convenient to cast the governing partialdifferential equations in the following general form [21]:

~rr �~JJ/ ¼ S/ ð4Þ

where / stands for a general variable, S/ is a volumetric generation rate or sourceterm, and~JJ/ is the flux of /. Equation (1) may be readily obtained from this generalequation by using ~JJ/ ¼ ~XXIðs; ~XXÞ and S/ ¼ �bIðs; ~XXÞ þ SI ðs; ~XXÞ.

2.2. Domain Discretization

In CVFEMs, the calculation domain is first spatially divided into elements. In asecond step of discretization, each element is divided into subcontrol volumes in sucha manner that, upon assembly of elements, complete control volumes are formedaround each node of the computational mesh. In two-dimensional andthree-dimensional formulations, three-node triangular and four-node tetrahedralelements are used, respectively.

With regard to angular discretization, discrete ordinates-type or azimuthaldiscretizations can be used. Discretization details are provided elsewhere [1].

It should be noted that FVMs and FEMs also need spatial and directionaldiscretizations.

2.3. CVFEM Approximation

An integral conservation equation corresponding to the RTE is obtained byapplying the conservation principle to control volumesV and solid angles xm such that

Zxm

ZA

~JJ/ �~nn dAdx ¼Zxm

ZV

S/ dV dx ð5Þ

where A is the surface area of the control volume and ~nn is a unit outward-pointingnormal to the differential area element dA.

In the suggested CVFEM, the radiative properties and the scattering phasefunction are nodal values and are assumed to prevail over the control volume asso-ciated with this point and over a discrete solid angle xm, while the source term S/

is linearized [21] and assumed constant over control volumes and solid angles.The radiant heat flux is assumed constant (average values are assumed) overcontrol-volume surfaces and solid angles [1].

The discretized integral conservation equation, corresponding to Eq. (5), in thedirection ~XXm for a control volume associated with node P and having N controlvolume faces, is finally,

XNn¼1

ImpnGmn An ¼ �bPI

mP VPxm þ Sm

IPVPxm ð6Þ

where ImPnis the radiative intensity evaluated on pn along ~XXm; G

mn is a geometric

quantity defined below, in Eq. (7); An is the surface area of a control-volume surface


pn; ImP is the radiative intensity evaluated at node P along ~XXm; S

mIPis the value of the

source term evaluated at node P in direction ~XXm; VP is the volume surrounding nodeP; and xm is the solid angle associated with direction ~XXm, respectively.

The geometric function Gmn is evaluated such that

Gmn ¼

Zxm

~XX �~nnn dx ð7Þ

where ~nnn is the unit outward-pointing normal to a panel pn. The rationale behindthese choices is discussed in [1].

To complete the CVFEM formulation, a relation between the value of theradiative intensity at control-volume surfaces and that of this same quantity at thegrid nodes of the finite-element mesh is required. This relation is established bythe prescription of an appropriate spatial interpolation function for intensities overthe elements, and this is the subject matter of the next section.

The issues of discretization equations, boundary conditions, and solutionprocedure are presented elsewhere [1] and are not repeated here.

3. SKEW UPWINDING SCHEME

The nature of the discretized RTE suggests that it is indeed highly desirable notonly to account for radiative intensity at upstream locations when closure is needed,but also to reflect the direction of propagation of radiation. Here, attention is limitedto those two features without regard to attenuation by absorption and out-scatteringor reinforcement by emission and in-scattering in the interpolation functions.

The convention adopted here is that depicted in Figure 1. Careful study of thisfigure should help the reader in the following sections. Discussions are limited to the

Figure 1. A typical two-dimensional element and its related notation.


two-dimensional context for simplicity and clarity, but are very exhaustive to helpthe reader in implementing these ideas, if need be.

With reference to Figure 1, N1, N2, N3 denote the nodes of the finite elementconsidered; O is the centroid of the element where a local Cartesian coordinate sys-tem will be translated; V1, V2, V3 are partial control volumes that will constitute thecomplete and nonoverlapping discretized volume of the calculation domain uponassembly of all triangular elements; p1, p2, p3 are the subcontrol volume surfaces;~nn is the unit outward-pointing normal to the control volume surfaces; and M isthe midpoint, between two nodes, along the edge of a triangular element side. It isworth noting, for further reference, that M1 is opposite to N1 and similarly for theother two pairs of nodes and midpoints.

3.1. An Exponential Scheme (ES)

The first scheme that was considered for implementation within the CVFEMwas the exponential scheme (ES) [1]. This scheme, based on a particularone-dimensional solution of the RTE within an element, yielded very satisfactorypredictions [19, 35]. However, this scheme suffers two major drawbacks: (1) Itrequires fix-up procedures to avoid computations of negative coefficients; and (2)exponentials are relatively expensive to compute. Hence, in the context of thedevelopment of comprehensive numerical methods for the solution of multiphaseturbulent reacting flow combined with radiative heat transfer, such a scheme couldlead to unacceptable computational time requirements.

3.2. An Upwind Scheme (US)

With reference to Figure 2, the implementation of this scheme is rather simple.If the dot product ~XXm �~nnp1 is positive (Figure 2a), the value of the radiative intensityat the integration point p1 is that of the node located immediately counterclockwisewith respect to the centroid O, here node N3. On the other hand, if this dot product isnegative (Figure 2b), the value of radiative intensity at the integration point p1 is thatof node N2.

For any of the three control volume surfaces (panels), p1, p2, p3, theimplementation is straightforward:

Impn ¼ wmn I

mNWðnþ2Þ

þ ð1� wmn ÞImNWðnþ1Þ

ð8Þ

where the weighting function wn is defined as

wmn � max

Gmn

Gmn

�� ; 0

!ð9Þ

and the integer function WðnÞ is such that W(n)¼ n, 8n¼ 1, 2, 3; and W(n)¼ n� 3,8n¼ 4, 5. The weighting functions wm

n for each element can be evaluated prior tothe iterative procedure, as they are geometric quantities that are independent ofthe radiation transfer problem.


The upwind scheme (US) is simple and yields rapid convergence of the solutionprocedure. Moreover, it ensures that no negative coefficients are produced in thediscretization equations. However, it is obvious that for several discrete directions,the directionality of radiant propagation is not very accurately taken into account.For discrete directions in which the node of influence, N3, is in a more or less directline of sight when observed from the integration point p1, such as in Figure 2c, thepredictions of intensity along that direction will be good. However, for the situationdepicted in Figure 2d, severe false scattering (numerical smearing) will occur, and forthat direction the predictions of radiant intensity will be far less accurate. Moreover,it is clear that this scheme will produce solutions that are very sensitive to the shapeof elements. Nevertheless, the upwind scheme was retained as an alternative becauseit is the simplest and therefore cheapest scheme and the inaccuracies it involvesare averaged over all directions when the radiative fluxes or radiant incident energyare evaluated. Hence, the results for the radiative fluxes and radiant incident energywere found to be in fair agreement with those obtained with higher-order schemes.

3.3. A Basic Skew Upwind Scheme (BSUS)

With reference to Figure 3, the basic skew upwind scheme (BSUS), applied to thetriangular element discretized into three equal subcontrol volumes, determined the value

Figure 2. An upwind scheme: (a) positive dot product; (b) negative dot product; (c) fair alignment; (d) poor

alignment.


of the intensity at the integration point p1 by tracking back along the projection of a dis-crete direction~XXm from p1 until the pencil (path) of radiation intersects the element edge ata reference point r. Here, the notation assumes that the edge number l is that of its corre-spondingmidpoint,M. This notation is used for ease of presentation and implementation.

The value taken for the radiative intensity at position r is then based on aninterpolated value, along this element edge, of the corresponding nodal values (forexample, nodes N2 and N3 in Figure 3b). In the context of upwinding (no attenu-ation), Impn ¼ Imrn . This leads to a value of the intensity at the integration point thatis an average of two nodal values.

When wm1 ¼ 1, see Eq. (9), the reference point will lie along either edge l¼ 2 or

l¼ 1. In Figure 3a, with respect to a local Cartesian coordinate system located at thecentroid of the element, O, which is oriented so as to have its axis x aligned with theprojection of ~XXm in the plane of the problem, the reference point r lies on the edgel¼ 2, for which yN3

� yp1 � yN1. In Figure 3b, the reference point r lies on the edge

l¼ 1, for which yN2� yp1 � yN3

. In either of these two cases, Imp1 can be expressed as alinear function of the intensity at two nodes such that

Imp1 ¼ f1l ImNWðlþ1Þ

þ ð1� f1lÞImNWðlþ2Þð10Þ

Figure 3. A basic skew upwind scheme: (a) reference node on edge l¼ 2 linking N3 and N1; (b) reference

node on edge l¼ 1 linking N2 and N3; (c) excellent alignment but reduces to the US; and (d) possibility of

negative coefficients.


where the weighting function f1l in terms of the local and aligned coordinatedirections is

f1l ¼yp1� yNWðlþ2Þ

yNWðlþ1Þ � yNWðlþ2Þ

ð11Þ

and l is either 1 or 2 according to the relative y coordinate of node N3 with respectto p1.

Similarly, when wm1 ¼ 0, the reference point will lie along either edge l¼ 3 or

l¼ 1. In Figure 3c, the reference point r lies on the edge l¼ 1, for whichyN2

� yp1 � yN3, and the above expressions apply. It can readily be observed that

the equations are also valid for the case shown in Figure 3d for whichyN1

� yp1 � yN2. Here, the relative y coordinate of node N2 with respect to p1

determines whether l¼ 3 or l¼ 1.Generalizing the former expressions for any of the three control volume

surfaces, p1, p2, p3, yields

Impn ¼ fnlImNWðlþ1Þ

þ ð1� fnlÞImNWðlþ2Þð12Þ

where the expression of fnl is

fnl ¼ypn � yNWðlþ2Þ

yNWðlþ1Þ � yNWðlþ2Þ

ð13Þ

However, this strategy can be challenged for several reasons. (1) For a givendirection, when both nodes are located upstream with respect to the integrationpoint p1, such as nodes N1 and N3 in Figure 3a, there is no guarantee that the inten-sity at reference point r will physically be a linear interpolation of the nodal values,except, in a highly scattering medium or when a fine spatial discretization is used. (2)For a particular direction, a node N could be located downstream with respect to theintegration point such as node N2 in Figure 3b, where N2 is found to have an influ-ence on Imp1 . This could lead to negative coefficients in the discretized algebraic equa-tions. (3) For some very limited cases only, such as that shown in Figure 3c, thescheme would propose a fair representation of the physics (without attenuation).In this case, however, it reduces to the US. (4) Finally, for the case represented inFigure 3d, the dependence on the intensity at node N1 constitutes an outflow of radi-ant energy from the control volume associated with node N2. Thus, if I

mN1

decreases,ImN2

will increase proportionately. And this clearly shows that negative coefficientscould be calculated in the numerical solutions.

Remedies to avoid negative coefficients could be thought as follows. (1) Whena node N is located downstream with respect to the integration point such as node N2

in Figure 3b, the weighting factor (here f11) could be set to zero (when the local xcoordinate difference between node N2 and point p1 is positive). Similarly, f11¼ 1when node N3 is located downstream of point p1. (2) To avoid the outflow problemillustrated in Figure 3d, the simplest remedy would be to have an influence of N2

only. Clearly, this overrides the properties of the basic skew upwind scheme (BSUS)


and reduces it more or less to the upwind scheme (US) but with more computationaltime requirement.

Nevertheless, having in mind its limitations, the second scheme retained forcomparison is the basic skew upwind scheme (BSUS) described by Eq. (12).

3.4. An Intermediate Skew Upwind Scheme (ISUS)

To avoid having downstream influences and negative coefficients, anintermediate skew upwinding scheme (ISUS) is now introduced. In this scheme,which embeds ideas discussed in the previous subsection, the value of the intensityalong the elements edges that link the nodes is assumed to be constant up to the mid-point of the edges; the value of intensity on the edges that delimit subcontrol Vn is I

mNn.

With reference to Figure 4a, line M2–a represents a line of discontinuity of theintensity originating from the volumes associated with N1 and N3. The relative influ-ence of both nodes on the value of the radiative intensity at p1 is then a weightedaverage of ImN3

and ImN1, the weights being the ratios jM2� bj=jc� bj and jM2� cj=

jc� bj, respectively. On panel p1, ImN1

prevails from the centroid O to point a, then

Figure 4. An intermediate skew upwind scheme: (a) weighted average of the intensity at nodes N3 and N1;

(b) influence of N1 only when yM1� yM2

; (c) influence of N3 only when yM2� 0; and (d) possibility of

negative coefficients.


ImN3prevails from a to M1. Segment jc� bj is the projection, normal to ~XXm, of the

control-volume surface p1 over edge l¼ 2.When wm

1 ¼ 1 [see Eq. (9)], with respect to coordinate axes located at thecentroid O and oriented such that the x-axis is aligned with the two-dimensional pro-jection of ~XXm, yM1

is positive and as long as yM2� 0, the node of influence for p1 will

be N3 only. The case for which yM2¼0 is shown in Figure 4b. When the situation

depicted in Figure 4a prevails,

Imp1 ¼ f þ1 ImN1þ ð1� f þ1 ÞImN3

ð14Þ

where f þ1 ¼yM2yM1

= . When yM2� yM1

, the node of influence for p1 will be N1 only.Figure 4c depicts the situation for which yM2

¼yM1.

To account for the above-described three possibilities, Eq. (14) will be valid ifthe weighting function is defined such that

f þ1 � min maxyM2

yM1

; 0

� �; 1

� �ð15Þ

When wm1 ¼ 0, as long as yM3

� 0, the node of influence for p1 will be N2 only.When the situation depicted in Figure 4d occurs,

Imp1 ¼ f �1 ImN1þ ð1� f �1 ÞImN2

ð16Þ

where f �1 ¼yM3yM1

= . When yM3� yM1

, the node of influence for p1 will be N1 only.Hence, with the following weighting function,

f �1 � min maxyM3

yM1

; 0

� �; 1

� �ð17Þ

Equation (16) provides the value of Imp1 for the three geometric possibilities withw1¼ 0.

Although the expressions for f are mathematical expressions evaluated throughthe determination of the local y coordinate of the element edges’ midpoints, theresulting expressions have a direct physical interpretation. For example, with respectto the element depicted in Figure 4a, it can be seen that f þ1 is the ratio of the radiative

flux that crosses control surface O–M2 into direction ~XXm over the radiative flux that

crosses control surface O–M1 along the same direction when there is no attenuation

or reinforcement within the element. Specifically,

f þ1 � yM2

yM1

¼~qqO�M2

~qqO�M1

ð18Þ

From this expression it can be concluded that the weighting factor will be equal to

this ratio as long as the medium in the element, for the purpose of interpolation only,

is assumed to be transparent.


Considering wm1 as defined in Eq. (9), it is possible to account for the negative

or positive value of ~XXm �~nnp1 by expressing Imp1 as

Imp1 ¼ wm1 ½f þ1 ImN1

þ ð1� f þ1 ÞImN3� þ ð1� wm

1 Þ½f �1 ImN1þ ð1� f �1 ÞImN2

� ð19Þ

Finally, with the use of the integer function W(n) defined previously, a general math-ematical expression of Impn at any of the three panels, p1, p2, p3, can be arrived at:

Impn ¼ wmn ½f mþ

n ImNmþ ð1� f mþ

n ÞImNWðnþ2Þ� þ ð1� wm

n Þ½f m�n ImNn

þ ð1� f m�n ÞImNWðnþ1Þ

� ð20Þ

where

f mþn � min max

yMWðnþ2Þ

yMn

; 0

� �; 1

� �f m�n � min max

yMWðnþ1Þ

yMn

; 0

� �; 1

� �ð21Þ

with n¼ 1, 2, 3 for panels p1, p2, p3, respectively. The superscript m is added to thedefinition of the weighting function f to permit calculation of these functions prior tothe iterative solution process.

In Figure 4a, the contribution of N1 is still seen by the equations for panel p1(between the centroid O and point a), but this contribution has no effect on theradiant energy conservation balance for the control volume associated with N3 afterthe contribution of p2 is accounted for. Because without attenuation Imp2 is ImN1

, allinfluence of N1 on p1, for this case, cancel. This is valid as long as there is no attenu-ation involved, at the interpolation level, along the propagation path within anelement. Otherwise, a net flux of radiant energy is still possible. That is, if the fluxentering p2 is different from that entering p1, between O and a, negative coefficientswould still be possible.

The value of the intensity at p1 in Figure 4a is simply a weighted average ofthat at nodes N1 and N3. On panel p1, I

mN1

prevails from the centroid to point a, then,from a to M1, I

mN3

prevails.

3.5. A Skew Upwinding Scheme (SUS)

The ideas embedded in the skew positive coefficient upwind scheme (SUS),which involve a flux-weighted average of the intensity over control-volume faces,are first outlined for control-volume surface p1 depicted in Figure 5a.

It has been seen that as long as there is no attenuation within the elements, theISUS should improve accuracy over the US and avoid negative coefficients in thediscretized algebraic equations. However, in a further refinement of the proposedscheme, it could be desirable to account for attenuation within the element. In thissection, for a positive value of the radiant heat flux at p1 (see Figure 5a), that is, whenwm1 ¼ 1, it is then mostly desirable to express the value of intensity at p1 directly in

terms of that at p2 and that at node N3 such that

Imp1 ¼ f mþ1 Imp2 þ ð1� f mþ

1 ÞImN3ð22Þ

with f mþ1 being defined appropriately.


This modification of the ISUS involves that: (1) negative coefficients cannot becomputed, since throughputs of radiant energy are correctly accommodated regardlessof the function f mþ

1 ; and (2) a simultaneous set of equations involving the threeintegration point intensities (that is, at p1, p2, and p3) must be solved to determine thesevalues in terms of the nodal values. To generalize this SPCUS, a similar definition off mþ1 to that used in Eq. (21) is employed with the obvious restriction that 0� f� 1.

To ensure a positive contribution to the coefficients when the radiative flux atp2 is negative—coming out of subcontrol volume V3 in Figure 5a—we need thatImp1 ¼ ImN3

; the values of Imp1 and Imp2 at the panels p1 and p2 for this situation shoulddepend only on the value of Im at node 3, which is located upstream of these two sur-faces, which implies that f mþ

1 ¼ 0 in Eq. (22). On the other hand, if Gm2 is positive—

coming into subcontrol volume 3 in Figure 5b—and greater than Gm1 , only Imp2 should

influence the value of Imp1 , because the amount of energy transported by radiation outof control volume V3 across panel p1 has to be greater than or equal to what comes inby radiation through panel p2, in order to ensure the positiveness of coefficients [1, 27,34]. For this case, f mþ

1 ¼ 1. In other situations, such as in Figure 5c, the function f mþ1 is

just the ratio of the directional integral of ~XX �~nnp over f mþ1 for control surface p2 to

that over p1. Consequently, a general expression for f mþ1 , for a positive Gm

1 , can be

Figure 5. A skew upwind scheme with positive coefficient: (a) influence of N3 only when yM2� 0; (b)

influence of p2 when yM1� yM2

; (c) weighted average of the intensity at nodes N3 and panel p2; and (d)

weighted average of the intensity at nodes N2 and panels p3 when Gm1 is negative.


defined as

f mþ1 � min max

Gm2

Gm1

; 0

� �; 1

� �ð23Þ

The subscript 1 stipulates the association with panel 1 and the superscriptþ indicatesthat Gm

1 is positive.If these same guidelines are followed when radiation is crossing control-volume

surface p1 in the negative direction such as in Figure 5d—Gm1 is negative—then an

equation, involving node 2 and panel p3, can readily be obtained such that

Imp1 ¼ f m�1 Imp3 þ ð1� f m�

1 ÞImN2ð24Þ

where f m�1 is defined as

f m�1 � min max

Gm3

Gm1

; 0

� �; 1

� �ð25Þ

Introducing the weighting function wm1 , defined as in Eq. (9), it is possible to

account for the negative or positive value of the radiative flux at p1 by expressingImp1 as

Imp1 ¼ wm1 ½f mþ

1 Imp2 þ ð1� f mþ1 ÞImN3

� þ ð1� wm1 Þ½f m�

1 Imp3 þ ð1� f m�1 ÞImN2

� ð26Þ

Finally, with the use of the integer function W(n) defined previously, a generalmathematical expression of Impn at any of the three panels, p1, p2, p3, can be arrived at:

Impn ¼ wmn ½f mþ

n ImpWðnþ1Þþ ð1� f mþ

n ÞImNWðnþ2Þ� þ ð1� wm

n Þ½f m�n ImpWðnþ2Þ

þ ð1� f m�n ÞImNWðnþ1Þ

�ð27Þ

where

f mþn � min max

GmWðnþ1ÞGm

n

; 0

� �; 1

� �f m�n � min max

GmWðnþ2ÞGm

n

; 0

� �; 1

� �ð28Þ

with n¼ 1, 2, 3 for panels p1, p2, p3, respectively.

4. CONCLUDING REMARKS

This article presents the detailed derivation of a skew, positive coefficient,upwind scheme for the computation of radiative transport in enclosures discretizedby use of triangular meshes. The exhaustive description was intended for the analystwho may want to implement such a procedure in his or her own numerical method.From the proposed descriptions, ideas can easily be extrapolated for the analyst whomay want to implement these ideas on rectangular meshes.


The proposed scheme is based on the application of sound physical arguments,resulting in (1) fast convergence of the algorithm, (2) inherent preclusion of the possi-bility of computing negative coefficients to the discretized algebraic equations, (3)relatively low levels of false scattering, (4) relative insensitivity to grid orientation,and (5) solutions completely free from undesirable oscillations. These attributesrender the scheme attractive, especially in the context of combined modes of heattransfer and fluid flow, for which computational time is a major concern. The per-formance of the proposed scheme is readily available in a companion article [36].

The development presented in this article, considers pure upwinding; that is,the effect of intensity attenuation and reinforcement are not considered within anelement. In this way, the specific features avoiding negative coefficients are readilyappreciated. One of the interesting features of the proposed scheme is that it canbe used in the context of combined-mode heat transfer and fluid flow problemsemploying the same procedure for the solution of the algebraic discretized conser-vation equations. Therefore, although implemented in the context of a CVFEM,the ideas are readily amenable for incorporation in a FVM or a FEM.

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Discrete-Ordinates Method, Numer. Heat Transfer B, vol. 30, pp. 23–43, 1996.18. N. Berour, D. Lacroix, and G. Jeandel, Radiative and Conductive Heat Exchanges in

High Temperature Glass Melt with the Finite Volume Method Approach. Influence ofSeveral Spatial Differencing Schemes on RTE Solution, Numer. Heat Transfer A,vol. 49, pp. 567–588, 2006.

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Skew Upwind Scheme, Numer. Heat Transfer, vol. 6, pp. 395–408, 1983.28. P. M. Gresho and R. L. Lee, Don’t Suppress the Wiggles—They’re Telling You

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APPENDIX: CLOSURES RELATIONS

Equations (8), (12), (20), and (27) are systems of equations for the unknownsImp . These systems of equations can conveniently be written in a matrix form, yielding

½A�3�3 Imp

n o3�1

¼ ½B�3�3 ImN�

3�1ð29Þ

or

a11 a12 a13a21 a22 a23a31 a32 a33

24

35 Imp1

Imp2Imp3

8><>:

9>=>; ¼

b11 b12 b13

b21 b22 b23

b31 b32 b33

264

375

ImN1

ImN2

ImN3

8><>:

9>=>; ð30Þ

A.1. Upwind Scheme

For a two-dimensional implementation of the upwind scheme (US), the A

matrix is the identity matrix, while the coefficients of the B matrix are

bjj ¼ 0 bjWðjþ1Þ ¼ ð1� wmj Þ bjWðjþ2Þ ¼ wm

j ð31Þ

A.2. Basic Skew Upwind Scheme

To implement the basic skew upwind scheme in two dimensions, the A matrixis also the identity and matrix B can be constructed such that, whenyNj

� ypi � yNWðjþ1Þ, we obtain

bij ¼ fij biWðjþ1Þ ¼ ð1� fijÞ biWðjþ2Þ ¼ 0 ð32Þ


A.3. Intermediate Skew Upwind Scheme

In this case, the A matrix is once again the identity matrix, and matrix B can beconstructed such that

bjj ¼ wmj f

mþj þ ð1� wm

j Þf m�j bjWðjþ1Þ ¼ ð1� wm

j Þð1� f m�j Þ bjWðjþ2Þ ¼ wm

j ð1� f mþj Þð33Þ

The full matrix coefficients for the ISUS are

wm1 f

mþ1 þ ð1� wm

1 Þf m�1 ð1� wm

1 Þð1� f m�1 Þ wm

1 ð1� f mþ1 Þ

wm2 ð1� f mþ

2 Þ wm2 f

mþ2 þ ð1� wm

2 Þf m�2 ð1� wm

2 Þð1� f m�2 Þ

ð1� wm3 Þð1� f m�

3 Þ wm3 ð1� f mþ

3 Þ wm3 f

mþ3 þ ð1� wm

3 Þf m�3

264

375

A.4. Skew Positive Coefficient Upwind Scheme

For the implementation of the SPCUS, the coefficients of matrices A and B are

ajj ¼ 1 ajWðjþ1Þ ¼ �wmj f

mþj ajWðjþ2Þ ¼ �ð1� wm

j Þf m�j

bjj ¼ 1 bjWðjþ1Þ ¼ ð1� wmj Þð1� f m�

j Þ bjWðjþ2Þ ¼ wmj ð1� f mþ

j Þð34Þ

The matrix coefficients for the SPCU scheme are

1 �w1fþ1 �ð1� w1Þf �1

�ð1� w2Þf �2 1 �w2fþ2

�w3fþ3 �ð1� w3Þf �3 1

264

375

0 ð1� w1Þð1� f �1 Þ w1ð1� f þ1 Þw2ð1� f þ2 Þ 0 ð1� w2Þð1� f �2 Þ

ð1� w3Þð1� f �3 Þ w3ð1� f þ3 Þ 0

264

375

These matrices involve geometric quantities only. They can be computed before theiterative solution procedure that solves for intensity begins.


a consistent interpolation function for the solution of radiative transfer on triangular meshes....

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