MULTI-SCALE METHODS FOR THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION

Transcription

1 MULTI-SCALE METHODS FOR THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION Pero J. Coelho *,, Nicolas Crouseilles **, Pero Pereira * an Maxime Roger *** * LAETA, IDMEC, Dept. of Mechanical Engineering, Instituto Superior Técnico, Universiae e Lisboa, Av. Rovisco Pais, Lisboa, Portugal ** INRIA-Rennes Bretagne-Atlantique (IPSO Project) an Université e Rennes 1 (IRMAR), Campus e Beaulieu, Rennes Ceex, France *** Université e Lyon, INSA e Lyon, CETHIL, UMR5008, F Villeurbanne Ceex Corresponence author: Phone: pero.coelho@tecnico.ulisboa.pt

2 ABSTRACT Various methos have been evelope an teste over the years to solve the raiative transfer equation () with ifferent results an trae-offs. Although the is extensively use, the approximate iffusion equation is sometimes preferre, particularly in optically thick meia, ue to the lower computational requirements. Recently, multi-scale moels, namely the omain ecomposition methos, the micro-macro moel an the hybri transportiffusion moel, have been propose as an alternative to the. In omain ecomposition methos, the omain is split into two subomains, namely a mesoscopic subomain where the is solve an a macroscopic subomain where the iffusion equation is solve. In the micro-macro an hybri transport-iffusion moels, the raiation intensity is ecompose into a macroscopic component an a mesoscopic one. In both cases, the aim is to reuce the computational requirements, while maintaining the accuracy, or to improve the accuracy for similar computational requirements. In this paper, these multi-scale methos are escribe, an the application of the micro-macro an hybri transport-iffusion moels to a threeimensional transient problem is reporte. It is shown that when the iffusion approximation is accurate, but not over the entire omain, the multi-scale methos may improve the solution accuracy in comparison with the solution of the. The orer of accuracy of the numerical schemes an the raiative properties of the meium play a key role in the performance of the multi-scale methos. 1

3 1. INTRODUCTION Many problems in mathematics, biology, physics, chemistry an engineering encompass ifferent spatial an/or time scales. The mathematical moelling of such problems requires a multi-scale approach. This may be efine as a general framework for formulation an esign of methos which moel a system s behaviour governe by a hierarchy of scales, both spatial an temporal, an their interactions, an provie a seamlessly couple platform through which the interacting scales mutually exchange their information [1]. The range of relevant length an time scales in materials science, for example, inclues the electronic, atomic, microscopic, mesoscopic an macroscopic, or continuum scales [2]. An efficient numerical solution of a multi-scale problem requires that appropriate numerical methos are use for every relevant scale, an couple to allow effective ata exchange between ifferent scales. In some multi-scale problems there is no clear scale separation, as in the case of turbulent flui flow an in raiative transfer. The moelling an numerical simulation of raiative transfer phenomena has been a very active fiel of research uring the last ecaes. Inee, raiative transfer unerlies numerous technological applications (e.g., combustion, optical tomography, solar energy), as well as more funamental research. Because of the multi-scale character of most raiative transfer phenomena, an since analytical solutions are available only in a few simplifie cases, the numerical simulation of raiative transfer phenomena is still a challenging task nowaays. Accoring to the physical context, raiative transfer phenomena can be moelle by means of two classes of mathematical moels: the raiative transfer equation () an the iffusion equation () (see [3]). At the top of the hierarchy of moels, the appears as the reference from a moelling point of view. The accurately escribes raiation transport in meia uner local thermal 2

4 equilibrium, yieling the time evolution of the raiation intensity, which epens on 6 imensions (space, irection an wavelength) plus time. However, the numerical solution of the may be computationally expensive, particularly when the optical thickness of the meium is high. The iscrete orinates metho (DOM), the finite volume metho (FVM) an the Monte Carlo metho become inefficient in the so-calle iffusive regime. Inee, the numerical parameters for the DOM or FVM methos, such as the gri size an time step, must satisfy severe constraints for stability reasons. As far as the Monte Carlo methos are concerne, the computational cost may be prohibitive ue to the high number of scattering events, an the numerical convergence becomes slower. At the bottom of the hierarchy, the escribes raiative transfer at the macroscopic scale, yieling the incient raiation, which epens on time, the three spatial imensions an wavelength. The is base on the assumption that the raiation intensity is nearly isotropic. Obviously, when this assumption is not vali, which is the case when the meium or part of the meium is not optically thick, or when bounary conitions or raiative sources have a strong influence on raiative transfer, the results are inaccurate. Moreover, fining accurate bounary conitions for the may be a complicate problem. The require computational time for the solution of the is much larger than that for the solution of the for realistic multi-imensional simulations. Hence, several computational methos have been evelope that combine the an the, aiming at the improvement of the computational efficiency of raiative transfer simulations without compromising the solution accuracy [4]. These methos are an interesting option in problems where iffusive an kinetic regimes coexist, e.g., meia with a low-scattering region an a strongly scattering region, or in problems where multiple spatial an temporal scales are present. This multi-scale 3

5 character makes the evelopment of numerical moels an the associate simulations a real challenge. In this work, our goal is to report recent avances on multi-scale moels for raiative transfer, namely a omain ecomposition strategy [5], a micro-macro moel [6] an a hybri transportiffusion moel [7], are presente. These approaches involve an interesting trae-off between accuracy an computational requirements an offer a goo alternative to the use of a full or. One possible strategy to eal with multi-scale moels is to couple the an the through a spatial omain ecomposition approach (DD), where the omain is ecompose into a mesoscopic subomain, in which the is solve, an a iffusive one, in which the is solve. Several variants of the DD metho have been reporte, epening on the numerical methos use to solve the an the, an on how the coupling between the two subomains is implemente [4, 8 11]. Most of these works have been evelope for light propagation in tissues for biomeical applications. The treatment of the interface between the macroscopic an the microscopic zones represents a critical issue. In fact, the bounary conitions of the must be consistent with the bounary conitions of the, which is not an easy task. Hence, a buffer zone between the kinetic an iffusive subomains was introuce to overcome this issue [5, 12], which avois the nee to efine bounary conitions for the. Another strategy is to apply a multi-scale moel in the entire omain. A first moel presente in this work is the micro-macro () moel, which was originally evelope in other research fiels [13 15], an recently applie to the raiative transfer equation [16]. It is base 4

6 on the ecomposition of the epenent variable of the governing equation into a macroscopic component an a mesoscopic one. The micro-macro moel satisfie by the incient raiation, G, plus a correction, ε, is equivalent to the. The solution of the is recovere by simply aing the contributions of the macro (G) an kinetic (ε) components. The resulting moel involves a two-way couple system, the numerical approximation of which nees artificial bounary conitions for the macroscopic unknown G. Moreover, ue to the two-way coupling, a Monte Carlo metho cannot be irectly applie to the system. However, it was shown that some improvements are observe when gri base methos are employe, compare to the numerical solution of the full, especially when the system is close to the iffusive regime [16]. The last moel escribe in this paper, referre to as hybri transport-iffusion () moel, relies also on a ecomposition of the raiation intensity [7]. In fact, expressing the raiation intensity as the sum of G, where G is the solution of the, plus a correction ε, one can write a one-way couple moel satisfie by G an ε. An approximate solution of the is calculate by aing the macroscopic an the kinetic components, as in the moel. However, in contrast to the moel, a Monte Carlo metho can be easily applie to the kinetic equation. Inee, since the oes not epen on ε, it can be solve inepenently on the whole time-space interval uner consieration. Then, a Monte Carlo metho can be easily use since the kinetic part of the system only iffers from the by the presence of a source term epening on G. Hence, the linear structure is preserve, which is a real avantage when using a Monte Carlo metho. Moreover, since bounary conitions are impose to the, e.g., using Marshak approximation [17], it is possible to recover the bounary conitions by simply consiering the ifference between the kinetic an the macro ones. 5

7 Previous stuies on the application of the an moels to raiative transfer problems were ite to one-imensional problems [7, 16]. They suggeste that the multi-scale moels coul be avantageous in the case of optically thick an strongly scattering meia. In the present work, the an the moels are extene to three-imensional transient problems, an applie to an optically thick meium with colate raiation from a short-pulse laser, as typically foun in optical tomography applications. The remainer of this paper is organize as follows. The next section is evote to recalling the main steps of the erivation of these three moels (DD, an ). Then, we focus on the numerical valiation an comparison of the an approaches for a threeimensional transient problem. We will pay attention to the omain of valiity an the efficiency (trae-off between the computational cost an the accuracy) with respect to the reference solution of the. The main conclusions are summarize in the last section. 2. THEORY 2.1. Raiative Transfer Moels Raiative transfer equation The transient raiative transfer equation () for emitting, absorbing an scattering meia can be written as: 1 I c t σ + s I = κi ( + ) I + s b κ σ s Φ( s s ) s I 4π π 4 (1) where I = I(r,s,t) is the raiation intensity at location r, propagation irection s an time t, c is the spee of light, κ is the absorption coefficient, σ s is the scattering coefficient, Φ is the scattering phase function, s is the incient irection, I b (r,t) is the blackboy raiation intensity an I 6

8 stans for I(, s,t) r Diffusion equation The can be erive by integrating the over a soli angle an applying the P 1 approximation expresse by ( r, s, t) I ( r, s, t) = G ( r, t) ( + 3q s) 4π I (2) where G is the incient raiation, q the raiative heat flux vector, an superscript enotes the iffusion it. The incient raiation is efine as follows: = G I s = I (3) 4π where symbol. represents the integral over the soli-angle space. The raiative heat flux in the iffusion it is efine as q = D G (4) where D is the iffusion coefficient, given by D = 1 [ 3( κ + σ s ( 1 g) )], an g is the asymmetry factor of the phase function. Inserting Eq. (2) into Eq. (1), integrating over all irections an applying the iffusion approximation, q t = 0, yiels the : 1 G c t ( D G ) = ( 4π Ib G ) κ (5) 2.2. Raiative Transfer Moels with Colate Irraiation Raiative Transfer Equation When the meium is subjecte to laser irraiance, it is convenient to ecompose the raiation 7

9 intensity into its colate component, I c, an its iffuse component, I, accoring to I(r,s,t) = I c (r,t) + I (r,s,t). Thus, two equations are obtaine: 1 I c t c + s I c ( s ) I c = κ + σ (6) 1 I c t + s I = κi b σ s s ( κ + σ s ) I + Φ( s s ) I s + Φ( s s c ) I c 4π σ 4π 4 π (7) Subscripts c an stan for the colate an iffusive components, respectively. The colate component is only ifferent from zero along the colate irection efine by the unit vector s c Diffusion equation Equation (6) is still solve in the macroscopic moel with colate irraiation. The in this case is evelope for the iffuse component of the raiation intensity: 1 G c t ( D G ) = κ ( 4π I G ) + σ I ( 3Dσ g I s ) b s c s c c (8) In orer to obtain Eq. (8), the P 1 approximation is applie to the iffuse component I : ( r, s, t) I ( r, s, t) = G ( r, t) ( + 3q s) 4π I (9) The raiative heat flux is now efine as: q = D G + 3Dσ g q (10) s c The multi-scale moels are presente below for the case of colate irraiation, since it is easy to euce the formulation without colate component by setting I c to zero. Consequently, the 8

10 multi-scale moels are evelope for the iffuse component I of the raiation intensity instea of I Domain-ecomposition methos In multi-scale raiation problems, one has to eal with the iffusive regime, where the seems more appropriate, an with the kinetic regime where the is the accurate moel. Therefore, one natural solution is to try to solve each moel wherever it is require using a omain-ecomposition metho. Even though our implementation of the DD metho, reporte in Roger an Crouseilles [12], has not been extene yet to 3D problems, being exclue from the analysis reporte below in section 3, a brief escription is inclue here for completeness purposes. One important issue in the DD methos is the treatment of the interface between the zone where each moel is solve. The classical approach for omain-ecomposition is to couple the moel through the bounary conitions (see Fig. 1) [4, 8 11]. The omain Ω is then split into macroscopic Ω m an kinetic Ω k subomains. In each subomain, the corresponing equation is solve. The kinetic subomain generally contains the bounaries, like in the example shown in Fig. 1, the raiation sources an the low-scattering regions. The couple moel is base on the (Eqs. (6) an (7)) for the kinetic subomain, an on the (Eqs. (6) an (8)) for the iffusive subomain. At the interface Γ=Ω m Ω k, the bounary conitions of the for the iffuse raiation intensity I (r,s,t), an the bounary conitions of the for the incient raiation G (r,t) must be consistent. Therefore, for r Γ, the bounary conitions of the are epenent on the solution an are given by ( + 3q s) 4π ( r, s, t) G ( r, t) I (11) = 9

11 On the other han, the bounary conitions of the for the incient raiation G (r,t) epen on the solution of the for r Γ [4, 11]: G ( r, t) I ( r, s, t) s (12) = 4π An iterative proceure must be evelope in orer to eal with the coupling of the an the in Eqs. (11) an (12). The bounary conitions at the interface Γ, i.e., Eqs. (11) an (12), require that the raiation intensity is almost isotropic. Therefore, the coupling can be complete only if the iffusion approximation is vali at the interface, which imposes a severe constraint on the choice of the interface location. In [4], the authors give the criterion that Γ must be locate at istances greater than the iffusion coefficient D from the bounaries, sources an low-scattering regions in orer to ensure that the coupling can be achieve efficiently. Another possible approach is to couple the an the through the equations, by means of a multi-scale moel between the macroscopic an the kinetic subomains, as propose in [12]. It avois the elicate issue of fining the interface location an to einate the iteration proceure from the omain-ecomposition strategy. Base on the iea of Degon an Jin [5], a buffer zone is introuce between the macroscopic an the kinetic subomains, as shown in Fig. 2. The an the are then couple through the equations insie the buffer zone. A transition function h(r,t) is propose to formalize the location of the ifferent subomains: h h h ( r,t) = 1 ( r,t) [ 0, 1] ( r,t) = 0 for r Ω for r B for r Ω k m (13) The multi-scale moel is then built by multiplying the by the transition function h, which ensures that the is solve in the kinetic omain where h=1, but not in the macroscopic 10

12 omain, where h=0. On the other han, the is multiplie by (1-h), which ensures that the is solve in the macroscopic subomain, but not in the kinetic subomain. In the buffer zone, a couple system of two equations, one at the macroscopic scale, the other at the mesoscopic scale, must be solve (all etails can be foun in [12]). One practical consequence of the metho is that if the bounaries of the omain are locate insie the kinetic subomain, as in Fig. 2, there is no nee to introuce bounary conitions for the. One main avantage of this approach is that no iteration proceure is neee if an explicit or semi-implicit strategy is use for the time iscretization. Moreover, the coupling between the an the is easily ealt with the transition function, which epens on time, an allows therefore a ynamic control of the coupling [12]. On the other han, the implementation of a numerical metho to solve the couple systems of equations may be a harer task than using a geometric interface Multi-scale moels A possible approach to eal with multi-scale raiation problems is the evelopment of a multi-scale moel for the whole omain. In these moels, such as the so-calle micro-macro moel () [6, 16] or the hybri transport-iffusion moel () [7], the raiation intensity is split into kinetic an macroscopic components. A couple system of two equations is then obtaine, one transport equation for each component. The an moels are equivalent to the, while in the omain-ecomposition approaches, the iffusion approximation is assume in the macroscopic subomain. Therefore, the an 11

13 moels may be an efficient alternative to the in strongly scattering meia, especially in cases where the iffusion approximation remains somewhat inaccurate Micro-macro moel In the moel, the iffuse component of the raiation intensity I is ecompose as follows: I ( r, s, t) G ( r, t) 4π + ε ( r, s t) = (14), where G = I is the iffuse component of the incient raiation. Inserting Eq. (9) into Eq. (7) an integrating over all irections yiels the following transport equation for G : 1 G c t + ( π Ib G) σ s Ic ε κ 4 + s = (15) G is the macroscopic component of the moel. The transport equation for the kinetic component, ε, is obtaine by subtracting Eq. (15) from Eq. (7), which yiels ε 1 c t + s ε = s ( κ + σ ) ε + Φ( s s ) σ s + Ic 4 π s ( Φ( s s ) 1) + ( s ε s G ) c σ 4π 4π 1 4π ε s + (16) The system of Eqs. (6), (15) an (16) is referre to as the micro-macro () moel an is equivalent to the. The macroscopic an the kinetic equations, (15) an (16), respectively, are couple. Eq. (15) may be solve using stanar numerical methos, such as the finite ifference or the finite volume methos, while Eq. (16) is mathematically similar to the, an may be solve using the classical methos usually employe for the, such as DOM or FVM. The coupling between Eq. (15) an Eq. (16) may foster the use of a semi-implicit or fully explicit time iscretization, as propose in [16], in orer to avoi an iterative proceure. 12

14 The exact bounary conitions of the raiation intensity are not conserve with the moel. When incoming bounary conitions are prescribe for I (r,s,t), they cannot be translate into a bounary conition for G (r,t), since I is not known for all irections. In this case, artificial bounary conitions of the Neumann type are generally use for Eq. (15). The bounary conitions for ε are then efine accoring to [16]: ( r, s, t) = I ( r, s, t) G ( r, t) π ε 4 (17) Hybri transport-iffusion moel The moel relies on the ecomposition of the iffuse component of the raiation intensity accoring to: I ( r, s, t) G ( r, t) 4π + ε ( r, s t) = (18), The macroscopic unknown in the moel is the incient raiation at the iffusive it, G, which satisfies the (Eq. (8)), in contrast to the moel, in which the macroscopic component is the exact incient raiation G = I. The transport equation for the kinetic component, ε, is erive by subtracting Eq. (8) from Eq. (7), yieling: ε 1 c t + s ε = 1 + 4π s 4π 4π 4 [ ( 3 Dσ g I s D G ) s G ] s s ( κ + σ ) ε + Φ( s s ) ε s + I ( Φ( s s ) 1) s σ c c σ π c c + (19) Unlike the moel, the three governing equations (6), (8) an (19) are not fully couple. Equation (8) oes not epen on ε, which means that the macroscopic Eq. (8) can be solve first, without any coupling with the kinetic component. Again, the kinetic equation is very similar to the an can be solve using classical methos for the. Since the equations for the moel are not couple, they are easier to solve than those of the moel. For 13

15 instance, the Monte Carlo metho can be envisage to solve the kinetic equation of the moel, while this was not possible with the moel ue to the coupling term with the macroscopic equation. Another avantage of the moel is that the exact bounary conitions for the raiation intensity are conserve. The bounary conitions for ε are efine accoring to Eq. (18), by =, but in this case they strictly compensate the error setting ε ( r, s,t) I ( r, s,t) G ( r,t) 4π mae on the bounary conitions for G (such as the frequently use Marshak bounary conitions). Since the value of ε has no influence on the value of (18) ensures that the bounary conitions on I are exactly matche. G at the bounary, Eq. 3. RESULTS AND DISCUSSION 3.1. Test Problem Our previous works on the an moels [7, 16] were restricte to one-imensional problems. An extension of that work to three-imensions is illustrate in the present problem. It consists of a cubic enclosure of size L submitte to an incient short-pulse laser on one of its faces, as schematically shown in Fig. 3. The intensity of the laser pulse has a Gaussian temporal istribution accoring to the following equation: 2 t t I 2 t p c ( 0,,t) Io ( ) exp 4 ln ( 2) c = s sc, 0 < t < tc s δ (20) where I o is the maximum raiative intensity of the pulse, which occurs at t = t c =3t p = 1.5/(β.c), an the irection of the colate irraiation is normal to the enclosure, i.e., s c = i. After 6t c, the meium is free from irraiation. The size of the cube is L=1 m, an it contains a scattering an 14

16 absorbing homogeneous meium with an optical thickness τ = β L an an albeo ω = σ s /β. The short-pulse laser irraiates the bounary x=0 accoring to Eq. (20). The bounaries y=0, y=l, z=0 an z=l are non-reflecting an col (non-emitting), as often consiere in problems of light propagation in biological tissues. The colate component of the raiation intensity was evaluate analytically (see [7] for etails). The macroscopic equation (Eq. (15) for the an Eq. (8) for the ) was solve using a stanar finite volume metho. The central ifference scheme was use to evaluate the secon-orer spatial erivatives. The transient term was iscretize using either a first-orer explicit Euler scheme or a secon-orer Runge-Kutta (RK2) scheme. The time step must be sufficiently small, ue to the stability requirements of explicit schemes. Details on the implementation of the bounary conitions are given in [7]. Equation (7) for the iffuse raiation intensity component of the, as well as Eqs. (16) an (19) for the mesoscopic component of the multi-scale moels, were solve using the DOM. In this stuy, an S 12 quarature was employe for the angular iscretization, while the spatial iscretization was carrie out using the finite volume metho. The convective-like term was iscretize using either the step scheme (first-orer accurate) or the CLAM scheme (seconorer accurate). The transient term was iscretize using either the fully-explicit Euler scheme or the RK2 scheme. The time step t for the DOM was efine as t=α. x/c, where the stability parameter α was set equal to 0.5. Therefore, when a finer gri is employe, the time step is also smaller. A previously valiate Monte Carlo algorithm [7] was use to provie benchmark results. 15

17 3.2. Moerate Albeo an Isotropic Scattering An isotropically scattering meium is stuie first, with an optical thickness τ =10 or 20 an an albeo ω=0.5. A uniform mesh with control volumes was use in the case of τ =10, while a finer mesh with control volumes was require for τ =20 in orer to obtain satisfactory results. Figures 4 an 5 show the preicte imensionless transmittance an reflectance signals, which are given by T R ( t) = I ( x = L, y, z, s, t) + I ( x = L, y, z, s, t)( s i) s yz I o 1 L L L c c π L L ( t) = I ( x = 0, y, z, s, t)( s i) I o 1 L π s y z (21) (22) The transmittance is initially equal to zero, since photons must travel throughout the meium before they reach the bounary x = L. The photons travel at the spee of light, an therefore those that are not scattere in the meium nee a time equal to L/c = 3.33 nanosecons to arrive at that bounary, so that only after this time the transmittance signal is ifferent from zero. However, the transmittance becomes only significant a little later, ue to the Gaussian nature of the incient pulse. The transmittance exhibits a sharp peak an a Gaussian like shape, ue to the colate raiation component that is able to travel across the meium without being fully absorbe or scattere. When the optical thickness of the meium increases, the magnitue of the transmittance signal is smaller, as well as the with of the Gaussian laser pulse, an therefore a finer gri in streamwise irection is neee to preict satisfactorily the sharp rise of the transmittance signal. Downstream of the peak of the transmittance, the signal exhibits a strong ecrease, but it oes not rop to zero. Instea, the transmittance remains positive for some more time, ue to the contribution of the iffuse component arising from scattering within the meium, i.e., from scattere photons, which 16

18 take more time to reach the bounary. The marke change in the slope of the transmittance signal at τ 3.7 nanosecons for τ=20 is ue to the ominant role of the colate raiation component before that time, in contrast to the ominant role of the iffuse component thereafter. When first-orer schemes are employe, the an the moels yiel a minor improvement of the preictions, but they o not reprouce well the initial increase of the transmittance. However, when secon-orers schemes are employe, all moels perform similarly an are in goo agreement with the Monte Carlo results, except the. The reflectance signal is entirely ue to photons that are scattere backwar. The intensity of the short-pulse laser only becomes significant after 0.3 nanosecons, an achieves its maximum value at 0.6 nanosecons for τ = 10 (0.2 an 0.3 nanosecons, respectively, for τ = 20). Therefore, the reflectance increases rapily after 0.3 nanosecons, an achieves a maximum shortly after the peak of the Gaussian pulse occurs. Then, it ecreases slowly, as the number of back scattere photons that reach the x = 0 bounary ecreases. The significantly overestimates the maximum reflectance. In the case τ=10, the moel overestimates the peak reflectance by about 7%, while both the an the are in close agreement with the reference Monte Carlo solution. There are only negligible ifferences between the moels for τ=20, apart from the Moerate Albeo an Anisotropic Scattering Figures 6 an 7 show the preictions obtaine for the same values of optical thickness, albeo an gri size, but now consiering anisotropic scattering. The Henyey-Greenstein phase function 17

19 was use with an asymmetry factor equal to 0.3. When forwar scattering is consiere (g > 0), the peak of the transmittance increases, an so oes the transmittance thereafter, ue to the greater number of photons that are scattere forwar. The preictions of the an moels are better than those of the for first-orer schemes, but the ifferences are negligible for secon-orer schemes. However, the first-orer schemes yiel large errors in comparison with the Monte Carlo solution, particularly for this case with anisotropic scattering, while the secon-orer schemes greatly improve the results an closely approach that reference solution. The always gives rather poor preictions. The multi-scale moels perform a little worst than the as far as the preiction of the reflectance is concerne, as shown in Fig. 7, probably ue to the very poor preictions of the iffusion approximation. In fact, the multi-scale moels are expecte to be useful when the solution of the macroscopic equation is a goo approximation of the raiation intensity, an the mesoscopic component is small. This is not the case of the reflectance for g=0.3, as illustrate in Fig. 7. Accoringly, the multi-scale moels are not recommene if the reflectance is a relevant quantity for the problem uner consieration Large Albeo an Isotropic Scattering Figure 8 shows the transmittance compute for τ=20, ω=0.9 an g=0 (isotropic scattering). In the present case, the contribution of the iffuse raiation component to the transmittance is ominant, while the colate component has a negligible contribution. The preicts well the transmittance up to the peak value, but overestimates the Monte Carlo reference solution thereafter. When first-orer iscretization schemes are employe, the DOM solution of the yiels rather poor preictions. The multi-scale moels significantly improve those preictions, but still epart markely from the reference solution. Gri refinement improves the preictions, but only the moel gives goo results for the finest mesh ( ). 18

20 When secon-orer schemes are use, the preictions of the an moels closely match the Monte Carlo results, an outperform the. The preicte reflectance is shown in Fig. 9 for the coarse gri ( ) along with firstorer schemes, an for the fine gri ( ) along with secon-orer schemes. The oes not provie a goo preiction of the reflectance when the coarse-gri is use, ue to the colate source. As a consequence of the poor preiction of the peak reflectance by the, the mesoscopic component in the multi-scale moels is of the orer of magnitue of the macroscopic component. Therefore, the accuracy of the an is comparable, but actually worst, than that of the. As the gri is refine, the, an give solutions in close agreement with the Monte Carlo metho. Therefore, there is no avantage of the multi-scale moels in comparison with the, as far as the preiction of the reflectance is concerne Large Albeo an Anisotropic Scattering We finally consier anisotropic scattering, escribe by the Henyey-Greenstein phasefunction, maintaining τ=20, ω=0.9. Figure 10 shows the results etermine for asymmetry factors g=0.3 an g=0.6. The peak of the transmittance increases when g increases, ue to the forwar-scattering, an occurs earlier than in the case of isotropic scattering. The behaviour of the various methos is ientical to that observe before. The oes not perform so well, since the raiation intensity fiel is more anisotropic, overpreicting the Monte Carlo solution both before an after the transmittance peak. Still, it contributes to improve the solution of the an moels in comparison with the for first-orer schemes. In the case of a fine gri, the results of the moel are in close agreement with the Monte Carlo results, while the an preictions are not so accurate. 19

21 Finally, the asymmetry factor is increase to 0.9. The transmittance is two orers of magnitue larger an the peak is attaine again at smaller times than in the previous case. In the case of g=0.9, the phase function exhibits a sharp forwar peak an must be normalize. If this is not one, then the preictions are rather poor for all methos, no matter the spatial an temporal iscretization schemes an gri size. This is ue to the strong anisotropy of the phase function, which causes a lack of conservation of scattere energy an asymmetry factor, as iscusse in Hunter an Guo [18]. The error inherent to this problem has been referre to as angular scattering error, but it is just a manifestation of the ray effect, since it is exclusively ue to the angular iscretization in the DOM. The normalization proceure evelope by Hunter an Guo [18] was use in the present work. The solutions obtaine using first orer schemes are very poor in the case of g=0.9, an are not shown here. The transmittance calculate using the along with secon-orer schemes is shown in Figure 11 for several meshes. The an moels give results similar to those of the when the secon-orer schemes are employe, an therefore their preictions are not shown here. The preictions converge to the Monte Carlo solution when the gri is refine, but the peak transmittance is unerpreicte even for the finest mesh, probably ue to the insufficient angular iscretization resolution. The angular iscretization errors were not relevant for the case of lower asymmetry factors, because the meium is homogeneous an there is no iscontinuity in the bounary conitions of the iffuse component of the raiation intensity in the present problem. In the case of g=0.9, however, an accurate angular iscretization is neee to accurately compute the in-scattering term. 20

22 3.6. Computational Requirements As far as the computational requirements for this test case are concerne, the CPU time require for the solution of the is typically lower than 1% of that of the other methos. The computational requirements of the an moels are comparable, an marginally excee those of the, by typically less than 5%. In fact, Eqs. (16) an (19) are mathematically similar to Eq. (7), an therefore the computational time neee to solve them is also similar, since they were solve using the same metho (DOM), but slightly higher for equations (16) an (19), ue to the nee to perform more calculations to evaluate the source term. In aition, the an moels require the solution of the macroscopic equation, but this is much faster to solve than the mesoscopic equation, because the macroscopic equation is irectionally inepenent, in contrast to the mesoscopic equation an to the. Hence, in the case of transient problems solve using an explicit time iscretization scheme, the avantage of using the or the moels instea of the lies in the higher accuracy that may be obtaine. As a consequence of the above comments on the CPU time requirements, it is better to solve irectly the instea of using the or the moels if the accuracy of the results is comparable. The an moels are only a goo option relatively to the when they yiel more accurate results than the. If the gives a goo estimation of the transmittance, e.g., when the optical thickness of the meium is high an there is a weak irectional epenence of the raiation intensity fiel, the mesoscopic component in Eqs. (15) an (19) is much smaller than the macroscopic one. Hence, numerical errors in the DOM solution of the mesoscopic equation, associate to the spatial, temporal an angular iscretization schemes, have a much lower impact on the preicte results when the mesoscopic component is small. This explains why the an moels perform better 21

23 than the in the case of first-orer schemes for the present problem. When secon-orer schemes are use, or a finer gri is employe, the errors in the DOM solution are greatly reuce, an the accuracy is significantly improve. Accoringly, in such a case there is no avantage in using the multi-scale moels relatively to the for most of the combinations of raiative properties consiere in the present test. 4. CONCLUSIONS Two multi-scale moels for the solution of raiative transfer problems have been presente, namely the moel an the moel. They are base on the ecomposition of the raiation intensity into two components: macroscopic an mesoscopic. The former satisfies the transport equation for the incient raiation in the moel, an the in the moel. In both moels, the mesoscopic component satisfies an equation similar to the, which has been solve using the DOM. The following conclusions may be rawn from the analysis carrie out: 1. The an moels, formerly applie to one-imensional problems, have been extene to three-imensional transient problems with colate raiation. These moels provie goo preictions of the transmittance for all stuie cases, provie that appropriate meshes an numerical schemes are use, but they only preict satisfactorily the reflectance in the case of isotropic scattering. 2. When first-orer spatial an temporal iscretization schemes are use, the transmittance preicte by the an moels is closer to the reference Monte Carlo solution than the transmittance calculate from the, for both isotropic an moerate anisotropic scattering. In such a case, the an moels are a goo alternative to the, since 22

24 the computational requirements of the multi-scale moels excee only marginally those of the. In general, the yiels more accurate preictions of the transmittance than the moel. 3. When secon-orer schemes or a rather fine spatial iscretization are neee to achieve goo preictions of the transmittance, the an moels are not an attractive alternative to the for almost all conitions investigate in the present work, since they all yiel similar results an the former are slightly more expensive an complex. 4. The an moels are useful if the is able to provie a goo estimation of the raiation intensity fiel, i.e., whenever the mesoscopic component is small. In the moel, the mesoscopic component may be evaluate using any metho employe to solve the, such as the finite volume, the finite element or the Monte Carlo methos, an oes not nee to be solve using the DOM, as in the present work. ACKOWLEDGEMENTS This work was financially supporte by national funs by FCT-Funação para a Ciência e Tecnologia uner the project PTDC/EMS-ENE/1028/2012, an through IDMEC, uner LAETA, project UID/EMS/50022/

25 REFERENCES [1] Mishra SK. Development of a multiscale an multiphysics simulation framework for reaction-iffusion-convection problems. Ph.D. Thesis, The University of Arizona; [2] Steinhauser MO. Computational multiscale moeling of fluis an solis, theory an applications. Berlin Heielberg: Springer-Verlag; [3] Moest M. Raiative heat transfer. 2 n e. New York: Acaemic Press; [4] Lehtikangas O, Tarvainen T. Hybri forwar-peake-scattering-iffusion approximations for light propagation in turbi meia with low-scattering regions. J Quant Spectrosc Raiat Transfer 2013;116: [5] Degon P, Jin S. A smooth transition moel between kinetic an iffusion equations. SIAM J Numer Anal 2005;42: [6] Lemou M, Mieussens L. A new asymptotic preserving scheme base on micro-macro formulation for linear kinetic equations in the iffusion it. SIAM J Sci Comput 2008;31: [7] Roger M, Caliot C, Crouseilles N, Coelho PJ. A hybri transport-iffusion moel for raiative transfer in absorbing an scattering meia. J Comput Phys 2014;275: [8] Wang L, Jacques S. Hybri moel of Monte Carlo simulation an iffusion theory for light reflectance by turbi meia. J Opt Soc Am A 1993;10: [9] Tarvainen T, Vauhkonen M, Kolehmainen V, Arrige S, Kaipio J. Couple raiative transfer equation an iffusion approximation moel for photon migration in turbi meium with low-scattering an non-scattering regions. Phys Me Biol 2005;50: [10] Gorpas D, Yova D, Politopoulos K. A three-imensional finite elements approach for 24

26 the couple raiative transfer equation an iffusion approximation moeling in fluorescence imaging. J Quant Spectrosc Raiat Transfer 2010;111: [11] Tarvainen T, Kolehmainen V, Arrige S, Kaipio J. Image reconstruction in iffuse optical tomography using the couple raiative transport-iffusion moel. J Quant Spectrosc Raiat Transfer 2011;112: [12] Roger M, Crouseilles N. A ynamic multi-scale moel for transient raiative transfer calculations. J Quant Spectrosc Raiat Transfer 2013;116: [13] Crouseilles N, Degon P, Lemou M. A hybri kinetic flui moel for solving the Vlasov BGK equation. J Comput Phys 2005;203: [14] Degon P, Dimarco G, Mieussens L. A multiscale kinetic flui solver with ynamic localization of kinetic effects. J. Comput Phys 2010;229: [15] Lemou M, Mieussens L. A New Asymptotic Preserving Scheme Base on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit. SIAM J Sci Comput 2008;31: [16] Roger M, Crouseilles N, Coelho PJ. The micro-macro moel for transient raiative transfer simulations. Proceeings of the 15 th International Heat Transfer Conference, August. Kyoto, Japan, [17] Marshak R. Note on the spherical harmonics metho as applie to a Milne problem for a sphere. Phys Rev 1947;71: [18] Hunter B, Guo Z. Conservation of asymmetry factor in phase function iscretization for raiative transfer analysis in anisotropic scattering meia. Int J Heat Mass Transfer 2012;55:

27 FIGURE CAPTIONS Figure 1 - Decomposition of the omain into macroscopic an kinetic subomains. Figure 2 - Introuction of a buffer zone B between the macroscopic an the kinetic subomains. Figure 3 - Schematic of the cube subjecte to an incient short-pulse laser. Figure 4 - Preicte transmittance for ω=0.5 an g=0. a) τ =10, gri , 1 st orer schemes. b) τ =10, gri , 2 n orer schemes. c) τ =20, gri , 1 st orer schemes. ) τ =20, gri , 2 n orer schemes. Figure 5 - Preicte reflectance for ω=0.5 an g=0. (a) τ =10, gri , 1 st orer schemes. (b) τ =20, gri , 1 st orer schemes. Figure 6 - Preicte transmittance for ω=0.5 an g=0.3. (a) τ =10, gri , 1 st orer schemes. (b) τ =10, gri , 2 n orer schemes. (c) τ =20, gri , 1 st orer schemes. () τ =20, gri , 2 n orer schemes. Figure 7 - Preicte reflectance for ω=0.5 an g=0.3. (a) τ =10, gri , 1 st orer schemes. (b) τ =20, gri , 1 st orer schemes. Figure 8 - Preicte transmittance for τ =20, ω=0.9 an g=0. (a) Gri , 1 st orer schemes. (b) Gri , 1 st orer schemes. 26

28 (c) Gri , 1 st orer schemes. () Gri , 2 n orer schemes. Figure 9 - Preicte reflectance for τ =20, ω=0.9 an g=0. (a) Gri , 1 st orer schemes. (b) Gri , 2 n orer schemes. Figure 10 - Preicte transmittance for τ =20 an ω=0.9. (a) g=0.3, gri , 1 st orer schemes. (b) g=0.3, gri , 1 st orer schemes. (c) g=0.6, gri , 1 st orer schemes. () g=0.6, gri , 1 st orer schemes. Figure 11 - Influence of the gri on the preicte transmittance for τ =20, ω=0.9 an g=

29 Figure 1 - Decomposition of the omain into macroscopic an kinetic subomains. 28

30 Figure 2 - Introuction of a buffer zone B between the macroscopic an the kinetic subomains. 29

31 Figure 3 - Schematic of the cube subjecte to an incient short-pulse laser. 30

32 8E-05 (a) 8E-05 (b) Transmittance 6E-05 4E-05 Transmittance 6E-05 4E-05 2E-05 2E t (nanosecons) t (nanosecons) Transmittance 4E-09 3E-09 2E-09 1E-09 (c) Transmittance 3E-09 2E-09 1E-09 () t(nanosecons) t(nanosecons) Figure 4 - Preicte transmittance for ω=0.5 an g=0. a) τ =10, gri , 1 st orer schemes. b) τ =10, gri , 2 n orer schemes. c) τ =20, gri , 1 st orer schemes. ) τ =20, gri , 2 n orer schemes. 31

33 Reflectance 5E-02 4E-02 3E-02 2E-02 Reflectance 4E-02 3E-02 2E-02 1E-02 (a) 1E-02 (b) t (nanosecons) t(nanosecons) Figure 5 - Preicte reflectance for ω=0.5 an g=0. (a) τ =10, gri , 1 st orer schemes. (b) τ =20, gri , 1 st orer schemes. 32

34 Transmittance 1E-04 8E-05 6E-05 4E-05 2E-05 (a) Transmittance 8E-05 6E-05 4E-05 2E-05 (b) t(nanosecons) t(nanosecons) Transmittance 7E-09 6E-09 5E-09 4E-09 3E-09 2E-09 1E-09 (c) Transmittance 5E-09 4E-09 3E-09 2E-09 1E-09 () t(nanosecons) t(nanosecons) Figure 6 - Preicte transmittance for ω=0.5 an g=0.3. (a) τ =10, gri , 1 st orer schemes. (b) τ =10, gri , 2 n orer schemes. (c) τ =20, gri , 1 st orer schemes. () τ =20, gri , 2 n orer schemes. 33

35 Reflectance 2.0E E E-02 (a) R(t)/4 Reflectance 2.0E E E E E-03 (b) t(nanosecons) R(t()/ t(nanosecons) Figure 7 - Preicte reflectance for ω=0.5 an g=0.3. (a) τ =10, gri , 1 st orer schemes. (b) τ =20, gri , 1 st orer schemes. 34

36 Transmittance 8E-07 6E-07 4E-07 (a) Transmittance 4E-07 3E-07 2E-07 (b) 2E-07 1E t(nanosecons) t(nanosecons) Transmittance 3E-07 2E-07 1E-07 (c) Transmittance 3E-07 2E-07 1E-07 () t(nanosecons) t(nanosecons) Figure 8 - Preicte transmittance for τ =20, ω=0.9 an g=0. (a) Gri , 1 st orer schemes. (b) Gri , 1 st orer schemes. (c) Gri , 1 st orer schemes. () Gri , 2 n orer schemes. 35

37 Reflectance 8E-02 6E-02 4E-02 Reflectance 8E-02 6E-02 4E-02 2E-02 (a) 2E-02 (b) t(nanosecons) t(nanosecons) Figure 9 - Preicte reflectance for τ =20, ω=0.9 an g=0. (a) Gri , 1 st orer schemes. (b) Gri , 2 n orer schemes. 36

38 Transmittance 3E-06 2E-06 1E-06 (a) Transmittance 2E-06 1E-06 (b) t(nanosecons) t(nanosecons) Transmittance 2E-05 1E-05 Transmittance 2E-05 1E-05 () t(nanosecons) t(nanosecons) Figure 10 - Preicte transmittance for τ =20 an ω=0.9. (a) g=0.3, gri , 1 st orer schemes. (b) g=0.3, gri , 1 st orer schemes. (c) g=0.6, gri , 1 st orer schemes. () g=0.6, gri , 1 st orer schemes. 37

39 Transmittance 1.5E E E-04 50x20x20 100x40x40 200x50x50 300x20x t (nanosecons) Figure 11 - Influence of the gri on the preicte transmittance for τ =20, ω=0.9 an g=