PARALLEL COMPUTING OF THREE-DIMENSIONAL MONTE CARLO SIMULATION OF TRANSIENT RADIATIVE TRANSFER IN PARTICIPATING MEDIA A. Sawetprawichkul, P.-f. Hsu, and K. Mitra Mechanical and Aerospace Engineering Department Florida Institute of Technology Melbourne, FL 32901 ABSTRACT A three-dimensional transient radiative transport model using Monte Carlo (MC) method was implemented in a Beowulf-class parallel computer system. The main purpose of this study is to simulate the light pulse transport inside the absorbing and scattering media such as biological tissues. The threedimensional results for the case of infinite height and width but finite thickness configuration have good agreement with our prior one-dimensional MC results. The major advantage of MC method is its flexibility and simplicity to simulate the photon movement in arbitrary geometry and complex boundary condition. Since the error bound of MC method is inversely proportional to the square root of the number of statistical samplings, it requires a large number of samples to reach the satisfactory accuracy. Therefore, the primary drawback of MC methods is that it is a computationally intensive method. However, the method is very adaptable to parallel computing, i.e., coarse grained algorithm. Parallel computing is introduced to improve the performance of this method. The parallel system is based on a cluster of commodity-class processors with a standard MassagePassing-Interface (MPI). MPI is a library of subprograms. It is chosen as a tool to implement parallel programming for its portability to other similar systems and the very low cost compared to the conventional supercomputer systems. a c G g I L q NOMENCLATURE absorption coefficient, 1/m propagation speed of radiation transport in the medium, m/s incident radiation or integrated intensity, W/m2 asymmetric factor radiation intensity, W/m2sr medium thickness, m radiative flux in x direction, W/m2 sˆ t t* tp x,y,z x* unit vector along a given direction time, s dimensionless time, tct pulse width, s position coordinate, m dimensionless distance, tx Greek symbols polar angle, rad t extinction coefficient, 1/m circumferential or azimuthal angle, rad s scattering coefficient, 1/m optical thickness of the medium scattering phase function solid angle, sr scattering albedo Superscript * dimensionless quantity Subscript p pulse INTRODUCTION The transient radiative transfer problems are difficult to solve analytically. Traditional radiation studies mostly neglect the transient term of radiative transfer equation (RTE).1,2 The transient radiation, however, must be considered in new applications such as the pulsed laser interaction within the media.3-6 Several numerical techniques have been studied and used to model the light transport in participating media. The model is related to the laser radiation in tissue for either therapeutic or diagnostic purposes. The simulated diffuse reflectance and transmittance are used to distinguish the tissues with cancer from normal tissues.7 The Monte Carlo algorithm was first introduced into the laser tissue interaction applications by Wilson and Adam.8 Many researchers have been developing the model to simulate the light distributions in tissue. Hasegawa, et al.9 as well as Brewster and Yamada6. The 1 American Institute of Aeronautics and Astronautics anisotropic scattering results were predicted based on the results of equivalent isotropic scattering media using isotropic scaling law.10 However, the results agreed for thick medium only. Most of the previous works were focused on steady state solutions. Recently, some numerical analysis of the transient radiative transfer has been studied. The results of pulse radiation source using different models have been reported.11 The conclusion is that the transient radiative signature is quite different among different models. Kienle, et al.12 studied the absorption and reduced scattering coefficients of twolayered turbid media by inverting the time-resolved reflectance. Their results, however, were limited to several constraints such as requirement of a known value of the first layer thickness and a close approximation of the property data. Integral equation formulations were recently developed by Tan and Hsu13 and Wu14 for transient problems. The results from integral equation formulation compare very well with the previous study of Monte Carlo simulation.15 The latter algorithm is based on that by Fleck.16 The algorithm used in this study is verified with other methods such as the discrete ordinate method based on a high order up-wind scheme17 and Volterra integral equation formulation13. By nature of the MC method, the accuracy can be improved by increasing the number of samplings. However, the drawback is that the computational time increases proportionally. Advancement in modern computers and parallel computing is a solution for this problem. The other possible alternatives are the reversed MC method and quasi-MC method. However, these are subjected to different limitations. Parallel computing has become a very attractive approach of the need for more computational power in many scientific and engineering applications.18 Weather forecast and computer animation are good examples for the need for parallel computing. The parallel computer architecture used in this study is based on the Beowulf concept developed in NASA/Goddard (Stirling, 1994). It is simply a collection of computers with commodity processors and parts working together to solve a problem efficiently and in less time and less cost. The coding is based on Single Program Multiple Data model (SPMD) and Message Passing Interface (MPI) is one of the two commonly used standard libraries to achieve parallelization. The system consists of one root or head node and the other computers are considered as slave or compute nodes. The communication among nodes, in our case, is through a private, fast Ethernet. Details of parallel computing will be discussed in the following section. Performance and efficiency of the algorithm are measured. The goal of this study is to understand the transient radiation in multi-dimensional participating media as well as implementing parallel computing to improve the performance and accuracy of the model. In the near future, the non-uniform property distribution will be considered and the model predictions will be compared with experimental data. The Radiative Transfer Equation The transient radiative transfer equation in direction ŝ is given as19-21 I x , sˆ, t I x , sˆ, t c t x t I x , sˆ, t aIb ( x , sˆ, t ) I x , sˆ, t sˆ, sˆ d 4 4 (1) The RTE is an integro-differential equation. The first term on the right hand side is the radiation loss due to absorption and scattering. The medium is assumed to be cold and its temperature distribution is not responsive to the pulse laser incidence. Hence the emission gain (the second term on the right hand side) may be neglected. Moreover, the radiative properties of the medium and the boundary are assumed to be constant through the transient process. The last term on the right hand side is the radiation gain by in-scattering. Equation (1) is a mathematical model of transient radiative transfer problems. Traditional approach is to solve the equation either analytically or numerically for the solutions. The Monte Carlo algorithm is different from traditional numerical methods such that it incorporates the probability distribution and physical laws to simulate the physical movement of the energy bundles and eventually to solve for solutions. Next section discusses the fundamental of the Monte Carlo algorithm. THE MONTE CARLO ALGORITHM The Monte Carlo simulation is a stochastic method that is based on the random sampling of variables from probability distribution. The history of the energy bundle is tracked as it propagates within the medium. Each energy bundle is assigned a fixed weight initially and traced along the moving path. The bundle is either absorbed or scattered during its interaction within the medium. The history of bundles is recorded until either it escapes the boundary or is totally absorbed within the medium. The bundle is emitted into the medium through the surface. The traveling distance to move the bundle before any interaction (absorption or scattering) is then calculated. After the bundle is moved according to the initial direction, the new location is determined. If the new location is still within the medium, it can be either absorbed or scattered. Otherwise, the reflectance and 2 American Institute of Aeronautics and Astronautics transmittance is updated when the bundle reaches the surface. In case of scattering, a new direction is calculated as well as the new traveling distance. The process is repeated until the last bundle has reached the surface or being absorbed. Followings are details of each calculation steps. Sampling random variables is the basis of the algorithm. The sampling method is discussed in details and the mapping technique is introduced. Sampling random variables The principle of Monte Carlo method is based on the random sampling of variables from probability distribution. A probability density function p(x) defines the distribution of random variable x over the interval (a,b). This variable x may be the variable step size, the angle of deflection, or the azimuth angle during scattering. The simulation of light propagation requires a value for x randomly and repeatedly based on pseudo-random numbers, generated by a computer. The random number, , is uniformly distributed over the interval (0,1). Though the variable x may be not uniformly distributed over (a,b). The non-uniform probability density function p(x) can be expressed as23 x p( x )dx for (0,1) (2) a This mapping technique will be used to represent a non-uniformly distributed variable, such as the step size or the deflection angle, by uniformly distributed random numbers. Initialize bundle The incident beam can be collimated or diffuse. For simplicity, the initial direction is assumed to be normal to the surface (orthogonal). The weight is initialized to unity. The effect of a refractive-indexmismatched interface between the medium and the ambient is neglected in this study. Generating the step size The step size or the bundle’s moving distance before next interaction, s, has the interval (0,). The probability density function of step size is the ratio between the power extinguished in distance ds and the ( s ) total power extinguished. It is proportional to e t 24 as : I e t s ds P( s )ds t o t e t s ds t s ds Ioe 0 Applying the mapping technique yields: s ln(1 ) t or s ln t Moving the bundles A bundle movement can be represented by three spatial coordinates (x,y,z) and two directional angles (). However, the deflection angle is with respect to the previous trajectory, which is arbitrary. It is more convenient to describe the bundle position by three Cartesian coordinates (x,y,z) and three directional cosines (x,y,z) A new position is updated by x x x si y y y si (5) z z z si Bundle absorption There are two ways to determine bundle absorption. First, the bundle weight is constant until being absorbed all in once. Another way is to split the bundle weight into parts (absorbed and scattered). The absorbed fraction is 1-, where is the albedo. The remained bundle is still scattering until its weight is below the threshold. In this study, the first technique is used. The albedo is compared with the random numbers to determine the absorption. If random number() > absorbed Otherwise scattered Bundle scattering After the bundle survives the absorption, it is ready to be scattered. There are two angles involving in scattering process: a deflection angle, (0 ) , and an azimuth angle, (0 2 ) . After a bundle is scattered, its trajectory is deflected from the previous direction by a deflection angle, The scattering phase function describes the probability distribution for the cosine of the deflection angle, cos. The Henyey-Greenstein phase function was originally proposed for galactic scattering approximates Mie scattering by particles comparable in size to the wavelengths of light.25 The scattering phase function is defined as21 ( sˆ' , sˆ) 1 g 1 g2 2 2 gsˆ'sˆ 32 (6) The asymmetric factor, g, varying between –1 and 1, characterizes the angular scattering distribution. Isotropic scattering has a value of g as zero. Highly forward and highly backward scattering has a value of g near 1 and –1 respectively. The same phase function is also described by Modest21, but the product of two unit vectors is used instead of the cosine of the polar angle. The latter expression is not a correct HG phase function. (4) The cosine of the deflection angle, cos, can be expressed in term of the random number, . The (3) 3 American Institute of Aeronautics and Astronautics derivation of anisotropic scattering by the mapping technique was discussed in details in previous study.26 In case of isotropic scattering (g=0), the probability density function becomes (½)sin. The mapping technique yields the cosine angle as cos1 (1 2 ) (7) The azimuth angle () is assumed to be uniformly distributed within the interval [0,2]. The probability density function of the azimuth angle, p(), is constant and equals to 1 / 2 . Again, the mapping technique yields the azimuth angle as: with 512 MB SDRAM. The total system memory space is 24 GB. The system is designed with one head node and 47 compute nodes. The interconnect between the nodes are channel-bonded fast Ethernet, i.e., double bandwidth of 200 Mbps. The job queuing is provided by the portable batch system (OpenPBS), which is built on top of the Maui job scheduler. The serial MC code was modified to run on parallel processors by implementing Message Passing Interface (MPI). Detailed hardware and software configuration information is given in website of http://olin.fit.edu/beowulf/. Message Passing Interface Message Passing Interface is a library of subprograms in C, C++ or FORTRAN language.18 The (8) 2 major advantage of MPI is its portability in parallel programming. The basic of MPI is to transmit the data Once both the deflection and the azimuth angles from one processor to another. There are two types of are found, the new direction may be calculated using MPI communication: point-to-point and collective coordinate transformation. Cashwell23 explained in communication. Point-to-point communication is details the transformation method using complex between one processor to another. Collective variables. The same results can be derived by using communication, however, can send or receive the data rotation matrix as well. The new directional cosine from/to one processor to/from many processors. (x',y',z') is calculated from the current direction In this study, we use Single Program Multiple Data (x,y,z) as: model, which means all the nodes will use the same sin program but may produce different data depending on x z cos y sin x cos x' the given input. Pseudo code of the Monte Carlo 2 1 z algorithm under the MPI environment is given below: sin (10) 'y y z cos x sin y cos Start the MPI function call by including the processor 1 z2 directive and determining how many processors z' sin cos 1 z2 z cos involved and the rank of each processor. If the process is root node (rank = 0) - Read the input data (such as medium thickness, However, if the bundle direction is close to the zscattering and absorption coefficient, total number axis, the above equations are no longer valid since of bundles and time steps). z~1. The following equations should be used instead: - Broadcast these values to all processors to be used for calculation. x' sin cos Each processor performs the calculation based on 'y sin sin (11) different random number sets generated according to z' z cos the rank of each processor. After all processors finish calculation, the output is gathered/collected by root node. PARALLEL COMPUTING Due to the nature of the Monte Carlo simulation, a Print the output from root node. stochastic method, it requires a number of samples to Terminate the execution by shutting down MPI call. satisfy the required accuracy. Increasing the samples or In parallel programming, it is also important to bundles can improve the accuracy, but also requires clarify the scope of global and local variables. Since all more time to compute. The previous MC work by the processors will perform the computation and authors were conducted on a 633 MHz 21164A Alpha communicate with the root or node zero to share data. processor workstation (Sawetprawichkul, et al. 2000). The quality of parallel algorithms can be measured by It could take several hours to complete the calculation speedup and efficiency. by the workstation. Recently, a 48-node IBM PC Quinn36 gave the definition of the speedup as “ratio cluster based was installed and used in this study. This between the time needed for the most efficient sequential system is capable to extend to 96-node. Each node is algorithm to perform a computation and the time needed an IBM x330 e-series, Pentium III 866 MHz processor to perform the same computation on a machine 4 American Institute of Aeronautics and Astronautics incorporating pipelining and/or parallelism”. The efficiency of the algorithm using p processors is the speedup divided by p. Pacheco18 also defined the speedup as the ratio between runtime of a serial solution and runtime of the parallel solution with p processes. The MC algorithm speedup typically scales up very well with the number of processor, i.e., linear. For most other parallel algorithms speedup, however, have lesser value due to overhead from communication time. RESULTS AND DISCUSSION In previous study37, the one-dimensional Monte Carlo simulation, based on the work by Hsu16, has been verified with the discrete ordinates method.17 The simulation compares the three linearly anisotropic scattering phase functions, i.e., forward, isotropic, and backward; respectively (Fig. 1). The media has the optical thickness of = 10, albedo of = 0.998, and an isotropic scattering phase function. The simulation involved several hours run time in workstation due to highly scattering media. Figure 1 shows a good agreement of transient solutions between the onedimensional Monte Carlo and the discrete ordinates method for all three cases. In this study, the MC algorithm discussed on the previous section was developed to treat threedimensional (3D) rectangular geometry. In the following 3D cases, the optical depth is defined to be the product of extinction coefficient and the plate thickness. The 3D rectangular medium has equal width and height to simplify analysis and the aspect ratio (AR) is defined to be the length ratio of width to thickness. The light irradiated at z = 0 plane or a node on the plane. For the case of AR = 1000, the 3D results have been validated with one-dimensional MC results as shown in Figure 2. Both reflectance and transmittance generated by 3D model agree very well with the 1D model. For long time solutions, both reflectance and transmittance merge to the same value as in the steady state solutions. The number of total energy bundles used in the simulation is 107. Figure 3 shows the convergence rate of the solution as the number of energy bundles increases. In the calculation of the RMS error, the MC solution with 108 energy bundles is used as the basis. The relative RMS error versus bundle number reflects the error bound of -1/2 slope as predicted by the central limit theorem. A parametric study of different aspect ratios is shown in Figs. 4 and 5. The 3D model with uniformly distributed source over the entire x-y plane was used to compare with the 1D results. Each case emitted ten millions energy bundles in total. It clearly shows that the as the aspect ratio increases, the transient signals are closer to those of the 1D geometry. However, it is surprising to find that one-dimensionality can only be approached at a very large AR = 1000. This is quite different from the steady state radiative transport, in which the AR = 10 case will produce close to onedimensional result. The calculated reflectance and transmittance at different locations, center of the plane, edge and corner, are presented in Figs. 6 and 7 for the case of AR = 10. Even though the source is uniformly distributed, but the location of exiting bundles affects the signal magnitude due to out-scattering near the boundary. At the corner, the transmittance and reflectance is the smallest of the three positions. Similar results are found for AR = 100 and 1000. At lower AR, the three-dimensional effect is most evident. The case of pulse irradiation at a single node in the center of the z = 0 plane is studied. The temporal distributions of reflectance and transmittance along the xaxis are given in Figs. 8 and 9. Reflectance gradually decreases as the time increases. However, the transmittance value increases from t* =15 to 30, then starts to decrease. The temporal trend is similar to those shown in Figs. 6 and 7. Even with 109 bundles used in the simulation, certain degree of fluctuation is still observed in the signals near the edge. Physically, very few photons will travel to the regions near the edges and corners, which leads to weak signal magnitude. It is difficulty to obtain adequate sampling unless the total bundle number is further increased. An attempt of 1010 bundles was used but didn't reduce the fluctuation much. Speedup is calculated to measure the efficiency of the algorithm. Figure 10 shows the speedups of the ideal case and actual performance at different processor number. The parallel efficiency (speedup divided by processor number) of actual calculation decreases slightly at large number of processor. At 40 processors, the parallel efficiency is 95%. It is suspected that the communication overhead increases as more processors are involved, thus the reduced efficiency. Typical computing time, for example, the case of Fig. 8, is 17 minutes with 40 processors for 10 9 energy bundle simulations. The computational time is proportional to the number of bundle at a fixed number of processor. CONCLUSION The three-dimensional Monte Carlo modeling has been developed to simulate the transient radiative transfer in participating media. The results were validated with the one-dimensional Monte Carlo solutions for the case of infinite height and width. Parallel computing was been implemented to improve the 5 American Institute of Aeronautics and Astronautics efficiency of the algorithm. It is found that the three dimensional effect is very evident even with large aspect ratio. It is only at aspect ratio of 1000, the three-dimensional results are close to that of onedimensional code. Nearly linear speedup was achieved with the algorithm except at large number of processors. At 40 processors, the parallel efficiency drops to 95% due to communication overhead. ACKNOWLEDGEMENTS This work is supported by Sandia National Laboratories with contract AW-9963 and Dr. Shawn P. Burns is the program manager of this project. The parallel system is funded by National Science Foundation MRI program grant EIA-0079710 and Dr. Rita V. Rodriguez is the program director. REFERENCES Sterling, T., D. J. Becker, D. Savarese, J. E. Dorband, U. A. Ranawak, C. V. Packer, "Beowulf: A Parallel Workstaion for Scientific Computation," Proceedings, Int. Conf. on Parallel Processing (1995). 1. 2. 3. 4. 5. 6. 7. Fiveland, W. A. “Three-Dimensional Radiative Heat Transfer Solution by the Discrete-Ordinates Method.” J. Thermophysics. Heat Transfer. 2.4 (1988): 309-316. Raithby, G. D., and E. H. Chui. “A Finite volume method for predicting a radiative heat transfer in enclosures with participating media.” J. Heat Transfer 112 (1990): 415-423. Mitra, K., and S. Kumar. “Application of Transient Radiative Transfer Equation to Oceanographic Lidar.” ASME HTD 353 (1997): 359-365. Gemert, M. J. C., an A. J. Welch. “Clinical Use of laser-Tissue Interaction.” IEEE Eng. Med. Biol. Mag. (1989): 10-13. Ishimaru, A. “Diffusion of Light in Turbid Material.” Applied Optics 28.12 (1989): 22102215. Brewster, M. Q., and Y. Yamada. “Optical Properties of Thick, Turbid Media from Picosecond Time-Resolved Light Scattering Measurements.” Int. J. Heat and Mass transfer 38 (1995): 2569-2581. Jacques, S. L., L. Wong, and A. H. Hielscher. “Time-Resolved Photon Propagation in Tissues.” Optical-Thermal Response of Laser Irradiated Tissue. Ed. Welch, A. J., and Martin J.C. van gemert. New York: Plenum Press, 1995. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Wilson, B. C., and G. Adam. “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue.” Med. Phys. 10 (1983): 824-830. Hasegawa, Y., et al. “Monte Carlo Simulation of Light Transmission through Living Tissues.” Appl. Opt. 30 (1991): 4515-4520. (all author's names) Guo, Z., S. Kumar, and S. Maruyama. “TimeResolved Evaluation of Two-Dimensional Scaled Isotropic Results for Transient Radiative Transfer in Anisotropic Scattering Media.” The 34th National Heat Transfer Conf., Pittsburgh, PA, August 2000. Mitra, K., and S. Kumar. “Development and Comparison of Models for Light-Pulse Transport Through Scattering-Absorbing Media.” Applied Optics 38.1 (1999): 188-196. Kienle, A., et al. “Investigation of two-layered Turbid Media with Time-Resolved Reflectance.” Appl. Opt. 37.28 (1998): 6852-6862. Tan, Z. M., and P.-f. Hsu. “An Integral Formulation of Transient Radiative Transfer.” ASME J. Heat transfer. 123(3): pp.466-475 (2000). Wu, C.-Y. “Propagation of scattered radiation in a participating planar medium with pulse irradiation.” J. Quant. Spectrosc. Radiat. Transfer, 64.5 (1999): 537-548. Hsu, P.-f. “Effects of Multiple Scattering and Reflective Boundary on the Transient Radiative Transfer Process.” Int. J. Thermal Sciences 40(6), pp. 539-549, (2000). Fleck, J. A. “The Calculation of Nonlinear Radiation Transport by a Monte Carlo Method.” Methods in Computational Physics. Ed. B. Alder, S. Fernbach and M. Rotenberg. New York and London: Academic Press 1 (1963): 43-65. Sakami, M., K. Mitra, and P.-f. Hsu. “Transient Radiative Transfer in Anisotropically Scattering Media using Momotonicity-Preserving Schemes.” The ASME Int. ME Cong. & Expo. Orlando, 2000. Pacheco, P. S. Parallel Programming with MPI. California: Morgan Kaufmann, 1997. Siegel, R., and J. R. Howell. Thermal radiative Heat transfer. 3rd ed. New York: McGraw-Hill, 1992: 864. Ozisik, M.N., Radiative Transfer and Interaction with Conduction and Convection. New York: Wiley, 1973: 251. Modest, M. F. Radiative Heat Transfer. New York: McGraw-Hill, 1993: 303. Lux, I. and K. Koblinger. Monte Carlo Particle Transport Methods: Neutron and Photon Calculations. Boca Raton, Florida: CRC Press, 1991. Cashwell, E. D., and C. J. Everett. Monte Carlo Method for Random Walk Problems. Great Britain: The Lewes Press, 1959. 6 American Institute of Aeronautics and Astronautics 24. Wang, L., S. L. Jacques, and L. Zheng. “MCML— Monte Carlo Modeling of Light Transport in Multi-layered tissues.” Computer Methods and Programs in Biomedicine. 47 (1995): 131-146. 25. Henyey, L. G., and J. L. Greenstein. “Diffuse Radiation in the Galaxy.” Astrophys. J. 93 (1941): 70-83. 26. Sawetprawichkul, A. “ Application of the Monte Carlo Method in the Solution of Radiative Transfer Problems.” M.S. Thesis. U. of Arizona, 1999. 27. Manke, J. W. “Parallel Computing in Aerospace.” Parallel Computing. 27.4 (2001): 329-336. 28. Rifai, S. M. “Automotive Design Applications of Fluid Flow Simulation on Parallel Computing Platforms.” Computer Methods in Applied Mechanics and Engineering. 184.2 (2000): 449466. 29. Thole, C. and K. Stueben. “Industrial Simulation on Parallel Computers.” Parallel Computing. 25.13 (1999): 2015-2037. 30. Drake, J. and I. Foster. “Introducing to the Special Issue on Parallel Computing in Climate and Weather Modeling.” Parallel Computing. 21.10 (1995): 1539-1544. 31. Turk, J. A. “Acceleration Techniques for the Radiative Analysis of General Computational Fluid Dynamics Solutions using Reverse MonteCarlo Ray Tracing.” Ph.D. Dissertation. Virginia Polytechnic Institute and State University, 1994. 32. Martino, R. L., et al. “Role of High Performance Parallel Computing in Biological Research.” Aunnual International Conference of the IEEE Engineering in Medicine and Biology Society. 16.2 (1994): 1386-1387. 33. Board, J. “Grand Challenges in Biomedical Computing.” Critical Reviews in Biomedical Engineering. 20.1. (1992): 1-24. 34. Wu, D., et al. “Monte Carlo Simulation of Light Propagation in Skin Tissue Phantoms using a Parallel Computing Method.” Proceedings of SPIE. 3914 (2000): 291-299. 35. Gropp, William., et al. MPI-The Complete Reference Vol 2. Massachusetts: MIT, 1998. 2 vols. 36. Quinn, Michael. Parallel Computing: Theory and Practice. 2nd ed. New York: McGraw-Hill, 1994. 37. Sawetprawichkul, A., P.-F. Hsu, K. Mitra., and M. Sakami “A Monte Carlo Study of the Transient Radiative transfer Within The One-Dimensional Multi-Layered Slab.” International Mechanical Engineering Congress and Exposition, Orlando, FL, November 2000: 4-9. 7 American Institute of Aeronautics and Astronautics 0 10 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 a =0.9 1 a =0 1 p Aspect ratio (AR) = 1000 10-3 a =-0.9 RMS error Transmittance 1 10 -1 10 -2 =10, =0.998 MC DOM -4 10 0 20 40 60 80 100 10 t* Fig. 1 Comparison of one-dimensional Monte Carlo and discrete ordinates solutions for linear anisotropic scattering media. 6 10 number of bundles 10 -1 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 p x:y:z = 1000:1000:1 Aspect ratio (AR) = 1000 -2 10 106 bundles 10 10 -3 7 -1 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 10 10 Fig. 3 RMS error of the 3D Monte Carlo solutions. Reflectance Reflectance & Transmittance 10 5 p -2 6 10 bundles 10 -3 1D RF 1D RF 3D TR 3D TR 1D -4 0 10 20 30 40 50 AR = 10 AR =100 10 60 70 Non-dimensional time, t* Fig. 2 Validation of the reflectance and transmittance of the 3D model with 1D model using infinite slab in 3D code. -4 AR = 1000 0 10 20 30 40 50 60 70 Non-dimensional time, t* Fig. 4 Parametric study of the reflectance for 3D using different aspect ratios to simulate 1D case. 8 American Institute of Aeronautics and Astronautics 10 -2 point A (center) point B (edge) point C (corner) Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 p 10 6 10 Transmittance Transmittance 10 bundles -5 -3 1D Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 AR = 10 10 p AR = 100 -4 9 10 AR = 1000 0 10 20 30 40 50 60 10 bundles, AR = 10 -6 0 70 10 Fig. 5 Parametric study of the transmittance for 3D using different aspect ratios to simulate 1D case. Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 -4 p p 9 10 bundles AR = 10 10 -5 10 -6 10 -7 Reflectance Reflectance 50 60 Fig. 7 Transmittance measurement at different locations. 10 9 10 40 -3 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 10 30 Non-dimensional time, t* Non-dimensional time, t* 10 20 -4 point A (center) point B (edge) point C (corner) 10 bundles AR = 10 t* = 15 t* = 30 t* = 45 t* = 60 -5 0 10 20 30 40 50 60 Non-dimensional time, t* Fig. 6 Reflectance measurement at different locations. 0 0.2 0.4 0.6 0.8 Non-dimensional distance, x* 1 Fig. 8 The distribution of the reflectance at different time (a single node irradiation). 9 American Institute of Aeronautics and Astronautics 40 10 -5 t* = 15 t* = 30 t* = 45 t* = 60 p 109 bundles AR = 10 35 Actual speedup 30 Ideal speedup 25 speedup Transmittance 10 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 -6 20 15 Optical depth, = 10 Albedo, = 0.998 pulse width, t * = 1 10 10 -7 p 5 1000 timesteps 0 0 0.2 0.4 0.6 0.8 1 Non-dimensional distance, x* Fig. 9 The distribution of the transmittance at different time (a single node irradiation). 0 5 10 15 20 25 30 35 40 number of processors Fig. 10 The speedup results using different number of processors. 10 American Institute of Aeronautics and Astronautics