Abstract

The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing scattering media subjected to a collimated short laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. The accuracy and computational efficiency of this algorithm are examined firstly. Afterwards, effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor, and the shapes of laser pulse on time-resolved signals of transmittance and reflectance are investigated. Results of the present method are found to compare very well with the data from the literature. For an oblique incidence, the LBM results in this paper are compared with those by Monte Carlo method generated by ourselves. In addition, transient radiative transfer in a two-Layer inhomogeneous media subjected to a short square pulse irradiation is investigated. At last, the LBM is further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Several trends on the time-resolved signals different from those for refractive index of 1 (i.e. refractive-index-matched boundary) are observed and analysed.

© 2013 Optical Society of America

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  1. A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer115(1), 7–16 (1993).
    [CrossRef]
  2. J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer124(6), 1176–1181 (2002).
    [CrossRef]
  3. T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer35(3), 719–726 (1992).
    [CrossRef]
  4. F. Liu, K. M. Yoo, and R. R. Alfano, “Ultrafast Laser-Pulse Transmission and Imaging Through Biological Tissues,” Appl. Opt.32(4), 554–558 (1993).
    [CrossRef] [PubMed]
  5. M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag.8(4), 10–13 (1989).
    [CrossRef] [PubMed]
  6. K. J. Grant, J. A. Piper, D. J. Ramsay, and K. L. Williams, “Pulsed lasers in particle detection and sizing,” Appl. Opt.32(4), 416–417 (1993).
    [CrossRef] [PubMed]
  7. S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer33, 187–294 (1999).
    [CrossRef]
  8. H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).
  9. Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
    [CrossRef]
  10. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer126(4), 621–627 (2004).
    [CrossRef]
  11. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf.93(1-3), 349–367 (2005).
    [CrossRef]
  12. M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express19(20), 19627–19642 (2011).
    [CrossRef] [PubMed]
  13. Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt.40(19), 3156–3163 (2001).
    [CrossRef] [PubMed]
  14. M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
    [CrossRef]
  15. Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer16, 289–296 (2002).
  16. J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer53(19-20), 3799–3806 (2010).
    [CrossRef]
  17. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer123(3), 466–475 (2001).
    [CrossRef]
  18. C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf.64(5), 537–548 (2000).
    [CrossRef]
  19. P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
    [CrossRef]
  20. J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B44(2), 187–208 (2003).
    [CrossRef]
  21. M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
    [CrossRef]
  22. L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
    [CrossRef]
  23. S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
    [CrossRef]
  24. L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer129(8), 1069–1074 (2007).
    [CrossRef]
  25. L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf.109(7), 1297–1308 (2008).
    [CrossRef]
  26. X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
    [CrossRef]
  27. S. Succi, The Lattice Boltzmann Method for Fluid Dynamics and Beyond (Oxford University, 2001).
  28. W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).
  29. S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).
  30. S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007).
    [CrossRef]
  31. S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).
  32. B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).
  33. P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).
  34. A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
    [CrossRef]
  35. Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
    [CrossRef] [PubMed]
  36. H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.86(1), 016706 (2012).
    [CrossRef] [PubMed]
  37. S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012).
    [CrossRef]
  38. R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer7, 624–630 (1993).

2012

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.86(1), 016706 (2012).
[CrossRef] [PubMed]

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012).
[CrossRef]

2011

M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express19(20), 19627–19642 (2011).
[CrossRef] [PubMed]

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
[CrossRef] [PubMed]

2010

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).

M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
[CrossRef]

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

2009

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).

2008

S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).

L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf.109(7), 1297–1308 (2008).
[CrossRef]

2007

L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer129(8), 1069–1074 (2007).
[CrossRef]

S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007).
[CrossRef]

2006

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

2005

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf.93(1-3), 349–367 (2005).
[CrossRef]

S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).

2004

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer126(4), 621–627 (2004).
[CrossRef]

2003

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B44(2), 187–208 (2003).
[CrossRef]

2002

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer16, 289–296 (2002).

J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer124(6), 1176–1181 (2002).
[CrossRef]

2001

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer123(3), 466–475 (2001).
[CrossRef]

Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt.40(19), 3156–3163 (2001).
[CrossRef] [PubMed]

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).

2000

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf.64(5), 537–548 (2000).
[CrossRef]

1999

S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer33, 187–294 (1999).
[CrossRef]

X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
[CrossRef]

1993

A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer115(1), 7–16 (1993).
[CrossRef]

K. J. Grant, J. A. Piper, D. J. Ramsay, and K. L. Williams, “Pulsed lasers in particle detection and sizing,” Appl. Opt.32(4), 416–417 (1993).
[CrossRef] [PubMed]

F. Liu, K. M. Yoo, and R. R. Alfano, “Ultrafast Laser-Pulse Transmission and Imaging Through Biological Tissues,” Appl. Opt.32(4), 554–558 (1993).
[CrossRef] [PubMed]

R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer7, 624–630 (1993).

1992

T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer35(3), 719–726 (1992).
[CrossRef]

1989

M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag.8(4), 10–13 (1989).
[CrossRef] [PubMed]

Aber, J.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

Alfano, R. R.

Asinari, P.

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).

Baek, S. W.

M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
[CrossRef]

Bindra, H.

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.86(1), 016706 (2012).
[CrossRef] [PubMed]

Borchiellini, R.

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).

Chai, J. C.

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B44(2), 187–208 (2003).
[CrossRef]

Chen, S.

X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
[CrossRef]

Chugh, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Costa, M.

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Cuccia, D.

Dong, S. K.

Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
[CrossRef] [PubMed]

Gardner, A.

Garetz, B. A.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

Grant, K. J.

Guo, Z.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

Guo, Z. X.

Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer16, 289–296 (2002).

Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt.40(19), 3156–3163 (2001).
[CrossRef] [PubMed]

Hayakawa, C.

He, X.

X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
[CrossRef]

Ho, J. R.

W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).

Hsu, P. F.

L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf.109(7), 1297–1308 (2008).
[CrossRef]

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf.93(1-3), 349–367 (2005).
[CrossRef]

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer126(4), 621–627 (2004).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer123(3), 466–475 (2001).
[CrossRef]

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

Jiaung, W. S.

W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).

Kim, M. Y.

M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
[CrossRef]

Kumar, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Kumar, S.

Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer16, 289–296 (2002).

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt.40(19), 3156–3163 (2001).
[CrossRef] [PubMed]

S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer33, 187–294 (1999).
[CrossRef]

Kuo, C. P.

W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).

Lankadasu, A.

S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).

Liu, F.

Liu, L. H.

L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf.109(7), 1297–1308 (2008).
[CrossRef]

L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer129(8), 1069–1074 (2007).
[CrossRef]

Liu, L. J.

L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer129(8), 1069–1074 (2007).
[CrossRef]

Lu, X. D.

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf.93(1-3), 349–367 (2005).
[CrossRef]

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer126(4), 621–627 (2004).
[CrossRef]

Ma, Y.

Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
[CrossRef] [PubMed]

Majumdar, A.

A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer115(1), 7–16 (1993).
[CrossRef]

Martinelli, M.

Mathur, S. R.

J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer124(6), 1176–1181 (2002).
[CrossRef]

Menon, S.

M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
[CrossRef]

Mishra, S. C.

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012).
[CrossRef]

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).

S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).

S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007).
[CrossRef]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).

Mitra, K.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer33, 187–294 (1999).
[CrossRef]

Mondal, B.

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).

S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).

Murthy, J. Y.

J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer124(6), 1176–1181 (2002).
[CrossRef]

Oliva, A.

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Patil, D. V.

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.86(1), 016706 (2012).
[CrossRef] [PubMed]

Pavan Kumar, T. B.

S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).

Piper, J. A.

Qi, H.

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

Qiu, T. Q.

T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer35(3), 719–726 (1992).
[CrossRef]

Ramsay, D. J.

Rienzo, A. F. D.

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

Roy, H. K.

S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007).
[CrossRef]

Ruan, L. M.

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

Sakami, M.

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

Schweiger, H.

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Segarra, C. D. P.

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Siegel, R.

R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer7, 624–630 (1993).

Spanier, J.

Tan, H. P.

Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
[CrossRef] [PubMed]

Tan, Z. M.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer123(3), 466–475 (2001).
[CrossRef]

Tien, C. L.

T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer35(3), 719–726 (1992).
[CrossRef]

van Gemert, M. C.

M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag.8(4), 10–13 (1989).
[CrossRef] [PubMed]

Venugopalan, V.

Vernekar, R. R.

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012).
[CrossRef]

Wang, D. L.

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

Wang, J. M.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

Wang, S. G.

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

Welch, A. J.

M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag.8(4), 10–13 (1989).
[CrossRef] [PubMed]

Williams, K. L.

Wu, C. Y.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf.64(5), 537–548 (2000).
[CrossRef]

Yoo, K. M.

Zhang, R. A.

X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
[CrossRef]

Adv. Heat Transfer

S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer33, 187–294 (1999).
[CrossRef]

Appl. Opt.

IEEE Eng. Med. Biol. Mag.

M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag.8(4), 10–13 (1989).
[CrossRef] [PubMed]

Int. J. Heat Mass Transfer

T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer35(3), 719–726 (1992).
[CrossRef]

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer53(19-20), 3799–3806 (2010).
[CrossRef]

M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer53(25-26), 5682–5691 (2010).
[CrossRef]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006).
[CrossRef]

Int. J. Numer. Methods Heat Fluid Flow

A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011).
[CrossRef]

Int. J. Therm. Sci.

P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001).
[CrossRef]

J. Comput. Phys.

S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007).
[CrossRef]

X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys.152(2), 642–663 (1999).
[CrossRef]

J. Heat Transfer

L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer129(8), 1069–1074 (2007).
[CrossRef]

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer123(3), 466–475 (2001).
[CrossRef]

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer126(4), 621–627 (2004).
[CrossRef]

A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer115(1), 7–16 (1993).
[CrossRef]

J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer124(6), 1176–1181 (2002).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf.93(1-3), 349–367 (2005).
[CrossRef]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 169–179 (2002).
[CrossRef]

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf.64(5), 537–548 (2000).
[CrossRef]

L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf.109(7), 1297–1308 (2008).
[CrossRef]

L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010).
[CrossRef]

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf.73(2-5), 159–168 (2002).
[CrossRef]

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012).
[CrossRef]

J. Thermophys, Heat Transfer

Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer16, 289–296 (2002).

J. Thermophys. Heat Transfer

R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer7, 624–630 (1993).

Numer. Heat Transf. B

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B44(2), 187–208 (2003).
[CrossRef]

H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B35, 113–136 (2001).

Numer. Heat Transfer, Part A

S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).

S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).

Numer. Heat Transfer, Part B

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).

W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).

Opt. Express

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1), 016704 (2011).
[CrossRef] [PubMed]

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.86(1), 016706 (2012).
[CrossRef] [PubMed]

Other

S. Succi, The Lattice Boltzmann Method for Fluid Dynamics and Beyond (Oxford University, 2001).

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Figures (10)

Fig. 1
Fig. 1

(a) A planar medium subjected to collimated short-pulse radiation with an incident angle θ0, and (b) the shape of the short pulse.

Fig. 2
Fig. 2

Comparison of the time-resolved signals of transmittance by LBM with those by FVM [23] for β = 1.0 m−1 and ω = 1.0

Fig. 3
Fig. 3

Comparison of the time-resolved signals of transmittance and reflectance by LBM with those by FVM [23] for different values of the scattering albedo ω and the extinction coefficient β.

Fig. 4
Fig. 4

Effects of the angle of incidence on the time-resolved reflectance and transmittance.

Fig. 5
Fig. 5

The time-resolved reflectance and transmittance for three values of the anisotropically scattering coefficient: (a) the transmittance, and (b) the reflectance. The optical thickness is τL = 10 and the albedo is ω = 0.998.

Fig. 6
Fig. 6

The time-resolved transmittance and reflectance for the Gauss pulse (a) the transmittance, and (b) the reflectance.

Fig. 7
Fig. 7

(a) The model of the two-layer media irradiated by the short square pulse laser and (b) the time-resolved signal of reflectance for the two-layer media.

Fig. 8
Fig. 8

The time-resolved transmittance and reflectance for the case with n = 1.5, β = 1.0 m−1 and ω = 1.0, (a) the transmittance, and (b) the reflectance.

Fig. 9
Fig. 9

Distributions of the diffuse intensities at different directions and different non-dimensional time t* at (a) x = 0, and (b) x = L.

Fig. 10
Fig. 10

Effects of the refractive index on the transient signals, (a) the reflectance, and (b) the transmittance.

Tables (2)

Tables Icon

Table 1 Computational Efficiency of the LBM

Tables Icon

Table 2 Internal and External Diffuse Reflectivity on the Semitransparent Surface for n = 1.2, 1.5 and 1.8

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

n c o I(x,μ,t) t +μ I(x,μ,t) x =βI(x,μ,t)+ σ s 2 1 1 I(x,μ,t)Φ(μ, μ )d μ .
I= I c + I d .
d I c ds =β I c .
n c o I d (x,μ,t) t +μ I d (x,μ,t) x =β I d (x,μ,t)+ S d + S c =β I d (x,μ,t)+ S t .
I w m =0.
ρ o ( n )= 1 2 + (3 n +1)( n 1) 6 ( n +1) 2 + n 2 ( n 2 1) 2 ( n 2 +1) 3 ln ( n 1) ( n +1) 2 n 3 ( n 2 +2 n 1) ( n 2 +1)( n 4 1) + 8 n 4 ( n 4 +1) ( n 2 +1) ( n 4 1) 2 ln( n ).
ρ I ( n )=1 1 n 2 [1 ρ o ( n )].
I w m =(1 ρ I ) n 2 I 0 + ρ I π n w s m >0 | n w s m | I w m w m , n w s m <0.
I c S (0,μ,t)= I 0 [H(t)H(t t p )]δ(μ μ 0 ).
I c G (0,μ,t)= I 0 exp[4ln2× ( t t c t p ) 2 ]δ(μ μ 0 ).
I c S (x,μ, t * )= I 0 exp(βs)×[ H( t * βs)H( t * βs t p * ) ]×δ(μ μ 0 ).
I c G (x,μ, t * )= I 0 exp(βs)exp[4ln2× ( t * βs t c * t p * ) 2 ]δ(μ μ 0 ).
nβ I d t * +μ I d x +β I d = S t .
nβ I d I ˜ d Δ t * +μ I d x +β I d = S t .
μ I d x + β B I d = S t + nβ Δ t * I ˜ d .
I d x = 1 μ ( S t + nβ Δ t * I ˜ d β B I d ).
I d m (x+ e m Δ t * , t * +Δ t * ) I d m (x, t * ) Δx = 1 μ ( S t + nβ Δ t * I ˜ d β B I d ) = 1 μ β B ( B β S t ( t * )+ nB Δ t * I ˜ d m I d m (x, t * )). m=1,2,...,M
I d m (x+ e m Δ t * , t * +Δ t * )= I d m (x, t * )+ Δx μ m β B [ B β S t ( t * )+ n n+Δ t * I ˜ d m I d m (x, t * ) ] = I d m (x, t * )+Δ t * e m β B [ B β S t ( t * )+ n n+Δ t * I ˜ d m I d m (x, t * ) ].
I d m (x+ e m Δ t * , t * +Δ t * )= I d m (x, t * )+ Δ t * τ ˜ m [ { I d m (x, t * ) } eq I d m (x, t * ) ].
τ ˜ m = B e m β .
{ I d m (x, t * ) } eq = B β S t ( t * )+ n n+Δ t * I ˜ d m .
S c = σ s 2 1 1 I c (x,μ,t)Φ(μ, μ )d μ = σ s 2 m'=1 M I c m' Φ m'm ω m' .
S d = σ s 2 1 1 I d (x,μ,t)Φ(μ, μ )d μ = σ s 2 m'=1 M I d m' Φ m'm ω m' .
q R ( t * )= 2π I d (0,μ, t * )| μ | dμ q 0 ,μ<0.
q T ( t * )= ( 2π I d (L,μ, t * ) μdμ+ I c (L, t * )cos θ 0 ) q 0 ,μ>0.
q R ( t * )= π ρ o I 0 +2π(1 ρ I ) I d (0,μ, t * ) | μ |dμ q 0 ,μ<0.
q T ( t * )= 2π(1 ρ I )( I d (L,μ, t * ) μdμ ) q 0 ,μ>0.
I(t)= I 0 exp[ 4ln2× ( t3 t p t p ) 2 ][ H(t)H(t6 t p ) ].

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