Spectral formulations to the solution of particle transport problems
Keywords:Analytical Discrete Ordinates Method, Rarefied Gas Dynamics, Neutron Transport, Spectral Methods
In this work, a deterministic approach to the solution of the integro differential linear Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. Mathematical aspects present in the solution of the two models are emphasized and extensions of the formulation to other applications are commented and referenced.
Azmy, Y. Y. (1988). The weighted diamond-difference form of nodal transport. Nuclear Science Engineering, 98, 29–40.
Badruzzaman, A. (1985). An efficient algorithm for nodal-transport solutions in multidimensional geometry. Nuclear Science and Engineering, 89, 281–290.
Barichello, L., Tres, A., Picoloto, C. B., Azmy, Y. Y. (2016). Recent studies on the asymptotic convergence of the spatial discretization for two-dimensional discrete ordinates solutions. Journal of Computational and Theoretical Transport, 45, 299–313.
Barichello, L. B. (2011). Explicit Formulations for Radiative Transfer Problems. In: H. R. B. Orlande, O. Fudyin, D. Maillet, R. M. Cotta (Org.) Thermal Measurements and Inverse Techniques. Boca Raton: CRC Press.
Barichello, L. B., Siewert, C. E. (1999a). A discrete-ordinates solution for a non-grey model with complete frequency redistribution. Journal of Quantitative Spectroscopy and Radiative Transfer, 62, 665–675.
Barichello, L. B., Siewert, C. E. (1999b). A discrete ordinates solution for a polarization model with complete frequency redistribution. The Astrophysical Journal, 512, 370–382.
Barichello, L. B., Siewert, C. E. (1999c). A discrete-ordinates solution for Poiseuille flow in a plane channel. Zeitschrift für Angewandte Mathematik und Physik, 50, 972–981.
Barichello, L. B., Siewert, C. E. (1999d). On the equivalence between the discrete-ordinates and the spherical-harmonics methods in radiative transfer. Nuclear Science and Engineering, 130, 79–84.
Barichello, L. B., Siewert, C. E. (2000). The temperature-jump problem in rarefied-gas dynamics. European Journal of Applied Mathematics, 11, 353–364.
Barichello, L. B., Siewert, C. E. (2001). A new version of the discrete-ordinates method. Em: 2nd International Conference on Heat and Mass Transfer, Computational Heat and Mass Transfer - CHMT 2001, E-Papers, Rio de Janeiro, vol 01, pp. 340–347.
Barichello, L. B., Siewert, C. E. (2003). Some comments on modeling the linearized Boltzmann equation. Journal of Quantitative Spectroscopy and Radiative Transfer, 77, 43–59.
Barichello, L. B., Camargo, M., Rodrigues, P., Siewert, C. E. (2001). Unified solution to classical flow problems based on the BGK model. Zeitschrift für Angewandte Mathematik und Physik, 52, 517–534.
Barichello, L. B., Cabrera, L. C., F. Prolo Filho, J. (2011). An analytical approach for a nodal scheme of two-dimensional neutron transport problems. Annals of Nuclear Energy, 38, 1310–1317.
Barichello, L. B., Picoloto, C. B., da Cunha, R. D. (2017). The ADO-Nodal method for solving two-dimensional discrete ordinates transport problems. Annals of Nuclear Energy, 108, 376–385.
Barichello, L. B., Pazinatto, C. B., Rui, K. (2020). An analytical discrete ordinates nodal solution to the two-dimensional adjoint transport problem. Annals of Nuclear Energy, 135, Article 106, 959.
Barros, R. C., Larsen, F. W. (1992). A spectral nodal method for one-group x,y-geometry discrete ordinates problems. Nuclear Science and Engineering, 111, 34–45.
Bhatnagar, P. L., Gross, E. P., Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys Rev, 94, 511.
Camargo, M., Rodrigues, P., Barichello, L. B. (2000). Discrete-ordinates solutions to some classical flow problems in the rarefied gas dynamics. 8th Brazilian Congress of Thermal Engineering and Sciences - ENCIT 2000.
Cercignani, C. (2006). Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford, Published to Oxford Scholarship Online: January 2010. Print ISBN-13: 9780198570646, DOI:10.1093/acprof:oso/9780198570646.001.0001.
Chalhoub, E., Garcia, R. D. M. (2000). The equivalence between two techniques of angular interpolation for the discrete ordinates method. Journal of Quantitative Spectroscopy & Radiative Transfer, 64, 517–535.
Chandrasekhar, S. (1950). Radiative Transfer. Oxford university Press, London.
Cromianski, S. R., Camargo, M., Rodrigues, P., Barichello, L. B. (2018). Avaliação de propriedades radiativas em meios homogêneos unidimensionais: reflectância e transmitância. TEMA – Tendências em Matemática Aplicada e Computacional, 18, 531–547.
Cromianski, S. R., Rui, K., Barichello, L. B. (2019). A study on boundary fluxes approximation in explicit nodal formulations for the solution of the two-dimensional neutron transport equation. Progress in Nuclear Energy, 110, 354–363.
Datta, B. N. (1995). Numerical Linear Algebra and Applications. Brooks/Cole Publishing Co, Pacific Grove, USA.
Dombrovsky, A. L., Timchenko, V., Jackson, M., Yeoh, G. H. (2011). A combined transient thermal model for laser hyperthermia of tumors with embedded gold nanoshells. International Journal of Heat and Mass Transfer, 54, 5459–5469.
Duderstadt, J. J., Martin, W. R. (1979). Transport Theory. John Wiley & Sons, Inc., New York.
Ebeling,W., Sokolov, I. M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems.World Scientific Publishing Co. Pte. Ltd., ISBN 978-90-277-1674-3.
Gad-el-Hak, M. (2005). MEMS: Introduction and Fundamentals. CRC Press.
Golub, G. H., Loan, C. F. V. (2013). Matrix Computations. 4th Edition, Johns Hopkins University Press, Baltimore.
Haghighat, A. (2014). Monte Carlo Methods for Particle Transport. CRC Press.
Klose, A. D., Netz, U., Beuthan, J., Hielscher, A. H. (2002). Optical tomography using the time-independent equation of radiative transfer‚ Part 1: forward model. Journal of Quantitative Spectroscopy and Radiative Transfer, 72, 691–713.
Knackfuss, R. F., Barichello, L. B. (2006). On the temperature-jump problem in rarefied gas dynamics: the effect of the Cercignani-Lampis boundary condition. SIAM Journal on Applied Mathematics, 66, 2149–2186.
Lathrop, K. D., Brinkley, F. W. (1970). Theory and use of the general-geometry TWOTRAN program. LA4432 - Los Alamos Scientific Lab Report, Univ of California.
Lewis, E. E., Miller, W. F. (1984). Computational Methods of Neutron Transport. John Wiley and Sons, New York.
Modest, M. F. (2003). Radiative Heat Transfer. 2nd Edition, Academic Press, San Diego.
Palmiotti, G., Rieunier, J. M., Gho, C., Salvatores, M. (1990). Optimized two-dimensional SN transport (BISTRO). Nuclear Science and Engineering, 104, 26–33.
Pazinatto, C. B., Barichello, L. (2018). Energy dependent source reconstructions via explicit formulations of the adjoint particles flux. Journal of Computational and Theoretical Transport, 47, 58–83.
Pazinatto, C. B., Barichello, L. (2019). On the use of the adjoint operator for source reconstruction in particle transport problems. Inverse Problems in Science and Engineering, 27, 513–539.
Picoloto, C. B., Tres, A., da Cunha, R. D., Barichello, L. B. (2013). Two-dimensional neutron transport problems with reflective boundary conditions: an analytical approach. Em: International Nuclear Atlantic Conference, Recife.
Picoloto, C. B., Tres, A., da Cunha, R. D., Barichello, L. B. (2015). Closed-form solutions for nodal formulations of two dimensional transport problems in heterogeneous media. Annals of Nuclear Energy, 86, 65–71.
Picoloto, C. B., da Cunha, R. D., Barros, R. C., Barichello, L. B. (2017). An analytical approach for solving a nodal formulation of two-dimensional fixed-source neutron transport problems with linearly anisotropic scattering. Progress in Nuclear Energy, 98, 193–201.
Poursalehi, N., Zolfaghari, A., Minuchehr, A. (2013). An adaptive mesh refinement approach for average current nodal expansion method in 2-D rectangular geometry. Annals of Nuclear Energy, 55, 61–70.
Prolo Filho, J. F., Barichello, L. B. (2014). General expressions for auxiliary equations of a nodal formulation for two-dimensional transport calculations. Journal of Computational and Theoretical Transport, 43, 1–22.
Rodrigues, P., Barichello, L. (2004). An integral equation approach for radiative transfer in a cylindrical media. Journal of Quantitative Spectroscopy and Radiative Transfer, 83, 765–776.
Scherer, C. S., Prolo Filho, J. F., Barichello, L. B. (2009a). An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. I. Flow problems. Zeitschrift für Angewandte Mathematik und Physik, 60, 70–115.
Scherer, C. S., Prolo Filho, J. F., Barichello, L. B. (2009b). An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. ii. heat transfer problems. Zeitschrift für Angewandte Mathematik und Physik, 60, 651–687.
Siewert, C. E. (2000). Poiseuille and thermal-creep flow in a cylindrical tube. Journal of Computational Physics, 160, 470–480.
Siewert, C. E. (2003). The linearized Boltzmann equation: a concise and accurate solution to basic flow problems. Zeitschrift für Angewandte Mathematik und Physik, 54, 273–303.
Tatsios, G., Valougeorgis, D. (2020). Uncertainty analysis of computed flow rates and pressure differences in rarefied pressure and temperature driven gas flows through long capillaries. European Journal of Mechanics / B Fluids, 51, 190–201.
Tres, A., Picoloto, C. B., F. Prolo Filho, J., da Cunha, R. D., Barichello, L. B. (2014). Explicit formulation of a nodal transport method for discrete ordinates calculations in two-dimensional fixed-source problems. Kerntechnik, 79, 155–162.
Wang, M., Lan, X., Li, Z. (2008). Analyses of gas flows in micro and nanochannels. International Journal of Heat and Mass Transfer, 51, 3630–3641.
Wick, G. C. (1943). über ebene Diffusionsprobleme. Zeitschrift für Physik, 121, 702–718.
Wu, L., Struchtrup, H. (2017). Assessment and development of the gas kinetic boundary conditions for the Boltzmann equation. J Fluid Mech, 823, 511–537.
Xu, L., Cao, L., Zheng, Y.,Wu, H. (2018). Hydra: a 3-D parallel discrete ordinates code for massive transport calculation. PHYSOR 2018, Cancun, Mexico, April 22-26.
Ziegenhein, P., Pierner, S., Kamerling, C. P., Oeilfke, U. (2015). Fast CPU-based Monte Carlo simulation for radiotherapy dose calculation. Physics in Medicine and Biology, 60, 6097–6111.
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