梯度折射率介质内辐射传递方程数值模拟的有限元法
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摘要
辐射传递是热量和信息传输的一种基本方式,有着广泛的应用背景。实际介质的折射率往往受其组分、温度等因素的影响,并不是通常辐射换热分析时假设的均匀分布。半透明梯度折射率介质内辐射传递在许多现象与过程中的重要作用引起了各国学者的重视。光线在梯度折射率介质内沿Fermat原理决定的曲线路径传播,因而梯度折射率介质内的辐射传递相对均匀介质内的辐射传递要复杂得多。传统上应用于均匀介质的数值方法并不能直接应用于求解梯度折射率介质内的辐射传递问题。
有限元法有着许多独特的优点,如可以灵活的处理不规则几何边界和复杂边界条件及具有高阶精度等。近来发展的基于微分形式辐射传递方程离散的有限元法还存在一些稳定性问题,对其在辐射换热中的应用仍待进一步研究。
本文研究发展适用于多维直角坐标系及圆柱坐标系下梯度折射率介质内辐射传递的有限元解法及其对于求解该类问题的性能。主要工作包括以下五个方面:
1.基于直角坐标系下梯度折射率介质内的辐射传递方程,发展了求解多维半透明梯度折射率介质内辐射传递的有限元法。研究比较了不同离散方案的有限元法对于求解直角坐标系下多维半透明梯度折射率介质内辐射传递的性能。基于最小二乘离散方案及流向迎风方案的有限元法可以消除常规伽辽金有限元法在求解辐射传递方程时结果会出现的非物理振荡。
2.给出了传统圆柱坐标系及一种新的圆柱坐标系下梯度折射率介质内的辐射传递方程,基于这两个方程发展了适用于求解复杂边界条件下多维半透明梯度折射率圆柱介质内辐射传递的有限元法。检验了方法对于求解圆柱坐标系下多维半透明梯度折射率介质内辐射传递的性能。基于新圆柱坐标系下梯度折射率介质内辐射传递方程的有限元法可以方便的求解镜反射及Fresnel壁面圆柱形介质内的辐射传递。
3.基于二阶扩散项具有很好的数值稳定性,给出了梯度折射率介质内二阶辐射传递方程并发展了基于该二阶辐射传递方程的有限元法,同时检验了方法对于求解多维梯度折射率介质内辐射传递的性能。该方法可以方便的用于求解吸收、发射及各向异性散射梯度折射率介质内的辐射传递,具有很好的数值特性。
4.发展了求解梯度折射率介质内辐射传递的非结构混合有限体积/有限元方法并研究了该方法求解梯度折射率介质内辐射传递的性能。该方法将角度坐标采用有限元法离散,而空间坐标采用有限体积法离散,可以有效地结合两种方法各自的优点。
5.基于圆柱坐标系下的梯度折射率介质内辐射传递方程,发展了适用于求解半透明多层及具有薄膜包覆的梯度折射率圆柱介质内辐射传递的有限元法。给出了多层圆柱相邻层间辐射强度角度插值采用径向基函数进行插值的方法。建立了具有薄膜包覆光纤热辐射特性有限元分析模型。
Radiative transfer is a basic way of heat and information transfer, and plays an important role in many scientific and engineering applications. Due to the structural characteristics of material or possible temperature dependency, the refractive index of medium may be not uniform as the hypothesis used in traditional radiative heat transfer analysis. The radiative heat transfer in semitransparent medium with graded index is of great importance in many phenomenon and processes, and has evoked the wide interest of many researchers. In graded index medium, the ray goes along a curved path determined by the Fermat principle. The solution of radiative transfer in graded index medium is more difficult than that in uniform index medium. The traditional methods used in solving radiative transfer equation in uniform index medium can not be directly used in graded index medium.
The finite element method has many special merits, such as, considerable flexibility to deal with the complex geometric shape, the higher approximation for the field variables in a volume or surface element and so on. The finite element method developed recently based on discretization of the differential form of radiative transfer equation suffer from stabilization problems, hence its application in solving radiative heat transfer need to be studied further.
This dissertation develops finite element methods and studies their characteristics and performance to solve radiative transfer in multidimensional graded index medium under both Cartesian and cylindrical coordinate systems. The scope of present research mainly contains five parts:
1. Based on the radiative transfer equation of graded index medium in Cartesian coordinate system, finite element methods are developed for solving radiative transfer in multidimensional semitransparent graded index medium. The performance of the finite element methods based on different stabilization schemes are studied and compared for solving radiative transfer in semitransparent graded index medium. The least square finite element method and streamline upwind based finite element method can successfully eliminate the nonphysical oscillatory of solutions that may appear in Galerkin scheme based finite element method.
2. The radiative transfer equations of graded index medium in the traditional cylindrical coordinate system and in a new cylindrical coordinate system are derived. Based on these two equations, finite element methods which are suitable to solve radiative transfer in multidimensional semitransparent cylindrical graded index medium with complex boundary conditions are developed. The performance of the methods is examined for solving radiative transfer in multidimensional semitransparent graded index medium under cylindrical coordinate system. The finite element method based on the radiative transfer equation under the new cylindrical coordinate system can conveniently solve the radiative heat transfer in cylindrical graded index medium with specular and Fresnel boundary.
3. Due to the second order diffusion term has good numerical stability, the second order radiative transfer equation in graded index medium is derived, and a finite element method is developed based on the derived second order equation. The performance of the finite element method for solving radiative transfer in multidimensional semitransparent graded index medium is examined. This second order radiative transfer equation can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. The numerical study verifies that the method has very good numerical property.
4. An unstructured hybrid finite volume/finite element method is presented to solve the radiative transfer problem in graded index medium. The performance of the hybrid method for solving radiative transfer in graded index medium is examined. In this method, the spatial discretization is carried out using the finite volume method, while the angular discretization is performed using the finite element method. The hybrid finite volume/finite element method can effectively combine the merits of the two methods.
5. Based on the radiative transfer equation in cylindrical coordinate system, the finite element methods which are suitable to solve radiative transfer in multilayer semitransparent graded index cylinder and in optical fiber coated with thin film are developed and studied. An interpolation method based on radial basis functions is presented to interpolate the radiative intensity between two adjacent layers. A finite element analysis model is established to obtain the radiative property of optical fiber coated with thin film.
有限元法有着许多独特的优点,如可以灵活的处理不规则几何边界和复杂边界条件及具有高阶精度等。近来发展的基于微分形式辐射传递方程离散的有限元法还存在一些稳定性问题,对其在辐射换热中的应用仍待进一步研究。
本文研究发展适用于多维直角坐标系及圆柱坐标系下梯度折射率介质内辐射传递的有限元解法及其对于求解该类问题的性能。主要工作包括以下五个方面:
1.基于直角坐标系下梯度折射率介质内的辐射传递方程,发展了求解多维半透明梯度折射率介质内辐射传递的有限元法。研究比较了不同离散方案的有限元法对于求解直角坐标系下多维半透明梯度折射率介质内辐射传递的性能。基于最小二乘离散方案及流向迎风方案的有限元法可以消除常规伽辽金有限元法在求解辐射传递方程时结果会出现的非物理振荡。
2.给出了传统圆柱坐标系及一种新的圆柱坐标系下梯度折射率介质内的辐射传递方程,基于这两个方程发展了适用于求解复杂边界条件下多维半透明梯度折射率圆柱介质内辐射传递的有限元法。检验了方法对于求解圆柱坐标系下多维半透明梯度折射率介质内辐射传递的性能。基于新圆柱坐标系下梯度折射率介质内辐射传递方程的有限元法可以方便的求解镜反射及Fresnel壁面圆柱形介质内的辐射传递。
3.基于二阶扩散项具有很好的数值稳定性,给出了梯度折射率介质内二阶辐射传递方程并发展了基于该二阶辐射传递方程的有限元法,同时检验了方法对于求解多维梯度折射率介质内辐射传递的性能。该方法可以方便的用于求解吸收、发射及各向异性散射梯度折射率介质内的辐射传递,具有很好的数值特性。
4.发展了求解梯度折射率介质内辐射传递的非结构混合有限体积/有限元方法并研究了该方法求解梯度折射率介质内辐射传递的性能。该方法将角度坐标采用有限元法离散,而空间坐标采用有限体积法离散,可以有效地结合两种方法各自的优点。
5.基于圆柱坐标系下的梯度折射率介质内辐射传递方程,发展了适用于求解半透明多层及具有薄膜包覆的梯度折射率圆柱介质内辐射传递的有限元法。给出了多层圆柱相邻层间辐射强度角度插值采用径向基函数进行插值的方法。建立了具有薄膜包覆光纤热辐射特性有限元分析模型。
Radiative transfer is a basic way of heat and information transfer, and plays an important role in many scientific and engineering applications. Due to the structural characteristics of material or possible temperature dependency, the refractive index of medium may be not uniform as the hypothesis used in traditional radiative heat transfer analysis. The radiative heat transfer in semitransparent medium with graded index is of great importance in many phenomenon and processes, and has evoked the wide interest of many researchers. In graded index medium, the ray goes along a curved path determined by the Fermat principle. The solution of radiative transfer in graded index medium is more difficult than that in uniform index medium. The traditional methods used in solving radiative transfer equation in uniform index medium can not be directly used in graded index medium.
The finite element method has many special merits, such as, considerable flexibility to deal with the complex geometric shape, the higher approximation for the field variables in a volume or surface element and so on. The finite element method developed recently based on discretization of the differential form of radiative transfer equation suffer from stabilization problems, hence its application in solving radiative heat transfer need to be studied further.
This dissertation develops finite element methods and studies their characteristics and performance to solve radiative transfer in multidimensional graded index medium under both Cartesian and cylindrical coordinate systems. The scope of present research mainly contains five parts:
1. Based on the radiative transfer equation of graded index medium in Cartesian coordinate system, finite element methods are developed for solving radiative transfer in multidimensional semitransparent graded index medium. The performance of the finite element methods based on different stabilization schemes are studied and compared for solving radiative transfer in semitransparent graded index medium. The least square finite element method and streamline upwind based finite element method can successfully eliminate the nonphysical oscillatory of solutions that may appear in Galerkin scheme based finite element method.
2. The radiative transfer equations of graded index medium in the traditional cylindrical coordinate system and in a new cylindrical coordinate system are derived. Based on these two equations, finite element methods which are suitable to solve radiative transfer in multidimensional semitransparent cylindrical graded index medium with complex boundary conditions are developed. The performance of the methods is examined for solving radiative transfer in multidimensional semitransparent graded index medium under cylindrical coordinate system. The finite element method based on the radiative transfer equation under the new cylindrical coordinate system can conveniently solve the radiative heat transfer in cylindrical graded index medium with specular and Fresnel boundary.
3. Due to the second order diffusion term has good numerical stability, the second order radiative transfer equation in graded index medium is derived, and a finite element method is developed based on the derived second order equation. The performance of the finite element method for solving radiative transfer in multidimensional semitransparent graded index medium is examined. This second order radiative transfer equation can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. The numerical study verifies that the method has very good numerical property.
4. An unstructured hybrid finite volume/finite element method is presented to solve the radiative transfer problem in graded index medium. The performance of the hybrid method for solving radiative transfer in graded index medium is examined. In this method, the spatial discretization is carried out using the finite volume method, while the angular discretization is performed using the finite element method. The hybrid finite volume/finite element method can effectively combine the merits of the two methods.
5. Based on the radiative transfer equation in cylindrical coordinate system, the finite element methods which are suitable to solve radiative transfer in multilayer semitransparent graded index cylinder and in optical fiber coated with thin film are developed and studied. An interpolation method based on radial basis functions is presented to interpolate the radiative intensity between two adjacent layers. A finite element analysis model is established to obtain the radiative property of optical fiber coated with thin film.
引文
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63 L.H. Liu, H.P. Tan and Q.Z. Yu. Internal Distribution of Radiation Absorption in One-Dimensional Semitransparent Medium. International Journal of Heat and Mass Transfer. 2002, 45(2): 417-424.
64 W.X. Tian, W. Huang and W.K.S. Chiu. Thermal Radiative Properties of a Semitransparent Fiber Coated with a Thin Absorbing Film. Journal of Heat Transfer. 2007, 129(6): 763-767.
65刘林华,谈和平.梯度折射率介质内热辐射传递的数值模拟.北京:科学出版社, 2006.
66 J.C. Chai, H.S. Lee and S.V. Patankar. Improved Treatment of Scattering Using the Discrete Ordinates Method. ASME Journal of Heat Transfer. 1994, 116(1): 260-263.
67 J.C. Chai, H.S. Lee and S.V. Patankar. Finite-Volume Method for Radiation Heat Transfer. New York: Taylor & Francis, 2000: 109-141.
68 T.J.R. Hughes and A. Brooks. Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Special Emphasis on the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1982, 32(1-3): 199-259.
69 A.N. Brooks and T.J.R. Hughes. Streamline Upwind/Petrov-Galerkin Formulation for Convection Dominated ?ows with Particular Emphasis on the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1990, 32(1-3): 199-259.
70章本照,印建安,张宏基.流体力学数值方法.北京:机械工业出版社, 2003.
71 M.F. Modest. Radiative Heat Transfer. 2nd ed. San Diego: Academic Press, 2003.
72 M.Y. Kim. Assessment of the Axisymmetric Radiative Heat Transfer in a Cylindrical Enclosure with the Finite Volume Method. International Journal of Heat and Mass Transfer. 2008, 51(21-22): 5144-5153.
73 M.F. Modest. Radiative Heat Transfer. New York: McGraw-Hill, 1993.
74 Y.Y. Kwan. Efficient Spectral-Galerkin Methods for Polar and Cylindrical Geometries. Applied Numerical mathematics. 2009, 59(1): 170-186.
75 J.M. Zhao and L.H. Liu. Least-Squares Spectral Element Method for Radiative Heat Transfer in Semitransparent Media. Numerical Heat Transfer Part B. 2006, 50(5): 473-489.
76 E.W. Marchand. Ray Tracing in Cylindrical Gradient-Index Media. Applied Optics. 1972, 11(5): 1104-1106.
77 J.M. Zhao and L.H. Liu. Second-Order Radiative Transfer Equation and Its Properties Of Numerical Solution Using The Finite-Element Method. Numerical Heat Transfer Part B. 2007, 51(4): 391-409.
78 S.T. Thynell and M.N. Ozisik. Radiation Transfer in Absorbing, Emitting, Isotropically Scattering, Homogeneous Cylindrical Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 1987, 38(6): 413-426.
79 Z. Altac. Radiative Transfer in Absorbing, Emitting and Linearly Anisotropic-Scattering Inhomogeneous Cylindrical Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2003, 77(2): 177-192.
80 H.P. Tan, Y. Huang and X.L. Xia. Solution of Radiative Heat Transfer in a Semitransparent Slab with an Arbitrary Refractive Index Distribution and Diffuse Gray Boundaries. International Journal of Heat and Mass Transfer. 2003, 46(11): 2005-2014.
81 L.H. Liu. Benchmark Numerical Solutions for Radiative Heat Transfer in Two-Dimensional Medium with Graded Index Distribution. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 102(2): 293-303.
82 L.H. Liu. Least-Squares Finite Element Method for Radiation Heat Transfer in Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 103(3): 536-544.
83 L. Wang, J. Yang, M.F. Modest and D.C. Haworth. Application of theFull-Spectrum K-Distribution Method to Photon Monte Carlo Solvers. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 104(2): 297-304.
84 Y. Chen and K.N. Liou. A Monte Carlo Method for 3D Thermal Infrared Radiative Transfer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 101(1): 166-178.
85 V.V. Buhmirov and D.V. Rakutina. Application of Zonal Methods to the Account of Complex Heat Transfer in Thermal Technological Installations. Heat Transfer Research. 2008, 39(3): 241-243.
86 W.F. Wu, Y.H. Feng and X.X. Zhang. Zonal Method Solution of Radiative Heat Transfer in a One-Dimensional Long Roller-Hearth Furnace in Csp. Journal of University of Science and Technology Beijing: Mineral Metallurgy Materials (Eng Ed). 2007, 14(4): 307-311.
87 K. Kim, E. Lee and T.H. Song. Discrete Ordinates Interpolation Method for Radiative Heat Transfer Problems in Three-Dimensional Enclosures Filled with Non-Gray or Scattering Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(15): 2579-2589.
88 M.Y. Kim, J.H. Cho and S.W. Baek. Radiative Heat Transfer between Two Concentric Spheres Separated by a Two-Phase Mixture of Non-Gray Gas and Particles Using the Modified Discrete-Ordinates Method. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(9): 1607-1621.
89 D.N. Trivic and C.H. Amon. Modeling the 3-D Radiation of Anisotropically Scattering Media by Two Different Numerical Methods. International Journal of Heat and Mass Transfer. 2008, 51(11-12): 2711-2732.
90 P.J. Coelho. A Hybrid Finite Volume/Finite Element Discretization Method for the Solution of the Solution of the Radiative Heat Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 93(1-3): 89-101.
91 R. Siegel and J.R. Howell. Thermal Radiation Heat Transfer. fourth ed. New York: Taylor and Francis, 2002.
92 M.E. Larsen and J.R. Howell. The Exchange Factor Method: An Alternative Zonal Formulation of Radiating Enclosure Analysis. ASME Journal of HeatTransfer. 1985, 107: 936-942.
93 R.D.M. Garcia, C.E. Siewert and A.M. Yacout. On the Use of Fresnel Boundary and Interface Conditions in Radiative-Transfer Calculations for Multilayered Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(5): 752-769.
94 Z.-M. WU and R. SCHABACK. Local Error Estimates for Radial Basis Function Interpolation of Scattered Data. IMA Journal of Numerical Analysis. 1993, 13(1): 13-27.
95 F.J. Narcowich. Recent Developments in Error Estimates for Scattered-Data Interpolation Via Radial Basis Functions. Numerical Algorithms. 2007, 39(1-3): 307-315.
96 J. Yoon. Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space SIAM J. Math. Anal. 2001, 33(4): 946-958.
97 J.-H. Jung and V.R. Durante. An Iterative Adaptive Multiquadric Radial Basis Function Method for the Detection of Local Jump Discontinuities Applied Numerical mathematics. 2009, 59(7): 1449-1466.
98 B. Fornberg and J. Zuev. The Runge Phenomenon and Spatially Variable Shape Parameters in Rbf Interpolation. Computers & Mathematics with Applications. 2007, 54(3): 379-398.
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61陈炳炎.光纤光缆的设计和制造.杭州:浙江大学出版社, 2003.
62彭吉虎,吴伯瑜.光纤技术及应用.北京:北京理工大学出版社, 1995.
63 L.H. Liu, H.P. Tan and Q.Z. Yu. Internal Distribution of Radiation Absorption in One-Dimensional Semitransparent Medium. International Journal of Heat and Mass Transfer. 2002, 45(2): 417-424.
64 W.X. Tian, W. Huang and W.K.S. Chiu. Thermal Radiative Properties of a Semitransparent Fiber Coated with a Thin Absorbing Film. Journal of Heat Transfer. 2007, 129(6): 763-767.
65刘林华,谈和平.梯度折射率介质内热辐射传递的数值模拟.北京:科学出版社, 2006.
66 J.C. Chai, H.S. Lee and S.V. Patankar. Improved Treatment of Scattering Using the Discrete Ordinates Method. ASME Journal of Heat Transfer. 1994, 116(1): 260-263.
67 J.C. Chai, H.S. Lee and S.V. Patankar. Finite-Volume Method for Radiation Heat Transfer. New York: Taylor & Francis, 2000: 109-141.
68 T.J.R. Hughes and A. Brooks. Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Special Emphasis on the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1982, 32(1-3): 199-259.
69 A.N. Brooks and T.J.R. Hughes. Streamline Upwind/Petrov-Galerkin Formulation for Convection Dominated ?ows with Particular Emphasis on the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1990, 32(1-3): 199-259.
70章本照,印建安,张宏基.流体力学数值方法.北京:机械工业出版社, 2003.
71 M.F. Modest. Radiative Heat Transfer. 2nd ed. San Diego: Academic Press, 2003.
72 M.Y. Kim. Assessment of the Axisymmetric Radiative Heat Transfer in a Cylindrical Enclosure with the Finite Volume Method. International Journal of Heat and Mass Transfer. 2008, 51(21-22): 5144-5153.
73 M.F. Modest. Radiative Heat Transfer. New York: McGraw-Hill, 1993.
74 Y.Y. Kwan. Efficient Spectral-Galerkin Methods for Polar and Cylindrical Geometries. Applied Numerical mathematics. 2009, 59(1): 170-186.
75 J.M. Zhao and L.H. Liu. Least-Squares Spectral Element Method for Radiative Heat Transfer in Semitransparent Media. Numerical Heat Transfer Part B. 2006, 50(5): 473-489.
76 E.W. Marchand. Ray Tracing in Cylindrical Gradient-Index Media. Applied Optics. 1972, 11(5): 1104-1106.
77 J.M. Zhao and L.H. Liu. Second-Order Radiative Transfer Equation and Its Properties Of Numerical Solution Using The Finite-Element Method. Numerical Heat Transfer Part B. 2007, 51(4): 391-409.
78 S.T. Thynell and M.N. Ozisik. Radiation Transfer in Absorbing, Emitting, Isotropically Scattering, Homogeneous Cylindrical Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 1987, 38(6): 413-426.
79 Z. Altac. Radiative Transfer in Absorbing, Emitting and Linearly Anisotropic-Scattering Inhomogeneous Cylindrical Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2003, 77(2): 177-192.
80 H.P. Tan, Y. Huang and X.L. Xia. Solution of Radiative Heat Transfer in a Semitransparent Slab with an Arbitrary Refractive Index Distribution and Diffuse Gray Boundaries. International Journal of Heat and Mass Transfer. 2003, 46(11): 2005-2014.
81 L.H. Liu. Benchmark Numerical Solutions for Radiative Heat Transfer in Two-Dimensional Medium with Graded Index Distribution. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 102(2): 293-303.
82 L.H. Liu. Least-Squares Finite Element Method for Radiation Heat Transfer in Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 103(3): 536-544.
83 L. Wang, J. Yang, M.F. Modest and D.C. Haworth. Application of theFull-Spectrum K-Distribution Method to Photon Monte Carlo Solvers. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 104(2): 297-304.
84 Y. Chen and K.N. Liou. A Monte Carlo Method for 3D Thermal Infrared Radiative Transfer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 101(1): 166-178.
85 V.V. Buhmirov and D.V. Rakutina. Application of Zonal Methods to the Account of Complex Heat Transfer in Thermal Technological Installations. Heat Transfer Research. 2008, 39(3): 241-243.
86 W.F. Wu, Y.H. Feng and X.X. Zhang. Zonal Method Solution of Radiative Heat Transfer in a One-Dimensional Long Roller-Hearth Furnace in Csp. Journal of University of Science and Technology Beijing: Mineral Metallurgy Materials (Eng Ed). 2007, 14(4): 307-311.
87 K. Kim, E. Lee and T.H. Song. Discrete Ordinates Interpolation Method for Radiative Heat Transfer Problems in Three-Dimensional Enclosures Filled with Non-Gray or Scattering Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(15): 2579-2589.
88 M.Y. Kim, J.H. Cho and S.W. Baek. Radiative Heat Transfer between Two Concentric Spheres Separated by a Two-Phase Mixture of Non-Gray Gas and Particles Using the Modified Discrete-Ordinates Method. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(9): 1607-1621.
89 D.N. Trivic and C.H. Amon. Modeling the 3-D Radiation of Anisotropically Scattering Media by Two Different Numerical Methods. International Journal of Heat and Mass Transfer. 2008, 51(11-12): 2711-2732.
90 P.J. Coelho. A Hybrid Finite Volume/Finite Element Discretization Method for the Solution of the Solution of the Radiative Heat Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 93(1-3): 89-101.
91 R. Siegel and J.R. Howell. Thermal Radiation Heat Transfer. fourth ed. New York: Taylor and Francis, 2002.
92 M.E. Larsen and J.R. Howell. The Exchange Factor Method: An Alternative Zonal Formulation of Radiating Enclosure Analysis. ASME Journal of HeatTransfer. 1985, 107: 936-942.
93 R.D.M. Garcia, C.E. Siewert and A.M. Yacout. On the Use of Fresnel Boundary and Interface Conditions in Radiative-Transfer Calculations for Multilayered Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 2008, 109(5): 752-769.
94 Z.-M. WU and R. SCHABACK. Local Error Estimates for Radial Basis Function Interpolation of Scattered Data. IMA Journal of Numerical Analysis. 1993, 13(1): 13-27.
95 F.J. Narcowich. Recent Developments in Error Estimates for Scattered-Data Interpolation Via Radial Basis Functions. Numerical Algorithms. 2007, 39(1-3): 307-315.
96 J. Yoon. Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space SIAM J. Math. Anal. 2001, 33(4): 946-958.
97 J.-H. Jung and V.R. Durante. An Iterative Adaptive Multiquadric Radial Basis Function Method for the Detection of Local Jump Discontinuities Applied Numerical mathematics. 2009, 59(7): 1449-1466.
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