求解辐射传递方程的谱元法
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摘要
对于半透明参与性介质内辐射换热的求解已经发展了很多方法,综合来看,基于微分形式辐射传递方程离散的方法具有很好的适应性。对于均匀折射率介质内辐射传递的求解,离散坐标法和有限体积法具有可以方便经济地处理多维问题、灵活地处理复杂介质和边界特性、易于与其他换热方式进行耦合计算等优点,是目前应用最多的方法。对于梯度折射率介质内辐射传递的求解,其中有限体积法和有限元法避免了复杂耗时的弯曲光线轨迹的计算,表现了很好的适应性。但是目前已发展的这些方法都是低阶方法,一般仅具有一阶或二阶精度,同时只具有h收敛特性,为了得到需要的计算精度,只能通过繁杂的对网格加密或重新划分来实现。
谱元法可以很好的克服低阶方法的不足,它兼具低阶有限元法和高阶谱方法的优点,将谱方法随着多项式阶数增加而收敛的p收敛特性与有限元法的易于适应复杂几何区域的优点及随着单元加密而收敛的h收敛特性充分结合。两种收敛方式使谱元法具有更优异的性能,其p收敛特性使谱元法可以在给定的网格划分情况下,仅通过增加单元近似多项式的阶数来达到收敛。
本文结合均匀及梯度折射率介质内辐射传递方程的数值特性,发展并研究了基于不同离散方案的谱元法用于求解半透明介质内辐射传递、辐射与导热耦合换热及瞬态辐射传递问题的特性及性能。主要工作包括以下五个方面:
1.发展了基于最小二乘稳定方案的谱元法用于求解均匀折射率介质内辐射传递方程,并给出了一种高效的数值实现算法。同时将该方法推广用于多维梯度折射率介质内辐射传递的求解。研究了该方法的数值稳定性及p收敛特性,其p收敛速率很快且满足指数规律。检验了该方法对于求解多维半透明均匀及梯度折介质内辐射传递的性能。给出了谱元法求解辐射传递方程的其他稳定方案及施加方式,研究并比较了基于其他稳定方案的谱元法求解半透明介质内辐射传递问题的性能。
2.采用一种与辐射传递方程偶宇称分解不同的处理方式推导了一个基于原始变量的二阶辐射传递方程。该二阶方程具有辐射传递偶宇称公式的主要优点同时克服了其绝大部分缺点,可以方便的用于求解吸收、发射及各向异性散射介质内的辐射传递。对一阶和二阶方程的误差扰动特性进行了对比分析,数值分析结果表明二阶辐射传递方程具有更好的数值特性。给出了二阶辐射传递方程伽辽金法离散的一般公式,检验并研究了该二阶辐射传递方程对于求解多维辐射传递问题的性能。3.发展了基于间断伽辽金方案的谱元法用来求解多维半透明介质内的辐射
传递。间断伽辽金方案不需要强加单元间的连续性,其离散得到的方程满足局部守恒性。该方法有效的消除了离散坐标方程对流特性造成的数值不稳定性。研究了该方法在结构及非结构网格时的p收敛特性。比较了单元边界数值通量的不同处理方式对该方法数值特性的影响。讨论了基于辐射传递方程离散的数值算法的两种射线效应,即边界热负荷不均及内部遮挡引起的射线效应。
4.基于二阶辐射传递方程发展了一种谱元法用于求解多维半透明介质内辐射与导热的耦合问题。检验了其求解多维半透明介质内辐射与导热耦合问题的性能。研究了该谱元法求解辐射与导热耦合问题的h和p收敛特性。在不同普朗克数下p收敛速率很快且服从指数规律并显著优于h收敛速率。数值实验验证表明谱元法对于扭曲网格有着很好的适应性。
5.基于间断谱元法具有局部守恒性、高阶精度及假扩散小等优点,将其推广用于瞬态辐射传递的求解。研究了平行光照射及瞬态漫射边界的特殊处理方法。给出了瞬态辐射传递方程间断谱元法离散和求解的实现过程。通过典型算例对提出的瞬态辐射传递求解的间断谱元法进行了验证,同时研究了其求解瞬态辐射传递问题的性能。
Many numerical methods have been developed for solving radiative heat transfer in semitransparent participant media. By comprehensive comparison, the methods based on discretization of the differential form of radiative transfer equation possess very good adaptabities. As for solving radiative transfer in uniform index media, discrete ordinates method and finite volume method have many advantages, such as, they are convenient and efficient to deal with multidimensional problem, flexible to deal with problem with complex media and boundary properties, and easy to be adapted to coupled solution with other heat transfer processes, etc., as a result, they are the mostly often used methods until recently. As for solving radiative transfer in graded index media, the finite volume method and the finite element method avoid the time-consuming computation of curved ray trajectories, so they show very good adaptabities. However, these already developed methods are all low order methods, generally with only first or second order of accuracy, furthermore they provide only h-convergence property, as a result, in order to gain wanted accuracy, cumbersome mesh refining or remeshing is necessary.
Spectral element method can effectively overcome the drawbacks of low order methods, which possesses both the advantages of low order finite element method and high order spectral method and best combines the p-convergence property of spectral methods, namely, convergence by increasing order of polynomial, and the advantage to be easy to applied to complex enclosures and the h-convergence property of finite element method, namely, convergence by mesh refining. Two convergence strategies make spectral element method are more effective, of which the p-convergence property make spectral element method can achieve convergence by just increasing the order of polynomial.
By considering the numerical properties of radiative transfer equation for uniform and graded index media, this paper develops and studies the characteristics and performances of spectral element methods based on different discretization schemes to solve radiative transfer in semitransparent participant media, coupled radiative and conductive heat transfer and transient radiative transfer. The scope of present research contains five parts:
1. A spectral element method based on least squares stabilization scheme is developed to solve radiative transfer in multidimensional uniform index media and an efficient implementation algorithm is presented. The proposed method is also extended to solve radiative transfer in graded index media. The stability and p-convergence characteristics of the method are studied and its p-convergence speed is very fast and follows exponential law. The performances of the method for solution of radiative transfer in multidimensional semitransparent uniform and graded index media are examined. Some other stabilization schemes of spectral element method and their imposing techniques are presented and the performances of spectral element method based on these stabilization schemes for solving radiative transfer in semitransparent media are studied and compared.
2. A second order radiative transfer equation of primitive variable is derived by using a different way than the even parity formulation. This second order equation possesses major advantages of the even parity formulation of radiative transfer, but overcomes most of its drawbacks, and can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. Perturbation error analysis are made for both the first order and the second order equation, by comparison, the second order radiative transfer equation shows better numerical properties. A general formulation of Galerkin discretization of the second order radiative transfer equation is presented, and the performances of the second order equation are examined for solving multidimensional radiative transfer problems.
3. A spectral element method based on discontinuous Galerkin scheme is developed to solve radiative transfer in multidimensional semitransparent media. The discontinuous Galerkin scheme need not to impose inter-elemental continuity and the resulting discretized equation own local conservativity. This method effectively eliminates the numerical instability caused by the convection property of discrete ordinates equation. The p-convergence characteristics of the proposed method are studied on both structured and unstructured meshes. Influences of different schemes for dealing with elemental boundary numerical flux on numerical properties of the proposed method are compared. Two kinds of ray effect are discussed for methods based on discretization of radiative transfer equation, namely, ray effect induced by boundary loading and ray effect induced by shielding of interior obstacle.
4. Based on the second order radiative transfer equation, a spectral element method is developed to solve coupled radiative and conductive transfer in multidimensional semitransparent media. The performances of the proposed method are examined for solving coupled radiative and conductive heat transfer in multidimensional semitransparent media. The h- and p-convergence characteristics of the proposed method are studied. The p-convergence speeds are very fast and follow exponential law under different values of Plank number and are superior to the h-convergence speed. Numerical examinations show that spectral element method possesses very good adaptability to skewed meshes.
5. Considering the discontinuous spectral element method has advantages, such as local conservativity, high order of accuracy and minimal artificial diffusion, it is extended to solve transient radiative transfer problems. Special treatment of the collimated irradiation and transient diffusive boundary are studied. The implementation of the discontinuous spectral element method for the discretization of transient radiative transfer equation is presented. The proposed discontinuous spectral element method for transient radiative transfer is examined by several classical examples, and its performances for solving transient radiative transfer are studied.
谱元法可以很好的克服低阶方法的不足,它兼具低阶有限元法和高阶谱方法的优点,将谱方法随着多项式阶数增加而收敛的p收敛特性与有限元法的易于适应复杂几何区域的优点及随着单元加密而收敛的h收敛特性充分结合。两种收敛方式使谱元法具有更优异的性能,其p收敛特性使谱元法可以在给定的网格划分情况下,仅通过增加单元近似多项式的阶数来达到收敛。
本文结合均匀及梯度折射率介质内辐射传递方程的数值特性,发展并研究了基于不同离散方案的谱元法用于求解半透明介质内辐射传递、辐射与导热耦合换热及瞬态辐射传递问题的特性及性能。主要工作包括以下五个方面:
1.发展了基于最小二乘稳定方案的谱元法用于求解均匀折射率介质内辐射传递方程,并给出了一种高效的数值实现算法。同时将该方法推广用于多维梯度折射率介质内辐射传递的求解。研究了该方法的数值稳定性及p收敛特性,其p收敛速率很快且满足指数规律。检验了该方法对于求解多维半透明均匀及梯度折介质内辐射传递的性能。给出了谱元法求解辐射传递方程的其他稳定方案及施加方式,研究并比较了基于其他稳定方案的谱元法求解半透明介质内辐射传递问题的性能。
2.采用一种与辐射传递方程偶宇称分解不同的处理方式推导了一个基于原始变量的二阶辐射传递方程。该二阶方程具有辐射传递偶宇称公式的主要优点同时克服了其绝大部分缺点,可以方便的用于求解吸收、发射及各向异性散射介质内的辐射传递。对一阶和二阶方程的误差扰动特性进行了对比分析,数值分析结果表明二阶辐射传递方程具有更好的数值特性。给出了二阶辐射传递方程伽辽金法离散的一般公式,检验并研究了该二阶辐射传递方程对于求解多维辐射传递问题的性能。3.发展了基于间断伽辽金方案的谱元法用来求解多维半透明介质内的辐射
传递。间断伽辽金方案不需要强加单元间的连续性,其离散得到的方程满足局部守恒性。该方法有效的消除了离散坐标方程对流特性造成的数值不稳定性。研究了该方法在结构及非结构网格时的p收敛特性。比较了单元边界数值通量的不同处理方式对该方法数值特性的影响。讨论了基于辐射传递方程离散的数值算法的两种射线效应,即边界热负荷不均及内部遮挡引起的射线效应。
4.基于二阶辐射传递方程发展了一种谱元法用于求解多维半透明介质内辐射与导热的耦合问题。检验了其求解多维半透明介质内辐射与导热耦合问题的性能。研究了该谱元法求解辐射与导热耦合问题的h和p收敛特性。在不同普朗克数下p收敛速率很快且服从指数规律并显著优于h收敛速率。数值实验验证表明谱元法对于扭曲网格有着很好的适应性。
5.基于间断谱元法具有局部守恒性、高阶精度及假扩散小等优点,将其推广用于瞬态辐射传递的求解。研究了平行光照射及瞬态漫射边界的特殊处理方法。给出了瞬态辐射传递方程间断谱元法离散和求解的实现过程。通过典型算例对提出的瞬态辐射传递求解的间断谱元法进行了验证,同时研究了其求解瞬态辐射传递问题的性能。
Many numerical methods have been developed for solving radiative heat transfer in semitransparent participant media. By comprehensive comparison, the methods based on discretization of the differential form of radiative transfer equation possess very good adaptabities. As for solving radiative transfer in uniform index media, discrete ordinates method and finite volume method have many advantages, such as, they are convenient and efficient to deal with multidimensional problem, flexible to deal with problem with complex media and boundary properties, and easy to be adapted to coupled solution with other heat transfer processes, etc., as a result, they are the mostly often used methods until recently. As for solving radiative transfer in graded index media, the finite volume method and the finite element method avoid the time-consuming computation of curved ray trajectories, so they show very good adaptabities. However, these already developed methods are all low order methods, generally with only first or second order of accuracy, furthermore they provide only h-convergence property, as a result, in order to gain wanted accuracy, cumbersome mesh refining or remeshing is necessary.
Spectral element method can effectively overcome the drawbacks of low order methods, which possesses both the advantages of low order finite element method and high order spectral method and best combines the p-convergence property of spectral methods, namely, convergence by increasing order of polynomial, and the advantage to be easy to applied to complex enclosures and the h-convergence property of finite element method, namely, convergence by mesh refining. Two convergence strategies make spectral element method are more effective, of which the p-convergence property make spectral element method can achieve convergence by just increasing the order of polynomial.
By considering the numerical properties of radiative transfer equation for uniform and graded index media, this paper develops and studies the characteristics and performances of spectral element methods based on different discretization schemes to solve radiative transfer in semitransparent participant media, coupled radiative and conductive heat transfer and transient radiative transfer. The scope of present research contains five parts:
1. A spectral element method based on least squares stabilization scheme is developed to solve radiative transfer in multidimensional uniform index media and an efficient implementation algorithm is presented. The proposed method is also extended to solve radiative transfer in graded index media. The stability and p-convergence characteristics of the method are studied and its p-convergence speed is very fast and follows exponential law. The performances of the method for solution of radiative transfer in multidimensional semitransparent uniform and graded index media are examined. Some other stabilization schemes of spectral element method and their imposing techniques are presented and the performances of spectral element method based on these stabilization schemes for solving radiative transfer in semitransparent media are studied and compared.
2. A second order radiative transfer equation of primitive variable is derived by using a different way than the even parity formulation. This second order equation possesses major advantages of the even parity formulation of radiative transfer, but overcomes most of its drawbacks, and can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. Perturbation error analysis are made for both the first order and the second order equation, by comparison, the second order radiative transfer equation shows better numerical properties. A general formulation of Galerkin discretization of the second order radiative transfer equation is presented, and the performances of the second order equation are examined for solving multidimensional radiative transfer problems.
3. A spectral element method based on discontinuous Galerkin scheme is developed to solve radiative transfer in multidimensional semitransparent media. The discontinuous Galerkin scheme need not to impose inter-elemental continuity and the resulting discretized equation own local conservativity. This method effectively eliminates the numerical instability caused by the convection property of discrete ordinates equation. The p-convergence characteristics of the proposed method are studied on both structured and unstructured meshes. Influences of different schemes for dealing with elemental boundary numerical flux on numerical properties of the proposed method are compared. Two kinds of ray effect are discussed for methods based on discretization of radiative transfer equation, namely, ray effect induced by boundary loading and ray effect induced by shielding of interior obstacle.
4. Based on the second order radiative transfer equation, a spectral element method is developed to solve coupled radiative and conductive transfer in multidimensional semitransparent media. The performances of the proposed method are examined for solving coupled radiative and conductive heat transfer in multidimensional semitransparent media. The h- and p-convergence characteristics of the proposed method are studied. The p-convergence speeds are very fast and follow exponential law under different values of Plank number and are superior to the h-convergence speed. Numerical examinations show that spectral element method possesses very good adaptability to skewed meshes.
5. Considering the discontinuous spectral element method has advantages, such as local conservativity, high order of accuracy and minimal artificial diffusion, it is extended to solve transient radiative transfer problems. Special treatment of the collimated irradiation and transient diffusive boundary are studied. The implementation of the discontinuous spectral element method for the discretization of transient radiative transfer equation is presented. The proposed discontinuous spectral element method for transient radiative transfer is examined by several classical examples, and its performances for solving transient radiative transfer are studied.
引文
1 王应时. 燃烧过程数值计算. 北京: 科学出版社, 1986.
2 R. Viskanta. Overview of Convection and Radiation in High Temperature Gas Flows. International Journal of Engineering Science. 1998, 36(12): 1677-1699.
3 W.H. Sutton and X.L. Chen. A General Integration Method for Radiative Transfer in 3-D Non-Homogeneous Cylindrical Media with Anisotropic Scattering. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 84: 65-103.
4 侯晓春, 季鹤鸣, 刘庆国, 严传俊, 赵坚行. 高性能航空燃气轮机燃烧技术. 北京: 国防工业出版社, 2002.
5 J. Abraham and V. Magi. Application of the Discrete Ordinates Method to Compute Radiant Heat Loss in a Diesel Engine. Numerical Heat Transfer Part A. 1997, 31(6): 597-610.
6 H. Miyama, H. Kaji, Y. Hirose and N. Arai. Heat Transfer Characteristics of a Rotary Regenerative Combustion System(RRX). Heat Transfer -Japanese Research. 1998, 27(8): 584-596.
7 N.G. Shah. A New Method of Computation of Radiant Heat Transfer in Combustion Chambers. 1979, Ph.D.
8 E.P. Keramidaa, H.H. Liakosa, M.A. Fountib, A.G. Boudouvisa and N.C. Markatos. Radiative Heat Transfer in Natural Gas-Fired Furnaces. International Journal of Heat and Mass Transfer. 2000, 43: 1801-1809.
9 B. Zheng, C.X. Lin and M.A. Ebadian. Combined Turbulent Forced Convection and Thermal Radiation in a Curved Pipe with Uniform Wall Temperature. Numerical Heat Transfer Part A. 2003, 44(2): 149-167.
10 郭印诚, 林文漪. 马蹄形火焰玻璃熔窑燃烧空间流动与传热的数值模拟. 燃烧科学与技术. 2000, 6(3): 244-248.
11 P.J. Coelho. Detailed Numerical Simulation of Radiative Transfer in a Nonluminous Turbulent Jet Diffusion Flame. Combustion and Flame. 2004, 136: 481-492.
12 张文普, 丰镇平. 级间分离的流场及热流分析研究. 推进技术. 2003, 24(3): 240-243.
13 谈和平, 夏新林, 刘林华, 阮立明. 红外辐射特性与传输数值计算. 哈尔滨: 哈尔滨工业大学出版, 2006.
14 S.S. Penner and D.B. Olfe. Radiation and Reentry. New York: Academic Press, 1968.
15 T. Nakamura and T. Kai. Combined Radiation–Conduction Analysis and Experiment of Ceramic Insulation for Reentry Vehicles Journal of Thermophysics and Heat Transfer. 2004, 18 (1): 24-29.
16 L. Marraffa, F. Mazoue, P. Reynier and C. Reimers. Some Aerothermodynamic Aspects of ESA Entry Probes. Chinese Journal of Aeronautics. 2006, 19 (2): 126-133.
17 K. Fujita, T. Abe and K. Suzuki. Uv Radiation Impact on Heating Rate of Super-Orbital Reentry Vehicle. International Symposium on Space Technology and Science, 21st, 1998, Omiya, Japan, 1: 963-968.
18 闵桂荣, 郭舜. 航天器热控制. 第二版. 北京: 科学出版社, 1998.
19 J.P. Longtin and C.L. Tien. Saturable Absorption During High-Intensity Laser Heating of Liquids. Journal of Heat Transfer-Transactions of the ASME. 1996, 118: 924-930.
20 强希文, 王铁良, 吴乃清. 多光束激光大气传输. 光电子技术. 1999, 19(3): 167-172.
21 李现勤, 程兆谷, 蒋金波. 激光束在大气中长距离传输聚焦特性的研究. 光学学报. 2001, 21(3): 324-329.
22 曹百灵, 邬承就, 饶瑞中, 魏合理, 袁怿谦, 龚知本. HF/DF 激光传输的大气衰减特性. 强激光与粒子束. 2003, 15(1): 17-20.
23 D.A. Lashof and D.R. Ahuja. Relative Contributions of Greenhouse Gas Emissions to Global Warming. Nature. 1990, 344: 529 - 531.
24 R.S. Lindzen. Some Coolness Concerning Global Warming. Bulletin of the American Meteorological Society. 1990, 71(3): 288-299.
25 S. Rahmstorf and A. Ganopolski. Long-Term Global Warming Scenarios Computed with an Efficient Coupled Climate Model. Climatic Change. 1999, 43(2).
26 S. Lahsasni, M. Kouhila, M. Mahrouz, A. Idlimam and A. Jamali. ThinLayer Convective Solar Drying and Mathematical Modeling of Prickly Pear Peel. Energy. 2004, 29: 211-224.
27 S. Kumar and K. Mitra. Microscale Aspects of Thermal Radiation Transport and Laser Application. Advances in Heat Transfer. 1999, 33: 187-294.
28 J.P. Longtin and C.L. Tien. Saturable Absorption During High-Intensity Laser Heating of Liquids. ASME Journal of Heat Transfer. 1996, 118: 924-930.
29 T.G. Qui and C.L. Tien. Short Pulse Laser Heating in Metals. International Journal of Heat and Mass Transfer. 1992, 35: 719-726.
30 Y. Yamada. Light-Tissue Interaction and Optical Imaging in Biomedicine. Annual Review of Heat Transfer 1995, 6: 1-59.
31 A.K. Popp, M.T. Valentine, P.D. Kaplan and D.A. Weitz. Microscopic Origin of Light Scattering in Tissue. Applied Optics. 2003, 42: 2871-2880.
32 M.S. Patterson, B.C. Wilson and D.R. Wyman. The Propagation of Optical Radiation in Tissue I. Models of Radiation Transport and Their Application Lasers in Medical Science. 1991, 6(2): 155-168.
33 E. Okada, M. Firbank, M. Schweiger, S.R. Arridge, M. Cope and D.T. Delpy. Theoretical and Experimental Investigation of near-Infrared Light Propagation in a Model of the Adult Head. Applied Optics. 1997, 36: 21-31.
34 E.L. Nussbaum, G.D. Baxter and L. Lilge. A Review of Laser Technology and Light-Tissue Interactions as a Background to Therapeutic Applications of Low Intensity Lasers and Other Light Sources. Physical Therapy Reviews. 2003, 8(1): 31-44.
35 J.C. Hebden, H. Veenstra, H. Dehghani, E.M.C. Hillman, M. Schweiger, S.R. Arridge and D.T. Delpy. Three-Dimensional Time-Resolved Optical Tomography of a Conical Breast Phantom. Applied Optics. 2001, 40: 3278-3287.
36 S.R. Arridge. Optical Tomography in Medical Imaging. Inverse Problem. 1999, 15: R41-R93.
37 A.F. Fercher, W. Drexler, C.K. Hitzenberger and T. Lasser. Optical Coherence Tomography - Principles and Applications. Rep. Prog. Phys. 2003, 66: 239-303.
38 M. Sakami, K. Mitra and T. Vo-Dinh. Analysis of Short-Pulse Laser PhotonTransport through Tissues for Optical Tomography. Opt. Lett. 2002, 27(5): 336-338.
39 R. Siegel and C.M. Spuckler. Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions. Journal of Thermophysics and Heat Transfer. 1993, 7(4): 624-630.
40 P. Ben Abdallah and V. Le Dez. Thermal Emission of a Semi-Transparent Slab with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 67(3): 185-198.
41 P. Ben Abdallah and V. Le Dez. Radiative Flux Field inside an Absorbing–Emitting Semi-Transparent Slab with Variable Spatial Refractive Index at Radiative Conductive Coupling. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 67(2): 125-137.
42 P. Ben Abdallah and V. Le Dez. Thermal Field inside an Absorbing-Emitting Semitransparent Slab at Radiative Equilibrium with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 65(4): 595-608.
43 X.L. Xia, Y. Huang and H.P. Tan. Thermal Emission and Volumetric Absorption of a Graded Index Semitransparent Medium Layer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 235-248.
44 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.
45 乔亚天. 梯度折射率光学. 北京: 科学出版社, 1991.
46 黄勇. 梯度折射率半透明介质内热辐射传递研究. 哈尔滨工业大学博士学位论文. 2002: 8-10.
47 H.C. Hottel and E.S. Cohen. Radiant Heat Exchange in a Gas-Filled Enclosure Allowance for Nonuniformity of Gas Temperature. AIChE Journal. 1958, 4: 3-14.
48 J.T. Farmer and J.R. Howell. Monte Carlo Prediction of Radiative Heat Transfer in Inhomogeneous, Anisotropic, Nongray Media. Journal of Thermophysics and Heat Transfer. 1994, 8(1): 133-139.
49 J.R. Howell. Application of Monte Carlo to Heat Transfer Problems. Advances in Heat Transfer. 1968, 5: 1-54.
50 J.R. Howell and M. Perlmutter. Monte Carlo Solution of Thermal Transfer through Radiant Media between Gray Walls. ASME Journal of Heat Transfer. 1964, 86(1): 116-122.
51 F.C. Lockwood and N.G. Shah. A New Radiation Solution Method for Incorporation in General Combustion Prediction Procedures Proc 18th Symposium (Int.) on Combustion, 1981, Pittsburgh, USA: 1403-1414.
52 M.F. Modest. Radiative Heat Transfer. New York: McGraw-Hill, 1993.
53 J.H. Jeans. The Equations of Radiative Transfer of Energy. Monthly Notices of the Royal Astronomical Society. 1917, 78: 28-36.
54 M.P. Mengue and R. Viskanta. Radiative Transfer in Three-Dimensional Rectangular Enclosures Containing Inhomogeneous, Anisotropically Scattering Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 1985, 33(6): 533-549.
55 M.F. Modest. The Improved Differential Approximation for Radiative Transfer in Multi-Dimensional Media. ASME Journal of Heat Transfer. 1990, 112(2): 819-821.
56 W.A. Fiveland. Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method. Journal of Thermophysics and Heat Transfer. 1988, 2: 309-316.
57 K.H. Lee and R. Viskanta. Two-Dimensional Combined Conduction and Radiation Heat Transfer: Comparison of the Discrete Ordinates Method and the Diffusion Approximation Methods. Numerical Heat Transfer Part A. 2001, 39(3): 205-225.
58 G. Krishnamoorthy, R. Rawat and P.J. Smith. Parallel Computations of Radiative Heat Transfer Using the Discrete Ordinates Method. Numerical Heat Transfer Part B. 2005, 47(1): 19-38.
59 H.S. Li, G. Flamant and J.D. Lu. Mitigation of Ray Effects in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 2003, 43(5): 445-466.
60 N. Sel and I. Ayranci. The Method of Lines Solution of the Discrete Ordinates Method for Radiative Heat Transfer in Enclosures Containing Scattering Media. Numerical Heat Transfer Part B. 2003, 43(2): 179-201.
61 J. Con and P.J. Coelho. Parallelization of the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1997, 32(2): 151-173.
62 K.B. Cheong and T.H. Song. An Alternative Discrete Ordinates Method with Interpolation and Source Differencing for Two-Dimensional Radiative Transfer Problems. Numerical Heat Transfer Part B. 1997, 32(1): 107-125.
63 J.C. Chai, H.O. Lee and S.V. Patankar. Ray Effect and False Scattering in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1993, 24(4): 373-389.
64 Z. Guo and S. Kumar. Three-Dimensional Discrete Ordinates Method in Transient Radiative Transfer. Journal of Thermophysics and Heat Transfer. 2002, 16(3): 289-296.
65 M. Sakami, A. Charette and V. Le Dez. Application of the Discrete Ordinates Method to Combined Conductive and Radiative Heat Transfer in a Two-Dimensional Complex Geometry. Journal of Quantitative Spectroscopy and Radiative Transfer. 1996, 56(4): 517-533.
66 J.C. Chai, H.S. Lee and S.V. Patankar. Ray Effect and False Scattering in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1993, 24: 373-389.
67 P.J. Coelho. A Modified Version of the Discrete Ordinates Method for Radiative Heat Transfer Modelling. Computational Mechanics. 2004, 33(5): 375-388.
68 K.D. Lathrop. Use of Discrete-Ordinate Methods for Solution of Photon Transport Problems. Nuclear Science and Engineering. 1966, 24: 381-388.
69 J.S. Truelove. Three-Dimensional Radiation in Absorbing-Emitting-Scattering Media Using Discrete-Ordinate Approximation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1988, 39(1): 27-31.
70 刘林华, 余其铮, 阮立明, 谈和平. 求解辐射传递方程的离散坐标法. 计算物理. 1998, 15(3): 337-343.
71 G.D. Raithby and E.H. Chui. A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media. ASME Journal of Heat Transfer. 1990, 112: 415-423.
72 J.C. Chai and H.S. Lee. Finite-Volume Method for Radiation Heat Transfer. Journal of Thermophysics and Heat Transfer. 1994, 8(32): 419-425.
73 J.Y. Murthy and S.R. Mathur. Finite Volume Method for Radiative HeatTransfer Using Unstructured Meshes. Journal of Thermophysics and Heat Transfer. 1998, 12(3): 313-321.
74 J.C. Chai, H.S. Lee and S.V. Patankar. Finite-Volume Method for Radiation Heat Transfer. New York: Taylor & Francis, 2000: 109-141.
75 J.C. Chai. One-Dimensional Transient Radiation Heat Transfer Modeling Using a Finite-Volume Method. Numerical Heat Transfer Part B. 2003, 44(2): 187-208.
76 J.C. Chai, H.O. Lee and S.V. Patankar. Treatment of Irregular Geometries Using a Cartesian Coordinates Finite-Volume Radiation Heat Transfer Procedure. Numerical Heat Transfer Part B. 1994, 26(2): 225-235.
77 G.D. Raithby. Discussion of the Finite-Volume Method for Radiation, and Its Application Using 3D Unstructured Meshes. Numerical Heat Transfer Part B. 1999, 35(4): 389-405.
78 G. Raithby. Evaluation of Discretization Errors in Finite Volume Radiant Heat Transfer Prediction. Numerical Heat Transfer Part B. 1999, 36(3): 241-264.
79 S. Chandrasekhar. Radiative Transfer. New York: Dover Publications Inc., 1960.
80 W.A. Fiveland. Discrete-Ordinates Solution of the Radiative Transport Equation for Rectangular Enclosures. Journal of Heat Transfer-Transactions of the ASME. 1984, 106: 699-706.
81 P. Ben Abdallah and V. Le Dez. Thermal Emission of a Two-Dimensional Rectangular Cavity with Spatial Affine Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 66(6): 555-569.
82 Y. Huang, X.L. Xia and H.P. Tan. Temperature Field of Radiative Equilibrium in a Semitransparent Slab with a Linear Refractive Index and Gray Walls. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 249-261.
83 Y. Huang, X.L. Xia and H.P. Tan. Radiative Intensity Solution and Thermal Emission Analysis of a Semitransparent Medium Layer with a Sinusoidal Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 217-233.
84 L.H. Liu. Discrete Curved Ray-Tracing Method for Radiative Transfer in anAbsorbing-Emitting Semitransparent Slab with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 83(2): 223-228.
85 L.H. Liu and H.P. Tan. Transient Temperature Response in Semitransparent Variable Refractive Index Medium Subjected to a Pulse Irradiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 83(3-4): 333-344.
86 D. Lemonnier and V. Le Dez. Discrete Ordinates Solution of Radiative Transfer across a Slab with Variable Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 73(2-5): 195-204.
87 L.H. Liu. Finite Volume Method for Radiation Heat Transfer in Graded Index Medium. Journal of Thermophysics and Heat Transfer. 2006, 20(1): 59-66.
88 L.H. Liu, L. Zhang and H.P. Tan. Finite Element Method for Radiation Heat Transfer in Multi-Dimensional Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 97: 436-445.
89 L.H. Liu. Finite Element Solution of Radiative Transfer across a Slab with Variable Spatial Refractive Index. International Journal of Heat and Mass Transfer. 2005, 48(11): 2260-2265.
90 D. Gottlieb and S.A. Orgszag. Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1977, Philadelphia, 26.
91 D. Gottlieb, M.Y. Hussaini and S.A. Orgszag. Theory and Application of Spectral Method. Spectral Methods for Partial Differential Equations, SIAM-CBMS Series, 1984, Philadelphia: 1-54.
92 B. Ritchie, P.G. Dykema and D. Braddy. Use of Fast-Fourier-Transform Computational Methods in Radiation Transport. Phys. Rev. E. 1997, 56(2): 2217.
93 A.D. Kim and A. Ishimaru. A Chebyshev Spectral Method for Radiative Transfer Equations Applied to Electromagnetic Wave Propagation and Scattering in a Discrete Random Medium. Journal of Computational Physics. 1999, 152(1): 264-280.
94 A.D. Kim and M. Moscoso. Chebyshev Spectral Methods for RadiativeTransfer. SIAM Journal on Scientific Computing. 2002, 23(6): 2075-2095.
95 A.T. Patera. A Spectral Element Method for Fluid Dynamics - Laminar Flow in a Channel Expansion. Journal of Computational Physics. 1984, 54: 468-488.
96 K.Z. Korczak and A.T. Patera. An Isoparametric Spectral Element Method for Solution of the Navier-Stokes Equations in Complex Geometry. Journal of Computational Physics. 1986, 62: 361-382.
97 E.M. Ronquist and A.T. Patera. Spectral Element Multigrid. I. Formulation and Numerical Results. Journal of Scientific Computing. 1987, 2: 389-406.
98 P.F. Fischer, E.M. Ronquist and A.T. Patera. Parallel Spectral Element Methods for Viscous Flow. Chichester, UK: Wiley, 1989: 223-232.
99 R.D. Henderson and G.E. Karniadakis. Unstructured Spectral Element Methods for Simulation of Turbulent Flows. Journal of Computational Physics. 1995, 122: 191-217.
100 G.E. Karniadakis and S.J. Sherwin. Spectral/hp Element Methods for CFD. New York: Oxford University Press, 1999.
101 C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral Methods in Fluid Dynamics. Springer Verlag, 1988.
102 L.J.P. Timmermans, P.D. Minev and F.N. van-de-Vosse. An Approximate Projection Scheme for Incompressible Flow Using Spectral Elements. International Journal for Numerical Methods in Fluids. 1996, 22(7): 673-688.
103 S.J. Sherwin and G.E. Karniadakis. A Triangular Spectral Element Method; Applications to the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1995, 123(1): 189-229.
104 M.O. Deville, P.F. Fischer and E.H. Mund. High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, 2002.
105 M. Rudman and H.M. Blackburn. Direct Numerical Simulation of Turbulent Non-Newtonian Flow Using a Spectral Element Method Applied Mathematical Modelling. 2006, 30(11): 1229-1248.
106 F.X. Giraldo. Strong and Weak Lagrange-Galerkin Spectral Element Methods for the Shallow Water Equations. Computers & Mathematics with Applications. 2003, 45(1-3): 97-121.
107 C. Chauvière and R.G. Owens. A New Spectral Element Method for the Reliable Computation of Viscoelastic Flow. Computer methods in applied mechanics and engineering. 2001, 190(31): 3999-4018.
108 C.H. Amon and B.B. Mikic. Spectral Element Simulations of Unsteady Forced Convective Heat Transfer. Application to Compact Heat Exchanger Geometries. Numerical Heat Transfer Part A. 1991, 19(1): 1-19.
109 L.J.P. Timmermans, F.N. van-de-Vosse and P.D. Minev. Taylor-Galerkin Based Spectral Element Methods for Convection-Diffusion Problems. International Journal for Numerical Methods in Fluids. 1994, 18: 853-870.
110 T. Warburton. Spectral/hp Methods on Polymorphic Multidomains: Algorithms and Applications. 1999.
111 M.H. Carpenter and D. Gottlieb. Spectral Methods on Arbitrary Grids. Journal of Computational Physics. 1996, 129(1): 74-86.
112 U. Lee, J. Kim and A.Y.T. Leung. The Spectral Element Method in Structural Dynamics. The Shock and Vibration Digest. 2000, 32(6): 451-465.
113 D. Komatitsch and J. Tromp. Introduction to the Spectral Element Method for Three-Dimensional Seismic Wave Propagation. Geophysical Journal International. 1999, 139(3): 806-822.
114 D. Komatitsch, Q. Liu, J. Tromp, P. Süss, C. Stidham and J.H. Shaw. Simulations of Ground Motion in the Los Angeles Basin Based Upon the Spectral-Element Method. Bulletin of the Seismological Society of America. 2004, 94(1): 187-206.
115 D. Komatitsch, C. Barnes and J. Tromp. Simulation of Anisotropic Wave Propagation Based Upon a Spectral Element Method. Geophysics. 2000, 65(4): 1251-1260.
116 J.S. Hesthaven and T. Warburton. Nodal High-Order Methods on Unstructured Grids: I. Time-Domain Solution of Maxwell's Equations Journal of Computational Physics. 2002, 181(1): 186-221.
117 J.S. Hesthaven and D. Gottlieb. Stable Spectral Methods for Conservation Laws on Triangles with Unstructured Grids. Computer methods in applied mechanics and engineering. 1999, 175(3-4): 361-381.
118 J.P. Pontaza and J.N. Reddy. Least-Squares Finite Element Formulations forOne-Dimensional Radiative Transfer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 95(3): 387-406.
119 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.
120 J.P. Berrut and L.N. Trefethen. Barycentric Lagrange Interpolation. SIAM Review. 2004, 46(3): 501-517.
121 N.J. Higham. The Numerical Stability of Barycentric Lagrange Interpolation. IMA Journal of Numerical Analysis. 2004, 24: 547-556.
122 R. Baltensperger and M.R. Trummer. Spectral Differencing with a Twist. SIAM Journal on Scientific Computing. 2003, 24(5): 1465-1487.
123 M.A. Heaslet and R.F. Warming. Radiative Transport and Wall Temperature Slip in an Absorbing Planar Medium. International Journal of Heat and Mass Transfer. 1965, 8(7): 979-994.
124 T.K. Kim and H. Lee. Effect of Anisotropic Scattering on Radiative Heat Transfer in Two-Dimensional Rectangular Enclosures. International Journal of Heat and Mass Transfer. 1988, 31(8): 1711-1721.
125 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.
126 D.Y. Byun, S.W. Baek and M.Y. Kim. Prediction of Radiative Heat Transfer in a 2D Enclosure with Blocked-Off, Multi-Block, and Embeded Boundary Treatments. Journal of Heat Transfer-Transactions of the ASME. 2000, 366: 119-126.
127 A.C. Ratzel and J.R. Howell. Two-Dimensional Radiation in Absorbing-Emitting Media Using the P-N Approximation. ASME Journal of Heat Transfer. 1983, 105(2): 333-340.
128 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.
129 L.H. Liu. Finite Element Simulation of Radiative Heat Transfer in Absorbing and Scattering Media. Journal of Thermophysics and Heat Transfer. 2004, 18: 555-557.
130 A.N. Brooks and T.J.R. Hughes. 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: 199-259.
131 C. Johnson. Numerical Solutions of Partial Differential Equations by the Finite Element Method. New York: Cambridge University Press, 1987.
132 李荣华, 冯果忱. 微分方程数值解法. 第三版. 北京: 高等教育出版社, 1996.
133 M.E. Larsen and J.R. Howell. The Exchange Factor Method: An Alternative Zonal Formulation of Radiating Enclosure Analysis. Journal of Heat Transfer-Transactions of the ASME. 1985, 107: 936-942.
134 V.B. Kisslelev, L. Roberti and G. Perona. An Application of the Finite-Element Method to the Solution of the Radiative Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1994, 51: 603-614.
135 E.E. Lewis and W.F. Miller. Computational Methods of Neutron Transport. New York Wiley, 1984.
136 W.A. Fiveland and J.P. Jessee. Finite Element Formulation of the Discrete-Ordinates Method for Multidimensional Geometries. Journal of Thermophysics and Heat Transfer. 1994, 8: 426-433.
137 W.A. Fiveland and J.P. Jessee. Comparison of Discrete Ordinates Formulations for Radiative Heat Transfer in Multidimensional Geometries. Journal of Thermophysics and Heat Transfer. 1995, 9(1): 47-54.
138 K.B. Cheong and T.H. Song. Examination of Solution Methods for the Second-Order Discrete Ordinate Formulation. Numerical Heat Transfer Part B. 1995, 27: 155-173.
139 J. Liu and Y.S. Chen. Examination of Conventional and Even-Parity Formulations of Discrete Ordinates Method in a Body-Fitted Coordinate System. Journal of Quantitative Spectroscopy and Radiative Transfer. 1999, 61(4): 417- 431.
140 M.Y. Kim, S.W. Baek and J.H. Park. Unstructured Finite-Volume Method for Radiative Heat Transfer in a Complex Two-Dimensional Geometry with Obstacles. Numerical Heat Transfer Part B. 2001, 39: 617-635.
141 X. Cui and B.Q. Li. Discontinuous Finite Element Solution of 2-D Radiative Transfer with and without Axisymmetry. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 96(3-4): 383-407.
142 X. Cui and B.Q. Li. A Discontinuous Finite-Element Formulation for Internal Radiation Problems. Numerical Heat Transfer Part B. 2004, 46(3): 223-242.
143 L.H. Liu and L.J. Liu. Discontinuous Finite Element Method for Radiative Heat Transfer in Semitransparent Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 105: 377-387.
144 F. Bassi and S. Rebay. High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations. Journal of Computational Physics. 1997, 138(2): 251-285.
145 B. Cockburn and C.W. Shu. The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems. Journal of Computational Physics. 1998, 141(2): 199-224.
146 C.E. Baumann and J.T. Oden. A Discontinuous hp Finite Element Method for Convection-Diffusion Problems. Computer methods in applied mechanics and engineering. 1999, 175(3): 311-341.
147 T.C. Warburton and G.E. Karniadakis. A Discontinuous Galerkin Method for the Viscous MHD Equations. Journal of Computational Physics. 1999, 152(2): 608-641.
148 B. Cockburn. Discontinuous Galerkin Methods. ZAMM. 2003, 83(11): 731-754.
149 B. Cockburn, G.E. Karniadakis and C.W. Shu. The Development of Discontinuous Galerkin Methods. Berlin: Springer Verlag, 2000: 3-50.
150 M.A. Ramankutty and A.L. Crosbie. Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures. Journal of Quantitative Spectroscopy and Radiative Transfer. 1997, 57(11): 107-140.
151 J.M. Zhao and L.H. Liu. Solution of Radiative Heat Transfer in Graded Index Media by Least Square Spectral Element Method. International Journal of Heat and Mass Transfer. 2007, 50: 2634-2642.
152 A.L. Crosbie and R.G. Schrenker. Radiative Transfer in a Two-DimensionalRectangular Medium Exposed to Diffuse Radiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1984, 31(4): 339-372.
153 R. Viskanta and R.J. Grosh. Heat Transfer by Simultaneous Conduction and Radiation in an Absorbing Medium. Journal of Heat Transfer. 1962, 84(1): 63-72.
154 W.W. Yuen and L.W. Wong. Heat Transfer by Conduction and Radiation in a One-Dimensional Absorbing, Emitting and Anisotropically-Scattering Medium. Journal of Heat Transfer. 1980, 102: 303-307.
155 S.P. Burns, J.R. Howell and D.E. Klein. Empirical Evaluation of an Important Approximation for Combined-Mode Heat Transfer in a Participating Medium Using the Finite-Element Method. Numerical Heat Transfer Part B. 1995, 27: 309-322.
156 W.W. Yuen and E.E. Takara. Analysis of Combined Conductive-Radiative Heat Transfer in a Two-Dimensional Rectangular Enclosure with a Gray Medium. ASME Journal of Heat Transfer. 1988, 110(2): 468-474.
157 M.M. Razzaque, J.R. Howell and D.E. Klein. Coupled Radiative and Conductive Heat Transfer in a Two-Dimensional Rectangular Enclosure with a Gray Participating Media Using Finite Elements. Journal of Heat Transfer. 1984, 106: 613-619.
158 T.Y. Kim and S.W. Baek. Analysis of Combined Conductive and Radiative Heat Transfer in a Two-Dimensional Rectangular Enclosure Using the Discrete Ordinates Method. International Journal of Heat and Mass Transfer. 1991, 34(9): 2265-2273.
159 S.C. Mishra, A. Lankadasu and K.N. Beronov. Application of the Lattice Boltzmann Method for Solving the Energy Equation of a 2-D Transient Conduction-Radiation Problem. International Journal of Heat and Mass Transfer. 2005, 48(17): 3648-3659.
160 M.L. Nice. Application of Finite Elements to Heat Transfer in a Participating Medium. Numerical Properties and Methodologies in Heat Transfer, Proceedings of the 2nd National Symposium, Hemisphere, 1983, Washington, DC: 497-514.
161 P. Talukdar and S.C. Mishra. Analysis of Conduction–Radiation Problem in Absorbing, Emitting and Anisotropically Scattering Media Using theCollapsed Dimension Method. International Journal of Heat and Mass Transfer. 2002, 45: 2159–2168.
162 P.-f. Hsu. Effects of Multiple Scattering and Reflective Boundary on the Transient Radiative Transfer Process. International Journal of Thermal Sciences. 2001, 40(6): 539-549.
163 Z. Guo, J. Aber, B.A. Garetz and S. Kumar. Monte Carlo Simulation and Experiments of Pulsed Radiative Transfer. 2002, 73: 159-168.
164 Z. Guo, S. Kumar and K. San. Multidimensional Monte Carlo Simulation of Short-Pulse Laser Transport in Scattering Media. Journal of Thermophysics and Heat Transfer. 2000, 14: 504-511.
165 B.C. Wilson and G. Adam. A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue. Medical Physics. 1983, 10: 824-830.
166 M. Sakami, K. Mitra and P.F. Hsu. Analysis of Light-Pulse Transport through Two-Dimensional Scattering and Absorbing Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 73: 169-179.
167 Y. Hasegawa, Y. Yamada, M. Tamura and Y. Nomura. Monte Carlo Simulation of Light Transmission through Living Tissues. Applied Optics. 1991, 30: 4515-4520.
168 N.A. Gentile. Implicit Monte Carlo Diffusion-an Acceleration Method for Monte Carlo Time-Dependent Radiative Transfer Simulations. Journal of Computational Physics. 2001, 172: 543-571.
169 M.Q. Brewster and Y. Yamada. Optical Properties of Thick Turbid Media from Picosecond Time-Resolved Light Scattering Measurements. International Journal of Heat and Mass Transfer. 1995, 38: 2569-2581.
170 Z.M. Tan and P.F. Hsu. An Integral Formulation of Transient Radiative Transfer. ASME Journal of Heat Transfer. 2001, 123: 466-475.
171 S.-H. Wu and C.-Y. Wu. Integral Equation Solutions for Transient Radiative Transfer in Nonhomogeneous Anisotropically Scattering Media. ASME Journal of Heat Transfer. 2000, 122: 818-822.
172 S.H. Wu and C.Y. Wu. Time-Resolved Spatial Distribution of Scattered Radiative Energy in a Two-Dimensional Cylindrical Medium with a Large Mean Free Path for Scattering. International Journal of Heat and Mass Transfer. 2001, 44: 2611-2619.
173 C.Y. Wu. Propagation of Scattered Radiation in a Participating Planar Medium with Pulse Irradiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 64: 537-548.
174 C.Y. Wu and S.H. Wu. Integral Equation Formulation for Transient Radiative Transfer in an Anisotropically Scattering Medium. International Journal of Heat and Mass Transfer. 2000, 43: 2009-2020.
175 K. Mitra and S. Kumar. Development and Comparison of Models for Light-Pulse Transport through Scattering-Absorbing Media. Applied Optics. 1999, 38(1): 188-196.
176 Z. Guo and S. Kumar. Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media. Applied Optics. 2001, 40(19): 3156-3163.
177 Z. Guo and K.H. Kim. Ultrafast-Laser-Radiation Transfer in Heterogeneous Tissues with the Discrete-Ordinates Method. Applied Optics. 2003, 42(16): 2897-2905.
178 K. Mitra, M.-S. Lai and S. Kumar. Transient Radiation Transport in Participating Media with a Rectangular Enclosure. Journal of Thermophysics and Heat Transfer. 1997, 11(3): 409-414.
179 J.C. Chai. Transient Radiative Transfer in Irregular Two-Dimensional Geometries. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 84: 281-294.
180 W. An, L.M. Ruan, H.P. Tan and H. Qi. Least-Squares Finite Element Analysis for Transient Radiative Transfer in Absorbing and Scattering Media. Journal of Heat Transfer. 2006, 128: 499-503.
181 L.H. Liu and P.-f. Hsu. Analysis of Transient Radiative Transfer in Semitransparent Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 105(3): 357-376.
182 陶文铨. 数值传热学. 第二版. 西安: 西安交通大学出版社, 2001: 82-86.
183 F.N. van de Vosse and P.D. Minev. Spectral Element Methods: Theory and Applications. http://www.mate.tue.nl/people/vosse/docs/vosse96b.pdf.
2 R. Viskanta. Overview of Convection and Radiation in High Temperature Gas Flows. International Journal of Engineering Science. 1998, 36(12): 1677-1699.
3 W.H. Sutton and X.L. Chen. A General Integration Method for Radiative Transfer in 3-D Non-Homogeneous Cylindrical Media with Anisotropic Scattering. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 84: 65-103.
4 侯晓春, 季鹤鸣, 刘庆国, 严传俊, 赵坚行. 高性能航空燃气轮机燃烧技术. 北京: 国防工业出版社, 2002.
5 J. Abraham and V. Magi. Application of the Discrete Ordinates Method to Compute Radiant Heat Loss in a Diesel Engine. Numerical Heat Transfer Part A. 1997, 31(6): 597-610.
6 H. Miyama, H. Kaji, Y. Hirose and N. Arai. Heat Transfer Characteristics of a Rotary Regenerative Combustion System(RRX). Heat Transfer -Japanese Research. 1998, 27(8): 584-596.
7 N.G. Shah. A New Method of Computation of Radiant Heat Transfer in Combustion Chambers. 1979, Ph.D.
8 E.P. Keramidaa, H.H. Liakosa, M.A. Fountib, A.G. Boudouvisa and N.C. Markatos. Radiative Heat Transfer in Natural Gas-Fired Furnaces. International Journal of Heat and Mass Transfer. 2000, 43: 1801-1809.
9 B. Zheng, C.X. Lin and M.A. Ebadian. Combined Turbulent Forced Convection and Thermal Radiation in a Curved Pipe with Uniform Wall Temperature. Numerical Heat Transfer Part A. 2003, 44(2): 149-167.
10 郭印诚, 林文漪. 马蹄形火焰玻璃熔窑燃烧空间流动与传热的数值模拟. 燃烧科学与技术. 2000, 6(3): 244-248.
11 P.J. Coelho. Detailed Numerical Simulation of Radiative Transfer in a Nonluminous Turbulent Jet Diffusion Flame. Combustion and Flame. 2004, 136: 481-492.
12 张文普, 丰镇平. 级间分离的流场及热流分析研究. 推进技术. 2003, 24(3): 240-243.
13 谈和平, 夏新林, 刘林华, 阮立明. 红外辐射特性与传输数值计算. 哈尔滨: 哈尔滨工业大学出版, 2006.
14 S.S. Penner and D.B. Olfe. Radiation and Reentry. New York: Academic Press, 1968.
15 T. Nakamura and T. Kai. Combined Radiation–Conduction Analysis and Experiment of Ceramic Insulation for Reentry Vehicles Journal of Thermophysics and Heat Transfer. 2004, 18 (1): 24-29.
16 L. Marraffa, F. Mazoue, P. Reynier and C. Reimers. Some Aerothermodynamic Aspects of ESA Entry Probes. Chinese Journal of Aeronautics. 2006, 19 (2): 126-133.
17 K. Fujita, T. Abe and K. Suzuki. Uv Radiation Impact on Heating Rate of Super-Orbital Reentry Vehicle. International Symposium on Space Technology and Science, 21st, 1998, Omiya, Japan, 1: 963-968.
18 闵桂荣, 郭舜. 航天器热控制. 第二版. 北京: 科学出版社, 1998.
19 J.P. Longtin and C.L. Tien. Saturable Absorption During High-Intensity Laser Heating of Liquids. Journal of Heat Transfer-Transactions of the ASME. 1996, 118: 924-930.
20 强希文, 王铁良, 吴乃清. 多光束激光大气传输. 光电子技术. 1999, 19(3): 167-172.
21 李现勤, 程兆谷, 蒋金波. 激光束在大气中长距离传输聚焦特性的研究. 光学学报. 2001, 21(3): 324-329.
22 曹百灵, 邬承就, 饶瑞中, 魏合理, 袁怿谦, 龚知本. HF/DF 激光传输的大气衰减特性. 强激光与粒子束. 2003, 15(1): 17-20.
23 D.A. Lashof and D.R. Ahuja. Relative Contributions of Greenhouse Gas Emissions to Global Warming. Nature. 1990, 344: 529 - 531.
24 R.S. Lindzen. Some Coolness Concerning Global Warming. Bulletin of the American Meteorological Society. 1990, 71(3): 288-299.
25 S. Rahmstorf and A. Ganopolski. Long-Term Global Warming Scenarios Computed with an Efficient Coupled Climate Model. Climatic Change. 1999, 43(2).
26 S. Lahsasni, M. Kouhila, M. Mahrouz, A. Idlimam and A. Jamali. ThinLayer Convective Solar Drying and Mathematical Modeling of Prickly Pear Peel. Energy. 2004, 29: 211-224.
27 S. Kumar and K. Mitra. Microscale Aspects of Thermal Radiation Transport and Laser Application. Advances in Heat Transfer. 1999, 33: 187-294.
28 J.P. Longtin and C.L. Tien. Saturable Absorption During High-Intensity Laser Heating of Liquids. ASME Journal of Heat Transfer. 1996, 118: 924-930.
29 T.G. Qui and C.L. Tien. Short Pulse Laser Heating in Metals. International Journal of Heat and Mass Transfer. 1992, 35: 719-726.
30 Y. Yamada. Light-Tissue Interaction and Optical Imaging in Biomedicine. Annual Review of Heat Transfer 1995, 6: 1-59.
31 A.K. Popp, M.T. Valentine, P.D. Kaplan and D.A. Weitz. Microscopic Origin of Light Scattering in Tissue. Applied Optics. 2003, 42: 2871-2880.
32 M.S. Patterson, B.C. Wilson and D.R. Wyman. The Propagation of Optical Radiation in Tissue I. Models of Radiation Transport and Their Application Lasers in Medical Science. 1991, 6(2): 155-168.
33 E. Okada, M. Firbank, M. Schweiger, S.R. Arridge, M. Cope and D.T. Delpy. Theoretical and Experimental Investigation of near-Infrared Light Propagation in a Model of the Adult Head. Applied Optics. 1997, 36: 21-31.
34 E.L. Nussbaum, G.D. Baxter and L. Lilge. A Review of Laser Technology and Light-Tissue Interactions as a Background to Therapeutic Applications of Low Intensity Lasers and Other Light Sources. Physical Therapy Reviews. 2003, 8(1): 31-44.
35 J.C. Hebden, H. Veenstra, H. Dehghani, E.M.C. Hillman, M. Schweiger, S.R. Arridge and D.T. Delpy. Three-Dimensional Time-Resolved Optical Tomography of a Conical Breast Phantom. Applied Optics. 2001, 40: 3278-3287.
36 S.R. Arridge. Optical Tomography in Medical Imaging. Inverse Problem. 1999, 15: R41-R93.
37 A.F. Fercher, W. Drexler, C.K. Hitzenberger and T. Lasser. Optical Coherence Tomography - Principles and Applications. Rep. Prog. Phys. 2003, 66: 239-303.
38 M. Sakami, K. Mitra and T. Vo-Dinh. Analysis of Short-Pulse Laser PhotonTransport through Tissues for Optical Tomography. Opt. Lett. 2002, 27(5): 336-338.
39 R. Siegel and C.M. Spuckler. Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions. Journal of Thermophysics and Heat Transfer. 1993, 7(4): 624-630.
40 P. Ben Abdallah and V. Le Dez. Thermal Emission of a Semi-Transparent Slab with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 67(3): 185-198.
41 P. Ben Abdallah and V. Le Dez. Radiative Flux Field inside an Absorbing–Emitting Semi-Transparent Slab with Variable Spatial Refractive Index at Radiative Conductive Coupling. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 67(2): 125-137.
42 P. Ben Abdallah and V. Le Dez. Thermal Field inside an Absorbing-Emitting Semitransparent Slab at Radiative Equilibrium with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 65(4): 595-608.
43 X.L. Xia, Y. Huang and H.P. Tan. Thermal Emission and Volumetric Absorption of a Graded Index Semitransparent Medium Layer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 235-248.
44 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.
45 乔亚天. 梯度折射率光学. 北京: 科学出版社, 1991.
46 黄勇. 梯度折射率半透明介质内热辐射传递研究. 哈尔滨工业大学博士学位论文. 2002: 8-10.
47 H.C. Hottel and E.S. Cohen. Radiant Heat Exchange in a Gas-Filled Enclosure Allowance for Nonuniformity of Gas Temperature. AIChE Journal. 1958, 4: 3-14.
48 J.T. Farmer and J.R. Howell. Monte Carlo Prediction of Radiative Heat Transfer in Inhomogeneous, Anisotropic, Nongray Media. Journal of Thermophysics and Heat Transfer. 1994, 8(1): 133-139.
49 J.R. Howell. Application of Monte Carlo to Heat Transfer Problems. Advances in Heat Transfer. 1968, 5: 1-54.
50 J.R. Howell and M. Perlmutter. Monte Carlo Solution of Thermal Transfer through Radiant Media between Gray Walls. ASME Journal of Heat Transfer. 1964, 86(1): 116-122.
51 F.C. Lockwood and N.G. Shah. A New Radiation Solution Method for Incorporation in General Combustion Prediction Procedures Proc 18th Symposium (Int.) on Combustion, 1981, Pittsburgh, USA: 1403-1414.
52 M.F. Modest. Radiative Heat Transfer. New York: McGraw-Hill, 1993.
53 J.H. Jeans. The Equations of Radiative Transfer of Energy. Monthly Notices of the Royal Astronomical Society. 1917, 78: 28-36.
54 M.P. Mengue and R. Viskanta. Radiative Transfer in Three-Dimensional Rectangular Enclosures Containing Inhomogeneous, Anisotropically Scattering Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 1985, 33(6): 533-549.
55 M.F. Modest. The Improved Differential Approximation for Radiative Transfer in Multi-Dimensional Media. ASME Journal of Heat Transfer. 1990, 112(2): 819-821.
56 W.A. Fiveland. Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method. Journal of Thermophysics and Heat Transfer. 1988, 2: 309-316.
57 K.H. Lee and R. Viskanta. Two-Dimensional Combined Conduction and Radiation Heat Transfer: Comparison of the Discrete Ordinates Method and the Diffusion Approximation Methods. Numerical Heat Transfer Part A. 2001, 39(3): 205-225.
58 G. Krishnamoorthy, R. Rawat and P.J. Smith. Parallel Computations of Radiative Heat Transfer Using the Discrete Ordinates Method. Numerical Heat Transfer Part B. 2005, 47(1): 19-38.
59 H.S. Li, G. Flamant and J.D. Lu. Mitigation of Ray Effects in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 2003, 43(5): 445-466.
60 N. Sel and I. Ayranci. The Method of Lines Solution of the Discrete Ordinates Method for Radiative Heat Transfer in Enclosures Containing Scattering Media. Numerical Heat Transfer Part B. 2003, 43(2): 179-201.
61 J. Con and P.J. Coelho. Parallelization of the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1997, 32(2): 151-173.
62 K.B. Cheong and T.H. Song. An Alternative Discrete Ordinates Method with Interpolation and Source Differencing for Two-Dimensional Radiative Transfer Problems. Numerical Heat Transfer Part B. 1997, 32(1): 107-125.
63 J.C. Chai, H.O. Lee and S.V. Patankar. Ray Effect and False Scattering in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1993, 24(4): 373-389.
64 Z. Guo and S. Kumar. Three-Dimensional Discrete Ordinates Method in Transient Radiative Transfer. Journal of Thermophysics and Heat Transfer. 2002, 16(3): 289-296.
65 M. Sakami, A. Charette and V. Le Dez. Application of the Discrete Ordinates Method to Combined Conductive and Radiative Heat Transfer in a Two-Dimensional Complex Geometry. Journal of Quantitative Spectroscopy and Radiative Transfer. 1996, 56(4): 517-533.
66 J.C. Chai, H.S. Lee and S.V. Patankar. Ray Effect and False Scattering in the Discrete Ordinates Method. Numerical Heat Transfer Part B. 1993, 24: 373-389.
67 P.J. Coelho. A Modified Version of the Discrete Ordinates Method for Radiative Heat Transfer Modelling. Computational Mechanics. 2004, 33(5): 375-388.
68 K.D. Lathrop. Use of Discrete-Ordinate Methods for Solution of Photon Transport Problems. Nuclear Science and Engineering. 1966, 24: 381-388.
69 J.S. Truelove. Three-Dimensional Radiation in Absorbing-Emitting-Scattering Media Using Discrete-Ordinate Approximation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1988, 39(1): 27-31.
70 刘林华, 余其铮, 阮立明, 谈和平. 求解辐射传递方程的离散坐标法. 计算物理. 1998, 15(3): 337-343.
71 G.D. Raithby and E.H. Chui. A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media. ASME Journal of Heat Transfer. 1990, 112: 415-423.
72 J.C. Chai and H.S. Lee. Finite-Volume Method for Radiation Heat Transfer. Journal of Thermophysics and Heat Transfer. 1994, 8(32): 419-425.
73 J.Y. Murthy and S.R. Mathur. Finite Volume Method for Radiative HeatTransfer Using Unstructured Meshes. Journal of Thermophysics and Heat Transfer. 1998, 12(3): 313-321.
74 J.C. Chai, H.S. Lee and S.V. Patankar. Finite-Volume Method for Radiation Heat Transfer. New York: Taylor & Francis, 2000: 109-141.
75 J.C. Chai. One-Dimensional Transient Radiation Heat Transfer Modeling Using a Finite-Volume Method. Numerical Heat Transfer Part B. 2003, 44(2): 187-208.
76 J.C. Chai, H.O. Lee and S.V. Patankar. Treatment of Irregular Geometries Using a Cartesian Coordinates Finite-Volume Radiation Heat Transfer Procedure. Numerical Heat Transfer Part B. 1994, 26(2): 225-235.
77 G.D. Raithby. Discussion of the Finite-Volume Method for Radiation, and Its Application Using 3D Unstructured Meshes. Numerical Heat Transfer Part B. 1999, 35(4): 389-405.
78 G. Raithby. Evaluation of Discretization Errors in Finite Volume Radiant Heat Transfer Prediction. Numerical Heat Transfer Part B. 1999, 36(3): 241-264.
79 S. Chandrasekhar. Radiative Transfer. New York: Dover Publications Inc., 1960.
80 W.A. Fiveland. Discrete-Ordinates Solution of the Radiative Transport Equation for Rectangular Enclosures. Journal of Heat Transfer-Transactions of the ASME. 1984, 106: 699-706.
81 P. Ben Abdallah and V. Le Dez. Thermal Emission of a Two-Dimensional Rectangular Cavity with Spatial Affine Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 66(6): 555-569.
82 Y. Huang, X.L. Xia and H.P. Tan. Temperature Field of Radiative Equilibrium in a Semitransparent Slab with a Linear Refractive Index and Gray Walls. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 249-261.
83 Y. Huang, X.L. Xia and H.P. Tan. Radiative Intensity Solution and Thermal Emission Analysis of a Semitransparent Medium Layer with a Sinusoidal Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 74(2): 217-233.
84 L.H. Liu. Discrete Curved Ray-Tracing Method for Radiative Transfer in anAbsorbing-Emitting Semitransparent Slab with Variable Spatial Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 83(2): 223-228.
85 L.H. Liu and H.P. Tan. Transient Temperature Response in Semitransparent Variable Refractive Index Medium Subjected to a Pulse Irradiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 83(3-4): 333-344.
86 D. Lemonnier and V. Le Dez. Discrete Ordinates Solution of Radiative Transfer across a Slab with Variable Refractive Index. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 73(2-5): 195-204.
87 L.H. Liu. Finite Volume Method for Radiation Heat Transfer in Graded Index Medium. Journal of Thermophysics and Heat Transfer. 2006, 20(1): 59-66.
88 L.H. Liu, L. Zhang and H.P. Tan. Finite Element Method for Radiation Heat Transfer in Multi-Dimensional Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2006, 97: 436-445.
89 L.H. Liu. Finite Element Solution of Radiative Transfer across a Slab with Variable Spatial Refractive Index. International Journal of Heat and Mass Transfer. 2005, 48(11): 2260-2265.
90 D. Gottlieb and S.A. Orgszag. Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1977, Philadelphia, 26.
91 D. Gottlieb, M.Y. Hussaini and S.A. Orgszag. Theory and Application of Spectral Method. Spectral Methods for Partial Differential Equations, SIAM-CBMS Series, 1984, Philadelphia: 1-54.
92 B. Ritchie, P.G. Dykema and D. Braddy. Use of Fast-Fourier-Transform Computational Methods in Radiation Transport. Phys. Rev. E. 1997, 56(2): 2217.
93 A.D. Kim and A. Ishimaru. A Chebyshev Spectral Method for Radiative Transfer Equations Applied to Electromagnetic Wave Propagation and Scattering in a Discrete Random Medium. Journal of Computational Physics. 1999, 152(1): 264-280.
94 A.D. Kim and M. Moscoso. Chebyshev Spectral Methods for RadiativeTransfer. SIAM Journal on Scientific Computing. 2002, 23(6): 2075-2095.
95 A.T. Patera. A Spectral Element Method for Fluid Dynamics - Laminar Flow in a Channel Expansion. Journal of Computational Physics. 1984, 54: 468-488.
96 K.Z. Korczak and A.T. Patera. An Isoparametric Spectral Element Method for Solution of the Navier-Stokes Equations in Complex Geometry. Journal of Computational Physics. 1986, 62: 361-382.
97 E.M. Ronquist and A.T. Patera. Spectral Element Multigrid. I. Formulation and Numerical Results. Journal of Scientific Computing. 1987, 2: 389-406.
98 P.F. Fischer, E.M. Ronquist and A.T. Patera. Parallel Spectral Element Methods for Viscous Flow. Chichester, UK: Wiley, 1989: 223-232.
99 R.D. Henderson and G.E. Karniadakis. Unstructured Spectral Element Methods for Simulation of Turbulent Flows. Journal of Computational Physics. 1995, 122: 191-217.
100 G.E. Karniadakis and S.J. Sherwin. Spectral/hp Element Methods for CFD. New York: Oxford University Press, 1999.
101 C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral Methods in Fluid Dynamics. Springer Verlag, 1988.
102 L.J.P. Timmermans, P.D. Minev and F.N. van-de-Vosse. An Approximate Projection Scheme for Incompressible Flow Using Spectral Elements. International Journal for Numerical Methods in Fluids. 1996, 22(7): 673-688.
103 S.J. Sherwin and G.E. Karniadakis. A Triangular Spectral Element Method; Applications to the Incompressible Navier-Stokes Equations. Computer methods in applied mechanics and engineering. 1995, 123(1): 189-229.
104 M.O. Deville, P.F. Fischer and E.H. Mund. High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, 2002.
105 M. Rudman and H.M. Blackburn. Direct Numerical Simulation of Turbulent Non-Newtonian Flow Using a Spectral Element Method Applied Mathematical Modelling. 2006, 30(11): 1229-1248.
106 F.X. Giraldo. Strong and Weak Lagrange-Galerkin Spectral Element Methods for the Shallow Water Equations. Computers & Mathematics with Applications. 2003, 45(1-3): 97-121.
107 C. Chauvière and R.G. Owens. A New Spectral Element Method for the Reliable Computation of Viscoelastic Flow. Computer methods in applied mechanics and engineering. 2001, 190(31): 3999-4018.
108 C.H. Amon and B.B. Mikic. Spectral Element Simulations of Unsteady Forced Convective Heat Transfer. Application to Compact Heat Exchanger Geometries. Numerical Heat Transfer Part A. 1991, 19(1): 1-19.
109 L.J.P. Timmermans, F.N. van-de-Vosse and P.D. Minev. Taylor-Galerkin Based Spectral Element Methods for Convection-Diffusion Problems. International Journal for Numerical Methods in Fluids. 1994, 18: 853-870.
110 T. Warburton. Spectral/hp Methods on Polymorphic Multidomains: Algorithms and Applications. 1999.
111 M.H. Carpenter and D. Gottlieb. Spectral Methods on Arbitrary Grids. Journal of Computational Physics. 1996, 129(1): 74-86.
112 U. Lee, J. Kim and A.Y.T. Leung. The Spectral Element Method in Structural Dynamics. The Shock and Vibration Digest. 2000, 32(6): 451-465.
113 D. Komatitsch and J. Tromp. Introduction to the Spectral Element Method for Three-Dimensional Seismic Wave Propagation. Geophysical Journal International. 1999, 139(3): 806-822.
114 D. Komatitsch, Q. Liu, J. Tromp, P. Süss, C. Stidham and J.H. Shaw. Simulations of Ground Motion in the Los Angeles Basin Based Upon the Spectral-Element Method. Bulletin of the Seismological Society of America. 2004, 94(1): 187-206.
115 D. Komatitsch, C. Barnes and J. Tromp. Simulation of Anisotropic Wave Propagation Based Upon a Spectral Element Method. Geophysics. 2000, 65(4): 1251-1260.
116 J.S. Hesthaven and T. Warburton. Nodal High-Order Methods on Unstructured Grids: I. Time-Domain Solution of Maxwell's Equations Journal of Computational Physics. 2002, 181(1): 186-221.
117 J.S. Hesthaven and D. Gottlieb. Stable Spectral Methods for Conservation Laws on Triangles with Unstructured Grids. Computer methods in applied mechanics and engineering. 1999, 175(3-4): 361-381.
118 J.P. Pontaza and J.N. Reddy. Least-Squares Finite Element Formulations forOne-Dimensional Radiative Transfer. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 95(3): 387-406.
119 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.
120 J.P. Berrut and L.N. Trefethen. Barycentric Lagrange Interpolation. SIAM Review. 2004, 46(3): 501-517.
121 N.J. Higham. The Numerical Stability of Barycentric Lagrange Interpolation. IMA Journal of Numerical Analysis. 2004, 24: 547-556.
122 R. Baltensperger and M.R. Trummer. Spectral Differencing with a Twist. SIAM Journal on Scientific Computing. 2003, 24(5): 1465-1487.
123 M.A. Heaslet and R.F. Warming. Radiative Transport and Wall Temperature Slip in an Absorbing Planar Medium. International Journal of Heat and Mass Transfer. 1965, 8(7): 979-994.
124 T.K. Kim and H. Lee. Effect of Anisotropic Scattering on Radiative Heat Transfer in Two-Dimensional Rectangular Enclosures. International Journal of Heat and Mass Transfer. 1988, 31(8): 1711-1721.
125 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.
126 D.Y. Byun, S.W. Baek and M.Y. Kim. Prediction of Radiative Heat Transfer in a 2D Enclosure with Blocked-Off, Multi-Block, and Embeded Boundary Treatments. Journal of Heat Transfer-Transactions of the ASME. 2000, 366: 119-126.
127 A.C. Ratzel and J.R. Howell. Two-Dimensional Radiation in Absorbing-Emitting Media Using the P-N Approximation. ASME Journal of Heat Transfer. 1983, 105(2): 333-340.
128 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.
129 L.H. Liu. Finite Element Simulation of Radiative Heat Transfer in Absorbing and Scattering Media. Journal of Thermophysics and Heat Transfer. 2004, 18: 555-557.
130 A.N. Brooks and T.J.R. Hughes. 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: 199-259.
131 C. Johnson. Numerical Solutions of Partial Differential Equations by the Finite Element Method. New York: Cambridge University Press, 1987.
132 李荣华, 冯果忱. 微分方程数值解法. 第三版. 北京: 高等教育出版社, 1996.
133 M.E. Larsen and J.R. Howell. The Exchange Factor Method: An Alternative Zonal Formulation of Radiating Enclosure Analysis. Journal of Heat Transfer-Transactions of the ASME. 1985, 107: 936-942.
134 V.B. Kisslelev, L. Roberti and G. Perona. An Application of the Finite-Element Method to the Solution of the Radiative Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1994, 51: 603-614.
135 E.E. Lewis and W.F. Miller. Computational Methods of Neutron Transport. New York Wiley, 1984.
136 W.A. Fiveland and J.P. Jessee. Finite Element Formulation of the Discrete-Ordinates Method for Multidimensional Geometries. Journal of Thermophysics and Heat Transfer. 1994, 8: 426-433.
137 W.A. Fiveland and J.P. Jessee. Comparison of Discrete Ordinates Formulations for Radiative Heat Transfer in Multidimensional Geometries. Journal of Thermophysics and Heat Transfer. 1995, 9(1): 47-54.
138 K.B. Cheong and T.H. Song. Examination of Solution Methods for the Second-Order Discrete Ordinate Formulation. Numerical Heat Transfer Part B. 1995, 27: 155-173.
139 J. Liu and Y.S. Chen. Examination of Conventional and Even-Parity Formulations of Discrete Ordinates Method in a Body-Fitted Coordinate System. Journal of Quantitative Spectroscopy and Radiative Transfer. 1999, 61(4): 417- 431.
140 M.Y. Kim, S.W. Baek and J.H. Park. Unstructured Finite-Volume Method for Radiative Heat Transfer in a Complex Two-Dimensional Geometry with Obstacles. Numerical Heat Transfer Part B. 2001, 39: 617-635.
141 X. Cui and B.Q. Li. Discontinuous Finite Element Solution of 2-D Radiative Transfer with and without Axisymmetry. Journal of Quantitative Spectroscopy and Radiative Transfer. 2005, 96(3-4): 383-407.
142 X. Cui and B.Q. Li. A Discontinuous Finite-Element Formulation for Internal Radiation Problems. Numerical Heat Transfer Part B. 2004, 46(3): 223-242.
143 L.H. Liu and L.J. Liu. Discontinuous Finite Element Method for Radiative Heat Transfer in Semitransparent Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 105: 377-387.
144 F. Bassi and S. Rebay. High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations. Journal of Computational Physics. 1997, 138(2): 251-285.
145 B. Cockburn and C.W. Shu. The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems. Journal of Computational Physics. 1998, 141(2): 199-224.
146 C.E. Baumann and J.T. Oden. A Discontinuous hp Finite Element Method for Convection-Diffusion Problems. Computer methods in applied mechanics and engineering. 1999, 175(3): 311-341.
147 T.C. Warburton and G.E. Karniadakis. A Discontinuous Galerkin Method for the Viscous MHD Equations. Journal of Computational Physics. 1999, 152(2): 608-641.
148 B. Cockburn. Discontinuous Galerkin Methods. ZAMM. 2003, 83(11): 731-754.
149 B. Cockburn, G.E. Karniadakis and C.W. Shu. The Development of Discontinuous Galerkin Methods. Berlin: Springer Verlag, 2000: 3-50.
150 M.A. Ramankutty and A.L. Crosbie. Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures. Journal of Quantitative Spectroscopy and Radiative Transfer. 1997, 57(11): 107-140.
151 J.M. Zhao and L.H. Liu. Solution of Radiative Heat Transfer in Graded Index Media by Least Square Spectral Element Method. International Journal of Heat and Mass Transfer. 2007, 50: 2634-2642.
152 A.L. Crosbie and R.G. Schrenker. Radiative Transfer in a Two-DimensionalRectangular Medium Exposed to Diffuse Radiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 1984, 31(4): 339-372.
153 R. Viskanta and R.J. Grosh. Heat Transfer by Simultaneous Conduction and Radiation in an Absorbing Medium. Journal of Heat Transfer. 1962, 84(1): 63-72.
154 W.W. Yuen and L.W. Wong. Heat Transfer by Conduction and Radiation in a One-Dimensional Absorbing, Emitting and Anisotropically-Scattering Medium. Journal of Heat Transfer. 1980, 102: 303-307.
155 S.P. Burns, J.R. Howell and D.E. Klein. Empirical Evaluation of an Important Approximation for Combined-Mode Heat Transfer in a Participating Medium Using the Finite-Element Method. Numerical Heat Transfer Part B. 1995, 27: 309-322.
156 W.W. Yuen and E.E. Takara. Analysis of Combined Conductive-Radiative Heat Transfer in a Two-Dimensional Rectangular Enclosure with a Gray Medium. ASME Journal of Heat Transfer. 1988, 110(2): 468-474.
157 M.M. Razzaque, J.R. Howell and D.E. Klein. Coupled Radiative and Conductive Heat Transfer in a Two-Dimensional Rectangular Enclosure with a Gray Participating Media Using Finite Elements. Journal of Heat Transfer. 1984, 106: 613-619.
158 T.Y. Kim and S.W. Baek. Analysis of Combined Conductive and Radiative Heat Transfer in a Two-Dimensional Rectangular Enclosure Using the Discrete Ordinates Method. International Journal of Heat and Mass Transfer. 1991, 34(9): 2265-2273.
159 S.C. Mishra, A. Lankadasu and K.N. Beronov. Application of the Lattice Boltzmann Method for Solving the Energy Equation of a 2-D Transient Conduction-Radiation Problem. International Journal of Heat and Mass Transfer. 2005, 48(17): 3648-3659.
160 M.L. Nice. Application of Finite Elements to Heat Transfer in a Participating Medium. Numerical Properties and Methodologies in Heat Transfer, Proceedings of the 2nd National Symposium, Hemisphere, 1983, Washington, DC: 497-514.
161 P. Talukdar and S.C. Mishra. Analysis of Conduction–Radiation Problem in Absorbing, Emitting and Anisotropically Scattering Media Using theCollapsed Dimension Method. International Journal of Heat and Mass Transfer. 2002, 45: 2159–2168.
162 P.-f. Hsu. Effects of Multiple Scattering and Reflective Boundary on the Transient Radiative Transfer Process. International Journal of Thermal Sciences. 2001, 40(6): 539-549.
163 Z. Guo, J. Aber, B.A. Garetz and S. Kumar. Monte Carlo Simulation and Experiments of Pulsed Radiative Transfer. 2002, 73: 159-168.
164 Z. Guo, S. Kumar and K. San. Multidimensional Monte Carlo Simulation of Short-Pulse Laser Transport in Scattering Media. Journal of Thermophysics and Heat Transfer. 2000, 14: 504-511.
165 B.C. Wilson and G. Adam. A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue. Medical Physics. 1983, 10: 824-830.
166 M. Sakami, K. Mitra and P.F. Hsu. Analysis of Light-Pulse Transport through Two-Dimensional Scattering and Absorbing Media. Journal of Quantitative Spectroscopy and Radiative Transfer. 2002, 73: 169-179.
167 Y. Hasegawa, Y. Yamada, M. Tamura and Y. Nomura. Monte Carlo Simulation of Light Transmission through Living Tissues. Applied Optics. 1991, 30: 4515-4520.
168 N.A. Gentile. Implicit Monte Carlo Diffusion-an Acceleration Method for Monte Carlo Time-Dependent Radiative Transfer Simulations. Journal of Computational Physics. 2001, 172: 543-571.
169 M.Q. Brewster and Y. Yamada. Optical Properties of Thick Turbid Media from Picosecond Time-Resolved Light Scattering Measurements. International Journal of Heat and Mass Transfer. 1995, 38: 2569-2581.
170 Z.M. Tan and P.F. Hsu. An Integral Formulation of Transient Radiative Transfer. ASME Journal of Heat Transfer. 2001, 123: 466-475.
171 S.-H. Wu and C.-Y. Wu. Integral Equation Solutions for Transient Radiative Transfer in Nonhomogeneous Anisotropically Scattering Media. ASME Journal of Heat Transfer. 2000, 122: 818-822.
172 S.H. Wu and C.Y. Wu. Time-Resolved Spatial Distribution of Scattered Radiative Energy in a Two-Dimensional Cylindrical Medium with a Large Mean Free Path for Scattering. International Journal of Heat and Mass Transfer. 2001, 44: 2611-2619.
173 C.Y. Wu. Propagation of Scattered Radiation in a Participating Planar Medium with Pulse Irradiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 2000, 64: 537-548.
174 C.Y. Wu and S.H. Wu. Integral Equation Formulation for Transient Radiative Transfer in an Anisotropically Scattering Medium. International Journal of Heat and Mass Transfer. 2000, 43: 2009-2020.
175 K. Mitra and S. Kumar. Development and Comparison of Models for Light-Pulse Transport through Scattering-Absorbing Media. Applied Optics. 1999, 38(1): 188-196.
176 Z. Guo and S. Kumar. Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media. Applied Optics. 2001, 40(19): 3156-3163.
177 Z. Guo and K.H. Kim. Ultrafast-Laser-Radiation Transfer in Heterogeneous Tissues with the Discrete-Ordinates Method. Applied Optics. 2003, 42(16): 2897-2905.
178 K. Mitra, M.-S. Lai and S. Kumar. Transient Radiation Transport in Participating Media with a Rectangular Enclosure. Journal of Thermophysics and Heat Transfer. 1997, 11(3): 409-414.
179 J.C. Chai. Transient Radiative Transfer in Irregular Two-Dimensional Geometries. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004, 84: 281-294.
180 W. An, L.M. Ruan, H.P. Tan and H. Qi. Least-Squares Finite Element Analysis for Transient Radiative Transfer in Absorbing and Scattering Media. Journal of Heat Transfer. 2006, 128: 499-503.
181 L.H. Liu and P.-f. Hsu. Analysis of Transient Radiative Transfer in Semitransparent Graded Index Medium. Journal of Quantitative Spectroscopy and Radiative Transfer. 2007, 105(3): 357-376.
182 陶文铨. 数值传热学. 第二版. 西安: 西安交通大学出版社, 2001: 82-86.
183 F.N. van de Vosse and P.D. Minev. Spectral Element Methods: Theory and Applications. http://www.mate.tue.nl/people/vosse/docs/vosse96b.pdf.