J.M. Zhao

Comparative Study on Accuracy and Solution Cost of the First/Second-Order Radiative Transfer Equations Using the Meshless Method

The direct collocation meshless (DCM) method is applied to solve and evaluate the performance of the second-order radiative transfer equation (SORTE) proposed by Zhao and Liu (Numer. Heat Transfer B, vol. 51, pp. 391–409, 2007). The SORTE transforms the original first-order radiative transfer equation (FORTE) into a form similar to a diffusion equation, so no additional artificial diffusion or upwinding treatment is needed in the numerical discretization for stabilization. In order to investigate the accuracy and cost of the direct collocation meshless method based on the SORTE, two typical radiative transfer problems are considered. These cases are also solved by the DCM approach and the least-squares collocation meshless (LSCM) approach based on the FORTE. Numerical results show that the DCM approach based on the SORTE is more accurate and stable than the DCM approach and the LSCM approach based on the FORTE. The convergence rate of the SORTE-based methods with increase of collocation point number is faster than that of the FORTE-based methods. For obtaining the same target accuracy, the DCM approach based on the SORTE is more efficient than the other two meshless methods based on the FORTE. In addition, the DCM approach based on the SORTE also exhibits higher accuracy in solving radiative transfer problems with complex geometries or discontinuous temperature distributions along the boundary.

A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media

A new second order form of radiative transfer equation (named MSORTE) is proposed, which overcomes the singularity problem of a previously proposed second order radiative transfer equation [J.E. Morel, B.T. Adams, T. Noh, J.M. McGhee, T.M. Evans, T.J. Urbatsch, Spatial discretizations for self-adjoint forms of the radiative transfer equations, J. Comput. Phys. 214 (1) (2006) 12-40 (where it was termed SAAI), J.M. Zhao, L.H. Liu, Second order radiative transfer equation and its properties of numerical solution using finite element method, Numer. Heat Transfer B 51 (2007) 391-409] in dealing with inhomogeneous media where some locations have very small/zero extinction coefficient. The MSORTE contains a naturally introduced diffusion (or second order) term which provides better numerical property than the classic first order radiative transfer equation (RTE). The stability and convergence characteristics of the MSORTE discretized by central difference scheme is analyzed theoretically, and the better numerical stability of the second order form radiative transfer equations than the RTE when discretized by the central difference type method is proved. A collocation meshless method is developed based on the MSORTE to solve radiative transfer in inhomogeneous media. Several critical test cases are taken to verify the performance of the presented method. The collocation meshless method based on the MSORTE is demonstrated to be capable of stably and accurately solve radiative transfer in strongly inhomogeneous media, media with void region and even with discontinuous extinction coefficient.

Meshless Method for Geometry Boundary Identification Problem of Heat Conduction

A geometry identification problem of two-dimensional heat conduction is solved by using the least-squares collocation meshless method and the conjugate gradient method. In the least-squares collocation meshless approach for solving the direct heat conduction problem, a number of collocation points and auxiliary points are used to discretize the problem domain, and the collocation points are taken to construct the trial function by moving least-squares approximation. Akima cubic interpolation is employed to transform the geometry boundary inverse problem to the discrete boundary points inverse problem and approximate the unknown boundary in an inverse iterative process. In order to illustrate the performance and verify the new solution method, four typical cases are considered. The numerical results show that the least-squares collocation meshless method combined with the conjugate gradient method is accurate and stable for solving the geometry identification problem of heat conduction.

Spectral element method for vector radiative transfer equation

A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.

Optical Properties of Sodium Chloride Solution Within the Spectral Range from 300 to 2500 nm at Room Temperature.

The optical properties of sodium chloride (NaCI) solution were experimentally determined by double optical pathlength transmission method in the spectral range from 300 to 2500 nm at the NaCl concentration range from 0 to 360 g/L. The results show that the refractive index of NaCl solution increases with NaCl concentrations and correlates nonlinearly with the concentration of NaCl solution. The absorption index of NaCl solution increases with NaCl concentrations in the visible spectral range of 300-700 nm, but varies little in the near-infrared spectral range of 700-2500 nm at room temperature. For the sake of applications, the fitted formulae of the refractive index and absorption index of NaCl solution as a function of wavelength and NaCl concentration are presented.

Monte Carlo method for polarized radiative transfer in gradient-index media

Abstract Light transfer in gradient-index media generally follows curved ray trajectories, which will cause light beam to converge or diverge during transfer and induce the rotation of polarization ellipse even when the medium is transparent. Furthermore, the combined process of scattering and transfer along curved ray path makes the problem more complex. In this paper, a Monte Carlo method is presented to simulate polarized radiative transfer in gradient-index media that only support planar ray trajectories. The ray equation is solved to the second order to address the effect induced by curved ray trajectories. Three types of test cases are presented to verify the performance of the method, which include transparent medium, Mie scattering medium with assumed gradient index distribution, and Rayleigh scattering with realistic atmosphere refractive index profile. It is demonstrated that the atmospheric refraction has significant effect for long distance polarized light transfer.

On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media

Abstract Light transport in graded index media follows a curved trajectory determined by Fermats principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.

A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media

Abstract The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM will cause a severe accuracy degradation, which compromises the performance of the LSFEM too much and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM using the one dimensional linear element is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme.

Improved transmission method for measuring the optical extinction coefficient of micro/nano particle suspensions

Extinction coefficients are fundamental for analyzing radiative transport in micro/nano particle suspensions. In the traditional transmission method for measuring the extinction coefficient of particles in a cuvette, a reference system is used to compensate for the influence of the cuvette and base fluid. However, the multiple reflections and refractions between the air-glass and liquid-glass interfaces cannot be sufficiently eliminated by using the reference system, and the induced measurement error increases significantly with increasing difference in refractive index between the two neighboring media at these interfaces. In this paper, an improved transmission method is proposed to measure the extinction coefficient of micro/nano particles. The extinction coefficient of the particles is determined based on an optical model, taking into account the multiple reflection and refraction at the glass-liquid interfaces. An experimental validation was conducted for suspensions with various mean particle sizes. By considering the higher-order transmission terms, the improved transmission method generally achieved high-accuracy improvement over the traditional transmission method for extinction coefficient measurement, especially for the case with a small optical thickness of particle suspensions. This work provides an alternative and more accurate way for measuring the extinction characteristics of micro/nano particle suspensions.

A Finite-Element Model for the Thermal Radiative Properties of Graded Index Fiber Coated with Thin Absorbing Film

A finite-element model in combination with the wave optical approach is developed based on the radiative transfer equation for graded index medium in cylindrical coordinate system to predict the total hemispherical thermal radiative properties of semitransparent graded index fiber coated with thin absorbing film. The film is made of a strong absorbing medium with thickness less than or on the order of the wavelength of peak magnitude of thermal radiation. Radiative absorptance of the fiber-film system is directly obtained by solving the radiation deposited in the system. Radiative transfer in the fiber is solved by a least squares finite-element method, while radiative transfer in the thin film is treated through wave optics, and the film is formulated as a special kind of semitransparent boundary condition for the fiber medium. The results obtained by the finite-element model for uniform index fiber are in good agreement with the results in the literature obtained through the ray tracing model. The effects of fiber refractive index distribution on predicted thermal radiative properties are investigated. For the fiber with or without film, the variation of refractive index distribution has a substantial influence on the effective emittance.